- Email: [email protected]

[m1+; v1.288; Prn:4/10/2018; 11:22] P.1 (1-36)

Available online at www.sciencedirect.com

ScienceDirect J. Differential Equations ••• (••••) •••–••• www.elsevier.com/locate/jde

Lyapunov stability for impulsive control affine systems Josiney A. Souza ∗ , Luana H. Takamoto Departamento de Matemática, Universidade Estadual de Maringá, Cx. Postal 331, 87.020-900, Maringá, Pr, Brazil Received 14 March 2018; revised 23 June 2018

Abstract This article studies several notions of Lyapunov stability for impulsive control affine systems in the setting of nonautonomous dynamical systems. It presents some relations between the stability of an impulsive control affine system and the stability of its adjacent control system. Stability of compact sets and their components are specially investigated. Lyapunov functionals are employed to characterize each type of stability of closed sets. © 2018 Published by Elsevier Inc. Keywords: Impulsive control system; Impulsive nonautonomous dynamical system; Lyapunov stability; Lyapunov functional

1. Introduction The study of impulsive dynamical systems has been developed intensively in the last few years. In this setting, several results on Lyapunov stability were presented, including the characterization of many types of stability by means of Lyapunov functionals (e.g. [6] and [11]). In the present paper, we extend the studies of Lyapunov stability to the setting of impulsive control affine systems by considering the theoretical concept of impulsive nonautonomous dynamical system. * Corresponding author.

E-mail addresses: [email protected] (J.A. Souza), [email protected] (L.H. Takamoto). https://doi.org/10.1016/j.jde.2018.09.033 0022-0396/© 2018 Published by Elsevier Inc.

JID:YJDEQ AID:9564 /FLA

2

[m1+; v1.288; Prn:4/10/2018; 11:22] P.2 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

We start with a control affine system x˙ (t) = X (x (t) , u (t)) = X0 (x (t)) +

n

ui (t) Xi (x (t)) ,

i=1

u ∈ Upc = {u : R → U : u piecewise constant} where the phase space is a connected d-dimensional C ∞ -manifold M, the control range U ⊂ Rn is compact and convex, and the vector fields X0 , ..., Xn on M are C ∞ . This type of control system determines a nonautonomous dynamical system ([12, Chapter IV]). Then we have three basic formulations of impulsive control system: addition of explicit jump function; perturbation with measure-driven dynamics; and dynamic changes by impulsive set and impulsive function. The first formulation of impulsive control system considers a system of the form

x˙ (t) = X (x (t) , u (t)) x (τ ) = g τ, x τ − , vτ

if t ∈ /T if τ ∈ T

where T ⊂ [a, b] is the set of impulsive times and g is the explicit jump function that determines the exit point x (τ ) of a jump at t = τ , based on input data that may include the entry point x τ − = limt→τ − x (t) and the jump policy choice vτ of the jump. In this impulsive control paradigm, the impulses are determined by the time position and every trajectory has an infinite number of impulses. The three possible cases: u absent, v absent, or both present, describe three types of impulsive control system, each one modeling different behavior. They are useful to model problems in economics, aerospace navigation, biological system, synchronization of chaotic circuits, etc. (as references sources we mention [15,21–23]). A measure-driven impulsive control system is defined as a perturbed system dx (t) = X (x (t) , u (t)) dt +

n

Yi (x (t)) μi (dt)

a.e. t ∈ [0, ∞)

i=1

where M = Rd and each μi is a real-valued regular Borel measure defined on Borel subsets of [0, ∞). The measures determine the activity of the impulsive dynamics. In the instant at which the total variation of the measures is nonzero, the impulsive dynamics run as a subsystem, and the state is observed to have jumped to the exit point of that subsystem. In certain sense, the measures determine the magnitude of the jumps. The difference from the previous case is that the measuredriven dynamics must provide a continuous dynamical system path connecting the entry and exit points of any jump in the state, while the explicit jump function g is under no such obligation (as references sources we mention [1–3,10,13,16]). Stability results for measure-driven impulsive control system are reported in [17,18]. The third formulation of impulsive control system is extracted from the set up of nonautonomous dynamical system, based on the recent works [4,5]. In this context, an impulsive nonautonomous dynamical system is determined by an impulsive set Mϕ and an impulsive function I : Mϕ → M. For each trajectory governed by a control function u, the positive impulsive semitrajectory is defined inductively by state jumps made by the impulsive function I as the trajectory meets the impulsive set Mϕ . In certain sense, the impulsive function represents an external force that interrupts the evolution of the system by abrupt changes of state. Many real world

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.3 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

3

problems in science and technology can be modeled by this type of impulsive systems, for instance, medicine intake, billiard-type systems, electronic trading systems, and financial markets (see [4,5] and references therein). The key distinguishing feature from the two previous impulsive control paradigms is that the state jumps depend on the impulsive state position, while in the other formulations the state jumps are determined by the impulsive time position. This means that an impulsive nonautonomous system may have trajectories with finite number of impulses or even with no impulse. For example, any stationary trajectory of the original system must be free of impulse. Moreover, the time instant φ (x, u) at which the impulse occurs depends on the initial point x and the control parameter u, and the time intervals between consecutive impulses do not have determined length. The aim of the present paper is to push this third formulation further, in order to add properties which describe the impulsive control affine system as a semigroup action, represented by its impulsive system semigroup. The idea is to reproduce the methodology of constructing the system semigroup of a conventional control affine system. We consider the vector fields which determine the control system and define the impulsive positive semitrajectory for each correspondent dynamical system. By composing the impulsive transition maps we obtain the impulsive system semigroup. Then we show that the impulsive positive semitrajectories of the system are determined by the impulsive system semigroup. With this semigroup description, we reproduce the notions of Lyapunov stability according to the semigroup theory established in [7–9]. By speaking about the results of the paper, we basically study notions of invariance, stability, and attraction. Although the dynamical behavior of an impulsive control system may be completely different from the dynamical behavior of its adjacent control system, we present situations where the invariance and stability for the original control system influences the invariance and stability for the associated impulsive control system, and vice-versa (Theorems 3.1, 3.2, 3.3, 3.4, and 3.5). After this previous discussion, we specialize the studies of impulsive stability for compact sets. We show that the compact impulsive stable sets correspond to the compact sets which have trivial impulsive prolongations (Theorem 4.1). Then we investigate the stability of a compact set by means of the stability of its components (Theorem 4.2). In the last section of the paper, we reproduce the Lyapunov functional method of studying stability. We follow the line of investigation of [7] to study a general case of functionals, where continuity is not required (Theorems 5.1, 5.2, 5.3, 5.4, and 5.5). In another case, we follow the strategy presented in [6] to discuss the existence and continuity of Lyapunov functional for each type of stability (Theorems 5.6, 5.7, 5.8, and 5.9). 2. Impulsive control systems In this section we present the basic formulation of impulsive control system and describe its semigroup structure. We follow the methodology of [4,5] to define the impulsive semitrajectories. We refer to [12] for the global theory of control systems. Consider the control affine system x˙ (t) = X (x (t) , u (t)) = X0 (x (t)) +

n

ui (t) Xi (x (t)) ,

i=1

()

u ∈ Upc = {u : R → U : u piecewise constant} on a connected d-dimensional C ∞ -manifold M, with compact and convex control range U ⊂ Rn and C ∞ vector fields X0 , ..., Xn . Assume that, for each u ∈ Upc and x ∈ M, the preceding equa-

JID:YJDEQ AID:9564 /FLA

4

[m1+; v1.288; Prn:4/10/2018; 11:22] P.4 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

tion has a unique solution ϕ (t, x, u), t ∈ R, with ϕ (0, x, u) = x, and the vector fields X (·, u) (u ∈ U ) are complete. For sequences of control functions u1 , ..., uk ∈ Upc and numbers s1 < s2 < ... < sk−1 , we define the (s1 , ..., sk−1 )-concatenation of u1 , ..., uk as the control function (u1 , ..., uk ) (s1 , ..., sk−1 ) given by ⎧ ⎪ ⎪ ⎪ ⎨

u1 (t), if t s1 u2 (t − s1 ), if s1 < t ≤ s2 . (u1 , ..., uk ) (s1 , ..., sk−1 ) (t) = .. ⎪ . ⎪ ⎪ ⎩ uk (t − sk−1 ), if t > sk−1 For s ≥ 0 and u, v ∈ Upc , one has ϕ (t, x, (u, v) (s)) =

ϕ (t, x, u) , if t ≤ s ϕ (t − s, ϕ (s, x, u) , v) , if t > s

for every x ∈ M. Piecing together the solutions for constant control functions we obtain all the solutions of the control system, which means that the control system () is determined by the set of vector fields F = {X (·, u) : u ∈ U } and the system semigroup S defined as

S = exp (tn Xn ) ◦ exp (tn−1 Xn−1 ) ◦ · · · ◦ exp (t1 X1 ) : Xj ∈ F, tj ≥ 0, n ∈ N . In other words, the positive orbit O+ (x) and the negative orbit O− (x) coincide with Sx and S −1 x, respectively, that is

Sx = O+ (x) = ϕ (t, x, u) : t ≥ 0, u ∈ Upc ,

S −1 x = O− (x) = ϕ (−t, x, u) : t ≥ 0, u ∈ Upc , for every x ∈ M. Thus a set A ⊂ M is positively invariant if Sx ⊂ A, for all x ∈ A, and it is negatively invariant if S−1 x ⊂ A for all x ∈ A. The closure U = cls Upc with respect to the weak* topology of L∞ (R, Rn ) is a compact metric space, and the phase map ϕ : R × M × U → M,

(t, x, u) → ϕ (t, x, u)

is continuous (see [12, Chapter 4] for details). For a given real number s ∈ R and a control function u : R → U , the s-shift of u is the function u · s : R → U given by u · s (t) = u (s + t) for all t ∈ R. This shift defines a dynamical system on U given by θ : U × R → U , θ (u, s) = u · s. The solutions of the control system satisfy the cocycle property ϕ (t + s, x, u) = ϕ (t, ϕ(s, x, u), u · s) that is, ϕ (t + s, x, u) = ϕ (t, ϕ(s, x, u), θ (u, s)). Thus ϕ and θ determine a nonautonomous dynamical system as defined in [4, Definition 2.1]. Then we can define an impulsive control system as the following.

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.5 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

5

Definition 2.1. An impulsive control affine system consists of a control system () together with a nonempty closed subset Mϕ ⊂ M such that for each x ∈ Mϕ and each u ∈ Upc there is > 0 such that {ϕ (t, x, u) : t ∈ (−, 0)} ∩ Mϕ = ∅

and

{ϕ (t, x, u) : t ∈ (0, )} ∩ Mϕ = ∅,

and a continuous map I : Mϕ → M. The set Mϕ is called the impulsive set and the map I is called the impulsive function. The impulsive set and the impulsive function play a fundamental role on the definition of impulsive semitrajectory. For each x ∈ M and each u ∈ U , we define the set Mϕ+ (x, u) = {ϕ (t, x, u) : t > 0} ∩ Mϕ . If Mϕ+ (x, u) = ∅, then there is s > 0 such that ϕ (s, x, u) ∈ Mϕ and ϕ (t, x, u) ∈ / Mϕ for all t ∈ (0, s). This property allows us to define the function φ : M × U → (0, +∞] by φ (x, u) =

if ϕ (s, x, u) ∈ Mϕ and ϕ (t, x, u) ∈ / Mϕ for t ∈ (0, s) , + +∞, if Mϕ (x, u) = ∅. s,

Note that the value φ (x, u) is the smallest positive time such that the positive semitrajectory of x with respect to the control u meets Mϕ . If φ (x, u) < +∞, the point ϕ (φ (x, u) , x, u) ∈ Mϕ is called impulsive point of x with respect to the control u. If t1 , ..., tk ≥ 0, u1 , ..., uk+1 ∈ Upc , and u = (u1 , ..., uk+1 ) t1 , t1 + t2 , ..., ki=1 ti , we have φ (x, u) = φ (x, u1 ), if φ (x, u1 ) ≤ t1 ; φ (x, u) = t1 + φ (ϕ (t1 , x, u1 ) , u2 ), if t1 < φ (x, u1 ) and φ (ϕ (t1 , x, u1 ) , u2 ) ≤ t2 ; and so on. As in [4, Definition 3.2], the impulsive positive semitrajectory of x ∈ M with respect to u is a map ϕ (·, x, u) defined in an interval J(x,u) ⊂ R+ , with 0 ∈ J(x,u) , given inductively by the following rule: if Mϕ+ (x, u) = ∅, then ϕ (t, x, u) = ϕ (t, x, u) for all t ∈ R+ , and in this case, + φ (x, u) = +∞. However, if Mϕ (x, u) = ∅, we define ϕ (·, x, u) on [0, φ (x, u)] by ϕ (t, x, u) =

ϕ (t, x, u) , I (ϕ (φ (x, u) , x, u)) ,

if 0 ≤ t < φ (x, u) , if t = φ (x, u) .

Then we denote x0+ = x, s0 = φ x0+ , u , x1 = ϕ s0 , x0+ , u , and x1+ = I ϕ s0 , x0+ , u . In this case s0 < +∞ and the process can go on, but now starting at x1+ with the control ϕ (t, x, u) = ϕ t − s0 , x1+ , u · s0 for all function u · s0 . If Mϕ+ x1+ , u · s0 = ∅, then we define + t ∈ [s0 , +∞), and in this case, φ x1 , u · s0 = +∞. However, if Mϕ+ x1+ , u · s0 = ∅, then we define ϕ (·, x, u) on s0 , s0 + φ x1+ , u · s0 by ϕ (t, x, u) =

ϕ t − s0 , x1+ , u · s0 , I ϕ φ x1+ , u · s0 , x1+ , u · s0 ,

if s0 ≤ t < s0 + φ x1+ , u · s0 , if t = s0 + φ x1+ , u · s0 .

JID:YJDEQ AID:9564 /FLA

6

[m1+; v1.288; Prn:4/10/2018; 11:22] P.6 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

We now denote s1 = φ x1+ , u · s0 , x2 = ϕ s1 , x1+ , u · s0 , and x2+ = I ϕ s1 , x1+, u · s0 , and so on. This process ends after a finite number of steps if Mϕ+ xn+ , u · n−1 i=0 si = ∅ for si = ∅ for all n ∈ N, some n ∈ N or it may proceed indefinitely, in this case, Mϕ+ xn+ , u · n−1 i=0 n−1 n−1 + +, u · +, u · s = ϕ s , x s = I ϕ s , x , x , x , sn = φ xn+ , u · n−1 i n+1 n i n n n i=0 i=0 i=0 si n+1 +∞ and ϕ (·, x, u) is defined in the interval [0, T (x, u)), T (x, u) = i=0 si , by ⎧ ⎨ ϕ t − n−1 si , x + , u · n−1 si , n i=0 i=0 ϕ (t, x, u) = ⎩ I ϕ sn , x + , u · n−1 si , n i=0

if

n−1

if t

i=0 si ≤ t < = ni=0 si .

n

i=0 si ,

As in [4,5], we assume hereon that T (x, u) = +∞ for all x ∈ M and u ∈ U . Then we obtain a map ϕ : R+ × M × U → M that preserves the cocycle property ϕ (t + s, x, u) = ϕ (t, ϕ (s, x, u) , u · s) for all x ∈ M and t, s ≥ 0 ([5, Corollary 2.13]). As in the conventional case, we shall provide the semigroup structure for the impulsive control xp (tXu ) : M → M given by system. For each constant u ∈ U and t ∈ R+ , we have the map e ϕ (t, x, u). The impulsive system semigroup S is defined by e xp (tXu ) (x) =

S = e xp (tn Xn ) ◦ · · · ◦ e xp (t1 X1 ) : Xj ∈ F, tj ≥ 0, n ∈ N . We claim that the positive impulsive semitrajectories of the control system () are determined by + (x) is the positive impulsive that is, Sx =O + (x), for every x ∈ M, where O the semigroup S, orbit of x given by

+ (x) = O ϕ (t, x, u) : t ≥ 0, u ∈ Upc . We need the following lemma. Lemma 2.1. Let k ≥ 2. For t1 , ..., tk ≥ 0 and u1 , ..., uk ∈ Upc , one has ϕ (tk , ϕ (tk−1 , ..., ϕ (t1 , x, u1 ) , ..., uk−1 ) , uk ) = ϕ (t1 + · · · + tk , x, v) for every x ∈ M, where v = (u1 , ..., uk ) t1 , t1 + t2 , ..., k−1 t . i i=1 Proof. It is enough to prove the equality for two times s, τ ≥ 0 and two control functions u, v ∈ Upc , since the general case follows by induction. Let u, v ∈ Upc and s ≥ 0. Since ϕ (t, x, (u, v) (s)) = ϕ (t, x, u) for all x ∈ M and t ∈ (−∞, s], we have ϕ (t, x, (u, v) (s)) = ϕ (t, x, u) for all x ∈ M and t ≤ s. In particular, ϕ (s, x, (u, v) (s)) = ϕ (s, x, u) for all x ∈ M. For t > s, we have ϕ (t, x, (u, v) (s)) = ϕ (t − s, ϕ (s, x, u) , v) for every x ∈ M. Hence

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.7 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

7

ϕ (t, x, (u, v) (s)) = ϕ (t − s + s, x, (u, v) (s)) = ϕ (t − s, ϕ (s, x, (u, v) (s)) , (u, v) (s) · s) = ϕ (t − s, ϕ (s, x, u) , (u · s, v) (0)) . Since ϕ (τ, y, (u · s, v) (0)) = ϕ (τ, ϕ (0, y, u · s) , v) = ϕ (τ, y, v) for every y ∈ M and τ > 0, we have ϕ (τ, ϕ (s, x, u) , (u · s, v) (0)) = ϕ (τ, ϕ (s, x, u) , v) for every τ > 0. It follows that ϕ (t, x, (u, v) (s)) = ϕ (t − s, ϕ (s, x, u) , (u · s, v) (0)) = ϕ (t − s, ϕ (s, x, u) , v) . Thus we have ϕ (t, x, (u, v) (s)) =

ϕ (t, x, u) , if 0 ≤ t ≤ s ϕ (t − s, ϕ (s, x, u) , v) , if t > s

for every x ∈ M. Now, since ϕ (s, x, u) = ϕ (s, x, (u, v) (s)), the equality holds for τ = 0. For τ > 0, we have τ + s > s, and then ϕ (τ + s, x, (u, v) (s)) = ϕ (τ, ϕ (s, x, u) , v). 2 We now prove the claim. Theorem 2.1. Let u ∈ Upc , u1 , ..., uN ∈ U , and 0 = t0 < t1 < ... < tN such that ⎧ u1 , for 0 ≤ t ≤ t1 ⎪ ⎪ ⎪ ⎪ u2 , for t1 < t ≤ t1 + t2 ⎪ ⎨ .. u (t) = . . ⎪ ⎪ N−1 N ⎪ ⎪ ⎪ ti < t ≤ ti ⎩ uN , for i=0

i=0

Then ϕ (t, x, u) = e xp

t−

k−1

ti X u k

xp t1 Xu1 (x) ◦ e xp tk−1 Xuk−1 ◦ · · · ◦ e

i=0

for every x ∈ M and

k−1

ti ≤ t ≤

i=0

k

+ (x) = Sx for every x ∈ M. ti , k = 1, ..., N . Thus O

i=0

Proof. Since u1 , ..., uN ∈ U represent a sequence N ofconstant control functions, we have u = N−1 ti . By Lemma 2.1, 0 ≤ t ≤ t1 implies (u1 , ..., uN ) t1 , ..., i=1 ti in the interval 0, i=0

ϕ (t, x, u) = ϕ (t, x, u1 ) = e xp tXu1 (x) for every x ∈ M. Hence the result holds for k = 1. Suppose by induction that the result holds for k k+1 ti < t ≤ ti , we have some k > 1. If i=0

i=0

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.8 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

8

ϕ (t, x, u) = ϕ t− = e xp

k

ti , ϕ

k

i=0

t−

ti , x, (u1 , ..., uk ) t1 , ...,

i=0

k

ti

k−1

ti

, uk+1

i=1

k k−1 ti , x, (u1 , ..., uk ) t1 , ..., ti Xuk+1 ϕ

i=0

i=0

i=1

By hypothesis, ϕ

k

ti , x, (u1 , ..., uk ) t1 , ...,

i=0

k−1

ti

xp tk−1 Xuk−1 ◦ · · ·◦ e xp t1 Xu1 (x) , = e xp tk Xuk ◦ e

i=1

hence ϕ (t, x, u) = e xp

t−

k

xp tk−1 Xuk−1 ◦ · · · ◦ e xp t1 Xu1 (x) . ti Xuk+1 ◦ e xp tk Xuk ◦ e

i=0

+ (x) = Sx. Therefore the expression holds for all k = 1, ..., N . We now prove the equality O Let u ∈ Upc and τ > 0. Then there are sequences u1 , ..., uN ∈ U and 0 = t0 < t1 < ... < tN such N ti and u = (u1 , ..., uN ) t1 , ..., N−1 t that τ = i=1 i in the interval [0, τ ]. By the first part of the i=0

proof, we have xp tN−1 XuN−1 ◦ · · · ◦ e xp t1 Xu1 (x) ∈ Sx. ϕ (τ, x, u) = e xp tN XuN ◦ e On the other hand, for t1 , ..., tk ≥ 0 and u1 , ..., uk ∈ U , there is v ∈ Upc such that xp t1 Xu1 (x) = ϕ (t1 + · · · + tk , x, v) e xp tk Xuk ◦ · · · ◦ e + (x). xp t1 Xu1 (x) ∈ O by Lemma 2.1. Hence e xp (tk ) Xuk ◦ · · · ◦ e

2

This theorem allows to study Lyapunov stability of impulsive control systems by the semigroup action point of view, based on the papers [7–9]. However, it should be observed that the action of the impulsive system semigroup admits discontinuities. Thus some theorems of stability, proved in the continuous case, may need alternative proof. In the following we give two illustrating examples of impulsive control system. Example 2.1. Consider the control system on M = R2 determined by the vector fields X =

∂ ∂x1

∂ , with Mϕ = {(x, 1/ |x|) : x = 0} and I ((x, 1/ |x|)) = (x, −1/ |x|). The trajectories ∂x2 of X are horizontal lines. The trajectories of Y are vertical lines. Piecing together these trajectories we obtain all the trajectories of the control system. Some impulsive positive semitrajectories are illustrated in Fig. 1. We will go back to this example in the next section. and Y =

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.9 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

9

Fig. 1. Examples of impulsive positive semitrajectories on the plane. On the left, the original positive semitrajectory; on the right, the correspondent impulsive positive semitrajectory.

Example 2.2. Let G be a simply connected nilpotent Lie group with Lie algebra g. Consider the control affine system on G determined by two nonzero vector fields X, Y ∈ g. It is well-known that G is isomorphic with g via the exponential map and the Campbell–Hausdorff product in g defined as A ∗ B = A + B + R2 + · · · Rk where Ri is given by the Lie brackets of A, B ∈ g and Ri ∈ gi , where 0 = gk+1 ⊂ gk ⊂ . . . ⊂ g2 ⊂ g is the descending central series of g. By identifying G with g, the corresponding flows of X and Y on g are respectively

JID:YJDEQ AID:9564 /FLA

10

[m1+; v1.288; Prn:4/10/2018; 11:22] P.10 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

X (t, Z) = tX ∗ Z,

Y (t, Z) = tY ∗ Z,

(t, Z) ∈ R × g.

Let j be the positive integer such that X ∈ gj but X ∈ / gj +1 . Consider a hyperplane hX of g j +1 containing g and such that X ∈ / hX . Then hX is a transversal section for the flow X , that is, there is a continuous map τX : g → R such that for every Z ∈ g, X (t, Z) ∈ hX if and only if t = τX (Z) ([14, Proposition 1]). Similarly, we have a transversal section hY for Y with map τY . Now set Mϕ = exp (hX ∩ hY ) and fix a map I = exp ◦T ◦ ln|Mϕ : Mϕ → G, where T : g → g is any affine transformation. We claim that Mϕ is an impulsive set according to Definition 2.1. Indeed, let g ∈ Mϕ and u ∈ Upc . There are Xu , Xv ∈ {X, Y } and > 0 such that ϕ (t, g, u) =

exp (tXu ) (g) , exp (tXv ) (g) ,

if t ∈ (−, 0] , if t ∈ [0, ) .

As ln (g) ∈ hX ∩ hY , we have τX (ln (g)) = 0 = τY (ln (g)). This means that exp (tXu ) (g) ∈ / Mϕ and exp (sXv ) (g) ∈ / Mϕ if respectively t ∈ (−, 0) and s ∈ (0, ), and the claim is proved. 3. Invariance and stability In this section we use methods of semigroup theory to study concepts of invariance and stability for impulsive control systems. We show how stability implies invariance and in which conditions the invariance and stability of an impulsive control system relates to the invariance and stability of its adjacent control system. Throughout, there is a fixed impulsive control system with impulsive set Mϕ and impulsive function I . Denote by d a Riemannian distance on the phase space M. Then B (x, ) will denote the ball with radius and center x. Let dA (x) denote the distance from x to A ⊂ M and then define B (A, ) = {x ∈ M : dA (x) < }. We also denote by A (A, δ, ε) the annulus {x ∈ M : δ < dA (x) < ε}. The following concepts of invariance and stability were introduced in the general setting of semigroup actions ([7]), except the notions of BH-equistability and I -invariance. Definition 3.1. Let A be a subset of M. (1) The set A is said to be positively S-invariant if Sx ⊂ A for all x ∈ A; (2) The set A is said to be I -invariant if I (x) ∈ A for each x ∈ Mϕ ∩ A; (3) The set A is said to be S-stable if for every > 0 and x ∈ A there exists a δ > 0 such that SB (x, δ) ⊂ B (A, ); (4) The set A is said to be S-uniformly stable if for every > 0 there exists a δ > 0 such that SB (A, δ) ⊂ B (A, ); (5) The set A is said to be S-orbitally stable if for every neighborhood U of A there exists a positively S-invariant neighborhood V of A with V ⊂ U ; (6) The set A is said to be S-BH-equistable if for every x ∈ A and y ∈ / A there is a neighborhood V of x and a neighborhood W of y such that W ∩ SV = ∅; (7) The set A is said to be S-equistable if for each y ∈ / A there exists an > 0 such that y ∈ / cls (SB (A, )). The BH-equistability means the stability in the sense of Bhatia and Hajek as named by Ciesielski ([11]).

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.11 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

11

It is easily seen that a set is S-stable whenever it is S-uniformly stable. On the other hand, any compact S-stable set is S-uniformly stable ([7, Theorem 3.2]). Moreover, any closed S-uniformly stable set is S-equistable ([7, Theorem 3.3]), and the converse holds for compact sets ([7, Theorem 3.4]). It is not difficult to check that an S-equistable set is S-BH-equistable and any compact S-BH-equistable set is S-equistable. All these notions of stability can be defined for impulsive control systems replacing S by S in Definitions 3.1. Any S-uniformly stable set is S-stable, and the converse holds for compact sets. These results do not depend on the continuity of the action (see [7, Theorem 3.2]). Analogously, an S-equistable set is S-BH-equistable and any compact S-BH-equistable set is S-equistable. We can also conclude that any closed S-uniformly stable set is S-equistable ([7, Theorem 3.3]), however, since [7, Theorem 3.4] requires the condition of connected orbits, we can not get the converse yet. The relationship between S-stability and S-equistability will be given in Theorem 4.1. We now recall another concept of stability related to the notion of attraction. For t > 0 we define the set ⎧ ⎫ n ⎨ ⎬ S≥t = e tj ≥ t, n ∈ N . xp (tn−1 Xn−1 ) ◦ · · · ◦ e xp (t1 X1 ) : Xj ∈ F, tj ≥ 0, xp (tn Xn ) ◦ e ⎩ ⎭ j =0

= S≥t : t > 0 is a filter basis on the subsets of S. This means that ∅ ∈ The family F / F and S≥t+s ⊂ S≥t ∩ S≥s for all t, s ≥ 0. The following concept of attraction was introduced in [8]. -attraction of A is the set Definition 3.2. Let A be a subset of M. The domain of weak F

= x ∈ M : S≥t x ∩ B (A, ) = ∅ for all , t > 0 Aw A, F x ∈ M : for each pair , t > 0 there are s ≥ t and u ∈ Upc = . such that dA ( ϕ (s, x, u)) < -attraction of A is the set The domain of F

= x ∈ M : for each ε > 0 there is t > 0 such that S≥t x ⊂ B (A, ε) A A, F x ∈ M : for each ε > 0 there is t > 0 such that = . dA ( ϕ (s, x, u)) < ε for all s ≥ t and u ∈ Upc ; it is -attractor if there is δ > 0 such that B (A, δ) ⊂ Aw A, F The set A is called a weak F -attractor if there is δ > 0 such that B (A, δ) ⊂ A A, F . called an F -asymptotically stable if it is an Definition 3.3. A subset A ⊂ M is said to be weak F -attractor; A is said to be F -asymptotically stable if it is an S-uniformly stable weak F -attractor. S-uniformly stable F Remark 3.1. In the case of dynamical system, the corresponding notions of asymptotic stability and weak asymptotic stability are equivalent. In the conventional case of control system, an asymptotically stable set is weak asymptotically stable, but the converse holds only under restrictive conditions (see [9, Section 3] for precise explanations).

JID:YJDEQ AID:9564 /FLA

12

[m1+; v1.288; Prn:4/10/2018; 11:22] P.12 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

As mentioned above, our purpose in this section is to discuss possible relations among the no although the relationship between the dynamics of S tions of invariance and stability for S and S, and the dynamics of S seems very weak. In fact, the dynamical behavior of the impulsive control system may be completely different from the dynamical behavior of its adjacent system. For ∂ instance, consider the control system on M = R2 determined by the vector fields X = and ∂x1 ∂ , with Mϕ = {(x, 1/ |x|) : x = 0} and I ((x, 1/ |x|)) = (x, −1/ |x|), as in Example 2.1. Y= ∂x2 Let A, B, C, D ⊂ R2 be the sets A = {(x, y) : x = 0, y < 1/ |x|} ∪ {0} × R, B = {(x, y) : y ≥ 0} , C = (x, y) : x ≥ 0, D = {(x, y) : x ≥ 0, y ≤ 0} . Note that A is positively S-invariant but not positively S-invariant; the set B is positively S-invariant but not positively S-invariant; the set C is both S-uniformly stable and S-uniformly -attractor but not an F -attractor. Note also that the conventional stable; and the set D is a weak F system has no weak attractor. In general, the stability of the conventional control system does not influence the stability of the associated impulsive control system, and vice-versa. In certain sense, the impulses are caused by an external force which may change the dynamics completely. See the examples in the end of this section. Thus, only under suitable conditions, the dynamics of S relates to the dynamics of S. We start proving a theorem that relates S-invariance and S-invariance. In order to prove positive S-invariance of a set A, we need to show that exp tn Xun ◦ · · · ◦ exp t1 Xu1 (x) ∈ A for all x ∈ A, Xuj ∈ F , tj ≥ 0, and n ∈ N. Then it is enough to consider any constant control function u ∈ U and prove that exp (tXu ) (x) ∈ A for all x ∈ A and t ≥ 0, since the composition mentioned above will belong to A in this case. The same strategy holds for S-invariance. Theorem 3.1. Let A ⊂ M be a nonempty subset. (1) If A is positively S-invariant and I -invariant then A is positively S-invariant. (2) If A is positively S-invariant and closed then A is positively S-invariant. Proof. (1) Let x ∈ A and u ∈ U . For t ∈ [0, φ (x, u)), we have e xp (tXu ) (x) = exp (tXu ) (x) ∈ A as A is positively S-invariant. Clearly, x1 = exp (φ (x, u) Xu ) (x) ∈ A. Since A is I -invariant, we have x1+ = I (x1 ) ∈ A. Hence e xp (tXu ) (x) = exp ((t − φ (x, u)) Xu ) x1+ ∈ A for t ∈ + φ (x, u) , φ (x, u) + φ x1 , u . We proceed in this way to obtain e xp (tXu ) (x) ∈ A for all t ≥ 0. Therefore A is positively S-invariant. / A for some s ≥ 0. De(2) Let x ∈ A and u ∈ U and suppose that exp (sXu ) (x) ∈ fine t = inf {s : exp (sXu ) (x) ∈ / A}. Then t ≥ φ (x, u) > 0 as A is positively S-invariant and exp (τ Xu ) (x) = e xp (τ Xu ) (x) ∈ A for all τ ∈ [0, φ (x, u)). Note that exp (τ Xu ) (x) ∈ A for all τ ∈ [0, t) and exp (tXu ) (x) ∈ cls (A) = A. We now need to prove that there exists a δ > 0 such that exp (τ Xu ) (x) ∈ A for all τ ∈ [0, t + δ). Indeed, for τ ∈ (t, t + φ (exp (tXu ) (x) , u)), we have

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.13 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

13

exp (τ Xu ) (x) = exp ((τ − t) Xu·t ) (exp (tXu ) (x)) = exp ((τ − t) Xu ) (exp (tXu ) (x)) = e xp ((τ − t) Xu ) (exp (tXu ) (x)) ∈ A, since τ − t∈ (0, φ (exp (tXu ) (x) , u)). Then exp (τ Xu ) (x) ∈ A for all τ ∈ [0, t + φ(exp (tXu ) (x) , u)), where φ (exp (tXu ) (x) , u) > 0, which contradicts the definition of t . 2 In the conventional case, any connected component of a closed positively invariant set is posi tively invariant. In the impulsive case, however, connected components of a positively S-invariant set need not be positively S-invariant, due to the discontinuities. Otherwise, any I -invariant (see the connected component of a closed positively S-invariant set is positively S-invariant appendix). to S-invariance and S-invariance. We now relate S-stability Theorem 3.2. Let A ⊂ M be a nonempty subset. and I -invariant. Furthermore, if A (1) If A is S-BH-equistable then A is positively S-invariant is also closed then A is positively S-invariant. (2) If A is closed and S-stable then A is positively S-invariant, positively S-invariant, and I -invariant. Proof. (1) Suppose that A ⊂ M is S-BH-equistable but not positively S-invariant. Then there are x ∈ A and s ∈ S such that sx ∈ / A. By hypothesis, there is a neighborhood V of x and a ∩ W = ∅, which is a contradiction since sx ∈ SV ∩ W. neighborhood W of sx such that SV Now suppose that A ⊂ M is S-BH-equistable but not I -invariant. Then there are x ∈ A ∩ Mϕ such that I (x) ∈ / A. Hence there is a neighborhood V of x and a neighborhood W of I (x) such ∩ W = ∅. For a given u ∈ U , we can find y ∈ M and t > 0 such that exp (tXu ) (y) = x, that SV exp ([0, t) Xu ) (y) ∩ Mϕ = ∅, and hence φ (y, u) = t . By the continuity of exp, we can find an s ∈ [0, t) such that exp (sXu ) (y) ∈ V . We have φ (exp (sXu ) (y) , u) = t − s. Indeed, note that exp ((t − s) Xu ) (exp (sXu ) (y)) = exp (tXu ) (y) = x ∈ Mϕ . Moreover, for τ ∈ (0, t − s), we have exp (τ Xu ) exp (sXu ) (y) = exp ((τ + s) Xu ) (y) ∈ / Mϕ since τ + s < t and φ (y, u) = t . Hence I (x) = I (exp (tXu ) (y)) = I (exp ((t − s) Xu ) (exp (sXu ) (y))) = e xp ((t − s) Xu ) (exp (sXu ) (y)) ∈ SV ∩ W = ∅. Finally, if A is also closed then it which contradicts the conditions I (x) ∈ W and SV is positively S-invariant, by Theorem 3.1. (2) Let x ∈ A and > 0. There exists a δ > 0 such that SB (x, δ) ⊂ B (A, ) as A is S-stable. In particular, we have Sx ⊂ B (A, ). Therefore Sx ⊂ {B (A, ) : > 0} = cls (A) = A. Hence A is positively S-invariant. By Theorem 3.1, we conclude that A is positively S-invariant. We now prove that A is I -invariant. Let x ∈ Mϕ ∩ A and u ∈ U . We can find y ∈ M and t > 0 such

JID:YJDEQ AID:9564 /FLA

14

[m1+; v1.288; Prn:4/10/2018; 11:22] P.14 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

that exp (tXu ) (y) = x, exp ([0, t) Xu ) (y) ∩ Mϕ = ∅, and hence φ (y, u) = t . For > 0 there (x, δ) ⊂ B (A, ) as A is S-stable. exists δ > 0 such that SB We can take s ∈ [0, t) such that exp (sXu ) (y) ∈ B (x, δ). As the proof for the previous theorem, we have φ (exp (sXu ) (y) , u) = t − s, and then I (x) = I (exp ((t − s) Xu ) (exp (sXu ) (y))) = e xp ((t − s) Xu ) (exp (sXu ) (y)) (x, δ) ⊂ B (A, ) . ∈ SB It follows that I (x) ∈

B (A, ) = cls (A) = A, and therefore A is I -invariant.

2

>0

The following theoFrom now on, we discuss relations between S-stability and S-stability. rems present sufficient conditions for S-uniform stability to imply S-uniform stability. Theorem 3.3. Assume that A is an S-uniformly stable subset of M. Assume also that there is a μ such that for every x ∈ B (A, μ) the following conditions hold: 1. if x ∈ Mϕ then dA (I (x)) ≤ dA (x); 2. if x ∈ I Mϕ , u ∈ Upc , and φ (x, u) < +∞ then dA (ϕ (φ (x, u) , x, u)) ≤ dA (x). Then A is S-uniformly stable. Proof. Take > 0 and assume that < μ. As A is S-uniformly stable, there is η > 0 such that SB (A, η) ⊂ B (A, ). Then we can find a δ > 0 such that SB (A, δ) ⊂ B (A, η/2). Let x ∈ B (A, η) and u ∈ Upc . If φ (x, u) = +∞ then ϕ (t, x, u) = ϕ (t, x, u) ∈ SB (A, η) ⊂ B (A, ) for all t ≥ 0. If φ (x, u) < +∞ then ϕ (t, x, u) = ϕ (t, x, u) ∈ SB (A, η) ⊂ B (A, ) for all t ∈ [0, φ (x, u)). Note that x1 = ϕ (φ (x, u) , x, u) ∈ SB (A, η) ⊂ B (A, ) ⊂ B (A, μ) . By hypothesis 1, we have dA x1+ = dA (I (x1 )) ≤ dA (x1 ) < η and hence x1+ ∈ B (A, η). Then we have ϕ (t, x, u) = ϕ t − s0 , x1+ , u · s0 ∈ SB (A, η) ⊂ B (A, ) for all t ∈ s0 , s0 + φ x1+ , u · s0 , where s0 = φ (x, u). Now, since x1+ ∈ B (A, μ) ∩ I Mϕ , we have dA (x2 ) = dA ϕ φ x1+ , u · s0 , x1+ , u · s0 ≤ dA x1+ < η by hypothesis 2. Thus we have dA x2+ = dA (I (x2 )) ≤ dA (x2 ) < η, by hypothesis 1, and hence x2+ ∈ B (A, η). It follows that ϕ (t, x, u) = ϕ t − (s0 + s1 ) , x2+ , u · (s0 + s1 ) ∈ SB (A, η) ⊂ B (A, )

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.15 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

15

for all t ∈ s0 + s1 , s0 + s1 + φ x2+ , u · (s0 + s1 ) , where s1 = φ x1+ , u · s0 . We proceed in (A, η) ⊂ B (A, ). Thus A is this way obtaining ϕ (t, x, u) ∈ B (A, ) for all t ≥ 0, that is SB S-uniformly stable. 2 Theorem 3.4. Assume that the set A ⊂ M is S-uniformly stable and there exists an η > 0 such stable. that I Mϕ ∩ B (A, η) ⊂ A. Then A is S-uniformly Proof. Take > 0 and assume that < η. There is δ > 0 such that SB (A, δ) ⊂ B (A, ). ϕ (t, x, u) = ϕ (t, x, u) ∈ B (A, ) for all Let x ∈ B (A, δ) and u ∈ Upc . If φ (x, u) = +∞ then t ≥ 0. If φ (x, u) < +∞ then ϕ (t, x, u) = ϕ (t, x, u) ∈ B (A, ) for all t ∈ [0, φ (x, u)). Note that x1 = ϕ (φ (x, u) , x, u) ∈ B (A, ) ⊂ B (A, η) and x1+ = I (x1 ) ∈ I Mϕ ∩ B (A, η) ⊂ A ⊂ B (A, δ). Then we have ϕ (t, x, u) = ϕ t − s0 , x1+ , u · s0 ∈ SB (A, δ) ⊂ B (A, ) for all t ∈ + s0 , s0 + φ x1 , u · s0 , where s0 = φ (x, u). We proceed in this manner obtaining ϕ (t, x, u) ∈ B (A, ) for all t ≥ 0, and therefore A is S-uniformly stable. 2 Since stability and uniform stability are equivalent for compact sets, Theorems 3.3 and 3.4 hold for a compact S-stable set. Besides, a simple statement of separability shows that a compact (see the appendix). S-stable set disjoined of Mϕ is S-stable We complete the amount of theorems of this section by presenting a simple criterium of stability for control system determined by finite set of vector fields. Theorem 3.5. Let F = {X1 , ..., Xn } be a finite set of commutative vector fields on the mani fold M. A set A ⊂ M is S-uniformly stable if and only if it is uniformly stable for each impulsive dynamical system determined by Xi . Proof. For simplicity, we assume n = 2 and denote F = {X, Y }. Define the dynamical systems ϕX and ϕY on M by ϕX (t, x) = exp (tX) (x) and ϕY (t, x) = exp (tY ) (x), t ∈ R and x ∈ M. For + a given subset B ⊂ M, denote by ϕX (B) the set of all positive impulsive semitrajectories through ϕY+ (B) analogously. Since X and Y are commutative the points of B with respect to ϕX . Define vector fields which determine the control system, we have = { Sx exp (tX) ◦ e xp (sY ) (x) : t, s ≥ 0} = { exp (sY ) ◦ e xp (tX) (x) : t, s ≥ 0} + + + = ϕY (B) for any subset B ⊂ M. Suppose that for all x ∈ M. Hence SB ϕY+ ϕ˜X ϕX (B) = (A, δ) ⊂ B (A, ). A ⊂ M is S-uniformly stable. For a given > 0, there is δ > 0 such that SB + (A, δ) ⊂ B (A, ), and therefore A is ϕX -uniformly stable. Then we have ϕX (B (A, δ)) ⊂ SB ϕX -uniformly Similarly, A is ϕY -uniformly stable. As to the converse, suppose that A is both ϕX (B (A, δX )) ⊂ stable and ϕY -uniformly stable. For a given > 0, there is δX > 0 such that (A, δY ) = ϕY (B (A, δY )) ⊂ B (A, δX ). It follows that SB B (A, ). Now there is δY > 0 such that ϕX (B (A, δX )) ⊂ B (A, ), and therefore A is S-uniformly stable. 2 ϕX ( ϕY (B (A, δY ))) ⊂ The following example illustrates an application of Theorem 3.5. Example 3.1. Consider the control affine system determined by the linear vector fields F = {X, Y } on M = R2 given by

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.16 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

16

X=

1 0 0 1

Y=

,

1 0 0 −1

.

The origin 0 ∈ R2 is unstable for both vector fields X and Y , and then it is unstable for the 1 n control system. Consider the impulsive set Mϕ = n∈N∗ C2n , where Ck = x ∈ R : x = , k x and the impulsive function I (x) = . Note that I (C2n ) = C2n+1 . Let the notations be 1 + x ϕY -uniformly stable. Indeed, for as in the proof for Theorem 3.5. The origin 0 is both ϕX and a given > 0, choose k > 0 such that cls (B (0, 1/2k)) ⊂ B (0, ). Since I (C2n ) = C2n+1 , the positive impulsive semitrajectories of X and Y , which start at B (0, 1/2k), do not go out of ϕY (B (0, 1/2k)) ⊂ B (0, ), which proves the assertion. B (0, 1/2k). Hence ϕX (B (0, 1/2k)) ∪ By Theorem 3.5, the origin is S-uniformly stable. To finish this section, we present some illustrating examples which show that the stability of the impulsive control system may be completely different from the stability of its adjacent system. Example 3.2. This example shows that S-stability does

not imply S-stability. Consider the control system on M = R2 \ x = (x1 , x2 ) : x12 + x22 < 1 determined by the vector fields {X, Y } given by ⎛

⎛

#

⎜ 1 ⎜ X (x1 , x2 ) = ⎝ ⎝x1 + x2 − 100

⎞ x12

+ x22 ⎟

2x1

⎠,

⎛

#

1 ⎜ ⎝−x1 + x2 − 100

⎞⎞ x12

+ x22 ⎟⎟

2x2

⎠⎠

and Y (x1 , x2 ) = (−x2 , x1 ) , with x2 + 99 I (0, x2 ) = 0, − , 100

Mϕ = {(0, x2 ) : x2 ∈ [1, +∞)} ,

for all x2 ∈ [1, +∞) .

Let A = (x1 , x2 ) : x12 + x22 = 1 . The trajectories of X have the following behavior: for x ∈ A, the trajectories move on the unit circle A; for x ∈ / A, the trajectories move on spirals from the unit circle A to infinity. The trajectories of Y move on circles centered at 0. Piecing together these trajectories we obtain all the trajectories of the control system. Clearly, A is not S-stable. Now, after an impulse, the trajectory will be closer to the unit circle. Therefore A is S-stable. Fig. 2 illustrates the trajectories of the control system. Example 3.3. This example shows that S-stability doesnot imply S-stability. Consider the con 1 1 2 2 2 trol system on M = x = (x1 , x2 ) : ≤ x1 + x2 ≤ ⊂ R determined by the vector fields 16 9

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.17 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

17

Fig. 2. Trajectories of the control system.

{X, Y } given by X (x1 , x2 ) = (x2 , −x1 ) and Y = (Y1 , Y2 ) , where

⎛

⎞

π ⎜ ⎟ Y1 (x1 , x2 ) = −x2 + x1 x12 + x22 sin ⎝ # ⎠ and 2 2 x1 + x2 ⎛ ⎞ π ⎜ ⎟ Y2 (x1 , x2 ) = x1 + x2 x12 + x22 sin ⎝ # ⎠, x12 + x22 with

1 7 1 Mϕ = (0, x2 ) : x2 ∈ , , I ((0, x2 )) = −2x2 + , 0 , 4 24 4 1 7 for all x2 ∈ , . The trajectories of X move on circles centered at 0. The trajecto4 24 ries of Y have the following behavior: if x = 14 then the trajectories move on the circle

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.18 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

18

Fig. 3. Trajectories of the control system.

1 1 C 1 = (x1 , x2 ) : x12 + x22 = ; if x = then the trajectories move on the circle C 1 = 3 3 16 4 1 1 1 2 2 ; and if < x < then the trajectories move on spirals that ap(x1 , x2 ) : x1 + x2 = 9 4 3 proach the circle C 1 in positive time and approach the circle C 1 in negative time. Clearly, the 4 3 1 7 , then I ((0, x2 )) > (0, x2 ), which implies circle C 1 is S-stable. However, if x2 ∈ 4 4 24 that C 1 is not S-stable. Fig. 3 illustrates the trajectories of the control system. 4

Example 3.4. In this example we clarify that the condition I Mϕ ∩ B (A, η) ⊂ A in Theorem 3.4 can not be ignored. A control system is called point transitive if cls (Sx) = M for some x ∈ M. Consider a control affine system determined by a set F of Killing vector fields on the Riemannian manifold M. Then the system semigroup S is a semigroup of isometries of M. For each x ∈ M, the closure cls (Sx) is S-uniformly stable. Indeed, for a given > 0, take y ∈ B (cls (Sx) , ). Then there is z ∈ cls (Sx) such that d (y, z) < . For any g ∈ S, it follows that d (g (y) , g (z)) = d (y, z) < , with g (z) ∈ cls (Sx). Hence SB (cls (Sx) , ) ⊂ B (cls (Sx) , ), and therefore cls (Sx) is S-uniformly stable. We now assume that the control system is not point transitive and consider a compact impulsive set Mϕ . If cls (Sx) ∩ Mϕ = ∅, we can find an η > 0 such that B (cls (Sx) , η) ∩ Mϕ = ∅. Then we can use Theorem 3.4 to show that cls (Sx) is S-uniformly stable. On the other hand, suppose that cls (Sx) ∩ Mϕ = ∅ but cls (Sx) ∩ stable. In fact, since I M is compact, I Mϕ = ∅. We claim that cls (Sx) is not S-uniformly ϕ we can find > 0 such that B (cls (Sx) , ) ∩ I Mϕ = ∅. For any δ > 0, we can take a point y ∈ B (cls (Sx) , δ) ∩ Mϕ . Take X ∈ F . Since B (cls (Sx) , δ) is an open set, we can find t > 0 such that z = exp (−tX) y ∈ B(cls (Sx) , δ). Then exp (tX) z = y ∈ Mϕ , and hence there is τ > 0 (cls (Sx) , δ) \ B (cls (Sx) , ), xp (τ X) z ∈ SB such that e xp (τ X) z ∈ I Mϕ . It follows that e and therefore cls (Sx) is not S-uniformly stable.

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.19 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

19

4. Stability of compact sets In this section we discuss stability in the special case of compact sets. We extend a classical result of conventional control system which characterizes stability of compact sets by means of prolongations. We also consider stable sets which are disconnected and discuss the stability of its connected components. We refer to [20] for the notion of prolongation for control systems. Throughout, there is a fixed impulsive control system with impulsive set Mϕ and impulsive function I . Definition 4.1. The S-prolongation of x ∈ M is the set + (x) = D

=

y ∈ M : there are sequences (gn ) in S and (xn ) in M such that xn → x and gn xn → y

y ∈ M : there are sequences (tn ) in R+ , (un ) in Upc , and (xn ) in M such that . ϕ (tn , xn , un ) → y xn → x and

+ (B) = For a given subset B ⊂ M, we define D

+ (x) : x ∈ B . D

The following is the main theorem of this section. It unifies the notions of Lyapunov stability in the compact case. Besides, it describes the compact S-stable sets as the compact subsets of M with trivial S-prolongation. Theorem 4.1. Assume that A is a compact subset of M. Then the following conditions are equivalent: 1. 2. 3. 4.

A is S-stable; A is S-orbitally stable; A is S-BH-stable; + (A) = A. D

there exists Proof. (1) ⇒ (2) Take a neighborhood U of A. Since A is compact and S-stable, (A, δ) ⊂ U . Clearly S SB (A, δ) ⊂ SB (A, δ). Then SB (A, δ) is the a δ > 0 such that SB required neighborhood of A. / A. Since A is compact, there are open sets W and V in M such (2) ⇒ (3) Let x ∈ A and y ∈ that A ⊂ W , y ∈ V , and W ∩ V = ∅. Then we can find a neighborhood U of A such that U ⊂ W ⊂ U , as A is S-orbitally and SU stable. Note that int (U ) is a neighborhood of x. We claim that ⊂ U , we have V ∩ S(int (U )) = ∅. Note that U ∩ V = ∅ as W ∩ V = ∅ and U ⊂ W . Since SU ∩ V = ∅, and then S(int (U )) ∩ V = ∅. Hence A is S-BH-stable. SU / A. There are neighborhoods U of x and V of y such that (3) ⇒ (4) Let x ∈ A and y ∈ = ∅, as A is S-BH-stable. V ∩ SU Take any sequences (tn ) in R+ , (xn ) in M, and (un ) in Upc such that xn → x. Since U is a neighborhood of x and xn → x, we may assume that xn ∈ U , it follows that for every n ∈ N. Since ϕ (tn , xn , un ) ∈ SU ϕ (tn , xn , un ) ∈ / V for every n. Hence + (x). This means that y ∈ + (A), and ϕ (tn , xn , un ) does not converge to y, and therefore y ∈ /D /D + thus D (A) ⊂ A. The inverse inclusion is obvious. Then we can find > 0, x ∈ A, (4) ⇒ (1) Suppose by contradiction that A is not S-stable. + ϕ (tn , xn , un ) ∈ / B (A, ) for ev(tn ) in R , (xn ) in M, and (un ) in Upc such that xn → x and ery n. By local compactness, we may assume that cls (B (A, )) is compact. If tn < φ (xn , un )

JID:YJDEQ AID:9564 /FLA

20

[m1+; v1.288; Prn:4/10/2018; 11:22] P.20 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

for infinitely many n, we may assume that tn < φ (xn , un ) for every n, by taking a subsequence. Then ϕ (tn , xn , un ) = ϕ (tn , xn , un ) for every n. Moreover, we may assume that xn ∈ B (A, ), for every n, as B (A, ) is a neighborhood of x and xn → x. Then for each n, there exists an sn ∈ [0, tn ] such that ϕ (sn , xn , un ) = ϕ (sn , xn , un ) ∈ ∂B (A, ). Since ∂B (A, ) is com+ (x) ⊂ D + (A) = A, pact, we may assume that ϕ (sn , xn , un ) → y ∈ ∂B (A, ). Thus y ∈ D which is a contradiction since y ∈ ∂B (A, ). This finishes the proof in the case where tn < φ (xn , un ) for infinitely many n. We now prove the case where tn < φ (xn , un ) only for finitely many n, that is, φ (xn , un ) ≤ tn for infinitely many n. We may assume that φ (xn , un ) ≤ tn for every n, by taking a subsequence. This means that each ϕ (tn , xn , un ) appears on the impulsive semitrajectory ϕ ([0, +∞) , xn , un ) later than the impulse point of xn . For each n, define n = {s ≥ 0 : ϕ (s, xn , un ) ∈ B (A, )}. Clearly 0 ∈ n as xn ∈ B (A, ). Moreover, define sn = inf {[0, φ (xn , un )) \n }. Clearly ϕ (sn , xn , un ) ∈ / B (A, ) since ϕ (·, xn , un ) is rightcontinuous. Take δ ∈ (0, ). We claim that there is n0 ∈ N such that for any n ≥ n0 we can find pn ∈ ϕ (n , xn , un ) and qn ∈ A such that d (pn , qn ) < δ. Indeed, suppose to the contrary that for infinitely many n there exists a τn ∈ n such that ϕ (τn , xn , un ) ∈ cls (A (A, δ, )). Since cls (A (A, δ, )) is compact, we may assume that ϕ (τn , xn , un ) → y ∈ cls (A (A, δ, )). Then y ∈ + (A) = A, which is a contradiction since y ∈ cls (A (A, δ, )) and therefore y ∈ + (x) ⊂ D / A. D Thus for any fixed n ≥ n0 we have ϕ ([0, sn ) , xn , un ) ⊂ B (A, δ). Since ϕ (sn , xn , un ) ∈ / B (A, ), it follows that ϕ (·, xn , un ) is not continuous at sn , and therefore ϕ (sn , xn , un ) = I (yn ) for some yn ∈ Mϕ . Since ϕ (s, xn , un ) → yn , as s sn , we have yn ∈ cls (B (A, δ)). Then we may assume that yn → y ∈ cls (B (A, δ)), as cls (B (A, δ)) is compact. Since I is continuous, we have I (yn ) → I (y), and therefore I (y) ∈ / A, as I (yn ) ∈ / B (A, ) for every n. This is a contradiction + (x) ⊂ D + (A) = A, as because I (y) ∈ D ϕ (sn , xn , un ) = I (yn ) → I (y). The proof is completed. 2 We use this result to study the connected components of a compact S-stable set. Sometimes we deal with disconnected sets, and then it is important to know about the stability of their components. It is not difficult to show that a reunion of S-stable sets is S-stable. Thus we concentrate the discussion on the converse, that is, what happens with the connected components of an S-stable set. For the following, we say that a connected component E of a closed set A is isolated in A if there are disjoint open sets V and W such that E ⊂ V and A \ E ⊂ W . For instance, if A has only finitely many connected components then each component is isolated in A. Theorem 4.2. Assume that A ⊂ M is a compact S-stable set. An isolated component E of A is S-stable if and only if it is I -invariant. then it is I -invariant by Theorem 3.2. As to the converse, suppose Proof. If E is S-stable that E is I -invariant. Since E is an isolated component of A, there are disjoint open sets V and W such that E ⊂ V and A\E ⊂ W . Moreover, E is compact as E is a component of the compact set A. By local compactness of M we may assume that cls (V ) is compact. Suppose by contradiction that E is not S-stable. Then we can find α > 0, x ∈ E, and a se quence (xn ) such that xn → x and Sxn is not contained in cls (B (E, α)). Take β > 0 such n is not that cls (B (E, β)) is compact and cls (B (E, β)) ⊂ V . Let γ = min {α, β}. Then Sx contained in cls (B (E, γ )), cls (B (E, γ )) is compact, and cls (B (E, γ )) ⊂ V . Choose a sequence (un ) in U such that ϕ ([0, +∞) , xn , un ) cls (B (E, γ )). We claim there exists n0

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.21 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

21

such that Mϕ+ (xn , un ) = ∅ for every n ≥ n0 . Indeed, suppose to the contrary that there are infinitely many n such that Mϕ+ (xn , un ) = ∅, that is, φ (xn , un ) = +∞. We may assume that ϕ (t, xn , un ) = ϕ (t, xn , un ) for all Mϕ+ (xn , un ) = ∅ for all n, by taking a subsequence. Then t ≥ 0 and n ∈ N. Since xn → x and x ∈ E ⊂ B (E, γ ), we may assume that xn ∈ B (E, γ ) for all n. Then there is a τn ≥ 0 such that ϕ (τn , xn , un ) = ϕ (τn , xn , un ) ∈ ∂B (E, γ ) ⊂ cls (B (E, γ )) as ϕ ([0, +∞) , xn , un ) is not contained in cls (B (E, γ )). We may assume that ϕ (τn , xn , un ) → + (x) ⊂ D + (A). By Theorem 4.1, D + (A) = A, hence y ∈ A. Then y ∈ ∂B (E, ). Then y ∈ D we have y ∈ A\E ⊂ W and y ∈ ∂B (E, γ ) ⊂ cls (B (E, γ )) ⊂ V , hence y ∈ V ∩ W , which contradicts V ∩ W = ∅. Thus there is n0 such that Mϕ+ (xn , un ) = ∅ for all n ≥ n0 . Now, by the x ⊂ B (E, γ ) ∪ W . We may asS-stability of A there exists a neighborhood Nx of x such that SN sume that xn ∈ Nx for all n as xn → x. Note that {t : ϕ (t, xn , un ) ∈ W } = ∅ for every n, because n is not contained in cls (B (E, γ )). Then, for each n, define tn = inf {t : ϕ (t, xn , un ) ∈ W }. Sx x ⊂ We have ϕ ([0, tn ) , xn , un ) ∩ W = ∅, and therefore ϕ ([0, tn ) , xn , un ) ⊂ B (E, γ ) as SN + B (E, γ ) ∪ W . Clearly, ϕ (tn , xn , un ) ∈ W and ϕ (tn , xn , un ) = xnk for some nk , according to the construction of impulsive positive semitrajectories. Then ϕ (tn , xn , un ) = I (qn ) for some qn ∈ cls (B (E, γ )). Now, for each n, take vn < tn such that there is no impulse point in the segment of trajectory ϕ ((vn , tn ) , xn , un ). Moreover, for each n, take rn ∈ (vn , tn ) such that tn − rn → 0. Since U is compact, there exists a w ∈ U such that un · rn → w. Define pn = ϕ (rn , xn , un ) ∈ cls (B (E, γ )). By of cls (B (E, )) we may that pn → 'the compactness assume nk−1 nk−2 nk−2 + ,u · p ∈ cls (B (E, )). Since rn ∈ s , s = φ x , where s nk−1 nk−1 n i=n0 i i=n0 i i=n0 si , nk−2 nk−2 + ,u · s , x we have pn = ϕ rn − i=n nk−1 n i=n0 si . Hence 0 i ⎛ qn = ϕ ⎝tn − ⎛

nk−2 i=n0

si , xn+k−1 , un ·

⎞ si ⎠

i=n0

⎛

= ϕ ⎝tn − rn , ϕ ⎝rn − ⎛

nk−2

⎛

nk−2

si , xn+k−1 , un ·

i=n0

nk−2

= ϕ ⎝tn − rn , pn , ⎝un ·

i=n0

⎞ ⎛ si ⎠ · ⎝rn −

nk−2 i=n0

nk−2

⎞ ⎛ si ⎠ , ⎝un · ⎞⎞

nk−2 i=n0

⎞ ⎛ si ⎠ · ⎝rn −

nk−2

⎞⎞ si ⎠⎠

i=n0

si ⎠⎠

i=n0

= ϕ (tn − rn , pn , un · rn ) and then qn → ϕ (0, p, w) = p. Thus p ∈ cls Mϕ = Mϕ , since qn ∈ Mϕ for all n. On the other + (x) ⊂ D + (A) = A since hand, we have p ∈ D ϕ (rn , xn , un ) = pn → p and xn → x. Note that cls (B (E, γ )) ∩ (A\E) = ∅ as cls (B (E, γ )) ⊂ V , A\E ⊂ W , and V ∩ W = ∅. Hence p ∈ / A\E, and therefore p ∈ Mϕ ∩ E. It follows that I (p) ∈ E as E is I -invariant. However I (qn ) → I (p) and I (qn ) = ϕ (tn , xn , un ) ∈ W , which implies that I (p) ∈ cls (W ), and therefore I (p) ∈ / E. This contradiction finishes the proof. 2 As an immediate consequence from Theorem 4.2, if a compact S-stable set A has a finite number of connected components then any I -invariant component of A is S-stable.

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.22 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

22

5. Lyapunov functionals In this section we reproduce the Lyapunov functional method of studying stability. We first consider a general case, as presented in [7], where the functionals need not be continuous. In another case, we follow the strategy presented in [6] to discuss the existence of continuous functionals. Each type of stability can be characterized by specific Lyapunov functional. In order to illustrate this relationship, consider the control system x˙ (t) = u (t) x (t) , u ∈ Upc = {u : R → [−10, −1] : u piecewise constant}, n : x = 1 , and imC , where C = x ∈ R k n∈N 2n+1 k n , x ∈ Mϕ . For u ∈ Upc and x ∈ R , the solution is of the form

on M = Rn , with impulsive set Mϕ = pulsive function I (x) =

x 1 + x

ϕ (t, x, u) = e where t =

k

i=1 ti .

(t 0

u(s)ds

x = et1 u1 +t2 u2 +···+tk uk x

If x > 1 we have

φ (x, u) = min t > 0 : t =

k

ti and

i=1

If

1 1 ≥ x > we have 2n + 1 2n + 3 φ (x, u) = min t > 0 : t =

k i=1

ti and

k

) ti ui = − ln x .

i=1

k

) ti ui = − ln (2n + 3) x .

i=1

The origin 0 is an S-stable equilibrium point, essentially because * * * (t * * * ϕ (t, x, u) = *e 0 u(s)ds x * ≤ *e−t x * ≤ x * * * x * * * < x if x ∈ Mϕ . This whenever 0 ≤ t ≤ φ (x, u) and x ∈ M \ Mϕ , and I (x) = * 1 + x * The function ψ (x) = x is a Lyapunov implies that σ (x) ≤ x for all x ∈ M and σ ∈ S. functional for the origin 0. 5.1. General case In the general case we do not assume any restrictive condition on the impulsive control system. The functionals associated to the stable sets need not be continuous. The following theorem describes closed S-stable set via Lyapunov functional. The proof for it is identical to the conventional case, and then it is omitted here. A complete proof can be found in the appendix or in [7].

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.23 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

23

Theorem 5.1. A closed set A ⊂ M is S-stable if and only if there exists a function ψ : M → R+ with the following properties: (1) (2) (3) (4)

ψ (x) = 0 if and only if x ∈ A; for every ε > 0 there exists δ > 0 such that ψ (x) ≥ δ whenever dA (x) ≥ ε; ψ (xn ) → 0 if (xn ) is a sequence in M with xn → x ∈ A (ψ is continuous at x ∈ A); ψ (σ x) ≤ ψ (x) for every σ ∈ S and x ∈ M.

A Lyapunov functional for a closed S-stable set A can be defined as dA (σ x) . 1 + dA (σ x) σ ∈S

ψ (x) = sup

(L)

If t ≥ s ≥ 0, the property 4 in the theorem above implies that ψ ( ϕ ϕ (t, x, u)) = ψ ( ϕ (t − s + s, x, u)) = ψ ( ϕ (t − s, ϕ˜ (s, x, u) , u · s)) ≤ ψ ( ϕ (s, x, u)) . Hence ψ is nonincreasing on positive impulsive semitrajectories. Moreover, properties 1 and 2 together imply that ψ separates A and the complement of any neighborhood B (A, ε). If A is a closed S-uniformly stable set, a Lyapunov functional for A has the additional property of isolating A from the complement of any neighborhood B (A, ε). It can be also defined as L. The characterization of closed S-uniformly stable set is formulated in the following theorem whose proof is also omitted (see the appendix or [7]). Theorem 5.2. A closed set A ⊂ M is S-uniformly stable if and only if there exists a function ψ : M → R+ with the following properties: (1) for every ε > 0 there exists δ > 0 such that ψ (x) ≥ δ whenever dA (x) ≥ ε; (2) for every ε > 0 there exists δ > 0 such that ψ (x) < ε whenever dA (x) < δ; (3) ψ (σ x) ≤ ψ (x) for every σ ∈ S and x ∈ M. The following theorem describes closed F-asymptotically stable set by means of specific functional. This case is a novelty, even for conventional control systems. As expected in this case, the Lyapunov functional has an asymptotic property in a neighborhood of the S-uniformly stable set. -asymptotically stable if and only if there exists a function Theorem 5.3. A closed set A ⊂ M is F + ψ : M → R with the following properties: (1) (2) (3) (4)

for every ε > 0 there exists δ > 0 such that ψ (x) ≥ δ whenever dA (x) ≥ ε; for every ε > 0 there exists δ > 0 such that ψ (x) < ε whenever dA (x) < ε; ψ (σ x) ≤ ψ (x) for every σ ∈ S and x ∈ M; ϕ (tn , x, un )) → 0 for all sequences (un ) there exists ε > 0 such that if x ∈ B (A, ε) then ψ ( in Upc and tn → +∞.

JID:YJDEQ AID:9564 /FLA

24

[m1+; v1.288; Prn:4/10/2018; 11:22] P.24 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

Proof. Suppose that such a function ψ : M → R+ exists. Properties 1, 2, and 3 together im -attractor. ply that A is S-uniformly stable, by Theorem 5.2. It remains to show that A is an F By property 4, there exists ε > 0 such that if x ∈ B (A, ε) then ψ ( ϕ (tn ,x, un )) → 0 for all se and therefore A is an quences (un ) in Upc and tn → +∞. We claim that B (A, ε) ⊂ A A, F -attractor. Suppose by contradiction that there is x ∈ B (A, ε) \ A A, F . Then there is some F δ > 0 such that S≥t x B (A, δ) for every t > 0. Hence we can find sequences (un ) in Upc and tn → +∞ such that ϕ (tn , x, un ) ∈ / B (A, δ) for every n ∈ N. As x ∈ B (A, ε), it follows that ψ ( ϕ (tn , x, un )) → 0. Nevertheless, by property 1, there is η > 0 such that ψ (x) ≥ η whenever dA (x) ≥ δ. This means that ψ ( ϕ (tn , x, un )) ≥ η for every n, a contradiction. Conversely, -asymptotically stable and define the function ψ : M → R+ by suppose that A is F dA (σ x) . 1 + dA (σ x) σ ∈S

ψ (x) = sup

We have seen that this function satisfies properties 1, 2, and 3. It remains to show property 4. -attractor, there exists ε > 0 such that B (A, ε) ⊂ A A, F . Let x ∈ B (A, ε) and Since A is an F take sequences (un ) in Upc and tn → +∞. For a given δ > 0 there is t > 0 such that S≥t x ⊂ ϕ (tn , x, un ) ⊂ S≥t x ⊂ B (A, δ), hence dA (σ B (A, δ). If tn > t then S ϕ (tn , x, un )) < δ for every δ ϕ (tn , x, un )) ≤ < δ. Thus σ ∈ S. This means that there is n0 such that n > n0 implies ψ ( 1+δ ψ ( ϕ (tn , x, un )) → 0, and the proof is finished. 2 -divergent if for each t > 0 there is n0 such Remark 5.1. We say that a sequence (σn ) in S is F that n > n0 implies σn ∈ S≥t (see [19]). The notation σn →F ∞ means that the sequence (σn ) is -divergent in S. Then the property 4 of Theorem 5.3 may be replaced by the following: F 4’. there exists ε > 0 such that if x ∈ B (A, ε) then ψ (σn x) → 0 as σn →F ∞. -asymptotic stability to the continuous case, in the next We let the discussion on weak F section (Theorem 5.9). We now discuss the more complicated case of S-equistability. A sufficient condition for a set A ⊂ M to be S-equistable is the existence of a lower semicontinuous function ψ : M → R+ satisfying (1) ψ −1 (0) = A; (2) for every ε > 0 there exists δ > 0 such that ψ (x) ≤ ε whenever The proof for this fact does not dA (x) ≤ δ; and (3) ψ (σ x) ≤ ψ (x) for all x ∈ M and σ ∈ S. depend on the continuity of the action and can be found in [7, Theorem 3.9]. However, the converse holds under continuous action (see [7, Theorem 3.10]). Then, to get a characterization of S-equistable set via Lyapunov functional, we require suitable conditions on the impulses and some modifications on the properties of the functional. The following theorem shows that the existence of a Lyapunov functional is sufficient for S-equistability. Theorem 5.4. Assume that I Mϕ ⊂ M \ Mϕ and I Mϕ \ A ⊂ M \ A. A closed set A ⊂ M is S-equistable if there exists a function ψ : M → R+ satisfying the following properties: (1) ψ (x) = 0 if and only if x ∈ A; (2) for every ε > 0 there exists δ > 0 such that ψ (x) ≤ ε whenever dA (x) ≤ δ;

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.25 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

25

(3) ψ is lower semicontinuous in M \ Mϕ \ A ; and ψ (I (x)) ≤ ψ (x) if x ∈ Mϕ \ A. (4) ψ (σ x) ≤ ψ (x) if x ∈ M \ Mϕ \ A and σ ∈ S, Proof. Let x ∈ / A and consider firstly the case x ∈ / Mϕ . By property 1, we have ψ (x) > 0. By property 2, there is δ > 0 such that ψ (B (A, δ)) ⊂ [0, ψ (x) /2]. Hence SB (A, δ) ⊂ −1 ψ ([0, ψ (x) /2]), by property 4. By lower semicontinuity in M \ Mϕ \ A , there is η > 0 such that d (x, y) < η, with y ∈ M \ Mϕ \ A , implies ψ (x) − ψ (y) < ψ (x) /2. Since Mϕ is closed, we can find α > 0 such that α < η and B (x, α) ⊂ M \ Mϕ . If y ∈ B (x, α), it fol (A, δ). This means that x ∈ (A, δ) . We lows that ψ (y) > ψ (x) /2, and hence y ∈ / SB / cls SB now suppose that x ∈ Mϕ . Then x ∈ Mϕ \ A and ψ (I (x)) ≤ ψ (x). We suppose by contradic (A, δ) for every δ > 0. For each pair (k, n) ∈ N × N, we can take y(k,n) ∈ tion that x ∈ cls SB (A, 1/n), which means there are t(k,n) ≥ 0, x(k,n) ∈ B (A, 1/n), and u(k,n) ∈ Upc B (x, 1/k) ∩ SB such that ϕ t(k,n) , x(k,n) , u(k,n) = y(k,n) . Let N × N be endowed with the product direction. Then the net y(k,n) (k,n)∈N×N converges to x. We consider two distinct cases: (a) there is a subnet y(ki ,ni ) i∈I such that φ y(ki ,ni ) , u(ki ,ni ) · t(ki ,ni ) < +∞ and φ y(ki ,ni ) , u(ki ,ni ) · t(ki ,ni ) −→ 0; (b) there is c > 0 such that c ≤ φ y(k,n) , u(k,n) · t(k,n) for all (k, n) ∈ N × N. In the case (a), the continuity of ϕ implies ϕ φ y(ki ,ni ) , u(ki ,ni ) · t(ki ,ni ) , y(ki ,ni ) , u(ki ,ni ) · t(ki ,ni ) −→ x. The continuity of I then implies ϕ φ y(ki ,ni ) , u(ki ,ni ) · t(ki ,ni ) , y(ki ,ni ) , u(ki ,ni ) · t(ki ,ni ) = I ϕ φ y(ki ,ni ) , u(ki ,ni ) · t(ki ,ni ) , y(ki ,ni ) , u(ki ,ni ) · t(ki ,ni ) −→ I (x) where ϕ φ y(ki ,ni ) , u(ki ,ni ) · t(ki ,ni ) , y(ki ,ni ) , u(ki ,ni ) · t(ki ,ni ) ϕ t(ki ,ni ) , x(ki ,ni ) , u(ki ,ni ) , u(ki ,ni ) · t(ki ,ni ) = ϕ φ y(ki ,ni ) , u(ki ,ni ) · t(ki ,ni ) , = ϕ t(ki ,ni ) + φ y(ki ,ni ) , u(ki ,ni ) · t(ki ,ni ) , x(ki ,ni ) , u(ki ,ni ) . On the other hand, it follows by hypothesis / A ∪ Mϕ . By the first part of the proof, that I (x) ∈ (A, δ) . By taking n0 ∈ N with 1/n0 < δ, there is there is some δ > 0 such that I (x) ∈ / cls SB i0 ∈ I such that ni0 > n0 . If i ≥ i0 then ni ≥ ni0 and hence x(ki ,ni ) ∈ B (A, 1/ni ) ⊂ B (A, δ). This means that (A, δ) ϕ t(ki ,ni ) + φ y(ki ,ni ) , u(ki ,ni ) · t(ki ,ni ) , x(ki ,ni ) , u(ki ,ni ) ∈ SB (A, δ) , a contradiction. In the case (b), we may whenever i ≥ i0 , and therefore I (x) ∈ cls SB assume that u(k,n) · t(k,n) → u, by compactness. If 0 ≤ t < min {φ (x, u) , c}, we have ϕ t, y(k,n) , u(k,n) · t(k,n) −→ ϕ (t, x, u) .

JID:YJDEQ AID:9564 /FLA

26

[m1+; v1.288; Prn:4/10/2018; 11:22] P.26 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

As A is closed and x ∈ / A, we can find t0 > 0, t0 < min {φ (x, u) , c}, such that ϕ (t0 , x, u) ∈ / A∪ Mϕ . Then there is some δ > 0 such that ϕ , x, u) ∈ / cls SB δ) . Nevertheless, we have (t (A, 0 ϕ t0 , y(k,n) , u(k,n) · t(k,n) → ϕ (t0 , x, u) and ϕ t0 , ϕ t0 , y(k,n) , u(k,n) · t(k,n) = ϕ t(k,n) , x(k,n) , u(k,n) , u(k,n) · t(k,n) = ϕ t0 + t(k,n) , x(k,n) , u(k,n) (A, δ) , a contradiction again. Thus x ∈ (A, δ) , for some hence ϕ (t0 , x, u) ∈ cls SB / cls SB δ > 0, and we conclude that A is S-equistable. 2 x ∈ Mϕ satisfies In order to show a converse to Theorem 5.4, we assume that each element the ϕ-strong tube condition (ϕ-STC), and hence φ is continuous on M \ Mϕ × U (see [5, Definition 3.3 and Theorem 3.5] for details). This condition yields a fundamental property of invariance, which is necessary to the construction of the Lyapunov functional. This property is given by the following lemma. Lemma 5.1. Assume that Mϕ ∩ I Mϕ = ∅ and each element x ∈ Mϕ satisfies the ϕ-strong tube condition. If X ⊂ M is positively S-invariant then S cls (X) \ Mϕ ⊂ cls (X). Proof. Let x ∈ cls (X) \ Mϕ , t > 0, and u ∈ U . Then there is a sequence (xn ) in X such that xn → x. Since φ is continuous on M \ Mϕ × U , we have φ (xn , u) → φ (x, u). If t < φ (x, u) then t < φ (xn , u) eventually, and hence ϕ (t, xn , u) = ϕ (t, xn , u) → ϕ (t, x, u) = ϕ (t, x, u) . As X is positively S-invariant, the sequence ( ϕ (t, xn , u)) lies in X, and therefore ϕ (t, x, u) ∈ ⊂ cls (X). If φ (x, u) < +∞ and t = φ (x, u) we have cls (X). If φ (x, u) = +∞, we have Sx ϕ (φ (xn , u) , xn , u) = I (ϕ (φ (xn , u) , xn , u)) → I (ϕ (φ (x, u) , x, u)) = ϕ (φ (x, u) , x, u) . Again the sequence ( ϕ (φ (xn , u) , xn , u)) lies in X, and therefore ϕ (t,x, u) ∈ cls (X). Now de+ note s0 = φ (x, u), x1+ = I (ϕ (s0 , x, u)), sn,0 = φ (xn , u), and xn,1 = I ϕ sn,0 , xn , u , for each + + + ∈ / Mϕ . As xn,1 → x1+ and sn,0 → s0 , we have φ xn,1 , u · sn,0 → n ∈ N. By hypothesis, x1+ , xn,1 + , u · sn,0 eventually, φ x1+ , u · s0 . If s0 < t < s0 + φ x1+ , u · s0 then sn,0 < t < sn,0 + φ xn,1 and hence + ϕ (t, x, u) . ϕ (t, xn , u) = ϕ t − sn,0 , xn,1 , u · sn,0 → ϕ t − s0 , x1+ , u · s0 = ⊂ cls (X). If φ x + , u · s0 < +∞ Thus ϕ (t, x, u) ∈ cls (X). If φ x1+ , u · s0 = +∞, we have Sx 1 + and t = s0 + φ x1 , u · s0 , we have + + + ϕ sn,0 + φ xn,1 , u · sn,0 , xn , u = I ϕ φ xn,1 , u · sn,0 , xn,1 , u · sn,0 → I ϕ φ x1+ , u · s0 , x1+ , u · s0

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.27 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

27

= ϕ s0 + φ x1+ , u · s0 , x, u = ϕ (t, x, u) . Therefore ϕ (t, x, u) ∈cls (X). Then we denote s1 = φ x1+ , u · s0 , x2+ = I ϕ s1 , x1+ , u · s0 , + + + + , u · sn,0 , and xn,2 = I ϕ sn,1 , xn,1 , u · sn,0 , where x2+ , xn,2 ∈ / Mϕ . This prosn,1 = φ xn,1 cess ends after a finite number of steps if φ xk+ , u · k−1 i=0 si = +∞ for some k ∈ N or it may k−1 + ⊂ cls (X) and proceed indefinitely if φ xk , u · i=0 si < +∞ for all k ∈ N. In any case, Sx therefore S cls (X) \ Mϕ ⊂ cls (X). 2 Putting together the conditions of Theorem 5.4 and Lemma 5.1, we can present a characteri zation of closed S-equistable set. Theorem 5.5. Assume that each element x ∈ Mϕ satisfies the ϕ-strong tube condition, I Mϕ ⊂ if and only if there exists M \ Mϕ , and I Mϕ \ A ⊂ M \ A. A closed set A ⊂ M is S-equistable + a function ψ : M → R satisfying the following properties: (1) (2) (3) (4)

ψ (x) = 0 if and only if x ∈ A; for every ε > 0 there exists δ > 0 such that ψ (x) ≤ ε whenever dA (x) ≤ δ; ψ is lower semicontinuous in M \ Mϕ \ A ; and ψ (I (x)) ≤ ψ (x) if x ∈ Mϕ \ A. ψ (σ x) ≤ ψ (x) if x ∈ M \ Mϕ \ A and σ ∈ S,

Proof. Suppose that A is S-equistable. Define the function ψ : M → R+ by (A, δ) ψ (x) = sup δ > 0 : x ∈ / cls SB ψ (x) = 0

if x ∈ / A ∪ Mϕ ,

if x ∈ A,

ψ (x) = ψ (I (x))

if x ∈ Mϕ \ A.

(A, δ0 ) , hence ψ (x) ≥ δ0 > 0. If If x ∈ / A ∪ Mϕ then there is δ0 > 0 such that x ∈ / cls SB x ∈ Mϕ \ A then I (x) ∈ / A ∪ Mϕ , hence ψ (x) = ψ (I (x)) > 0. We now prove property 4. It is If x ∈ A then σ x ∈ A, enough to show that ψ (σ x) ≤ ψ (x) for x ∈ M \ Mϕ \ A and σ ∈ S. since A is positively S-invariant by Theorem 3.2. Hence ψ (σ x) = 0 = ψ (x). If x ∈ / A ∪ Mϕ (A, δ) then x ∈ (A, δ) , by Lemma 5.1, and hence ψ (σ x) ≤ ψ (x). and σ x ∈ / cls SB / cls SB ε To show property 2, consider any ε > 0 and take δ = . Suppose that dA (x) ≤ δ. If x ∈ / A ∪ Mϕ 2 then ψ (x) ≤ dA (x) ≤ δ < ε. If x ∈ A then ψ (x) = 0 < ε. If x ∈ Mϕ \ A then I (x) ∈ / A ∪ Mϕ , by the hypothesis. Since A is closed, we can find y ∈ / A ∪ Mϕ and u ∈ Upc such that d (y, x) ≤ δ, ϕ (φ (y, u) , y, u) = x, and ϕ (t, y, u) ∈ / A ∪ Mϕ whenever 0 ≤ t ≤ φ (y, u). Then we have dA (y) ≤ ε and ψ (x) = ψ (I (x)) = ψ (I (ϕ (φ (y, u) , y, u))) = ψ (ϕ˜ (φ (y, u) , y, u)) ≤ ψ (y) ≤ dA (y) ≤ ε.

JID:YJDEQ AID:9564 /FLA

28

[m1+; v1.288; Prn:4/10/2018; 11:22] P.28 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

Finally, we show property 3. Let x ∈ M \ Mϕ \ A and ε > 0. If x ∈ A and y ∈ M then ψ (x) − ψ (y) = −ψ (y) / A ∪ Mϕ then ψ (x) > 0and there is η > 0 such that η > < ε. If x ∈ (A, η) . Since M \ cls SB (A, η) is a neighborhood of x, there is ψ (x) − ε and x ∈ / cls SB δ > 0 such that B (x, δ) ∩ cls SB (A, η) = ∅. If y ∈ M \ Mϕ \ A and d (x, y) < δ, it follows (A, η) and y ∈ that y ∈ / cls SB ϕ . Hence ψ (y) ≥ η > ψ (x) − ε. This proves that ψ is /A∪M lower semicontinuous in M \ Mϕ \ A . The converse follows by Theorem 5.4. 2 5.2. Continuous case We now consider a special case of impulsive control system in which continuous Lyapunov functionals can be provided. We assume the following hypothesis on the impulsive control system: 1. Each element x ∈ Mϕ satisfies ϕ-STC and hence φ is continuous on M \ Mϕ × U ([5]); 2. Mϕ ∩ I Mϕ = ∅; 3. For all x ∈ M, k ≥ 1, and u ∈ Upc , xk+ is defined and Mϕ+ xk+ , u = ∅. The hypothesis 3 implies that the control system has no equilibrium point, that is, there is no stationary trajectory. Then the results of this section are limited to this class of control systems. set. The following theorem characterizes closed S-stable Theorem 5.6. A closed set A ⊂ M is S-stable if and only if there exists a functional ψ : M → R+ with the following properties: 1. 2. 3. 4.

ψ is continuous in M \ Mϕ \ A ; For every > 0 there exists a δ > 0 such that ψ (x) ≥ δ if dA (x) ≥ and x ∈ / Mϕ ; For any sequence (wn ) in M such that wn → x ∈ A, one has ψ (wn ) → 0; ψ (ϕ (t, x, u)) ≤ ψ (x) if 0 ≤ t ≤ φ (x, u), x ∈ M \ Mϕ and u ∈ Upc , and ψ (I (x)) ≤ ψ (x) if x ∈ Mϕ .

Proof. Suppose that there exists a functional ψ : M → R+ satisfying properties 1, 2, 3, and 4.

Let > 0 and x ∈ A. Define μ = inf ψ (w) : w ∈ / Mϕ and dA (w) ≥ /2 . By property 2, there exists δ > 0 such that μ ≥ δ > 0. Therefore μ > 0. Since x ∈ A, we have that x ∈ int (A) or x ∈ ∂A. Suppose x ∈ int (A) and take α > 0 such that B (x, α) ⊂ A. Since ψ is continuous at x, there exists β > 0 such that |ψ (y) − ψ (x)| < μ for all y ∈ B (x, β). Take δ1 = min {α, β}. Then B (x, δ1 ) ⊂ A and |ψ (y) − ψ (x)| < μ for all y ∈ B (x, δ1 ). We claim that ψ (x) = 0. Indeed, take wn = x for all n ∈ N. Then wn → x, and hence ψ (wn ) → 0, by property 3. Since ψ is continuous at x, we have ψ (wn ) → ψ (x), and therefore ψ (x) = 0, as desired. we It follows that ψ (y) < μ for all y ∈ B (x, δ1 ). In order to prove that A is S-stable, show that SB (x, δ1 ) ⊂ B (A, ). Indeed, suppose to the contrary that there are t1 ∈ (0, +∞), ϕ (t1 ,z, u1 ) ∈ z ∈ B (x, δ1 ), and u1 ∈ Upc such that / B (A, ). Then dA ( ϕ (t1 , z, u1 )) ≥ > /2. ϕ (t1 , z, u1 )) ≥ μ. We now analyze / Mϕ , as Mϕ ∩ I Mϕ = ∅. Hence ψ ( Clearly, ϕ (t1 , z, u1 ) ∈ two cases: z ∈ / Mϕ and z ∈ Mϕ . ϕ (t, z, v)) = Suppose that z ∈ / Mϕ . Note that ψ (z) < μ as z ∈ B (x, δ1 ). Then we have ψ ( ψ (ϕ (t, z, v)) ≤ ψ (z) < μ for all t ∈ [0, φ (z, v)) and v ∈ Upc , by property 4. For t = φ (z, v), we have

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.29 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

29

ψ ( ϕ (t, z, v)) = ψ ( ϕ (φ (z, v) , z, v)) = ψ (I (z1 )) ≤ ψ (z1 ) = ψ (ϕ (φ (z, v) , z, v)) ≤ ψ (z) < μ. If t ∈ φ (z, v) , φ (z, v) + φ z1+ , v · s0 , we have ψ ( ϕ (t, z, v)) = ψ ϕ t − s0 , z1+ , v · s0 ≤ ψ z1+ = ψ (I (z1 )) < μ. We proceed in this manner obtaining ψ ( ϕ (t, z, v)) < μ for all t ≥ 0 and v ∈ Upc . In particular, (x, δ1 ) ⊂ B (A, ). we have ψ ( ϕ (t1 , z, u1 )) < μ, which is a contradiction. Hence SB Now, suppose that z ∈ Mϕ . We can find a τ > 0 such that τ < t1 and ϕ (τ, z, u1 ) = ϕ (τ, z, u1 ) ∈ ϕ (t, ϕ (τ, z, u1 ) , v)) ϕ (τ, z, u1 ) ∈ B (x, δ1 ) \ Mϕ . Since / Mϕ , we can prove that ψ ( < μ for all t ≥ 0 and v ∈ Upc , as we did before in the case z ∈ / Mϕ . In particular, we have ψ ( ϕ (t1 , z, u1 )) = ψ ( ϕ (τ, z, u1 ) , u1 · τ )) < μ, which is a contradiction. Hence ϕ (t1 − τ, (x, δ1 ) ⊂ B (A, ). SB Now, suppose x ∈ ∂A. As we did before, since ψ is continuous at x, we can find δ2 > 0 (x, δ2 ) ⊂ B (A, ). such that δ2 < and ψ (y) < μ for all y ∈ B (x, δ2 ). We claim that SB Indeed, suppose to the contrary that there are t2 ∈ (0, +∞), z ∈ B (x, δ2 ), and u2 ∈ Upc such that ϕ (t2 , z,u2 ) ∈ / B (A, ). Then we have dA ( ϕ (t2 , z, u2 )) ≥ . Clearly, ϕ (t2 , z, u2 ) ∈ / Mϕ , as Mϕ ∩ I Mϕ = ∅. Hence ψ ( ϕ (t2 , z, u2 )) ≥ μ. We then analyze when z ∈ / Mϕ and when z ∈ Mϕ . Suppose that z ∈ / Mϕ . Analogously, we can show that ψ ( ϕ (t, z, v)) < μ for all t ≥ 0 and v ∈ Upc . In particular, we have ψ ( ϕ (t2 , z, u2 )) < μ, which is a contradiction. Hence (x, δ2 ) ⊂ B (A, ). Suppose now that z ∈ Mϕ . We can find a τ > 0 such that ϕ (τ, z, u2 ) = SB ϕ (τ, z, u2 ) ∈ B (x, δ2 ) \ Mϕ . Since ϕ (τ, z, u2 ) ∈ ϕ (t, ϕ (τ, z, u2 ) , v)) / Mϕ , we can prove that ψ ( / Mϕ . In particular, we < μ for all t ≥ 0 and v ∈ Upc , as we did before in the case z ∈ have ψ ( ϕ (t2 , z, u2 )) = ψ ( ϕ (τ, z, u2 ) , u2 · τ )) < μ, which is a contradiction. Hence ϕ (t2 − τ, (x, δ2 ) ⊂ B (A, ). SB By Theorem 3.2, A is positively S-invariant. As to the converse, suppose that A is S-stable. Define the function ψ : M → [0, 1] by ⎧ ⎫ k−1 ⎨ dA ϕ t, xk+ , u · k−1 ⎬ i=0 si + : 0 ≤ t ≤ φ x ψ (x) = sup sup sup , u · s , i k ⎩ 1 + d ϕ t, x + , u · k−1 s ⎭ u∈U k≥0 A i i=0 i=0 k if x ∈ M \ Mϕ , ψ (x) = ψ (I (x)) ,

if x ∈ Mϕ .

It is easily seen that ψ (x) = 0 if and only if x ∈ A or I (x) ∈ A. This function satisfies the properties 1, 2, and 3. prove that ψ satisfies property 1. Indeed, since M \ Mϕ \ A = M \ Mϕ ∪ We firstly A ∩ Mϕ , we show that ψ is continuous in M \ Mϕ and in A ∩ Mϕ . Let x ∈ M \ Mϕ . We can find η > 0 such that B (x, η) ∩ Mϕ = ∅, as Mϕ is closed. Take a sequence (wn ) in M such that wn → x and take u ∈ U . We may assume that wn ∈ B (x, η) for all n. By the continuity of I in Mϕ and the continuity of φ in M \ Mϕ × U , we have + (wn )+ 1 = I (ϕ (φ (wn , u) , wn , u)) −→ I (ϕ (φ (x, u) , x, u)) = x1 .

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.30 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

30

Clearly, x1+ ∈ / Mϕ , as Mϕ ∩ I Mϕ = ∅. Then we can find η1 > 0 such that B x1+ , η1 ∩ Mϕ = ∅. + + + 1 1 Since (wn )+ n )1 ∈ B x1 , η1 for n > n0 . By using the no1 → x1 , there is n0 ∈ N such that (w tations of Section 2, we define sk = φ xk+ , u · k−1 i=0 si for all k ∈ N. Furthermore, for each k−1 + n ∈ N, we define sˆn k = φ (wn )k , u · i=0 sˆn i for all k ∈ N. By the continuity of φ + in M \ Mϕ × U , we have φ (wn )+ 1 , u · sˆn 0 → φ x1 , u · s0 , since sˆn 0 = φ (wn , un ) → φ (x, u) = s0 . Thus the value ) dA ϕ t, (wn )+ 1 , un · sˆn 0 + : 0 ≤ t ≤ φ (wn )1 , u · sˆn 0 sup 1 + dA ϕ t, (wn )+ 1 , un · sˆn 0

converges to the value ) + dA ϕ t, x1+ , u · s0 , 0 ≤ t ≤ φ x1 , u · s0 . sup 1 + dA ϕ t, x1+ , u · s0

Now note that + + (wn )+ 2 = I ϕ φ (wn )1 , u · sˆn 0 , (wn )1 , u · sˆn 0 → I ϕ φ x1+ , u · s0 , x1+ , u · s0 = x2+ . Clearly, x2+ ∈ / Mϕ as Mϕ ∩ I Mϕ = ∅. Then we can find an η2 > 0 such that B x2+ , η2 ∩ + + exists an n20 > 0 such that (wn )+ x 2 , η2 Mϕ = ∅ as Mϕ is closed. Since (wn )+ 2 → x2 , there 2 ∈B → , u · s ˆ + s ˆ for n > n20 . By the continuity of φ in M \ Mϕ × U , we have φ (wn )+ n n 2 0 1 + φ x2 , u · (s0 + s1 ) , since sˆn 0 + sˆn 1 = φ (wn , u) + φ (wn )+ → φ u) + , u · s ˆ (x, 1 n 0 + dA ϕ t, (wn )2 , u · sˆn 0 + sˆn 1 : 0 ≤ t ≤ φ x1+ , u · s0 = s0 + s1 . Thus the value sup 1 + dA ϕ t, (wn )+ 2 , u · sˆn 0 + sˆn 1 dA ϕ t, x2+ , u · (s0 + s1 ) + : converges to the value sup φ (wn )2 , u · sˆn 0 + sˆn 1 1 + dA ϕ t, x2+ , u · (s0 + s1 ) / Mϕ for all k ∈ N, we can proceed in this manner 0 ≤ t ≤ φ x2+ , u · (s0 + s1 ) . Since xk+ ∈ obtaining (wn )+ k

=I ϕ φ

+ ,u · −→ I ϕ φ xk−1

(wn )+ k−1 , u · k−2 i=0

and

k−2 k−2 + sˆn i , (wn )k−1 , u · sˆn i i=0

+ si , xk−1 ,u ·

k−2 i=0

i=0

si

= xk+

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.31 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

dA ϕ 0≤t≤φ

sup k−1 sˆn i (wn )+ k ,u· i=0

t, (wn )+ k ,u ·

1 + dA ϕ

−→

sup

0≤t≤φ xk+ ,u·

k−1

si

i=0

t, xk+ , u ·

k−1

sˆn

i

i=0

t, (wn )+ k ,u ·

dA ϕ

k−1

31

k−1

sˆn

i=0

i

si

i=0

1 + dA ϕ t, xk+ , u ·

k−1

. si

i=0

Hence

⎫ ⎪ ⎪ ⎪ sˆn i dA ϕ ⎪ ⎬ i=0 sup sup ⎪ k−1 k≥0 ⎪ ⎪ ⎪ k−1 + ⎪ ⎪ ⎪ ⎪ sˆn i ⎭ ⎩0≤t≤φ (wn )+k ,u· sˆn i 1 + dA ϕ t, (wn )k , u · ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

t, (wn )+ k ,u ·

i=0

k−1

i=0

⎫ k−1 ⎪ + ⎪ ⎪ dA ϕ t, xk , u · si ⎪ ⎬ i=0 −→ sup , sup ⎪ k−1 k≥0 ⎪ ⎪ ⎪ k−1 + ⎪ ⎪ ⎪ ⎪ si ⎭ ⎩0≤t≤φ xk+ ,u· si 1 + dA ϕ t, xk , u · ⎧ ⎪ ⎪ ⎪ ⎪ ⎨

i=0

i=0

and therefore ψ (wn ) → ψ (x). It follows that ψ is continuous in M \ Mϕ . We now prove that ψ is continuous in A ∩ Mϕ . Let x ∈ A ∩ Mϕ and take a sequence (x, δ) ⊂ B (A, ). zn → x. Note that ψ (x) = 0. For a given > 0, take δ > 0 such that SB There is n0 such that zn ∈ B δ) for every n > n . Then Sz ⊂ B ) for every n > n0 . (x, (A, 0 n k−1 k−1 + + Take u ∈ U . We have ϕ t, (zn )k , u · i=0 sˇn i = ϕ t, (zn )k , u · i=0 sˇn i ∈ B (A, ) ' k−1 + k−1 + and hence dA ϕ t, (zn )+ s ˇ , u · , u · < for all t ∈ 0, φ (z ) n n i=0 i=0 sˇn i , k k i k−1 k ∈ N, and n > n0 , where sˇn k = φ (zn )+ i=0 sˇn i for all k. Thus ψ (zn ) < , for k ,u · every n > n0 , and hence ψ (zn ) → 0 = ψ (x). Therefore ψ is continuous in A ∩ Mϕ . We now prove that ψ satisfies property 2. Take > 0 and x ∈ / Mϕ . Let δ = . If dA (x) ≥ 1+ dA (x) then ≥ δ and hence ψ (x) ≥ δ. For the property 3, assume that x ∈ A and x ∈ / Mϕ . We 1 + dA (x) can find δ > 0 such that B (x, δ) ∩ Mϕ = ∅. Take a sequence (pn ) in M such that pn → x. Then there exists an N > 0 such that pn ∈ B (x, δ) for n > N . By the continuity of ψ in M \ Mϕ \ A , we have ψ (pn ) → ψ (x). Now, suppose that x ∈ Mϕ . Take a sequence (zn ) in M \ Mϕ such that zn → x. By the continuity of ψ in M \ Mϕ \ A , we have ψ (zn ) → ψ (x) = 0. Take a sequence / Mϕ (yn ) in Mϕ such that yn → x. Note that I (yn) → I (x) as I is continuous. Clearly, I (yn ) ∈ for all n ∈ N and I (x) ∈ / Mϕ as Mϕ ∩ I Mϕ = ∅. By the continuity of ψ in M \ Mϕ \ A , we have ψ (I (yn )) → ψ (I (x)) and therefore ψ (yn ) → ψ (x) = 0.

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.32 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

32

Finally, we prove the property 4. If x ∈ Mϕ then ψ (I (x)) = ψ (x) by definition. Take x ∈ M \ Mϕ , u, v ∈ Upc , and s ∈ [0, φ (x, v)]. Let y = ϕ (s, x, v) and take w ∈ Upc given by w (t) =

v (t) , if t ≤ s, u (t − s) , if t > s.

We firstly suppose that s < φ (x, v). Then we have φ (x, w) > s. In fact, if φ (x, w) ≤ s we would have ϕ (φ (x, w) , x, v) = ϕ (φ (x, w) , x, w) ∈ Mϕ , and hence φ (x, v) ≤ φ (x, w) ≤ s < φ (x, v), a contradiction. We claim that φ (x, w) = φ (y, u) + s. Indeed, note that w · s (t) =

v · s (t) , u (t) ,

if t ≤ 0, if t > 0.

Since φ (x, w) − s > 0, we have ϕ (φ (x, w) − s, y, u) = ϕ (φ (x, w) − s, ϕ (s, x, w) , w · s) = ϕ (φ (x, w) , x, w) ∈ Mϕ and hence φ (y, u) ≤ φ (x, w) − s. On the other hand, since ϕ (φ (y, u) , y, u) ∈ Mϕ and ϕ (φ (y, u) , y, u) = ϕ (φ (y, u) + s, x, w), we have φ (x, w) ≤ φ (y, u) + s, and therefore φ (x, w) = φ (y, u) + s. Now consider the impulsive semitrajectories through x with re y and spect to u and w, respectively. Define s0 = φ (y, u), sˆ0 = φ (x, w), sk = φ yk+ , u · k−1 i=0 si , s ˆ and sˆk = φ xk+ , w · k−1 i=0 i for all k ≥ 1. We have sˆ0 = φ (x, w) = φ (y, u) + s = s0 + s. For t ≥ 0, it follows that w · sˆ0 (t) = w · s (s0 + t) = u · s0 (t). Analogously, we have w · k−1 i=0 sˆi = k−1 u · i=0 si for k ≥ 1. Moreover, we have y1+ = I (ϕ (s0 , y, u)) = I (ϕ (s0 , ϕ (s, x, w) , w · s)) = I ϕ sˆ0 , x, w = x1+ . Analogously, yk+ = xk+ for all k ≥ 1, and then sk = sˆk for all k ≥ 1. Thus, for 0 ≤ t ≤ φ y0+ , u , + we have ϕ t, y0 , u = ϕ t + s, x0+ , w , with s ≤ t + s ≤ φ x0+ , w . This implies that dA ϕ t, y0+ , u dA ϕ t, x0+ , w + ≤ + sup sup 0≤t≤φ y + ,u 1 + dA ϕ t, y0 , u 0≤t≤φ x + ,w 1 + dA ϕ t, x0 , w 0

0

and hence ⎛

⎞ dA ϕ t, yk+ , u · k−1 si i=0 ⎜ ⎟ sup sup ⎝ k−1 ⎠ + k≥0 0≤t≤φ y + ,u· k−1 s 1 + dA ϕ t, y , u · i=0 si k i=0 i k ⎛ ⎞ dA ϕ t, xk+ , w · k−1 sˆi i=0 ⎜ ⎟ sup ≤ sup ⎝ k−1 ⎠ . + k≥0 0≤t≤φ x + ,w· k−1 sˆ 1 + dA ϕ t, x , w · i=0 sˆi k i=0 i k Therefore ψ (ϕ (s, x, v)) = ψ (y) ≤ ψ (x).

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.33 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

33

Now we suppose that s = φ (x, v). Then ϕ (s, x, w) = ϕ (s, x, v) ∈ Mϕ , and therefore s = φ (x, w). This means that the impulsive semitrajectory through I (ϕ (s, x, v)) with respect to u coincides with the impulsive semitrajectory through x with respect to w from x1+ = I (ϕ (s, x, v)), since x1 = ϕ (s, x, w) = ϕ (s, x, v). It follows that ψ (I (ϕ (s, x, v))) ≤ ψ (x). By definition we have ψ (ϕ (s, x, v)) = ψ (I (ϕ (s, x, v))) ≤ ψ (x). 2 For the case of S-equistable set, we have the following result. Theorem 5.7. Assume that A ⊂ M is a closed set and I Mϕ \ A ⊂ (M \ A) \ Mϕ . Then A is S-equistable if and only if there exists a functional ψ : M → R+ with the following properties: 1. 2. 3. 4.

ψ is continuous in M \ Mϕ \ A ; ψ (x) = 0 for x ∈ A, ψ (x) > 0 for x ∈ / A ∪ Mϕ ; For every > 0 there exists a δ > 0 such that ψ (x) ≤ if dA (x) ≤ δ; ψ (ϕ (t, x, u)) ≤ ψ (x) if 0 ≤ t ≤ φ (x, u), x ∈ M \ Mϕ , and u ∈ Upc , and ψ (I (x)) ≤ ψ (x) if x ∈ Mϕ .

Proof. The proof is just a combination of Theorems 5.5 and 5.6, and then it is omitted. A complete proof can be found in the appendix. 2 stable set, we need the following lemma whose In order to prove the case of S-uniformly proof can be found in the appendix. Lemma 5.2. Assume that A ⊂ M is a closed set. Let ψ : M → R+ be a continuous function on M \ Mϕ \ A with the following properties: / Mϕ ; 1. For every > 0 there exists a δ > 0 such that ψ (x) ≥ δ if dA (x) ≥ and x ∈ 2. For every > 0 there exists a δ > 0 such that ψ (x) ≤ if dA (x) ≤ δ. Assume also ϕ (t, w, u)) ≤ ψ (w) for all t ≥ 0, w that there exists a δ > 0 such that ψ ( ∈ A, δ ⊂ cls B A, δ \ Mϕ , and u ∈ Upc . Then there exists δ > 0, 0 < δ ≤ δ , such that SB B A, δ . We have the following theorem for the case of closed S-uniformly stable set. Theorem 5.8. A closed set A ⊂ M is S-uniformly stable if and only if there exists a functional ψ : M → R+ with the following properties: 1. 2. 3. 4.

ψ is continuous in M \ Mϕ \ A ; For every > 0 there exists a δ > 0 such that ψ (x) ≥ δ if dA (x) ≥ and x ∈ / Mϕ ; For every > 0 there exists a δ > 0 such that ψ (x) ≤ if dA (x) ≤ δ; ψ (ϕ (t, x, u)) ≤ ψ (x) if 0 ≤ t ≤ φ (x, u), x ∈ M \ Mϕ , and u ∈ Upc , and ψ (I (x)) ≤ ψ (x) if x ∈ Mϕ .

Proof. The proof follows immediately from Lemma 5.2, by noticing that, for a given > 0, we have ψ ( ϕ (t, x, u)) ≤ ψ (x) for all t ≥ 0, x ∈ cls (B (A, )) \Mϕ , and u ∈ Upc . The reader can find the complete proof in the appendix. 2

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.34 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

34

We finally present a characterization of compact weak S-asymptotically stable set. -asymptotically stable if and only if there exists a Theorem 5.9. A compact set A ⊂ M is weak F functional ψ : M → R+ with the following properties: ψ is continuous in M \ Mϕ \ A ; For every > 0 there exists a δ > 0 such that ψ (x) ≤ if dA (x) ≤ δ; For every > 0 there exists a δ > 0 such that ψ (x) ≥ δ if dA (x) ≥ and x ∈ / Mϕ ; ψ (ϕ (t, x, u)) ≤ ψ (x) if 0 ≤ t ≤ φ (x, u), x ∈ M \ Mϕ , and u ∈ Upc , and ψ (I (x)) ≤ ψ (x) if x ∈ Mϕ ; ϕ (t, x, u)) → 0 as 5. There exists a δ > 0 such that if x ∈ B (A, δ) \ A and u ∈ Upc then ψ ( t → +∞.

1. 2. 3. 4.

Proof. Suppose that there exists a functional ψ : M → R+ satisfying properties 1, 2, 3, 4, and 5. By Theorem 5.8, it follows that A is S-uniformly stable. We need to prove that A is -weak attractor. By item 5, there exists a δ > 0 such that if x ∈ B (A, δ) \ A and u ∈ Upc an F then ψ ( ϕ (t, x, stability of exists δ > 0, δ < δ, 0 as t → +∞. By the S-uniform A, there u)) → . Indeed, suppose to the A, δ ⊂ B (A, δ). We claim that B A, δ ⊂ Aw A, F such that SB . Since A is positively S-invariant by Thecontrary that there exists an x ∈ B A, δ \ Aw A, F / A. Since x ∈ / Aw A, F , there is a pair , τ > 0 orem 3.2, we have A ⊂ Aw A, F , hence x ∈ such that S≥τ x ∩ B (A, ) = ∅. By taking u ∈ Upc , we have ϕ˜ (t, x, u) ∈ S≥τ x \ B (A, ) for all t ≥ τ . Note that ϕ˜ (t, x, u) ∈ / Mϕ for all t > 0, according to the construction of impulsive positive semitrajectories. By item 3, there is δ > 0 such that ψ (ϕ˜ (t, x, u)) ≥ δ for every t > 0. On the other ϕ (t, x, u)) → 0 as t → +∞, which is a contradiction. Hence by item 5, we have ψ ( hand, , and therefore A is an F -weak attractor. B A, δ ⊂ Aw A, F To prove the converse, consider the functional ψ defined as in Theorem 5.6. 2 It is interesting to observe that the Lyapunov functional is continuous in the whole phase space M provided Mϕ ⊂ A, in all Theorems 5.6, 5.7, 5.8, and 5.9. The following example applies the results of this section. Example 5.1. Consider the control affine system x˙ = X (x, u (t)) = X0 (x) + u (t) X1 (x) , u ∈ Upc = {u : R → [0, 4] : u piecewise constant} # on M = x = (x1 , x2 ) ∈ R2 : x = x12 + x22 ≥ 1 , where X0 and X1 are given by X0 (x1 , x2 ) = (−x2 , x1 ) , X1 (x1 , x2 ) =

1 − x2 (x1 , x2 ) , x2

with Mϕ = {(0, x2 ) : x2 ∈ [1, 2]} ,

I ((0, x2 )) = (x2 , 0) ,

for all x2 ∈ [1, 2] .

JID:YJDEQ AID:9564 /FLA

[m1+; v1.288; Prn:4/10/2018; 11:22] P.35 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

35

x − 1 Let A = x ∈ R2 : x = 1 and define the function ψ : M → [0, 1] by ψ (x) = . We x claim that ψ satisfies the four properties stated in Theorem 5.6 and therefore A is S-stable. Indeed, ψ satisfies property 1, since ψ is continuous (differentiable) in M and, in particular, in M \ Mϕ \ A . Now, take > 0 and x ∈ / Mϕ . Let δ = . If dA (x) ≥ then 1+ ψ (x) =

x − 1 dA (x) = δ. ≥ ≥ x 1 + dA (x) 1 +

This proves the property 2. For the property 3, take a sequence (wn ) in M such that wn → x ∈ A. Since x ∈ A, it follows that wn → 1 and hence ψ (wn ) → 0. Finally, we have the derivative x1 dψx (X (x, u (t))) = x3 = u (t) = u (t)

1 − x2 x2 1 − x2 u (t) x1 − x2 + u (t) x2 + x1 x2 x3 x2

x12 + x22 1 − x2 x3

x2

−

x1 x2 x1 x2 + 3 x x3

1 − x2 ≤ 0. x3

Hence, for x ∈ M \ Mϕ and u ∈ Upc , ψ (ϕ (t, x, u)) is decreasing for t ∈ [0, φ (x, u)]. If x ∈ Mϕ , we have ψ (I (x)) = ψ (x), and therefore ψ also satisfies the property 4 of Theorem 5.6. 6. Conclusion and last comments The proposal of this paper is to study Lyapunov stability of impulsive control system in the set up of semigroup actions. This is possible by considering a control affine system with piecewise constant control functions and compact convex control range. This type of control system determines a nonautonomous dynamical system, and then an impulsive control system can be defined by using the formulation established in [4,5]. We show that the resulting impulsive control system is determined by the action of its impulsive system semigroup. Then we study several concepts of Lyapunov stability for semigroup actions. Because of the general formulation with impulsive set and impulsive function, the invariance and stability of the impulsive control system may be completely different respectively from the invariance and stability of its adjacent control system. Then suitable conditions should be assumed to get a relationship between the dynamics of the impulsive system and the dynamics of the original system. Even with discontinuities, it is possible to reproduce some characterizations of stability for impulsive control systems, similarly to the conventional case. In fact, as expected from the semigroup point of view, the compact impulsive stable sets correspond to the compact sets with trivial impulsive prolongations. Moreover, each type of stability of closed sets is characterized by specific Lyapunov functional. The strategy of describing an impulsive control system as a semigroup action is possible in the nonautonomous paradigm of impulsive system. It would be interesting to know if there exist other formulations of impulsive control system which have a semigroup action interpretation. The ideas presented in this paper might be explored to study recursive and dispersive concepts of impulsive control systems ([20]). We think that Poisson instability and Lyapunov stability along impulsive semitrajectories are the ingredients to dispersiveness.

JID:YJDEQ AID:9564 /FLA

36

[m1+; v1.288; Prn:4/10/2018; 11:22] P.36 (1-36)

J.A. Souza, L.H. Takamoto / J. Differential Equations ••• (••••) •••–•••

Appendix A. Supplementary material Supplementary material related to this article can be found online at https://doi.org/10.1016/ j.jde.2018.09.033. References [1] A. Arutyunov, D. Karamzin, F.L. Pereira, A nondegenerate maximum principle for the impulsive control problem with state constraints, SIAM J. Control Optim. 43 (2005) 1812–1843. [2] A. Arutyunov, D. Karamzin, F.L. Pereira, On constrained impulsive control problems: controlling system jumps, J. Math. Sci. 165 (2010) 654–688. [3] A. Arutyunov, D. Karamzin, F.L. Pereira, A generalization of the impulsive control concept: controlling system jumps, Discrete Contin. Dyn. Syst. Ser. A 29 (2011) 403–415. [4] E.M. Bonotto, M.C. Bortolan, T. Caraballo, R. Collegari, Attractors for impulsive non-autonomous dynamical systems and their relations, J. Differential Equations 262 (2017) 3524–3550. [5] E.M. Bonotto, M.C. Bortolan, T. Caraballo, R. Collegari, Impulsive nonautonomous dynamical systems and impulsive cocycle attractors, Math. Methods Appl. Sci. 40 (2017) 1095–1113. [6] E.M. Bonotto, N.G. Grulha Jr., Lyapunov stability of closed sets in impulsive semidynamical systems, Electron. J. Differential Equations 2010 (2010), Paper No. 78, 18 pp. [electronic only]. [7] C.J. Braga Barros, V.H.L. Rocha, J.A. Souza, Lyapunov stability for semigroup actions, Semigroup Forum 88 (2014) 227–249. [8] C.J. Braga Barros, V.H.L. Rocha, J.A. Souza, Lyapunov stability and attraction under equivariant maps, Canad. J. Math. 67 (2015) 1247–1269. [9] C.J. Braga Barros, V.H.L. Rocha, J.A. Souza, On attractors and stability for semigroup actions and control systems, Math. Nachr. 289 (2016) 272–287. [10] A. Bressan, F. Rampazzo, Impulsive control systems with commutative vector fields, J. Optim. Theory Appl. 71 (1991) 67–83. [11] K. Ciesielski, On stability in impulsive dynamical systems, Bull. Pol. Acad. Sci. Math. 52 (2004) 81–91. [12] F. Colonius, W. Kliemann, The Dynamics of Control, Birkhäuser, Boston, 2000. [13] D. Karamzin, V. Oliveira, F.L. Pereira, G.N. Silva, On the properness of the extension of dynamic optimization problems to allow impulsive controls, ESAIM Control Optim. Calc. Var. 21 (2015) 867–875. [14] C. Kawan, O.G. Rocio, A.J. Santana, On topological conjugacy of left invariant flows on semisimple and affine Lie groups, Proyecciones 30 (2011) 175–188. [15] V. Lakshmikantham, D.D. Ba˘ınov, P.S. Simeonov, Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, vol. 6, World Scientific Publishing Co. Inc., Teaneck, NJ, 1989. [16] F.L. Pereira, G.N. Silva, Necessary conditions of optimality for vector-valued impulsive control problems, Systems Control Lett. 40 (2000) 205–215. [17] F.L. Pereira, G.N. Silva, Stability for impulsive control systems, Dyn. Syst. 17 (2002) 421–434. [18] F.L. Pereira, G.N. Silva, Lyapunov stability of measure driven differential inclusions, J. Differential Equations 40 (2004) 1122–1130. [19] S.A. Raminelli, J.A. Souza, Global attractors for semigroup actions, J. Math. Anal. Appl. 407 (2013) 316–327. [20] J.A. Souza, H.V.M. Tozatti, Prolongational limit sets of control systems, J. Differential Equations 254 (2013) 2183–2195. [21] S. Tang, R.A. Cheke, State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J. Math. Biol. 50 (2005) 257–292. [22] T. Yang, Impulsive Control Theory, Lecture Notes in Control and Information Sciences, vol. 272, Springer-Verlag, Berlin, 2001. [23] R. Zhang, Z. Xu, S.X. Yang, X. He, Generalized synchronization via impulsive control, Chaos Solitons Fractals 38 (2008) 97–105.

Copyright © 2022 COEK.INFO. All rights reserved.