haddad.gatech.edu · 224 s.g. nersesov, w.m. haddad / nonlinear analysis: hybrid systems 1 (2007)...

Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243www.elsevier.com/locate/nahs

Control vector Lyapunov functions for large-scale impulsivedynamical systems

Sergey G. Nersesova,∗, Wassim M. Haddadb,1

a Department of Mechanical Engineering, Villanova University, Villanova, PA 19085-1681, United Statesb School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, United States

Received 15 May 2006; accepted 15 May 2006

Abstract

Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory in order to enlarge theclass of Lyapunov functions that can be used for analyzing system stability. In this paper, we provide generalizations to therecent extensions of vector Lyapunov theory for continuous-time systems to address stability and control design of impulsivedynamical systems via vector Lyapunov functions. Specifically, we provide a generalized comparison principle involving hybridcomparison dynamics that are dependent on the comparison system states as well as the nonlinear impulsive dynamical systemstates. Furthermore, we develop stability results for impulsive dynamical systems that involve vector Lyapunov functions andhybrid comparison inequalities. Based on these results, we show that partial stability for state-dependent impulsive dynamicalsystems can be addressed via vector Lyapunov functions. Furthermore, we extend the recently developed notion of control vectorLyapunov functions to impulsive dynamical systems. Using control vector Lyapunov functions, we construct a universal hybriddecentralized feedback stabilizer for a decentralized affine in the control nonlinear impulsive dynamical system that possessesguaranteed gain and sector margins in each decentralized input channel. These results are then used to develop hybrid decentralizedcontrollers for large-scale impulsive dynamical systems with robustness guarantees against full modeling and input uncertainty.c© 2006 Elsevier Ltd. All rights reserved.

Keywords: Vector Lyapunov functions; Hybrid comparison principle; Partial stability; Control vector Lyapunov functions; Hybrid decentralizedcontrol; Large-scale impulsive systems

1. Introduction

The mathematical descriptions of many hybrid dynamical systems can be characterized by impulsive differentialequations [1–6]. Impulsive dynamical systems can be viewed as a subclass of hybrid systems and consist ofthree elements—namely, a continuous-time differential equation, which governs the motion of the dynamicalsystem between impulsive or resetting events; a difference equation, which governs the way the system states areinstantaneously changed when a resetting event occurs; and a criterion for determining when the states of the systemare to be reset. Since impulsive systems can involve impulses at variable times, they are in general time-varying

∗ Corresponding author. Tel.: +1 610 519 8977; fax: +1 610 519 7312.E-mail addresses: [email protected] (S.G. Nersesov), [email protected] (W.M. Haddad).

1 Tel.: +1 404 894 1078; fax: +1 404 894 2760.

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224 S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243

systems, wherein the resetting events are both a function of time and the system’s state. In the case where the resettingevents are defined by a prescribed sequence of times which are independent of the system state, the equations areknown as time-dependent differential equations [1,2,4,7–9]. Alternatively, in the case where the resetting events aredefined by a manifold in the state space that is independent of time, the equations are autonomous and are known asstate-dependent differential equations [1,2,4,7–9].

Even though impulsive dynamical systems were first formulated by Mil’man and Myshkis [10,11], the fundamentaltheory of impulsive differential equations is developed in the monographs by Bainov, Lakshmikantham, Perestyuk,Samoilenko, and Simeonov [1,2,4,5,12]. These monographs develop qualitative solution properties, existence ofsolutions, asymptotic properties of solutions, and stability theory of impulsive dynamical systems. In the recent seriesof papers [8,9,13,14], stability, dissipativity, and optimality results have been developed for impulsive dynamicalsystems which include invariant set stability theorems, partial stability, dissipativity theory, and optimal controldesign.

In this paper, we provide generalizations to the recent extensions of vector Lyapunov theory for continuous-time systems [15] to address stability and control design of impulsive dynamical systems via vector Lyapunovfunctions. Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory inorder to enlarge the class of Lyapunov functions that can be used for analyzing system stability. Lyapunov methodshave also been used by control system designers to obtain stabilizing feedback controllers for nonlinear systems.In particular, for smooth feedback, Lyapunov-based methods were inspired by Jurdjevic and Quinn [16] who givesufficient conditions for smooth stabilization based on the ability of constructing a Lyapunov function for the closed-loop system. More recently, Artstein [17] introduced the notion of a control Lyapunov function whose existenceguarantees a feedback control law which globally stabilizes a nonlinear dynamical system. In general, the feedbackcontrol law is not necessarily smooth, but can be guaranteed to be at least continuous at the origin in addition to beingsmooth everywhere else. Even though for certain classes of nonlinear dynamical systems a universal constructionof a feedback stabilizer can be obtained using control Lyapunov functions [18,19], there does not exist a unifiedprocedure for finding a Lyapunov function candidate that will stabilize the closed-loop system for general nonlinearsystems.

In an attempt to simplify the construction of Lyapunov functions for the analysis and control design of nonlineardynamical systems, several researchers have resorted to vector Lyapunov functions as an alternative to scalarLyapunov functions. Vector Lyapunov functions were first introduced by Bellman [20] and Matrosov [21], and furtherdeveloped in [22–25], with [22–24,26–29] exploiting their utility for analyzing large-scale systems. The use of vectorLyapunov functions in dynamical system theory offers a very flexible framework since each component of the vectorLyapunov function can satisfy less rigid requirements as compared to a single scalar Lyapunov function. Weakeningthe hypothesis on the Lyapunov function enlarges the class of Lyapunov functions that can be used for analyzingsystem stability. In particular, each component of a vector Lyapunov function need not be positive definite with anegative or even negative-semidefinite derivative. Alternatively, the time derivative of the vector Lyapunov functionneed only satisfy an element-by-element inequality involving a vector field of a certain comparison system. Since inthis case the stability properties of the comparison system imply the stability properties of the dynamical system, theuse of vector Lyapunov theory can significantly reduce the complexity (i.e., dimensionality) of the dynamical systembeing analyzed. Extensions of vector Lyapunov function theory that include relaxed conditions on standard vectorLyapunov functions as well as matrix Lyapunov functions appear in [28–30].

The results of this paper closely parallel those in [15] and include a generalized comparison principle involvinghybrid comparison dynamics that are dependent on the comparison system states as well as the nonlinear impulsivedynamical system states. Next, we develop stability theorems based on hybrid comparison inequalities as well aspartial stability results for impulsive systems using vector Lyapunov functions. Furthermore, we extend the newlydeveloped notion of control vector Lyapunov functions presented in [15] to impulsive dynamical systems and showthat in the case of a scalar comparison system the definition of a control vector Lyapunov function collapses into acombination of the classical definition of a control Lyapunov function for continuous-time dynamical systems givenin [17] and the definition of a control Lyapunov function for discrete-time dynamical systems given in [31,32]. Inaddition, using control vector Lyapunov functions, we present a universal hybrid decentralized feedback stabilizer fora decentralized affine in the control nonlinear impulsive dynamical system with guaranteed gain and sector margins.These results are then used to develop hybrid decentralized controllers for large-scale impulsive dynamical systemswith robustness guarantees against full modeling and input uncertainty.

S.G. Nersesov, W.M. Haddad / Nonlinear Analysis: Hybrid Systems 1 (2007) 223–243 225

2. Mathematical preliminaries

In this section we introduce notation and definitions needed for developing stability analysis and synthesis resultsfor nonlinear impulsive dynamical systems via vector Lyapunov functions. Let R denote the set of real numbers, Z+

denote the set of nonnegative integers, Rn denote the set of n × 1 column vectors, and (·)T denote transpose. Forv ∈ Rq we write v ≥≥ 0 (respectively, v � 0) to indicate that every component of v is nonnegative (respectively,positive). In this case, we say that v is nonnegative or positive, respectively. Let Rq

+ and Rq+ denote the nonnegative

and positive orthants of Rq , that is, if v ∈ Rq , then v ∈ Rq+ and v ∈ Rq

+ are equivalent, respectively, to v ≥≥ 0 and

v � 0. Furthermore, let◦

D andD denote the interior and the closure of the setD ⊂ Rn , respectively. Finally, we write‖·‖ for an arbitrary spatial vector norm in Rn , V ′(x) for the Frechet derivative of V at x , Bε(α), α ∈ Rn , ε > 0, for theopen ball centered at α with radius ε, e ∈ Rq for the ones vector, that is, e , [1, . . . , 1]

T, and x(t) → M as t → ∞

to denote that x(t) approaches the set M, that is, for each ε > 0 there exists T > 0 such that dist (x(t),M) < ε forall t > T , where dist (p,M) , infx∈M ‖p − x‖.

The following definition introduces the notion of classW functions involving quasimonotone increasing functions.

Definition 2.1 ([26]). A function w = [w1, . . . , wq ]T

: Rq×V → Rq , where V ⊆ Rs , is of classWc if for every fixed

y ∈ V ⊆ Rs , wi (z′, y) ≤ wi (z′′, y), i = 1, . . . , q , for all z′, z′′∈ Rq such that z′

j ≤ z′′

j , z′

i = z′′

i , j = 1, . . . , q, i 6= j ,where zi denotes the i th component of z.

If w(·, y) ∈ Wc we say that w satisfies the Kamke condition [33,34]. Note that if w(z, y) = W (y)z, whereW : V → Rq×q , then the function w(·, y) is of class Wc if and only if W (y) is essentially nonnegative for all y ∈ V ,that is, all the off-diagonal entries of the matrix function W (·) are nonnegative. Furthermore, note that it follows fromDefinition 2.1 that any scalar (q = 1) function w(z, y) is of class Wc.

Finally, we introduce the notion of class Wd functions involving nondecreasing functions.

Definition 2.2 ([25]). A function w = [w1, . . . , wq ]T

: Rq× V → Rq , where V ⊆ Rs , is of class Wd if for every

fixed y ∈ V ⊆ Rs , w(z′, y) ≤≤ w(z′′, y) for all z′, z′′∈ Rq such that z′

≤≤ z′′.

If w(z, y) = W (y)z, where W : V → Rq×q , then the function w(·, y) is of class Wd if and only if W (y) isnonnegative for all y ∈ V , that is, all the entries of the matrix function W (·) are nonnegative. Note that if w(·, y) ∈ Wd,then w(·, y) ∈ Wc.

Next, consider the nonlinear comparison system given by

z(t) = w(z(t), y(t)), z(t0) = z0, t ∈ Iz0 , (1)

where z(t) ∈ Q ⊆ Rq , t ∈ Iz0 , is the comparison system state vector, y : T → V ⊆ Rs is a given continuousfunction, Iz0 ⊆ T ⊆ R+ is the maximal interval of existence of a solution z(t) of (1), Q is an open set, 0 ∈ Q, andw : Q× V → Rq . We assume that w(·, y(t)) is continuous in t and satisfies the Lipschitz condition

‖w(z′, y(t)) − w(z′′, y(t))‖ ≤ L‖z′− z′′

‖, t ∈ T , (2)

for all z′, z′′∈ Bδ(z0), where δ > 0 and L > 0 is a Lipschitz constant. Hence, it follows from Theorem 2.2 of [35]

that there exists τ > 0 such that (1) has a unique solution over the time interval [t0, t0 + τ ].

Theorem 2.1 ([15]). Consider the nonlinear comparison system (1). Assume that the function w : Q × V → Rq iscontinuous and w(·, y) is of classWc. If there exists a continuously differentiable vector function V = [v1, . . . , vq ]

T:

Iz0 → Q such that

V (t) � w(V (t), y(t)), t ∈ Iz0 , (3)

then V (t0) � z0, z0 ∈ Q, implies

V (t) � z(t), t ∈ Iz0 , (4)

where z(t), t ∈ Iz0 , is the solution to (1).

Next, we present a stronger version of Theorem 2.1 where the strict inequalities are replaced by soft inequalities.


Theorem 2.2 ([15]). Consider the nonlinear comparison system (1). Assume that the function w : Q × V → Rq iscontinuous and w(·, y) is of class Wc. Let z(t), t ∈ Iz0 , be the solution to (1) and [t0, t0 + τ ] ⊆ Iz0 be a compactinterval. If there exists a continuously differentiable vector function V : [t0, t0 + τ ] → Q such that

V (t) ≤≤ w(V (t), y(t)), t ∈ [t0, t0 + τ ], (5)

then V (t0) ≤≤ z0, z0 ∈ Q, implies V (t) ≤≤ z(t), t ∈ [t0, t0 + τ ].

Next, consider the nonlinear dynamical system given by

x(t) = f (x(t)), x(t0) = x0, t ∈ Ix0 , (6)

where x(t) ∈ D ⊆ Rn, t ∈ Ix0 , is the system state vector, Ix0 is the maximal interval of existence of a solution x(t)of (6), D is an open set, 0 ∈ D, and f (·) is Lipschitz continuous on D. The following result is a direct consequenceof Theorem 2.2.

Corollary 2.1 ([15]). Consider the nonlinear dynamical system (6). Assume there exists a continuously differentiablevector function V : D → Q ⊆ Rq such that

V ′(x) f (x) ≤≤ w(V (x), x), x ∈ D, (7)

where w : Q×D → Rq is a continuous function, w(·, x) ∈ Wc, and

z(t) = w(z(t), x(t)), z(t0) = z0, t ∈ Iz0,x0 , (8)

has a unique solution z(t), t ∈ Iz0,x0 , where x(t), t ∈ Ix0 , is a solution to (6). If [t0, t0 + τ ] ⊆ Ix0 ∩ Iz0,x0 is acompact interval, then V (x0) ≤≤ z0, z0 ∈ Q, implies V (x(t)) ≤≤ z(t), t ∈ [t0, t0 + τ ].

If in (6) f : Rn→ Rn is globally Lipschitz continuous, then (6) has a unique solution x(t) for all t ≥ t0. A more

restrictive sufficient condition for global existence and uniqueness of solutions to (6) is continuous differentiabilityof f : Rn

→ Rn and uniform boundedness of f ′(x) on Rn . Note that if the solutions to (6) and (8) are globallydefined for all x0 ∈ D and z0 ∈ Q, then the result of Corollary 2.1 holds for any arbitrarily large but compact interval[t0, t0 + τ ] ⊂ R+. For the remainder of this paper we assume that the solutions to the systems (6) and (8) are definedfor all t ≥ t0. Continuous differentiability of f (·) and w(·, ·) provides a sufficient condition for the existence anduniqueness of solutions to (6) and (8) for all t ≥ t0.

3. Stability of impulsive systems via vector Lyapunov functions

In this section, we consider state-dependent impulsive dynamical systems [8] given by

x(t) = fc(x(t)), x(t0) = x0, x(t) 6∈ Z, t ∈ Ix0 , (9)

∆x(t) = fd(x(t)), x(t) ∈ Z, (10)

where x(t) ∈ D ⊆ Rn , t ∈ Ix0 , is the system state vector, Ix0 is the maximal interval of existence of a solution x(t)to (9) and (10), D is an open set, 0 ∈ D, fc : D → Rn is Lipschitz continuous and satisfies fc(0) = 0, fd : D → Rn

is continuous, ∆x(t) , x(t+) − x(t), and Z ⊂ D ⊆ Rn is the resetting set. For a particular trajectory x(t), t ≥ 0, welet tk = τk(x0), x0 ∈ D, denote the kth instant of time at which x(t) intersects Z . To ensure the well-posedness of theresetting times we make the following assumptions [8]:

A1. If x(t) ∈ Z \ Z , then there exists ε > 0 such that, for all 0 < δ < ε, x(t + δ) 6∈ Z .A2. If x(tk) ∈ ∂Z ∩Z , then the system states reset to x+(tk) , x(tk)+ fd(x(tk)), according to the resetting law (10),

which serves as the initial condition for the continuous-time dynamics (9).

Assumption A1 ensures that, if a trajectory reaches the closure of Z at a point that does not belong to Z , then thetrajectory must be directed away fromZ , that is, a trajectory cannot enterZ through a point that belongs to the closureof Z but not to Z . Furthermore, A2 ensures that when a trajectory intersects the resetting set Z , it instantaneouslyexits Z . Finally, we note that if x0 ∈ Z then the system initially resets to x+

0 = x0 + fd(x0) 6∈ Z , which serves as theinitial condition for continuous-time dynamics (9). Since the resetting times are well defined and distinct, and sincethe solution to (9) exists and is unique, it follows that the solution of the impulsive dynamical system (9) and (10)


also exists and is unique over a forward time interval. However, it is important to note that the analysis of impulsivedynamical systems can be quite involved. In particular, such systems can exhibit Zenoness and beating, as well asconfluence, wherein solutions exhibit infinitely many resettings in a finite time, encounter the same resetting surfacea finite or infinite number of times in zero time, and coincide after a certain point in time. In this paper we allow forthe possibility of confluence and Zeno solutions; however, A2 precludes the possibility of beating. Furthermore, sincenot every bounded solution of an impulsive dynamical system over a forward time interval can be extended to infinitydue to Zeno solutions, we assume that existence and uniqueness of solutions are satisfied in forward time. For detailssee [1,2,4].

The next result presents a generalization of the comparison principle given in Corollary 2.1 to impulsive dynamicalsystems.

Theorem 3.1. Consider the impulsive dynamical system (9) and (10). Assume there exists a continuouslydifferentiable vector function V : D → Q ⊆ Rq such that

V ′(x) fc(x) ≤≤ wc(V (x), x), x 6∈ Z, (11)

V (x + fd(x)) ≤≤ V (x) + wd(V (x), x), x ∈ Z, (12)

where wc : Q× D → Rq and wd : Q× Z → Rq are continuous functions, wc(·, x) ∈ Wc, wd(·, x) ∈ Wd, and thecomparison impulsive dynamical system

z(t) = wc(z(t), x(t)), z(t0) = z0, x(t) 6∈ Z, t ∈ Iz0,x0 , (13)

∆z(t) = wd(z(t), x(t)), x(t) ∈ Z, (14)

has a unique solution z(t), t ∈ Iz0,x0 , where x(t), t ∈ Ix0 , is a solution to (9) and (10). If [t0, t0 + τ ] ⊆ Ix0 ∩ Iz0,x0

is a compact interval, then

V (x0) ≤≤ z0, z0 ∈ Q, (15)

implies

V (x(t)) ≤≤ z(t), t ∈ [t0, t0 + τ ]. (16)

Proof. Without loss of generality, let x0 6∈ Z , x0 ∈ D. If x0 ∈ Z , then by Assumption A2, x0 + fd(x0) 6∈ Zserves as the initial condition for the continuous-time dynamics. If for x0 6∈ Z the solution x(t) 6∈ Z for allt ∈ [t0, t0 + τ ], then the result follows from Corollary 2.1. Next, suppose the interval [t0, t0 + τ ] contains the resettingtimes τk(x0) < τk+1(x0), k ∈ {1, 2, . . . , m}. Consider the compact interval [t0, τ1(x0)] and let V (x0) ≤≤ z0. Then itfollows from (11) and Corollary 2.1 that

V (x(t)) ≤≤ z(t), t ∈ [t0, τ1(x0)], (17)

where z(t), t ∈ Iz0 , is the solution to (13). Now, since wd(·, x) ∈ Wd it follows from (12) and (17) that

V (x(τ+

1 (x0))) ≤≤ V (x(τ1(x0))) + wd(V (x(τ1(x0))), x(τ1(x0)))

≤≤ z(τ1(x0)) + wd(z(τ1(x0)), x(τ1(x0)))

= z(τ+

1 (x0)). (18)

Consider the compact interval [τ+

1 (x0), τ2(x0)]. Since V (x(τ+

1 (x0))) ≤≤ z(τ+

1 (x0)), it follows from (11) that

V (x(t)) ≤≤ z(t), t ∈ [τ+

1 (x0), τ2(x0)]. (19)

Repeating the above arguments for t ∈ [τ+

k (x0), τk+1(x0)], k = 3, . . . , m, yields (16). Finally, in the case ofinfinitely many resettings over the time interval [t0, t0 + τ ], let limk→∞ τk(x0) = τ∞(x0) ∈ (t0, t0 + τ ]. In this case,[t0, τ∞(x0)] = [t0, τ1(x0)] ∪

[⋃∞

k=1[τk(x0), τk+1(x0)]]. Repeating the above arguments, the result can be shown for

the interval [t0, τ∞(x0)]. �

Note that if the solutions to (9), (10), (13) and (14) are globally defined for all x0 ∈ D and z0 ∈ Q, then the result ofTheorem 3.1 holds for any arbitrarily large but compact interval [t0, t0 + τ ] ⊂ R+. For the remainder of this paper we


assume that the solutions to the systems (9), (10), (13) and (14) are defined for all t ≥ t0. Next, consider the cascadenonlinear impulsive dynamical system given by

z(t) = wc(z(t), x(t)), z(t0) = z0, x(t) 6∈ Z, t ≥ t0, (20)

x(t) = fc(x(t)), x(t0) = x0, x(t) 6∈ Z, (21)

∆z(t) = wd(z(t), x(t)), x(t) ∈ Z, (22)

∆x(t) = fd(x(t)), x(t) ∈ Z, (23)

where z0 ∈ Q ⊆ Rq , x0 ∈ D ⊆ Rn , [zT(t), xT(t)]T, t ≥ t0, is the solution to (20)–(23), wc : Q × D → Rq andwd : Q × Z → Rq are continuous, wc(·, x) ∈ Wc, wd(·, x) ∈ Wd, wc(0, 0) = 0, fc : D → Rn is Lipschitzcontinuous on D, fc(0) = 0, and fd : D → Rn is continuous. The following definition introduces several types ofpartial stability of the nonlinear state-dependent impulsive dynamical system (20)–(23).

Definition 3.1 ([6]). (i) The nonlinear impulsive dynamical system (20)–(23) is Lyapunov stable with respect to z if,for every ε > 0 and x0 ∈ Rn , there exists δ = δ(ε, x0) > 0 such that ‖z0‖ < δ implies that ‖z(t)‖ < ε for all t ≥ 0.

(ii) The nonlinear impulsive dynamical system (20)–(23) is Lyapunov stable with respect to z uniformly in x0 if, forevery ε > 0, there exists δ = δ(ε) > 0 such that ‖z0‖ < δ implies that ‖z(t)‖ < ε for all t ≥ 0 and for all x0 ∈ Rn .

(iii) The nonlinear impulsive dynamical system (20)–(23) is asymptotically stable with respect to z if it is Lyapunovstable with respect to z and, for every x0 ∈ Rn , there exists δ = δ(x0) > 0 such that ‖z0‖ < δ implies thatlimt→∞ z(t) = 0.

(iv) The nonlinear impulsive dynamical system (20)–(23) is asymptotically stable with respect to z uniformly inx0 if it is Lyapunov stable with respect to z uniformly in x0 and there exists δ > 0 such that ‖z0‖ < δ implies thatlimt→∞ z(t) = 0 uniformly in z0 and x0 for all x0 ∈ Rn .

(v) The nonlinear impulsive dynamical system (20)–(23) is globally asymptotically stable with respect to z if it isLyapunov stable with respect to z and limt→∞ z(t) = 0 for all z0 ∈ Rq and x0 ∈ Rn .

(vi) The nonlinear impulsive dynamical system (20)–(23) is globally asymptotically stable with respect to zuniformly in x0 if it is Lyapunov stable with respect to z uniformly in x0 and limt→∞ z(t) = 0 uniformly in z0and x0 for all z0 ∈ Rq and x0 ∈ Rn .

(vii) The nonlinear impulsive dynamical system (20)–(23) is exponentially stable with respect to z uniformly in x0if there exist scalars α, β, δ > 0 such that ‖z0‖ < δ implies that ‖z(t)‖ ≤ α‖z0‖e−βt , t ≥ 0, for all x0 ∈ Rn .

(viii) The nonlinear impulsive dynamical system (20)–(23) is globally exponentially stable with respect to zuniformly in x0 if there exist scalars α, β > 0 such that ‖z(t)‖ ≤ α‖z0‖e−βt , t ≥ 0, for all z0 ∈ Rq and x0 ∈ Rn .

Theorem 3.2. Consider the impulsive dynamical system (9) and (10). Assume that there exist a continuouslydifferentiable vector function V : D → Q ∩ Rq

+ and a positive vector p ∈ Rq+ such that V (0) = 0, the scalar

function v : D → R+ defined by v(x) , pTV (x), x ∈ D, is such that v(x) > 0, x 6= 0, and

V ′(x) fc(x) ≤≤ wc(V (x), x), x 6∈ Z, (24)

V (x + fd(x)) ≤≤ V (x) + wd(V (x), x), x ∈ Z, (25)

where wc : Q×D → Rq and wd : Q× Z → Rq are continuous, wc(·, x) ∈ Wc, wd(·, x) ∈ Wd, and wc(0, 0) = 0.Then the following statements hold:

(i) If the nonlinear impulsive dynamical system (20)–(23) is Lyapunov stable with respect to z uniformly in x0, thenthe zero solution x(t) ≡ 0 to (9) and (10) is Lyapunov stable.

(ii) If the nonlinear impulsive dynamical system (20)–(23) is asymptotically stable with respect to z uniformly in x0,then the zero solution x(t) ≡ 0 to (9) and (10) is asymptotically stable.

(iii) If D = Rn , Q = Rq , v : Rn→ R+ is radially unbounded, and the nonlinear impulsive dynamical system

(20)–(23) is globally asymptotically stable with respect to z uniformly in x0, then the zero solution x(t) ≡ 0 to(9) and (10) is globally asymptotically stable.

(iv) If there exist constants ν ≥ 1, α > 0, and β > 0 such that v : D → R+ satisfies

α‖x‖ν

≤ v(x) ≤ β‖x‖ν, x ∈ D, (26)

and the nonlinear impulsive dynamical system (20)–(23) is exponentially stable with respect to z uniformly in x0,then the zero solution x(t) ≡ 0 to (9) and (10) is exponentially stable.


(v) If D = Rn , Q = Rq , there exist constants ν ≥ 1, α > 0, and β > 0 such that v : Rn→ R+ satisfies (26), and

the nonlinear impulsive dynamical system (20)–(23) is globally exponentially stable with respect to z uniformlyin x0, then the zero solution x(t) ≡ 0 to (9) and (10) is globally exponentially stable.

Proof. Assume there exist a continuously differentiable vector function V : D → Q ∩ Rq+ and a positive vector

p ∈ Rq+ such that v(x) = pTV (x), x ∈ D, is positive definite, that is, v(0) = 0 and v(x) > 0, x 6= 0. Since

v(x) = pTV (x) ≤ maxi=1,...,q{pi }eTV (x), x ∈ D, where e , [1, . . . , 1]T, the function eTV (x), x ∈ D, is also

positive definite. Thus, there exist r > 0 and class K functions α, β : [0, r ] → R+ such that Br (0) ⊂ D and

α(‖x‖) ≤ eTV (x) ≤ β(‖x‖), x ∈ Br (0). (27)

(i) Let ε > 0 and choose 0 < ε < min{ε, r}. It follows from Lyapunov stability of the nonlinear impulsivedynamical system (20)–(23) with respect to z uniformly in x0 that there exists µ = µ(ε) = µ(ε) > 0 such that if‖z0‖1 < µ, where ‖z‖1 ,

∑qi=1 |zi | and zi is the i th component of z, then ‖z(t)‖1 < α(ε), t ≥ t0, for any x0 ∈ D.

Now, choose z0 = V (x0) ≥≥ 0, x0 ∈ D. Since V (x), x ∈ D, is continuous, the function eTV (x), x ∈ D, is alsocontinuous. Hence, for µ = µ(ε) > 0 there exists δ = δ(µ(ε)) = δ(ε) > 0 such that δ < ε and if ‖x0‖ < δ, theneTV (x0) = eTz0 = ‖z0‖1 < µ, which implies that ‖z(t)‖1 < α(ε), t ≥ t0. Now, with z0 = V (x0) ≥≥ 0, x0 ∈ D,and the assumption that wc(·, x) ∈ Wc and wd(·, x) ∈ Wd, it follows from (24) and (25), and Theorem 3.1 that0 ≤≤ V (x(t)) ≤≤ z(t) on any compact interval [t0, t0 + τ ], and hence, eTz(t) = ‖z(t)‖1, [t0, t0 + τ ]. Let τ > t0 besuch that x(t) ∈ Br (0), t ∈ [t0, t0 + τ ] for all x0 ∈ Bδ(0). Thus, using (27), it follows that for ‖x0‖ < δ,

α(‖x(t)‖) ≤ eTV (x(t)) ≤ eTz(t) < α(ε), t ∈ [t0, t0 + τ ], (28)

which implies ‖x(t)‖ < ε < ε, t ∈ [t0, t0 + τ ]. Now, suppose, ad absurdum, that for some x0 ∈ Bδ(0) there existst > t0 + τ such that ‖x(t)‖ ≥ ε. Then, for z0 = V (x0) and the compact interval [t0, t] it follows from (24) and (25),and Theorem 3.1 that V (x(t)) ≤≤ z(t), which implies that α(ε) ≤ α(‖x(t)‖) ≤ eTV (x(t)) ≤ eTz(t) < α(ε). This isa contradiction, and hence, for a given ε > 0 there exists δ = δ(ε) > 0 such that for all x0 ∈ Bδ(0), ‖x(t)‖ < ε, t ≥ t0,which implies Lyapunov stability of the zero solution x(t) ≡ 0 to (9) and (10).

(ii) It follows from (i) and the asymptotic stability of the nonlinear impulsive dynamical system (20)–(23) withrespect to z uniformly in x0 that the zero solution x(t) ≡ 0 to (9) and (10) is Lyapunov stable, and there exists µ > 0such that if ‖z0‖1 < µ, then limt→∞ z(t) = 0 for any x0 ∈ D. As in (i), choose z0 = V (x0) ≥≥ 0, x0 ∈ D. It followsfrom Lyapunov stability of the zero solution x(t) ≡ 0 to (9) and (10), and the continuity of V : D → Q ∩ Rq

+ thatthere exists δ = δ(µ) > 0 such that if ‖x0‖ < δ, then ‖x(t)‖ < r, t ≥ t0, and eTV (x0) = eTz0 = ‖z0‖1 < µ. Thus, byasymptotic stability of (20)–(23) with respect to z uniformly in x0, for any arbitrary ε > 0 there exists T = T (ε) > t0such that ‖z(t)‖1 < α(ε), t ≥ T . Thus, it follows from (24) and (25), and Theorem 3.1 that 0 ≤≤ V (x(t)) ≤≤ z(t)on any compact interval [T, T + τ ], and hence, eTz(t) = ‖z(t)‖1, t ∈ [T, T + τ ], and, by (27),

α(‖x(t)‖) ≤ eTV (x(t)) ≤ eTz(t) < α(ε), t ∈ [T, T + τ ]. (29)

Now, suppose, ad absurdum, that for some x0 ∈ Bδ(0), limt→∞ x(t) 6= 0, that is, there exists a sequence {tn}∞

n=1,with tn → ∞ as n → ∞, such that ‖x(tn)‖ ≥ ε, n ∈ Z+, for some 0 < ε < r . Choose ε = ε and the interval[T, T + τ ] such that at least one tn ∈ [T, T + τ ]. Then it follows from (29) that α(ε) ≤ α(‖x(tn)‖) < α(ε), which isa contradiction. Hence, there exists δ > 0 such that for all x0 ∈ Bδ(0), limt→∞ x(t) = 0 which along with Lyapunovstability implies asymptotic stability of the zero solution x(t) ≡ 0 to (9) and (10).

(iii) Suppose D = Rn , Q = Rq , v : Rn→ R+ is radially unbounded, and the nonlinear impulsive dynamical

system (20)–(23) is globally asymptotically stable with respect to z uniformly in x0. In this case, V : Rn→ Rq

+

satisfies (27) for all x ∈ Rn , where the functions α, β : R+ → R+ are of class K∞ [35]. Furthermore, Lyapunovstability of the zero solution x(t) ≡ 0 to (9) and (10) follows from (i). Next, for any x0 ∈ Rn and z0 = V (x0) ∈ Rq

+,identical arguments as in (ii) can be used to show that limt→∞ x(t) = 0, which proves global asymptotic stability ofthe zero solution x(t) ≡ 0 to (9) and (10).

(iv) Suppose (26) holds. Since p ∈ Rq+, then

α‖x‖ν

≤ eTV (x) ≤ β‖x‖ν, x ∈ D, (30)


where α4= α/ maxi=1,...,q{pi } and β

4= β/ mini=1,...,q{pi }. It follows from the exponential stability of the nonlinear

impulsive dynamical system (20)–(23) with respect to z uniformly in x0 that there exist positive constants γ , µ, and η

such that if ‖z0‖1 < µ, then

‖z(t)‖1 ≤ γ ‖z0‖1e−η(t−t0), t ≥ t0, (31)

for all x0 ∈ D. Choose z0 = V (x0) ≥≥ 0, x0 ∈ D. By continuity of V : D → Q ∩ Rq+, there exists δ = δ(µ) > 0

such that for all x0 ∈ Bδ(0), eTV (x0) = eTz0 = ‖z0‖1 < µ. Furthermore, it follows from (24), (25), (30) and (31),and Theorem 3.1 that for all x0 ∈ Bδ(0) the inequality

α‖x(t)‖ν≤ eTV (x(t)) ≤ eTz(t) ≤ γ ‖z0‖1e−η(t−t0) ≤ γ β‖x0‖

νe−η(t−t0) (32)

holds on any compact interval [t0, t0 + τ ]. This in turn implies that for any x0 ∈ Bδ(0),

‖x(t)‖ ≤

(γ β

α

) 1ν

‖x0‖e−ην(t−t0), t ∈ [t0, t0 + τ ]. (33)

Now, suppose, ad absurdum, that for some x0 ∈ Bδ(0) there exists t > t0 + τ such that

‖x(t)‖ >

(γ β

α

) 1ν

‖x0‖e−ην(t−t0). (34)

Then for the compact interval [t0, t], it follows from (33) that ‖x(t)‖ ≤

(γ β

α

) 1ν‖x0‖e−

ην(t−t0), which is a

contradiction. Thus, inequality (33) holds for all t ≥ t0 establishing exponential stability of the zero solution x(t) ≡ 0to (9) and (10).

(v) The proof is identical to the proof of (iv). �

If V : D → Q ∩ Rq+ satisfies the conditions of Theorem 3.2 we say that V (x), x ∈ D, is a vector Lyapunov

function [26]. Note that for stability analysis each component of a vector Lyapunov function need not be positivedefinite, nor does it need to have a negative definite time derivative along the trajectories of (9) and (10) betweenresettings and negative semi-definite difference across the resettings. This provides more flexibility in searching fora vector Lyapunov function as compared to a scalar Lyapunov function for addressing the stability of impulsivedynamical systems.

Next, we use the vector Lyapunov stability results of Theorem 3.2 to develop partial stability analysis results fornonlinear impulsive dynamical systems [14,36]. Specifically, consider the nonlinear impulsive dynamical system (9)and (10) with partitioned dynamics2 given by

xI(t) = fIc(xI(t), xII(t)), xI(t0) = xI0, x(t) 6∈ Z, t ≥ t0, (35)

xII(t) = fIIc(xI(t), xII(t)), xII(t0) = xII0, x(t) 6∈ Z, (36)

∆xI(t) = fId(xI(t), xII(t)), x(t) ∈ Z, (37)

∆xII(t) = fIId(xI(t), xII(t)), x(t) ∈ Z, (38)

where xI(t) ∈ DI, t ≥ t0, DI ⊆ RnI is an open set such that 0 ∈ DI, xII(t) ∈ RnII , t ≥ t0, ∆xI(t) = xI(t+) − xI(t),∆xII(t) = xII(t+) − xII(t), fIc : DI × RnII → RnI is such that for all xII ∈ RnII , fIc(0, xII) = 0 and fIc(·, xII) islocally Lipschitz in xI, fIIc : DI × RnII → RnII is such that for every xI ∈ DI, fIIc(xI, ·) is locally Lipschitz in xII,fId : DI × RnII → RnI is continuous and fId(0, xII) = 0 for all xII ∈ RnII , fIId : DI × RnII → RnII is continuous,Z ∈ DI × RnII , x(t) , [xT

I (t), xTII(t)]

T∈ D = DI × RnII ⊆ Rn, t ≥ t0, x0 , [xT

I0, xTII0]

T, and nI + nII = n. For thenonlinear impulsive dynamical system (35)–(38) the definitions of partial stability given in [14] hold. Furthermore,for a particular trajectory x(t) = (xI(t), xII(t)), t ≥ 0, we let tk(=τk(xI0, xII0)) denote the kth instant of time at

2 Here we use the Roman subscripts I and II as opposed to Arabic subscripts 1 and 2 for denoting the partial states of x in order not to confusethe partial states with the component states of the vector Lyapunov function.


which x(t) intersects Z and we assume that Assumptions 1 and 2 hold for x(t) = (xI(t), xII(t)), t ≥ 0. Note thatfor the impulsive dynamical system (9) and (10), fc(xI, xII) = [ f T

Ic(xI, xII), f TIIc(xI, xII)]

T, (xI, xII) ∈ DI × RnII , andfd(xI, xII) = [ f T

Id(xI, xII), f TIId(xI, xII)]

T, (xI, xII) ∈ DI × RnII . For the following result define

∆V (xI, xII) , V (xI + fId(xI, xII), xII + fIId(xI, xII)) − V (xI, xII), (39)

for a given vector function V : DI × RnII → Rq .

Theorem 3.3. Consider the nonlinear impulsive dynamical system (35)–(38). Assume that there exist a continuouslydifferentiable vector function V : DI × RnII → Q ∩ Rq

+, a positive vector p ∈ Rq+, and class K functions α(·) and

β(·) such that the scalar function v : DI × RnII → R+ defined by v(xI, xII) , pTV (xI, xII) satisfies

α(‖xI‖) ≤ v(xI, xII) ≤ β(‖xI‖), (xI, xII) ∈ DI × RnII , (40)

and

V ′(xI, xII) f (xI, xII) ≤≤ wc(V (xI, xII), xI, xII), (xI, xII) 6∈ Z, (41)

∆V (xI, xII) ≤≤ wd(V (xI, xII), xI, xII), (xI, xII) ∈ Z, (42)

where wc : Q×DI × RnII → Rq and wd : Q×Z → Rq , where Z ⊂ DI × RnII , are continuous, wc(·, xI, xII) ∈ Wc,wd(·, xI, xII) ∈ Wd, and wc(0, 0, 0) = 0. Then the following statements hold:

(i) If the nonlinear impulsive dynamical system (20), (22) and (35)–(38) is Lyapunov (respectively, asymptotically)stable with respect to z uniformly in (xI0, xII0), then the nonlinear impulsive dynamical system (35)–(38) isLyapunov (respectively, asymptotically) stable with respect to xI uniformly in xII0.

(ii) If DI = RnI , Q = Rq , the functions α(·) and β(·) are class K∞, and the nonlinear impulsive dynamical system(20), (22) and (35)–(38) is globally asymptotically stable with respect to z uniformly in (xI0, xII0), then thenonlinear impulsive dynamical system (35)–(38) is globally asymptotically stable with respect to xI uniformly inxII0.

(iii) If there exist constants ν ≥ 1, α > 0, and β > 0 such that v : DI × RnII → R+ satisfies

α‖xI‖ν

≤ v(xI, xII) ≤ β‖xI‖ν, (xI, xII) ∈ DI × RnII , (43)

and the nonlinear impulsive dynamical system (20), (22) and (35)–(38) is exponentially stable with respect toz uniformly in (xI0, xII0), then the nonlinear impulsive dynamical system (35)–(38) is exponentially stable withrespect to xI uniformly in xII0.

(iv) If DI = RnI , Q = Rq , there exist constants ν ≥ 1, α > 0, and β > 0 such that v : RnI × RnII → R+ satisfies(43), and the nonlinear impulsive dynamical system (20), (22) and (35)–(38) is globally exponentially stablewith respect to z uniformly in (xI0, xII0), then the nonlinear impulsive dynamical system (35)–(38) is globallyexponentially stable with respect to xI uniformly in xII0.

Proof. Since p ∈ Rq+ is a positive vector it follows from (40) that

α(‖xI‖)/ maxi=1,...,q

{pi } ≤ eTV (xI, xII) ≤ β(‖xI‖)/ mini=1,...,q

{pi }, (xI, xII) ∈ DI × RnII . (44)

Next, let ε > 0 and note that it follows from Lyapunov stability of the nonlinear impulsive dynamical system (20),(22) and (35)–(38) with respect to z uniformly in (xI0, xII0) that there exists µ = µ(ε) > 0 such that if ‖z0‖1 < µ,where ‖z‖1 ,

∑qi=1 |zi | and zi is the i th component of z, then ‖z(t)‖1 < α(ε)/ maxi=1,...,q{pi }, t ≥ t0, for any

(xI0, xII0) ∈ DI × RnII . Now, choose z0 = V (xI0, xII0) ≥≥ 0, (xI0, xII0) ∈ DI × RnII . Since V (·, ·) is continuous, thefunction eTV (·, ·) is also continuous. Moreover, it follows from the continuity of β(·) that for µ = µ(ε) there existsδ = δ(µ(ε)) = δ(ε) > 0 such that δ < ε and if ‖xI0‖ < δ, then β(‖xI0‖)/ mini=1,...,q{pi } < µ which, by (44), impliesthat eTV (xI0, xII0) = eTz0 = ‖z0‖1 < µ for all xII0 ∈ RnII , and hence, ‖z(t)‖1 < α(ε)/ maxi=1,...,q{pi }, t ≥ t0. Inaddition, it follows from (41) and (42), and Theorem 3.1 that 0 ≤≤ V (xI(t), xII(t)) ≤≤ z(t) on any compact interval[t0, t0 + τ ], and hence, eTz(t) = ‖z‖1, [t0, t0 + τ ]. Thus, it follows from (44) that for all ‖xI0‖ < δ, xII0 ∈ RnII , andt ∈ [t0, t0 + τ ],

α(‖xI(t)‖)/ maxi=1,...,q

{pi } ≤ eTV (xI(t), xII(t)) ≤ eTz(t) < α(ε)/ maxi=1,...,q

{pi }, (45)


which implies that ‖xI(t)‖ < ε, t ∈ [t0, t0 + τ ].Next, suppose, ad absurdum, that for some xI0 ∈ DI with ‖xI0‖ < δ and for some xII0 ∈ RnII there exists

t > t0 + τ such that ‖xI(t)‖ ≥ ε. Then, for z0 = V (xI0, xII0) and the compact interval [t0, t] it follows fromTheorem 3.1 that V (xI(t), xII(t)) ≤≤ z(t) which implies that α(ε)/ maxi=1,...,q{pi } ≤ α(‖xI(t)‖)/ maxi=1,...,q{pi }

≤ eTV (xI(t), xII(t)) ≤ eTz(t) < α(ε)/ maxi=1,...,q{pi }. This is a contradiction and hence, for a given ε > 0, thereexists δ = δ(ε) > 0 such that for all xI0 ∈ DI with ‖xI0‖ < δ and for all xII0 ∈ RnII , ‖xI(t)‖ < ε, t ≥ t0, whichimplies Lyapunov stability of the nonlinear impulsive dynamical system (35)–(38) with respect to xI uniformly in xII0.

The remainder of the proof involves similar arguments as above and as in the proof of parts (ii)–(v) of Theorem 3.2and, hence, is omitted. �

Remark 3.1. Note that Theorem 3.3 allows us to address stability of time-dependent nonlinear impulsive dynamicalsystems via vector Lyapunov functions. In particular, with xI(t) ≡ x(t), xII(t) ≡ t , nI = n, nII = 1, fIc(xI, xII) =

fc(x(t), t), fIIc(xI(t), xII(t)) = 1, fId(xI, xII) = fd(x(t), t), fIId(xI(t), xII(t)) = 0, and Z = D × T , withT , {t1, t2, . . .}, Theorem 3.3 can be used to establish stability results for the nonlinear time-dependent impulsivedynamical system given by

x(t) = fc(x(t), t), x(t0) = x0, t 6= tk, t ≥ t0, (46)

∆x(t) = fd(x(t), t), t = tk, (47)

where x0 ∈ D ⊆ Rn . For details on the unification between partial stability of state-dependent impulsive systems andstability theory for time-dependent impulsive systems see [14].

4. Control vector Lyapunov functions for impulsive systems

In this section, we consider a feedback control problem and generalize the notion of a control vector Lyapunovfunction introduced in [15] to nonlinear impulsive dynamical systems. Specifically, consider the nonlinear controlledimpulsive dynamical system given by

x(t) = Fc(x(t), uc(t)), x(t0) = x0, x(t) 6∈ Z, t ≥ t0, (48)

∆x(t) = Fd(x(t), ud(t)), x(t) ∈ Z, (49)

where x0 ∈ D, D ⊆ Rn is an open set with 0 ∈ D, uc(t) ∈ Uc ⊆ Rmc , t ≥ t0, ud(tk) ∈ Ud ⊆ Rmd , tk denotes the kthinstant of time at which x(t) intersects Z for a particular trajectory x(t) and input (uc(·), ud(·)), Fc : D × Uc → Rn

is Lipschitz continuous for all (x, uc) ∈ D × Uc and satisfies Fc(0, 0) = 0, and Fd : D × Ud → Rn is continuous.Here, we assume that uc(·) and ud(·) are restricted to the class of admissible control inputs consisting of measurablefunctions such that (uc(t), ud(tk)) ∈ Uc × Ud for all t ≥ t0 and k ∈ Z[t0,t) , {k : t0 ≤ tk < t}, where the constraintset Uc × Ud is given with (0, 0) ∈ Uc × Ud. Furthermore, we assume that uc(·) and ud(·) satisfy sufficient regularityconditions such that the nonlinear impulsive dynamical system (48) and (49) has a unique solution forward in time.Let φc : D → Uc be such that φc(0) = 0 and let φd : Z → Ud. If (uc(t), ud(tk)) = (φc(x(t)), φd(x(tk))), where x(t),t ≥ t0, satisfies (48) and (49), then (uc(·), ud(·)) is called a hybrid feedback control.

Definition 4.1. If there exist a continuously differentiable vector function V = [v1, . . . , vq ]T

: D → Q ∩ Rq+,

continuous functions wc = [wc1, . . . , wcq ]T

: Q × D → Rq and wd = [wd1, . . . , wdq ]T

: Q × Z → Rq ,and a positive vector p ∈ Rq

+ such that V (0) = 0, v(x) , pTV (x), x ∈ D, is positive definite, wc(·, x) ∈ Wc,wd(·, x) ∈ Wd, wc(0, 0) = 0, Fc(x) , ∩

qi=1 Fci (x) 6= Ø, x ∈ D, x 6∈ Z, x 6= 0, Fd(x) , ∩

qi=1 Fdi (x) 6= Ø, x ∈ Z ,

where Fci (x) , {uc ∈ Uc : v′

i (x)Fc(x, uc) < wci (V (x), x)}, x ∈ D, x 6∈ Z, x 6= 0, i = 1, . . . , q, andFdi (x) , {ud ∈ Ud : vi (x + Fd(x, ud)) − vi (x) ≤ wdi (V (x), x)}, x ∈ Z , i = 1, . . . , q, then the vector functionV : D → Q ∩ Rq

+ is called a control vector Lyapunov function candidate.

It follows from Definition 4.1 that if there exists a control vector Lyapunov function candidate, then there exists ahybrid feedback control law φc : D → Uc and φd : Z → Ud such that

V ′(x)Fc(x, φc(x)) � wc(V (x), x), x ∈ D, x 6∈ Z, x 6= 0, (50)

V (x + Fd(x, φd(x))) ≤≤ V (x) + wd(V (x), x), x ∈ Z. (51)


Moreover, if the nonlinear impulsive dynamical system


x(t) = Fc(x(t), φc(x(t))), x(t0) = x0, x(t) 6∈ Z, (53)

∆z(t) = wd(z(t), x(t)), x(t) ∈ Z, (54)

∆x(t) = Fd(x(t), φd(x(t))), x(t) ∈ Z, (55)

where z0 ∈ Q and x0 ∈ D, is asymptotically stable with respect to z uniformly in x0, then it follows from Theorem 3.2that the zero solution x(t) ≡ 0 to (53) and (55) is asymptotically stable. In this case, the vector function V : D → Rq

+

given in Definition 4.1 is called a control vector Lyapunov function for impulsive dynamical system (48) and (49).This is a generalization of the notion of control vector Lyapunov functions introduced in [15] for continuous-timesystems. Furthermore, if D = Rn , Q = Rq , Uc = Rmc , Ud = Rmd , v : Rn

→ R+ is radially unbounded, and thesystem (52)–(55) is globally asymptotically stable with respect to z uniformly in x0, then the zero solution x(t) ≡ 0to (48) and (49) is globally asymptotically stabilizable.

Remark 4.1. If in Definition 4.1 wc(z, x) = wc(z), wd(z, x) = wd(z), and the zero solution z(t) ≡ 0 to

z(t) = wc(z(t)), z(t0) = z0, x(t) 6∈ Z, t ≥ t0, (56)

∆z(t) = wd(z(t)), x(t) ∈ Z, (57)

where z0 ∈ Q, is asymptotically stable, then it follows from Theorem 3.2, with wc(z, x) = wc(z), wd(z, x) = wd(z),that V : D → Q∩Rq

+ is a control vector Lyapunov function. In this case, the nonlinear impulsive comparison system(56) and (57) is a time-dependent impulsive dynamical system [8]. To see this, note that the resetting times τk(x0),k ∈ Z+, where x(τk(x0)) ∈ Z and x(t) is the solution to (48) and (49), are determined by the state of (48) and (49),and hence provide a prescribed sequence of the resetting times for (56) and (57) since the dynamics of the impulsivesystem (48) and (49) and the comparison system (56) and (57) are decoupled.

In the case where q = 1, wc(z, x) ≡ wc(z), and wd(z, x) ≡ wd(z), Definition 4.1 implies the existence of apositive-definite continuously differentiable function v : D → Q ∩ R+ and continuous functions wc : Q → R andwd : Q → R, where Q ⊆ R, such that wc(0) = 0, Fc(x) = {uc ∈ Uc : v′(x)Fc(x, uc) < wc(v(x))} 6= Ø, x ∈ D,x 6∈ Z , x 6= 0, and Fd(x) = {ud ∈ Ud : v(x + Fd(x, ud)) − v(x) ≤ wd(v(x))} 6= Ø, x ∈ Z , which implies

infuc∈Uc

v′(x)Fc(x, uc) < wc(v(x)), x 6∈ Z, x 6= 0, (58)

infud∈Ud

[v(x + Fd(x, ud)) − v(x)] ≤ wd(v(x)), x ∈ Z. (59)

Now, the fact that Fc(x) 6= Ø, x ∈ D, x 6∈ Z , x 6= 0, and Fd(x) 6= Ø, x ∈ Z , implies the existence of a hybridfeedback control law φc : D → Uc and φd : Z → Ud such that v′(x)Fc(x, φc(x)) < wc(v(x)), x ∈ D, x 6∈ Z, x 6= 0,and v(x + Fd(x, φd(x))) − v(x) ≤ wd(v(x)), x ∈ Z . Moreover, if v : D → R+ is a control vector Lyapunovfunction (with q = 1), then it follows from Remark 4.1 that the zero solution z(t) ≡ 0 to the system (56) and (57) isasymptotically stable and, since q = 1, this implies that wc(z) < 0, z ∈ Q ∩ R+, z 6= 0. Thus, since v(·) is positivedefinite, (58) and (59) can be rewritten as

infuc∈Uc

v′(x)Fc(x, uc) < 0, x 6∈ Z, x 6= 0, (60)

infud∈Ud

[v(x + Fd(x, ud)) − v(x)] ≤ wd(v(x)), x ∈ Z, (61)

which can be regarded as a definition of a scalar control Lyapunov function for impulsive dynamical systems. Eventhough such a result has never been addressed in the literature, it can be seen that this is a combination of the classicaldefinition of a control Lyapunov function for continuous-time dynamical systems given in [17] and the definition of acontrol Lyapunov function for discrete-time dynamical systems given in [31,32].

Next, consider the case where the control input to (48) and (49) possesses a decentralized control architecture sothat the dynamics of (48) and (49) are given by

xi (t) = Fci (x(t), uci (t)), x(t) 6∈ Z, t ≥ t0, i = 1, . . . , q, (62)


∆xi (t) = Fdi (x(t), udi (t)), x(t) ∈ Z, i = 1, . . . , q, (63)

where xi (t) ∈ Rni , x(t) = [xT1 (t), . . . , xT

q (t)]T, uci (t) ∈ Uci ⊆ Rmci , t ≥ t0, udi (tk) ∈ Udi ⊆ Rmdi , k ∈ Z+,tk denotes the kth resetting time for a particular trajectory of (62) and (63),

∑qi=1 ni = n,

∑qi=1 mci = mc, and∑q

i=1 mdi = md. Note that xi (t) ∈ Rni , t ≥ t0, i = 1, . . . , q, as long as x(t) ∈ D, t ≥ t0, and the sets of controlinputs are given by Uc = Uc1 × · · · × Ucq ⊆ Rmc and Ud = Ud1 × · · · × Udq ⊆ Rmd . In the case of a componentdecoupled control vector Lyapunov function candidate, that is, V (x) = [v1(x1), . . . , vq(xq)]T, x ∈ D, it suffices torequire in Definition 4.1 that, for all i = 1, . . . , q ,

Fci (x) = {uc ∈ Uc : v′

i (xi )Fci (x, uci ) < wci (V (x), x)} 6= Ø, x ∈ D, x 6∈ Z, x 6= 0, (64)

Fdi (x) = {ud ∈ Ud : vi (xi + Fdi (x, udi )) − vi (x) ≤ wdi (V (x), x)} 6= Ø, x ∈ Z, (65)

to ensure that Fc(x) =⋂q

i=1 Fci (x) 6= Ø, x ∈ D, x 6∈ Z, x 6= 0, and Fd(x) =⋂q

i=1 Fdi (x) 6= Ø, x ∈ Z . Note that

for a component decoupled control vector Lyapunov function V : D → Q ∩ Rq+, (64) is equivalent to

infuc∈Uc

V ′(x)Fc(x, uc) � wc(V (x), x), x ∈ D, x 6∈ Z, x 6= 0, (66)

and (65) implies

infud∈Ud

[V (x + Fd(x, ud)) − V (x)] ≤≤ wd(V (x), x), x ∈ Z, (67)

where the infimum in (66) and (67) is taken componentwise, that is, for each component of (66) and (67) the infimum iscalculated separately. It follows from the fact thatFc(x) 6= Ø, x ∈ D, x 6∈ Z , x 6= 0, andFd(x) 6= Ø, x ∈ Z , that thereexists a hybrid feedback control law φc : D → Uc and φd : Z → Ud such that φc(x) = [φT

c1(x), . . . , φTcq(x)]T, x ∈ D,

φd(x) = [φTd1(x), . . . , φT

dq(x)]T, x ∈ Z , where φci : D → Uci , φdi : Z → Udi , v′

i (xi )Fci (x, φci (x)) <

wci (V (x), x), x ∈ D, x 6∈ Z, x 6= 0, and vi (xi + Fdi (x, φdi (x))) − vi (xi ) ≤ wdi (V (x), x), x ∈ Z , i = 1, . . . , q.

Remark 4.2. If wci (V (x), x) = 0 for x ∈ D with xi = 0, then condition (64) holds for all x ∈ D such that xi 6= 0.

Next, we consider the special case of a nonlinear impulsive dynamical system of the form (62) and (63) with affinecontrol inputs given by

xi (t) = fci (x(t)) + Gci (x(t))uci (t), x(t) 6∈ Z, t ≥ t0, i = 1, . . . , q, (68)

∆xi (t) = fdi (x(t)) + Gdi (x(t))udi (t), x(t) ∈ Z, i = 1, . . . , q, (69)

where fci : Rn→ Rni satisfying fci (0) = 0 and Gci : Rn

→ Rni ×mci are smooth functions (at least continuouslydifferentiable mappings) for all i = 1, . . . , q , fdi : Rn

→ Rni and Gdi : Rn→ Rni ×mdi are continuous for all

i = 1, . . . , q , uci (t) ∈ Rmci , t ≥ t0, and udi (tk) ∈ Rmdi , k ∈ Z+, for all i = 1, . . . , q.

Theorem 4.1. Consider the controlled nonlinear impulsive dynamical system given by (68) and (69). If thereexist a continuously differentiable, component decoupled vector function V : Rn

→ Rq+, continuous functions

P1ui : Rn→ R1×mdi , P2ui : Rn

→ Rmdi ×mdi , i = 1, . . . , q, wc = [wc1, . . . , wcq ]T

: Rq+ × Rn

→ Rq ,

wd = [wd1, . . . , wdq ]T

: Rq+ × Z → Rq , and a positive vector p ∈ Rq

+ such that V (0) = 0, the scalar functionv : Rn

→ R+ defined by v(x) , pTV (x), x ∈ Rn , is positive definite and radially unbounded, wc(·, x) ∈ Wc,wd(·, x) ∈ Wd, wc(0, 0) = 0, and, for all i = 1, . . . , q,

vi (xi + fdi (x) + Gdi (x)udi ) = vi (xi + fdi (x)) + P1ui udi + uTdi P2ui (x)udi , x ∈ Rn, udi ∈ Rmdi , (70)

v′

i (xi ) fci (x) < wci (V (x), x), x ∈ Ri , (71)

vi (xi + fdi (x)) − vi (xi ) −14

P1ui (x)PĎ2ui (x)PT

1ui (x) ≤ wdi (V (x), x), x ∈ Z, (72)

where Ri , {x ∈ Rn, x 6= 0 : v′

i (xi )Gci (x) = 0}, i = 1, . . . , q, and PĎ2ui (·) is Moore–Penrose generalized inverse of

P2ui (·) [37], then V : Rn→ Rq

+ is a control vector Lyapunov function candidate. If, in addition, there exist φc : Rn→

Rmc and φd : Z → Rmd such that φc(x) = [φTc1(x), . . . , φT

cq(x)]T, x ∈ Rn , φd(x) = [φTd1(x), . . . , φT

dq(x)]T, x ∈ Z ,


and the system (52)–(55) is globally asymptotically stable with respect to z uniformly in x0, then the zero solutionx(t) ≡ 0 to (53) and (55) is globally asymptotically stable and V : Rn

→ Rq+ is a control vector Lyapunov function.

Proof. Note that for all i = 1, . . . , q ,

infuci ∈Rmci

v′

i (xi )( fci (x) + Gci (x)uci ) =

{−∞, x 6∈ Ri ,

v′

i (xi ) fci (x), x ∈ Ri ,

< wci (V (x), x), x ∈ Rn, x 6= 0, (73)

which implies that Fci (x) 6= Ø, x ∈ Rn , x 6= 0, i = 1, . . . , q. Next, note that it follows from a Taylor series

expansion about x∗

i = xi + fdi (x) that P1ui (x) = v′

i (x∗

i )Gdi (x) and P2ui (x) =12 GT

di (x)∂2vi∂x2

i|xi =x∗

iGdi (x). Since

V (·) is continuously differentiable it follows that ∂2vi∂x2

i|xi =x∗

iis symmetric, and hence, P2ui (·) is symmetric for all

i = 1, . . . , q . Next, with udi = −12 PĎ

2ui (x)PT1ui (x), x ∈ Z , i = 1, . . . , q, it follows from (70) and (72) that

vi (xi + fdi (x) + Gdi (x)udi ) − vi (xi ) = vi (xi + fdi (x)) − vi (xi ) + P1ui (x)udi + uTdi P2ui (x)udi

= vi (xi + fdi (x)) − vi (xi ) −14


1ui (x)

≤ wdi (V (x), x), x ∈ Z, i = 1, . . . , q, (74)

which implies that Fdi (x) 6= Ø, x ∈ Z . Now, the result is a direct consequence of the definition of a control vectorLyapunov function by noting that, for component decoupled vector Lyapunov functions, (64) and (65) are equivalentto Fc(x) 6= Ø, x ∈ Rn , x 6= 0, and Fd(x) 6= Ø, x ∈ Z , respectively. �

Using Theorem 4.1 we can construct an explicit feedback control law that is a function of the control vectorLyapunov function V (·). Specifically, consider the hybrid feedback control law φc(x) = [φT

c1(x), . . . , φTcq(x)]T, x ∈

Rn , and φd(x) = [φTd1(x), . . . , φT

dq(x)]T, x ∈ Z , given by

φci (x) =

−

c0i +

(αi (x) − wci (V (x), x)) +

√(αi (x) − wci (V (x), x))2 + (βT

i (x)βi (x))2

βTi (x)βi (x)

βi (x), βi (x) 6= 0,

0, βi (x) = 0,

(75)

and

φdi (x) = −12

PĎ2ui (x)PT

1ui (x), x ∈ Z, (76)

where αi (x) , v′

i (xi ) fci (x), x ∈ Rn , βi (x) , GTci (x)v′T

i (xi ), x ∈ Rn , and c0i > 0, i = 1, . . . , q. The derivative V (·)

along the trajectories of the dynamical system (68), with uc = φc(x), x ∈ Rn , given by (75), is given by

vi (xi ) = v′

i (xi )( fci (x) + Gci (x)φci (x))

= αi (x) + βTi (x)φci (x)

=

−c0iβ

Ti (x)βi (x) −

√(αi (x) − wci (V (x), x))2 + (βT

i (x)βi (x))2

+ wci (V (x), x), βi (x) 6= 0,

αi (x), βi (x) = 0,

< wci (V (x), x), x ∈ Rn . (77)

In addition, using (70), the difference of V (x) at the resetting instants with ud = φd(x), x ∈ Z , given by (76), is givenby

∆vi (xi ) = vi (xi + fdi (x) + Gdi (x)φdi (x)) − vi (xi )

= vi (xi + fdi (x)) − vi (xi ) −14


1ui (x)

≤ wdi (V (x), x), x ∈ Z. (78)


Thus, if the zero solution z(t) ≡ 0 to (52)–(55) is globally asymptotically stable with respect to z uniformly inx0, then it follows from Theorem 3.2 that the zero solution x(t) ≡ 0 to (68) and (69) with uc = φc(x) =

[φTc1(x), . . . , φT

cq(x)]T, x ∈ Rn , given by (75) and ud = φd(x) = [φTd1(x), . . . , φT

dq(x)]T, x ∈ Z , given by (76) isglobally asymptotically stable.

Remark 4.3. If in Theorem 4.1 wc(z, x) = wc(z), wd(z, x) = wd(z), and the zero solution z(t) ≡ 0 to (56) and(57) is globally asymptotically stable, then it follows from Theorem 3.2 that the hybrid feedback control law givenby (75) and (76) is a globally asymptotically stabilizing controller for the nonlinear impulsive dynamical system (68)and (69).

Remark 4.4. In the case where q = 1, the functions w(·, ·) and wd(·, ·) in Theorem 4.1 can be set to be identicallyzero, that is, wc(z, x) ≡ 0 and wd(z, x) ≡ 0. In this case, the feedback control law (75) specializes to Sontag’suniversal formula [18], the feedback control law (76) specializes to the discrete-time control law given in [32], andthe hybrid feedback control law (75) and (76) is a global stabilizer for the nonlinear impulsive dynamical system (68)and (69).

Since fci (·) and Gci (·) are smooth and vi (·) is continuously differentiable for all i = 1, . . . , q, it follows thatαi (x) and βi (x), x ∈ Rn , i = 1, . . . , q , are continuous functions, and hence, φci (x) given by (75) is continuous forall x ∈ Rn if either βi (x) 6= 0 or αi (x) − wci (V (x), x) < 0 for all i = 1, . . . , q. Hence, the feedback control lawgiven by (75) is continuous everywhere except for the origin. The following result provides necessary and sufficientconditions under which the feedback control law given by (75) is guaranteed to be continuous at the origin in additionto being continuous everywhere else.

Proposition 4.1 ([15]). The feedback control law φc(x) given by (75) is continuous on Rn if and only if for everyε > 0, there exists δ > 0 such that for all 0 < ‖x‖ < δ there exists uci ∈ Rmci such that ‖uci‖ < ε andαi (x) + βT

i (x)uci < wci (V (x), x), i = 1, . . . , q.

Remark 4.5. If the conditions of Proposition 4.1 are satisfied, then the feedback control law φc(x) given by (75) iscontinuous on Rn . However, it is important to note that for a particular trajectory x(t), t ≥ 0, of (68) and (69), φc(x(t))is a piecewise continuous function of time due to state resettings.

5. Stability margins and inverse optimality

In this section, we construct a hybrid feedback control law that is robust to sector bounded input nonlinearities.Specifically, we consider the nonlinear impulsive dynamical system (68) and (69) with nonlinear uncertainties in theinput so that the impulsive dynamics of the system are given by

xi (t) = fci (x(t)) + Gci (x(t))σci (uci (t)), x(t) 6∈ Z, t ≥ t0, i = 1, . . . , q, (79)

∆xi (t) = fdi (x(t)) + Gdi (x(t))σdi (udi (t)), x(t) ∈ Z, i = 1, . . . , q, (80)

where σci (·) ∈ Φci , {σci : Rmci → Rmci : σci (0) = 0 and 12 uT

ci uci ≤ σTci (uci )uci < ∞, uci ∈ Rmci }, σdi (·) ∈ Φdi ,

{σdi : Rmdi → Rmdi : σdi (0) = 0 and αdu2di j ≤ σdi j (udi )udi j < βdu2

di j , udi ∈ Rmdi }, i = 1, . . . , q, σdi j (·) ∈ Rand udi j ∈ R, j = 1, . . . , mdi , are the j th components of σdi (·) and udi , respectively, and 0 ≤ αd < 1 < βd < ∞.In addition, we show that for the dynamical system (68) and (69) the hybrid feedback control law to be defined inTheorem 5.1 is inverse optimal in the sense that it minimizes a derived hybrid performance functional over the set ofstabilizing controllers S(x0) , {(uc(·), ud(·)) : uc(·) and ud(·) are admissible andx(t) → 0 as t → ∞}.

Theorem 5.1. Consider the nonlinear impulsive dynamical system (68) and (69) and assume that the conditions ofTheorem 4.1 hold with (72) replaced by

vi (xi + fdi (x)) − vi (xi ) −14

P1ui (x)(R2di (x) + P2ui (x))−1 PT1ui (x) ≤ wdi (V (x), x), x ∈ Z, (81)

where R2di : Z → Rmdi ×mdi is positive definite, wc(z, x) ≡ wc(z), wd(z, x) ≡ wd(z), and with the zero solutionz(t) ≡ 0 to (56) and (57) being globally asymptotically stable. Then the hybrid feedback control law (φci (·), φdi (·))


given by (75) and

φdi (x) = −12(R2di (x) + P2ui (x))−1 PT

1ui (x), x ∈ Z, i = 1, . . . , q, (82)

minimizes the performance functional given by

J (x0, uc(·), ud(·)) =

∫∞

t0

q∑i=1

[L1ci (x(t)) + uTci (t)R2ci (x(t))uci (t)]dt

+

∑k∈Z[t0,∞)

q∑i=1

[L1di (x(tk)) + uTdi (tk)R2di (x(tk))udi (tk)] (83)

in the sense that

J (x0, φc(x(·)), φd(x(·))) = min(uc(·),ud(·))∈S(x0)

J (x0, uc(·), ud(·)), x0 ∈ Rn, (84)

where Z[t0,∞) = {k : t0 ≤ tk < ∞}, L1ci (x) , −αi (x) +γi (x)

2 βTi (x)βi (x), x ∈ Rn ,

R2ci (x) ,

1

2γi (x)Imci , βi (x) 6= 0,

0, βi (x) = 0,

(85)

γi (x) ,

c0i +

(αi (x) − wci (V (x))) +

√(αi (x) − wci (V (x)))2 + (βT

i (x)βi (x))2

βTi (x)βi (x)

> 0, βi (x) 6= 0,

0, βi (x) = 0,

(86)

L1di (x) , φTdi (x)(R2di (x) + P2ui (x))φdi (x) − vi (xi + fdi (x)) + vi (xi ), x ∈ Z , and P2ui (·) and vi (·) are given in

Theorem 4.1. Furthermore, J (x0, φc(x(·)), φd(x(·))) = eTV (x0), x0 ∈ Rn , where V : Rn→ Rq

+ is a control vectorLyapunov function for the impulsive dynamical system (68) and (69). In addition, the nonlinear impulsive dynamicalsystem (79) and (80) with the hybrid feedback control law given by (75) and (82) is globally asymptotically stable

for all σci (·) ∈ Φci and σdi (·) ∈ Φdi , i = 1, . . . , q, where αd = maxi=1,...,q

(1

1+θdi

), βd = mini=1,...,q

(1

1−θdi

),

θdi ,√

γdi

γ di, γ

di, infx∈Z σmin(R2di (x)), γ di , supx∈Z σmax(R2di (x) + P2ui (x)), i = 1, . . . , q.

Proof. To show that the hybrid feedback control law (75) and (82) minimizes (83) in the sense of (84), define theHamiltonian

H(x, uc, ud) = Hc(x, uc) + Hd(x, ud), (87)

where

Hc(x, uc) ,q∑

i=1

[L1ci (x) + uTci R2ci (x)uci + v′

i (xi )( fci (x) + Gci (x)uci )], (88)

Hd(x, ud) ,q∑

i=1

[L1di (x) + uTdi R2di (x)udi + vi (xi + fdi (x) + Gdi (x)udi ) − vi (xi )], (89)

and note that H(x, φc(·), φd(·)) = 0 and H(x, uc, ud) ≥ 0, x ∈ Rn , uc ∈ Rmc , ud ∈ Rmd , since

H(x, uc, ud) =

q∑i=1

(uci − φci (x))T R2ci (x)(uci − φci (x))

+

q∑i=1

(udi − φdi (x))T(R2di (x) + P2ui (x))(udi − φdi (x)),

x ∈ Rn, uc ∈ Rmc , ud ∈ Rmd . (90)


Thus,

J (x0, uc(·), ud(·)) =

∫∞

t0

[Hc(x(t), uc(t)) −

q∑i=1

v′

i (xi (t))( fci (x(t)) + Gci (x(t))uci (t))

]dt

+

∑k∈Z[t0,∞)

[Hd(x(tk), ud(tk))

+

q∑i=1

(vi (xi (tk))) − vi (xi (tk) + fdi (x(tk)) + Gdi (x(tk))udi (tk))

]

= − limt→∞

eTV (x(t)) + eTV (x0) +

∫∞

t0Hc(x(t), uc(t))dt +

∑k∈Z[t0,∞)

Hd(x(tk), ud(tk))

≥ eTV (x0)

= J (x0, φc(x(·)), φd(x(·))), (91)

which yields (84).Next, we show that the uncertain nonlinear impulsive dynamical system (79) and (80) with the hybrid feedback

control law (75) and (82) is globally asymptotically stable for all σci (·) ∈ Φci (·) and σdi (·) ∈ Φdi (·). It followsfrom Theorem 4.1 that the hybrid feedback control law (75) and (82) globally asymptotically stabilizes the impulsivedynamical system (68) and (69) and the vector function V : Rn

→ Rq+ is a control vector Lyapunov function for the

impulsive dynamical system (68) and (69). Note that with (86) the continuous-time feedback control law (75) can berewritten as φci (x) = −γi (x)βi (x), x ∈ Rn, i = 1, . . . , q. Let the control vector Lyapunov function V : Rn

→ Rq+

for (68) and (69) be a vector Lyapunov function candidate for (79) and (80). Then the vector Lyapunov derivativecomponents along the trajectories of (79) are given by

vi (xi ) = v′

i (xi )( fci (x) + Gci (x)σci (φci (x))) = αi (x) + βTi (x)σci (φci (x)), x 6∈ Z, i = 1, . . . , q. (92)

Note that φci (x) = 0, and hence, σci (φci (x)) = 0 whenever βi (x) = 0 for all i = 1, . . . , q. In this case, itfollows from (71) that vi (xi ) < wci (V (x)), x 6∈ Z, βi (x) = 0, x 6= 0, i = 1, . . . , q. Next, consider the case whereβi (x) 6= 0, i = 1, . . . , q . In this case, note that

αi (x) − wci (V (x)) −γi (x)

2βT

i (x)βi (x) =−c0iβ

Ti (x)βi (x)

2

+

(αi (x) − wci (V (x))) −

√(αi (x) − wci (V (x)))2 + (βT

i (x)βi (x))2

2< 0, x 6∈ Z, βi (x) 6= 0, (93)

for all i = 1, . . . , q . Thus, the vector Lyapunov derivative components given by (92) satisfy

vi (xi ) < wci (V (x)) +γi (x)

2βT

i (x)βi (x) + βTi (x)σci (φci (x))

= wci (V (x)) +1

2γi (x)φT

ci (x)φci (x) −1

γi (x)φT

ci (x)σci (φci (x))

= wci (V (x)) +1

γi (x)

[φT

ci (x)φci (x)

2− φT

ci (x)σci (φci (x))

]≤ wci (V (x)), x 6∈ Z, βi (x) 6= 0, (94)

for all σci (·) ∈ Φci and i = 1, . . . , q .Next, consider the Lyapunov difference of each component of V (·) at the resetting instants for the resetting

dynamics (80) with udi = φdi (x) given by (82). Note that it follows from (81) that L1di (x) + wdi (V (x), x) ≥ 0,x ∈ Z , i = 1, . . . , q , and hence, since σdi (·) ∈ Φdi it follows that


∆vi (xi ) = vi (xi + fdi (x) + Gdi (x)σdi (φdi (x))) − vi (xi )

≤ vi (xi + fdi (x)) − vi (xi ) + P1ui (x)σdi (φdi (x))

+ σTdi (φdi (x))P2ui (x)σdi (φdi (x)) + L1di (x) + wdi (V (x), x)

= P1ui (x)σdi (φdi (x)) + σTdi (φdi (x))P2ui (x)σdi (φdi (x))

+ φTdi (x)(R2di (x) + P2ui (x))φdi (x) + wdi (V (x), x)

= φTdi (x)(R2di (x) + P2ui (x))φdi (x) + σT

di (φdi (x))P2ui (x)σdi (φdi (x))

− 2φTdi (x)(R2di (x) + P2ui (x))σdi (φdi (x)) + wdi (V (x), x)

= [σdi (φdi (x)) − φdi (x)]T(R2di (x) + P2ui (x))[σdi (φdi (x)) − φdi (x)]

− σTdi (φdi (x))R2di (x)σdi (φdi (x)) + wdi (V (x), x)

≤ γ di [σdi (φdi (x)) − φdi (x)]T[σdi (φdi (x)) − φdi (x)] − γ

diσT

di (φdi (x))σdi (φdi (x)) + wdi (V (x), x)

= γ di (1 − θ2di )

[σdi (φdi (x)) −

11 + θdi

φdi (x)

]T [σdi (φdi (x)) −

11 − θdi

φdi (x)

]+ wdi (V (x), x)

= γ di (1 − θ2di )

mdi∑j=1

(σdi j (φdi (x)) −

11 + θdi

φdi j (x)

)(σdi j (φdi (x)) −

11 − θdi

φdi j (x)

)+ wdi (V (x), x)

≤ wdi (V (x), x), x ∈ Z, i = 1, . . . , q. (95)

Since the impulsive dynamical system (56) and (57) is globally asymptotically stable it follows from Theorem 3.2that the nonlinear impulsive dynamical system (79) and (80) is globally asymptotically stable for all σci (·) ∈ Φci andσdi (·) ∈ Φdi , i = 1, . . . , q . �

Remark 5.1. It follows from Theorem 5.1 that with the hybrid feedback stabilizing control law (75) and (82) thenonlinear impulsive dynamical system (68) and (69) has a sector (and hence gain) margin (( 1

2 , ∞), ( 11+θdi

, 11−θdi

)),i = 1, . . . , q , in each decentralized input channel. For details on stability margins for nonlinear dynamical systems,see [38–40].

6. Decentralized control for large-scale impulsive dynamical systems

In this section, we apply the proposed hybrid control framework to decentralized control of large-scale nonlinearimpulsive dynamical systems. Specifically, we consider the large-scale dynamical system G involving energy exchangebetween n interconnected subsystems. Let xi : [0, ∞) → R+ denote the energy (and hence a nonnegative quantity)of the i th subsystem, let uci : [0, ∞) → R denote the control input to the i th subsystem, let σci j : Rn

+ → R+, i 6=

j, i, j = 1, . . . , n, denote the instantaneous rate of energy flow from the j th subsystem to the i th subsystem betweenresettings, let σdi j : Rn

+ → R+, i 6= j , i, j = 1, . . . , n, denote the amount of energy transferred from the j thsubsystem to the i th subsystem at the resetting instant, and let Z ⊂ Rn

+ be a resetting set for the large-scale impulsivedynamical system G.

An energy balance for each subsystem Gi , i = 1, . . . , q, yields [6,41]

xi (t) =

n∑j=1, j 6=i

[σci j (x(t)) − σc j i (x(t))] + Gci (x(t))uci (t), x(t0) = x0, x(t) 6∈ Z, t ≥ t0, (96)

∆xi (t) =

n∑j=1, j 6=i

[σdi j (x(t)) − σd j i (x(t))] + Gdi (x(t))udi (t), x(t) ∈ Z, (97)

or, equivalently, in vector form for the large-scale impulsive dynamical system G

x(t) = fc(x(t)) + Gc(x(t))uc(t), x(t0) = x0, x(t) 6∈ Z, t ≥ t0, (98)

∆x(t) = fd(x(t)) + Gd(x(t))ud(t), x(t) ∈ Z, (99)


where x(t) = [x1(t), . . . , xn(t)]T, t ≥ t0, fci (x) =∑n

j=1, j 6=i φci j (x), where φci j (x) , σci j (x) − σc j i (x), x ∈ Rn+,

i 6= j, i, j = 1, . . . , q , denotes the net energy flow from the j th subsystem to the i th subsystem between resettings,Gc(x) = diag[Gc1(x), . . . , Gcn(x)] = diag[x1, . . . , xn], x ∈ Rn

+, Gd(x) = diag[Gd1(x), . . . , Gdn(x)], x ∈ Z ,Gdi : Rn

→ R, i = 1, . . . , n, uc(t) ∈ Rn, t ≥ t0, ud(tk) ∈ Rn , k ∈ Z+, fdi (x) =∑n

j=1, j 6=i φdi j (x), whereφdi j (x) , σdi j (x) − σd j i (x), x ∈ Z , i 6= j , i, j = 1, . . . , q, denotes the net amount of energy transferredfrom the j th subsystem to the i th subsystem at the instant of resetting. Here, we assume that σci j : Rn

+ → R+,i 6= j , i, j = 1, . . . , n, are locally Lipschitz continuous on Rn

+, σci j (0) = 0, i 6= j , i, j = 1, . . . , n, anduc = [uc1, . . . , ucn]

T: R → Rn is such that uci : R → R, i = 1, . . . , n, are bounded piecewise continuous

functions of time. Furthermore, we assume that σci j (x) = 0, x ∈ Rn+, whenever x j = 0, i 6= j , i, j = 1, . . . , n,

and xi +∑n

j=1, j 6=i φdi j (x) ≥ 0, x ∈ Z . In this case, fc(·) is essentially nonnegative [41,42] (i.e., fci (x) ≥ 0 for all

x ∈ Rn+ such that xi = 0, i = 1, . . . , n) and x + fd(x), x ∈ Z ⊂ Rn

+, is nonnegative (i.e. xi + fdi (x) ≥ 0 for all x ∈ Z ,i = 1, . . . , n). The above constraints imply that if the energy of the j th subsystem of G is zero, then this subsystemcannot supply any energy to its surroundings between resettings and the i th subsystem of G cannot transfer moreenergy to its surroundings than it possesses at the instant of resetting. Finally, in order to ensure that the trajectories ofthe closed-loop system remain in the nonnegative orthant of the state space for all nonnegative initial conditions, weseek a hybrid feedback control law (uc(·), ud(·)) that guarantees the continuous-time closed-loop system dynamics(98) are essentially nonnegative and the closed-loop system states after the resettings are nonnegative [43].

For the dynamical system G, consider the control vector Lyapunov function candidate V (x) =

[v1(x1), . . . , vn(xn)]T, x ∈ Rn+, given by

V (x) = [x1, . . . , xn]T, x ∈ Rn

+. (100)

Note that V (0) = 0 and v(x) , eTV (x), x ∈ Rn+, is such that v(0) = 0, v(x) > 0, x 6= 0, x ∈ Rn

+, and v(x) → ∞ as‖x‖ → ∞ with x ∈ Rn

+. Also, note that since vi (xi ) = xi , x ∈ Rn , i = 1, . . . , n, are linear functions of x , it followsthat P2ui (x) ≡ 0, i = 1, . . . , n, and hence, by (76), φdi (x) ≡ 0, i = 1, . . . , n. Furthermore, consider the functions

wc(V (x), x) =

[−σc11(v1(x1)) +

n∑j=1, j 6=1

φc1 j (x), . . . ,−σcnn(vn(xn)) +

n∑j=1, j 6=n

φcnj (x)

]T

,

x ∈ Rn+, (101)

wd(V (x), x) =

[n∑

j=1, j 6=1

φd1 j (x), . . . ,

n∑j=1, j 6=n

φdnj (x)

]T

, x ∈ Z, (102)

where σci i : R+ → R+, i = 1, . . . , n, are positive definite functions, and note that wc(·, x) ∈ Wc, x ∈ Rn+,

wc(0, 0) = 0, wd(·, ·) does not depend on V (·), and hence wd(·, x) ∈ Wd, x ∈ Z . Also note that it follows fromRemark 4.2 that Ri , {x ∈ Rn

+, xi 6= 0 : V ′

i (xi )Gci (x) = 0} = {x ∈ Rn+, xi 6= 0 : xi = 0} = Ø, and hence,

condition (71) is satisfied for V (·) and wc(·, ·) given by (100) and (101), respectively, and condition (72) is satisfiedas an equality for wd(·, ·) given by (102) and P2ui (x) ≡ 0, i = 1, . . . , n.

To show that the impulsive dynamical system


∆z(t) = wd(z(t), x(t)), x(t) ∈ Z, (104)

where z(t) ∈ Rn+, t ≥ t0, x(t), t ≥ t0, is the solution to (98) and (99), the i th component of wc(z, x) is given

by wci (z, x) = −σci i (zi ) +∑n

j=1, j 6=i φci j (x), z ∈ Rn+, x ∈ Rn

+, and the i th component of wd(z, x) is given bywdi (z, x) =

∑nj=1, j 6=i φdi j (x), x ∈ Z , is globally asymptotically stable with respect to z uniformly in x0, consider

the partial Lyapunov function candidate v(z) = eTz, z ∈ Rn+. Note that v(·) is radially unbounded, v(0) = 0, v(z) > 0,

z ∈ Rn+, z 6= 0, ˙v(z) = −

∑ni=1 σci i (zi ) +

∑ni=1

∑nj=1, j 6=i φci j (x) = −

∑ni=1 σci i (zi ) < 0, z ∈ Rn

+, z 6= 0, x ∈ Rn+,

and ∆v(z) =∑n

i=1∑n

j=1, j 6=i φdi j (x) = 0, z ∈ Rn+, x ∈ Z . Thus, it follows from Theorem 2.1 of [14] that the

impulsive dynamical system (103), (104), (98) and (99) is globally asymptotically stable with respect to z uniformly


Fig. 6.1. Controlled system states versus time.

in x0. Hence, it follows from Theorem 4.1 that V (x), x ∈ Rn+, given by (100) is a control vector Lyapunov function

for the dynamical system (98) and (99).Next, using (75) with αi (x) = v′

i (xi ) fci (x) =∑n

j=1, j 6=i φci j (x), βi (x) = xi , x ∈ Rn+, and c0i > 0, i = 1, . . . , n,

we construct a globally stabilizing hybrid decentralized feedback controller for (98) and (99) given by

φci (x) =

−

c0i +

σci i (xi ) +

√σ 2

ci i (xi ) + x2i

x2i

xi , xi 6= 0,

0, xi = 0,

i = 1, . . . , n, (105)

and

φdi (x) ≡ 0, i = 1, . . . , n. (106)

It can be seen from the structure of the feedback control law (105) and (106) that the continuous-time closed-loopsystem dynamics are essentially nonnegative and the closed-loop system states after the resettings are nonnegative.Furthermore, since αi (x) − wci (V (x), x) = σci i (vi (xi )), x ∈ Rn

+, i = 1, . . . , n, the continuous-time feedbackcontroller φc(·) is fully independent from fc(x) which represents the internal interconnections of the large-scalesystem dynamics, and hence, is robust against full modeling uncertainty in fc(x).

For the following simulation we consider (98) and (99) with σci j (x) = σci j xi x j , σci i (x) = σci i x2i , and

σdi j (x) = σdi j x j , where σci j ≥ 0, i 6= j , i, j = 1, . . . , n, σci i > 0, i = 1, . . . , n, and σdi j ≥ 0, i 6= j , i, j = 1, . . . , n,and 1 ≥

∑nj=1, j 6=i σd j i , i = 1, . . . , n. Note that in this case the conditions of Proposition 4.1 are satisfied, and hence

the continuous-time feedback control law (105) is continuous on Rn+. For our simulation we set n = 3, σc11 = 0.1,

σc22 = 0.2, σc33 = 0.01, σc12 = 2, σc13 = 3, σc21 = 1.5, σc23 = 0.3, σc31 = 4.4, σc32 = 0.6, σd12 = 0.75,σd13 = 0.33, σd21 = 0.2, σd23 = 0.5, σd31 = 0.66, σd32 = 0.2, c01 = 1, c02 = 1, c03 = 0.25, the resetting setZ = {x ∈ R3

: x1 + x2 −0.75 = 0}, with initial condition x0 = [3, 4, 1]T. Fig. 6.1 shows the states of the closed-loop

system versus time and Fig. 6.2 shows continuous-time control signal for each decentralized control channel as afunction of time.

7. Conclusion

In this paper we developed vector Lyapunov stability results for nonlinear impulsive dynamical systems.Specifically, we addressed the generalized comparison principle involving hybrid comparison dynamics that aredependent on comparison system states as well as the nonlinear impulsive dynamical system states. We presentedLyapunov stability theorem to provide sufficient conditions for several types of stability using vector Lyapunov


Fig. 6.2. Control signals in each decentralized control channel versus time.

functions. In addition, we addressed partial stability of nonlinear impulsive dynamical systems via vector Lyapunovfunctions. Furthermore, we generalized the novel notion of control vector Lyapunov functions to impulsive dynamicalsystems. Using this notion, we developed universal decentralized hybrid feedback stabilizer for a decentralizedaffine in control nonlinear impulsive dynamical system with robustness guarantees against full modeling and inputuncertainty. Finally, we designed decentralized controllers for large-scale impulsive dynamical systems and presenteda numerical example.

Acknowledgments

This research was supported in part by the Air Force Office of Scientific Research under Grant F49620-03-1-0178and by the Naval Surface Warfare Center under Contract N65540-05-C-0028.

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