Global Stability of Polytopic Linear Time-Varying Dynamic Systems under Time-Varying Point Delays and Impulsive Controls

Abstract

This paper investigates the stability properties of a class of dynamic linear systems possessing several linear time-invariant parameterizations (or configurations) which conform a linear time-varying polytopic dynamic system with a finite number of time-varying time-differentiable point delays. The parameterizations may be timevarying and with bounded discontinuities and they can be subject to mixed regular plus impulsive controls within a sequence of time instants of zero measure. The polytopic parameterization for the dynamics associated with each delay is specific, so that (𝑞+1) polytopic parameterizations are considered for a system with 𝑞 delays being also subject to delay-free dynamics. The considered general dynamic system includes, as particular cases, a wide class of switched linear systems whose individual parameterizations are timeinvariant which are governed by a switching rule. However, the dynamic system under consideration is viewed as much more general since it is time-varying with timevarying delays and the bounded discontinuous changes of active parameterizations are generated by impulsive controls in the dynamics and, at the same time, there is not a prescribed set of candidate potential parameterizations.

1. Introduction

The stabilization of dynamic systems is a very important question since it is the first requirement for most of applications. Powerful techniques for studying the stability of dynamic systems are Lyapunov stability theory and fixed point theory which can be easily extended from the linear time-invariant case to the time-varying one as well as to functional differential equations, as those arising, for instance, from the presence of internal delays, and to certain classes of nonlinear systems, [1, 2]. Dynamic systems which are of increasing interest are the so-called switched systems which consist of a set of individual parameterizations and a switching law which selects along time, which parameterization is active. Switched systems are essentially timevarying by nature even if all the individual parameterizations are timeinvariant. The interest of such systems arises from the fact that some existing systems in the real world modify their parameterizations to better adapt to their environments. Another important interest of some of such systems relies on the fact that changes of parameterizations through time can lead to benefits in certain applications, [3–13]. The natural way of modelling these situations lies in the definition of appropriate switched dynamic systems. For instance, the asymptotic stability of Liénard-type equations with Markovian switching is investigated in [4, 5]. Also, time-delay dynamic systems are very important in the real life for appropriate modelling of certain biological and ecology systems and they are present in physical processes implying diffusion, transmission, tele-operation, population dynamics, war and peace models, and so forth. (see, e.g., [1, 2, 12–18]). Linear switched dynamic systems are a very particular case of the dynamic system proposed in this paper. Switched systems are very important in practical applications since their parameterizations are not constant. A switched system can result, for instance, from the use of a multimodel scheme, a multicontroller scheme, a buffer system or a multiestimation scheme. For instance, a (nonexhaustive) list of papers deal with some of these questions related to switched systems follow(1)In [15], the problem of delay-dependent stabilization for singular systems with multiple internal and external incommensurate delays is focused on. Multiple memoryless state-feedback controls are designed so that the resulting closed-loop system is regular, independent of delays, impulsefree and asymptotically stable. A relevant related problem for obtaining sufficiency-type conditions of asymptotic stability of a time-delay system is the asymptotic comparison of its solution trajectory with its delayfree counterpart provided that this last one is asymptotically stable, [19].(2)In [20], the problem of the 𝑁-buffer switched flow networks is discussed based on a theorem on positive topological entropy.(3)In [21], a multi-model scheme is used for the regulation of the transient regime occurring between stable operation points of a tunnel diode-based triggering circuit.(4)In [22, 23], a parallel multi-estimation scheme is derived to achieve close-loop stabilization in robotic manipulators whose parameters are not perfectly known. The multi-estimation scheme allows the improvement of the transient regime compared to the use of a single estimation scheme while achieving at the same time closed-loop stability.(5)In [24], a parallel multi-estimation scheme allows the achievement of an order reduction of the system prior to the controller synthesis so that this one is of reducedorder (then less complex) while maintaining closed-loop stability. (6)In [25], the stabilization of switched dynamic systems is discussed through topologic considerations via graph theory.(7)The stability of different kinds of switched systems subject to delays has been investigated in [11–13, 17, 26–28].(8)The stability switch and Hopf bifurcation for a diffusive prey-predator system is discussed in [6] in the presence of delay. (9)A general theory with discussed examples concerning dynamic switched systems is provided in [3].(10)Some concerns of time-delay impulsive models are of increasing interest in the areas of stabilization, neural networks, and Biological models with particular interest in positive dynamic systems. See, for instance, [29–40] and references therein.

The dynamic system under investigation is a linear polytopic system subject to internal point delays and feedback state-dependent impulsive controls. Both parameters and delays are assumed to be timevarying in the most general case. The control impulses can occur as separate events from possible continuous-time or bounded-jump type parametrical variations. Furthermore, each delayed dynamics is potentially parameterized in its own polytope. Those are the main novelties of this paper since it combines a time-varying parametrical polytopic nature with individual polytopes for the delay-free dynamics with time-varying parameters which are unnecessarily smooth for all time with a potential presence of delayed dynamics with point time-varying delays. The case of switching between parameterizations at certain time instants, what is commonly known as a switched system, [3, 17, 20–28], is also included in the developed formalism as a particular case as being equivalent to define the whole systems as a particular parameterization of the polytopic system at one of its vertices. The delays are assumed to be time differentiable of bounded time-derivative for some of the presented stability results but just bounded for the rest of results. An important key point is that if the system is stabilizable, then it can be stabilized via impulsive controls without requiring the delay-free dynamics of the system as it is then shown in some of the given examples. Usually, for a given interimpulse time interval, the impulsive amplitudes are larger as the instability degree becomes larger, and the signs of the control components also should be appropriate, in order to compensate it by the stabilization procedure. Such a property also will hold for nonpolytopic parameterizations. The design philosophy adopted in the paper is that stabilization might be achieved through appropriate impulsive controls at certain impulsive time instants without requiring the design of a standard regular controller. The paper is organized as follows. Section 2 discusses the various evolution operators valid to build the state-trajectory solutions in the presence of impulsive feedback state-dependent controls. Analytic expressions are given to define such operators. In particular, an important operator defined and discussed in this paper is the so-called impulsive evolution operator. Such an evolution operator is sufficiently smooth within open time intervals between each two consecutive impulsive times, but it also depends on impulses at time instants with hose ones happen. Section 3 discusses new global stability and global asymptotic stability issues based on Krasovsky-Lyapunov functionals taking account of the feedback state-dependent control impulses. The relevance of the impulsive controls towards stabilization is investigated in the sense that the most general results do not require stability properties of the impulse-free system (i.e., that resulting as a particular case of the general one in the absence of impulsive controls). Some included very conservative stability results follow directly from the structures of the state-trajectory solution and the evolution operators of Section 2 without invoking Lyapunov stability theory. It is proven that stabilization is achievable if impulses occur at certain intervals and with the appropriate amplitudes. Finally, two application examples are given in Section 4.

Notation 1.1. Z, R, and C are the sets of integer, real, and complex numbers, respectively.
𝐙₊ and 𝐑₊ denote the positive subsets of 𝐙, respectively, and 𝐂₊ denotes the subset of C of complex numbers with positive real part, and 𝑛∶={1,2,…,𝑛}⊂𝐙₊, forall𝑛∈𝐙₊.
𝐙₋ and 𝐑₋ denote the negative subsets of 𝐙, respectively, and 𝐂₋ denotes the subset of C of complex numbers with negative real part.𝐙𝐙₀₊∶=𝐙₊∪{0},𝐑₀₊∶=𝐑₊∪{0},𝐂₀₊∶=𝐂₊∪{0}₀₋∶=𝐙₋∪{0},𝐑₀₋∶=𝐑₋∪{0},𝐂₀₋∶=𝐂₋∪{0}(1.1)
Given some linear space 𝑋 (usually R or C), then C^(𝑖)(𝐑₀₊,𝑋) denotes the set of functions of class 𝐶^(𝑖). Also, BPC^(𝑖)(𝐑₀₊,𝑋)and PC^(𝑖)(𝐑₀₊,𝑋) denote the set of functions in 𝐶^(𝑖−1)(𝐑₀₊,𝑋) which, furthermore, possess bounded piecewise continuous or, respectively, piecewise continuous 𝑖th derivative on 𝑋.
𝐿(𝑋) denotes the set of linear operators from 𝑋 to 𝑋. In particular, the linear space denoted by 𝑋 denotes the state space of the dynamic system with controls in the linear space 𝑈.
𝐼_𝑛 denotes the 𝑛th identity matrix.
The symbols 𝑀≻0,𝑀≺0,𝑀≼0,and𝑀≼0 stand for positive definite, negative definite, positive semidefinite, and negative semidefinite square real matrices 𝑀, respectively. The notations 𝑀≻𝐷,𝑀≺𝐷,𝑀≼𝐷,and𝑀≼𝐷 stand correspondingly for (𝑀−𝐷)≻0,(𝑀−𝐷)≺0,(𝑀−𝐷)≼0,and(𝑀−𝐷)≼0, and Superscript “𝑇” stands for transposition of matrices and vectors.
𝜆_max(𝑀) and 𝜆_min(𝑀)stand for the maximum and minimum eigenvalues of a definite square real matrix 𝑀=(𝑚_𝑖𝑗).
A finite or infinite strictly ordered sequence of impulsive time instants is defined by Imp∶={𝑡_𝑖∈𝐑₀₊∶𝑡_𝑖+1>𝑡_𝑖}, where an impulsive control 𝑢(𝑡_𝑖)𝛿(𝑡−𝑡_𝑖) occurs with 𝛿(⋅) being the Dirac delta of the Dirac distribution.

2. The Dynamic System Subject to Time Delays and Impulsive Controls

Consider the following polytopic linear time-differential system of state vector and control of respective dimensions 𝑛 and 𝑚 and being subject to 𝑞 time-varying point delays:𝑥̇(𝑡)=^𝑞_𝑁𝑖=0_𝑗=1𝜆_𝑖𝑗(𝑡)𝐴_𝑖𝑗(𝑡)𝑥𝑡−ℎ_𝑖(𝑡)+𝐵_𝑖𝑗(𝑡)𝑢_𝑖𝑗(𝑡)=𝜆^𝑇(𝑡)𝑥(𝑡)+𝑢(𝑡),(2.1) where the incommensurate time-varying delays are ℎ₀(𝑡)=0forall𝑡∈𝐑₀₊, ℎ_𝑖∈PC⁽¹⁾(𝐑₀₊,𝐑₀₊), forall𝑖∈𝑞∶={1,2,…,𝑞} (i.e., the delays are continuous time differentiable of bounded time derivative), and𝜆^𝑇(𝑡)=𝜆₀₁(𝑡)𝜆₀₂(𝑡)⋯𝜆_0𝑁(𝑡)⋯𝜆_𝑞1(𝑡)𝜆_𝑞2(𝑡)⋯𝜆_𝑞𝑁(𝑡),𝑥^𝑇(𝑡)=𝑥^𝑇(𝑡)𝐴^𝑇01(𝑡)⋯𝐴^𝑇0𝑁(𝑡)⋯𝑥^𝑇𝑡−ℎ_𝑞(𝑡)𝐴^𝑇𝑞1(𝑡)⋯𝐴^𝑇𝑞𝑁(𝑡),𝑢^𝑇(𝑡)=𝑢^𝑇01(𝑡)𝐵^𝑇01(𝑡)⋯𝑢^𝑇0𝑁(𝑡)𝐵^𝑇0𝑁(𝑡)⋯𝑢^𝑇𝑞1(𝑡)𝐵^𝑇𝑞1(𝑡)⋯𝑢^𝑇𝑞𝑁(𝑡)𝐵^𝑇𝑞𝑁(𝑡)(2.2) are vector functions from 𝐑₀₊ to 𝐑^(𝑞+1)𝑁, 𝐑^{(𝑞+1)𝑁𝑛} and 𝐑^{(𝑞+1)𝑁𝑚}, respectively, and

(i) 𝑥∶𝐑₀₊∪[−ℎ,0)→𝑋⊂𝐑^𝑛 is the state vector, which is almost everywhere time differentiable on 𝐑₀₊ satisfying (2.1), subject to bounded piecewise continuous initial conditions on [−ℎ,0),that is, 𝑥=𝜑∈BPC⁽⁰⁾([−ℎ,0],𝐑^𝑛), where ℎ=ℎ(0)=max_𝑖∈𝑞sup(ℎ_𝑖(0))≤ℎ∶=max_𝑖∈𝑞sup_{𝑡∈𝐑0+}(ℎ(𝑡)), and 𝑢_𝑖𝑗∶𝐑₀₊→𝑈⊂𝐑^𝑚 are the control vectors forall𝑖∈𝑞∪{0}forall𝑗∈𝑁 and 𝐴_𝑖𝑗∈BPC⁽⁰⁾(𝐑₀₊,𝐑^𝑛×𝑛) and 𝐵_𝑖𝑗∈BPC⁽⁰⁾(𝐑₀₊,𝐑^𝑛×𝑚) parameterize the dynamic system.

(ii) 𝜆_𝑖𝑗∈BPC⁽⁰⁾(𝐑₀₊,𝐑₀₊), subject to the constraint ∑^𝑞𝑖=0∑^𝑁𝑗=0𝜆_𝑖𝑗(𝑡)∈[𝑐₁,𝑐₂]⊂𝐑₊,forall𝑡∈𝐑₀₊ with ∞>𝜀₂≥𝑐₂≥𝑐₁≥𝜀₁≥0 are the weighting scalar functions defining the polytopic system in the various delayed dynamics and parameterizations which are not all simultaneously zero at any time for some given lower-bound and upper-bound scalars 𝜀₁ and 𝜀₂. Note that there exist two summations in (2.1) related to these scalar functions, one them referring to the contribution of delayed dynamics for the various delays and the second one related to the system parameterization within the polytopic structure. It will be not assumed through the paper that the delay-free auxiliary system is stable. Note that the dynamic system can be seen as a convex polytopic dynamic system𝑥̇(𝑡)=^𝑞_𝑁𝑖=0_𝑗=1𝜆_𝑖𝑗(𝑡)𝑥̇_𝑖𝑗(𝑡)(2.3) formed with subsystems of the form 𝑥̇_𝑖𝑗(𝑡)=𝐴_𝑖𝑗(𝑡)𝑥(𝑡−ℎ_𝑖(𝑡))+𝐵_𝑖𝑗(𝑡)𝑢_𝑖𝑗(𝑡). The controls 𝑢_𝑖𝑗∶𝐑₀₊→𝑈⊂𝐑^𝑚 are generated from the state-feedback impulsive controller as follow: 𝑢_𝑖𝑗(𝑡)=𝐾_𝑖𝑗𝑢(𝑡)𝑥𝑡−ℎ_𝑖(𝑡)∀𝑖∈𝑞={1,2,…,𝑞};∀𝑡∈𝐑₀₊,for𝑖=0,𝑡∉Imp,_0𝑗𝑡⁺=𝐾_0𝑗𝑡⁺+𝐾^0𝑗(𝑡)𝑥𝑡⁺for𝑖=0,𝑡∈Imp,(2.4) where the strictly ordered Imp:={𝑡_𝑖∈𝐑₀₊∶𝑡_𝑖+1>𝑡_𝑖,𝑖∈𝐙₊} is the so-called sequence of impulsive time instants where the control impulses occur whose elements form a monotonically increasing sequence; that is, for any well posed test function 𝑓∶𝐑→𝐑, 𝑓(𝑡)=^∞−∞𝑓(𝜏)𝛿(𝑡−𝜏)𝑑𝜏=^𝑡+_𝑡𝑓(𝜏)𝛿(𝑡−𝜏)𝑑𝜏=lim_𝜀→0+^{𝑡+𝜀𝑡−𝜀}𝑓(𝜏)𝛿(𝑡−𝜏)𝑑𝜏,(2.5) where 𝛿(𝑡) is the Dirac distribution at time 𝑡=0 with the following notational convention being used:𝑔(𝑡⁺)=lim_𝜀→0⁺𝑔(𝑡+𝜀)≠𝑔(𝑡)=lim_𝜀→0⁺𝑔(𝑡−𝜀) either if 𝑡∈Imp or if 𝑔 is bounded having left and right limits at a discontinuity point 𝑡∈𝐑₀₊, and 𝑔(𝑡⁺)=𝑔(𝑡)if 𝐑₀₊∋𝑡∉Imp since the functions used are all left-continuous functions. Partial sequences of impulsive time instants are denoted by specifying the time intervals they refer to, as for instance, Imp[𝑇₁,𝑇₂]={𝑡∈Imp∶𝑡∈[𝑇₁,𝑇₂]} and Imp[𝑇₁,𝑇₂)={𝑡∈Imp∶𝑡∈[𝑇₁,𝑇₂)}. Note that Imp=Imp[0,∞). The regular and impulsive controller gain matrices are, respectively, 𝐾_𝑖𝑗∈BPC⁽⁰⁾(𝐑₀₊,𝐑^𝑚×𝑛) and 𝐾^𝑖𝑗∶Imp→𝐑^𝑚×𝑛 being a discrete sequence of bounded matrices. Note that, if 𝐾_0𝑗(𝑡) is discontinuous at the time instant 𝑡, then 𝐾_0𝑗(𝑡⁺)≠𝐾_0𝑗(𝑡) even if 𝑡∉Imp. The extensions to vector and matrix test functions are obvious by using respective appropriate zero components or entries if impulses do not occur at time 𝑡, a particular component or matrix entry. The substitution of the control law (2.4) into the open-loop system equation (2.1) leads to the closed-loop functional dynamic system as follows:𝑥̇(𝑡)=∑^𝑁𝑗=1𝜆_0𝑗(𝑡)𝐴^∗0𝑗(𝑡𝑥)+𝑡𝐵_0𝑗(𝑡)𝐾^0𝑗(𝑡)𝛿(0)𝑥(𝑡)+∑^𝑞𝑖=1∑^𝑁𝑗=1𝜆_𝑖𝑗(𝑡)𝐴^∗𝑖𝑗(𝑡)𝑥𝑡−ℎ_𝑖(𝑡)₊,=𝐼_𝑛+∑^𝑁𝑗=1𝜆_0𝑗(𝑡)𝐵_0𝑗(𝑡)𝐾^0𝑗(𝑡)𝑥(𝑡);(2.6)

forall𝑡∈𝐑₀₊ with 𝐾^0𝑗(𝑡)=0; forall𝑡∉Imp, where𝐴^∗𝑖𝑗(𝑡)=𝐴_𝑖𝑗(𝑡)+𝐵_𝑖𝑗(𝑡)𝐾_𝑖𝑗(𝑡),∀𝑖∈𝑞∪{0}∀𝑗∈𝑁,(2.7)

Equation (2.6) becomes𝑥̇(𝑡)=^𝑞_𝑁𝑖=0_𝑗=1𝜆_𝑖𝑗(𝑡)𝐴^∗𝑖𝑗(𝑡)𝑥𝑡−ℎ_𝑖(𝑡),(2.8) for all 𝑡∉Imp and also at the left limits for all 𝑡∈Imp, and 𝑥(𝑡⁺)−𝑥(𝑡)=∑^𝑁𝑗=1𝜆_0𝑗(𝑡)𝐵_0𝑗(𝑡)𝐾^0𝑗(𝑡)𝑥(𝑡), which is zero if 𝑡∉Imp, with 𝑥̇𝑡⁺=^𝑁_𝑗=1𝜆_0𝑗(𝑡)𝐴^∗0𝑗𝑡⁺𝑥𝑡⁺+^𝑞_𝑁𝑖=1_𝑗=1𝜆_𝑖𝑗(𝑡)𝐴^∗𝑖𝑗𝑡⁺𝑥𝑡⁺−ℎ_𝑖(𝑡)(2.9) for the right limits of all 𝑡∈Imp. Define 𝐷∶=Imp∪𝐷_𝑝, where𝐷_𝑝∶=⎛⎜⎜⎝_{𝑖∈𝑞∪{0},𝑗∈𝑁}𝐷_𝐴𝑖𝑗⎞⎟⎟⎠∪⎛⎜⎜⎝_{𝑖∈𝑞∪{0},𝑗∈𝑁}𝐷_𝐵𝑖𝑗⎞⎟⎟⎠∪⎛⎜⎜⎝_{𝑖∈𝑞∪{0},𝑗∈𝑁}𝐷_𝜆𝑖𝑗⎞⎟⎟⎠∪⎛⎜⎜⎝_{𝑖∈𝑞∪{0},𝑗∈𝑁}𝐷_𝐾𝑖𝑗⎞⎟⎟⎠(2.10) is the total set of discontinuities on 𝐑₀₊ of 𝐴_𝑖𝑗∈BPC⁽⁰⁾(𝐑₀₊,𝐑^𝑛×𝑛), 𝐵_𝑖𝑗∈BPC⁽⁰⁾(𝐑₀₊,𝐑^𝑛×𝑚), 𝜆_𝑖𝑗∈BPC⁽⁰⁾(𝐑₀₊,𝐑₀₊), and 𝐾_𝑖𝑗∈BPC⁽⁰⁾(𝐑₀₊,𝐑^𝑚×𝑛)forall𝑖∈𝑞∪{0}, forall𝑗∈𝑁 which are in the respective sets 𝐷_𝐴𝑖𝑗, 𝐷_𝐵𝑖𝑗, 𝐷_𝜆𝑖𝑗, and 𝐷_𝐾𝑖𝑗. The following technical assumptions are made.

Assumption. thereexist𝜐∈𝐑₊ such that 𝑡_𝑘+1−𝑡_𝑘≥𝜐,forall𝑡_𝑘,𝑡_𝑘+1(>𝑡_𝑘)∈Imp.

Assumption. ((⋃_𝑗∈𝑁𝐷_𝐵0𝑗)∪(⋃_𝑗∈𝑁𝐷_𝜆𝑖𝑗))∩Imp=∅.
Assumption 2.1 implies that the sequence of impulsive time instants is a real sequence with no accumulation points. It is a technical assumption to guarantee the existence and uniqueness of an almost everywhere time-differentiable state-trajectory solution. Assumption 2.2 is needed for all the functions 𝜆_0𝑗(𝐵_0𝑗)_𝑘ℓ∈BPC⁽⁰⁾(𝐑₀₊,𝐑₀₊)forall𝑗∈𝑁, forall𝑘∈𝑛 and forallℓ∈𝑚, build with the entries 𝐵_0𝑗∈BPC⁽⁰⁾(𝐑₀₊,𝐑^𝑛×𝑚). This follows since they are piecewise continuous on 𝐑₀₊ and, furthermore, continuous at any small neighborhood around any point of the sequence of impulsive time instants where control impulses occur. From Picard-Lindeloff theorem, there is a unique solution for any vector function of initial conditions 𝜑∈BPC⁽⁰⁾([−ℎ,0],𝐑^𝑛) and 𝑥∈BPC⁽¹⁾(𝐑₊,𝐑^𝑛). The state-trajectory solution of the closed-loop system (2.8)-(2.9) for initial conditions  𝜑∈BPC⁽⁰⁾([−ℎ,0],𝐑^𝑛) is given by𝑥(𝑡)=Ψ(𝑡)Ψ⁻¹(0)𝑥(0)+^𝑞_𝑁𝑖=1_𝑗=1^𝑡0+Ψ⁻¹(𝜏)𝜆_𝑖𝑗(𝜏)𝐴^∗𝑖𝑗(𝜏)𝑥𝜏−ℎ_𝑖(𝜏)𝑑𝜏_{𝑡𝑘∈Imp[𝑁0,𝑡)}_𝑗=1=Ψ_𝑠𝜆_0𝑗𝑡_𝑘Ψ⁻¹𝑡_𝑘𝐵_0𝑗𝑡_𝑘𝐾^0𝑗𝑡_𝑘𝑥𝑡_𝑘⎤⎥⎥⎦+𝑡,𝑡₀𝑥𝑡₀+^𝑞_𝑁𝑖=1_𝑗=1^𝑡0𝜆_𝑖𝑗(𝜏)Ψ_𝑠(𝑡,𝜏)𝐴^∗𝑖𝑗(𝜏)𝑥𝜏−ℎ_𝑖(𝜏)𝑑𝜏_{𝑡𝑘∈Imp𝑡0𝑁,𝑡}_𝑗=1𝜆_0𝑗𝑡_𝑘Ψ_𝑠𝑡,𝑡_𝑘𝐵_0𝑗𝑡_𝑘𝐾^0𝑗𝑡_𝑘𝑥𝑡_𝑘⎤⎥⎥⎦,(2.11) subject to 𝑥(𝑡)=𝜑(𝑡),forall𝑡∈[−ℎ,0], where(1)Ψ(𝑡)∈𝐶⁽⁰⁾(𝐑₀₊,𝐑^𝑛×𝑛) is an almost everywhere differentiable matrix function on 𝐑₊ (being time differentiable on the non connected real set ⋃_{𝑡𝑖∈Imp}(𝑡_𝑖+1−𝑡_𝑖)) with unnecessarily continuous time derivatives which satisfies Ψ̇(𝑡)=∑^𝑁𝑗=1𝜆_0𝑗(𝑡)𝐴^∗0𝑗(𝑡)Ψ(𝑡) on 𝐑₊ with Ψ(0)=𝐼_𝑛. If 𝐴_𝑖𝑗, 𝐵_𝑖𝑗, 𝜆_𝑖𝑗,and 𝐾_𝑖𝑗forall𝑖∈𝑞∪{0}, forall𝑗∈𝑁 are everywhere continuous on 𝐑₊,then Ψ(𝑡)∈𝐶⁽¹⁾(𝐑₀₊,𝐑^𝑛×𝑛), Ψ_𝑠(⋅,⋅)∶𝐑²⁰⁺→𝐑^𝑛×𝑛as Ψ_𝑠(𝑡,𝜏)=Ψ(𝑡)Ψ⁻¹(𝜏)forall𝑡≥𝜏, and (2)Imp[𝑡₀,𝑡):={𝑡_𝑘∈𝐑₀₊∶𝑡₀≤𝑡_𝑘(∈Imp)<𝑡}⊂Imp is the strictly ordered sequence of impulsive time instants with input impulses occurred on [𝑡₀,𝑡) for any 𝑡₀∈𝐑₊. Also, Imp(𝑡₀,𝑡):={𝑡_𝑘∈Imp∶𝑡₀<𝑡_𝑘<𝑡}⊂Imp;Imp(𝑡₀,𝑡]:={𝑡_𝑘∈Imp∶𝑡₀<𝑡_𝑘≤𝑡}⊂Imp are defined in a closed way.The solution (2.11) is identically defined by 𝑥(𝑡)=𝑍(𝑡)+𝑍⁻¹(0)𝑥(0)+^0−ℎ𝑍⁻¹(𝜏)𝜑(𝜏)𝑑𝜏_{𝑡𝑘∈Imp𝑁(0,𝑡)}_𝑗=1𝜆_0𝑗𝑡_𝑘𝑍⁻¹𝑡_𝑘𝐵_0𝑗𝑡_𝑘𝐾^0𝑗𝑡_𝑘𝑥𝑡_𝑘⎤⎥⎥⎦,(2.12) where 𝑍(𝑡)∈𝐶⁽⁰⁾(𝐑₀₊,𝐑^𝑛×𝑛) is an almost everywhere differentiable matrix function on 𝐑₊, with unnecessarily continuous time derivatives, which satisfies (2.8) on 𝐑₊ with Z (0)=𝐼_𝑛, 𝑍(𝑡)=0forall𝑡∈𝐑₋. Defining the matrix function 𝑍_𝑠(⋅,⋅)∶𝐑²⁰⁺→𝐑^𝑛×𝑛 as 𝑍_𝑠(𝑡,𝜏)=𝑍(𝑡)𝑍⁻¹(𝜏)forall𝑡≥𝜏, one has from (2.12) for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1] for any two consecutive given𝑡_𝑘,𝑡_𝑘+1∈Imp as follow: 𝑥(𝑡)=𝑍_𝑠𝑡,𝑡_𝑘𝑥𝑡^+𝑘+^𝑞_𝑖=1^0−ℎ_𝑖+𝑍_𝑠𝑡,𝑡_𝑘+𝜏𝜑𝑡_𝑘+𝜏𝑑𝜏_𝑁_𝑗=1𝑍_𝑠𝑡,𝑡_𝑘𝜆_0𝑗𝑡_𝑘𝐵_0𝑗𝑡_𝑘+1𝑥𝑡^+𝑘+1𝐾^0𝑗𝑡_𝑘+1,(2.13) which becomes for 𝑡=𝑡^+𝑘+1 as follow: 𝑥𝑡^+𝑘+1=𝐼_𝑛+^𝑁_𝑗=1=𝑍_𝑠𝑍_𝑠𝑡_𝑘+1,𝑡_𝑘𝐵_0𝑗𝑡_𝑘+1𝐾^0𝑗𝑡_𝑘+1𝑥𝑡_𝑘+1+𝑡_𝑘+1,𝑡_𝑘𝑥𝑡^+𝑘+^𝑞_𝑖=1^0−ℎ_𝑖𝑍_𝑠𝑡,𝑡_𝑘+𝜏𝜑𝑡_𝑘+𝜏𝑑𝜏_𝑁_𝑗=1𝑍_𝑠𝑡_𝑘+1,𝑡_𝑘𝜆_0𝑗𝑡_𝑘𝐵_0𝑗𝑡_𝑘+1𝑥𝑡_𝑘+1𝐾^0𝑗𝑡_𝑘+1𝛿𝑡,𝑡_𝑘+1,(2.14) where 𝛿(𝑡,𝑡_𝑘+1)=1 if 𝑡=𝑡_𝑘+1 and zero otherwise is the Kronecker delta. In view of (2.12), the state-trajectory solution can be defined by the impulsive evolution operator {𝑇(𝑡,𝑡_𝑘)∶𝑡∈[𝑡_𝑘,𝑡_𝑘+1],forall𝑡_𝑘∈Imp}, associated with {𝑍(𝑡)∶𝑡∈𝐑₀₊} where 𝑇(⋅,⋅)∶{([𝑡_𝑘,𝑡_𝑘+1]∶𝑡_𝑘∈Imp∪{0})}→𝐿(𝑋), which is represented by 𝑥(𝑡)=𝑇(𝑡,𝑡_𝑘)𝑥_𝑡^+𝑘; forall𝑡∈[𝑡_𝑘,𝑡_𝑘+1],forall𝑡_𝑘∈Impso that: 𝑥𝑡_+𝑘+1𝑥(𝑡)=𝑇𝑡,𝑡_𝑘𝑥_𝑡^+𝑘,=𝑇𝑡^+𝑘+1,𝑡_𝑘𝑥_𝑡^+𝑘=𝐼_𝑛+^𝑁_𝑗=1𝜆_0𝑗𝑡_𝑘+1𝐵_0𝑗𝑡_𝑘+1𝐾^0𝑗𝑡_𝑘+1𝑇𝑡_𝑘+1,𝑡_𝑘𝑥_𝑡^+𝑘,(2.15)forall𝑡∈[𝑡_𝑘,𝑡_𝑘+1],forall𝑡_𝑘∈Imp, where 𝑥_𝑡 and 𝑥_𝑡+ denote the strings of state solution trajectory and {𝑥(𝜏)∶𝜏∈[𝑡−ℎ,𝑡)} and {𝑥(𝜏)∶𝜏∈[𝑡−ℎ,𝑡]}, respectively. The subsequent result follows directly for the state-trajectory solution from (2.11) for any initial conditions 𝜑∈BPC⁽⁰⁾([−ℎ,0],𝐑^𝑛).

Theorem 2.3. The following properties hold.
(i) The state-trajectory solution satisfies the following equations on any interval [𝜁,𝑡)⊂𝐑₀₊for any 𝜑∈BPC⁽⁰⁾([−ℎ,0],𝐑^𝑛): 𝑥𝑡^+𝑘+1=𝐼_𝑛+^𝑁_𝑗=1=Ψ_𝑠Ψ_𝑠𝑡_𝑘+1,𝑡_𝑘+1𝜆_0𝑗𝑡_𝑘+1𝐵_0𝑗𝑡_𝑘+1𝐾^0𝑗𝑡_𝑘+1𝑥𝑡_𝑘+1(2.16)+𝑡_𝑘+1,𝜁𝑥𝜁⁺+^𝑡𝑘+1_𝜁+Ψ_𝑠𝑡_𝑘+1,𝜏^𝑞_𝑁𝑖=1_𝑗=1𝜆_𝑖𝑗(𝜏)𝐴^∗𝑖𝑗(𝜏)𝑥𝜏−ℎ_𝑖(𝜏)𝑑𝜏_{𝑡𝑖∈Imp𝜁,𝑡𝑘+1𝑁}_𝑗=1=𝐼𝜆_0𝑗𝑡_𝑖Ψ_𝑠𝑡_𝑘+1,𝑡_𝑖𝐵_0𝑗𝑡_𝑖𝐾^0𝑗𝑡_𝑖𝑥𝑡_𝑖(2.17)_𝑛+^𝑁_𝑗=1=Ζ_𝑠Ζ_𝑠𝑡_𝑘+1,𝑡_𝑘+1𝐵_0𝑗𝑡_𝑘+1𝐾^0𝑗𝑡_𝑘+1𝑥𝑡_𝑘+1(2.18)+𝑡_𝑘+1,𝜁𝑥𝜁⁺+^𝑞_𝑖=1^0−ℎ_𝑖𝑍_𝑠𝑡_𝑘+1,𝜁+𝜏𝑥(𝜁+𝜏)𝑑𝜏_{𝑡𝑖∈Imp𝜁,𝑡𝑘+1𝑁}_𝑗=1=𝑇𝑡_+𝑘+1𝜆_0𝑗𝑡_𝑖𝑍_𝑠𝑡_𝑘+1,𝑡_𝑖𝐵_0𝑗𝑡_𝑖𝑥𝑡_𝑖𝐾^0𝑗𝑡_𝑖(2.19)=,𝜁𝑥_𝜁⁺(2.20)_{𝑡𝑖,𝑡𝑖+1∈Imp𝜁,𝑡𝑘+1}𝐼^𝑛+^𝑁_𝑗=1𝜆_0𝑗𝑡_𝑖+1𝐵_0𝑗𝑡_𝑖+1𝐾^0𝑗𝑡_𝑖+1𝑇𝑡_𝑖+1,𝑡_𝑖𝑥_𝜁⁺,(2.21)forall𝑡_𝑘+1(>𝜁)∈Imp,forall𝜁∈𝐑₀₊ with 𝑇(𝑡_𝑘+1,𝑡_𝑘+1)=𝑍_𝑠(𝑡_𝑘+1,𝑡_𝑘+1)=Ψ_𝑠(𝑡_𝑘+1,𝑡_𝑘+1)=𝐼_𝑛. Equations (2.17) and (2.19) are also valid by replacing 𝑡_𝑘+1→𝑡, forall𝑡∈(𝑡_𝑘+1,𝑡_𝑘+2) if 𝑡_𝑘+2∈Impand forall𝑡∈(𝑡_𝑘+1,∞) if (𝑡_𝑘+1,∞)∩Imp=∅, that is, if the sequence of impulsive time instants is finite with the last impulsive time instant being 𝑡_𝑘+1. Equation (2.21) has to be modified by replacing𝑡_𝑘+1→𝑡and then by premultiplying it by 𝑇(𝑡,𝑡_𝑘+1).
(ii) Assume that ‖‖‖‖‖_{𝑡𝑖,𝑡𝑖+1∈Imp𝜁,𝑡𝑘+1}𝐼^𝑛+^𝑁_𝑗=1‖‖‖‖𝐼𝜆_0𝑗𝑡_𝑖+1𝐵_0𝑗𝑡_𝑖+1𝐾^0𝑗𝑡_𝑖+1𝑇𝑡_𝑖+1,𝑡_𝑖‖‖‖‖‖≤𝑀_𝑇≤1(2.22)_𝑛+^𝑁_𝑗=1𝜆_0𝑗(𝑡)𝐵_0𝑗(𝑡)𝐾^0𝑗(𝑡)𝑇𝑡,𝑡_𝑐imp‖‖‖‖≤𝑀^𝑇≤1,(2.23)forall𝑡_𝑘+1(>𝜁)∈Imp, forall𝜁∈𝐑₀₊, and forall𝑡≥𝑡_𝑐imp provided that 𝑐_imp∶=cardImp[0,∞)<∞, then ‖Γ𝑥‖_L^p_(𝐑+,𝑋)≤𝐶_Γ, where Γ∶Dom(Γ)≡𝑋→L^𝑝(𝐑₊,𝑋) is defined by (Γ𝑥)(𝑡)=𝑇(𝑡,𝜃)𝑥. for all 𝑥∈𝑋.

Proof. (i) It follows directly for the state-trajectory solution from (2.11), (2.14), and (2.15) for any time interval [𝜁−ℎ,𝜁) of initial conditions 𝜑∈BPC⁽⁰⁾([−ℎ,0],𝐑^𝑛).
(ii) The first part follows from the definition of the impulsive evolution operator. If, in addition, 𝑀_𝑇<1, then it follows from the following given constraints: ∃lim_𝑡→∞𝑇(𝑡,𝜃)𝜉=0,∀𝑡(>𝜃)∈𝐑₊,𝜃∈𝐑₀₊,∀𝜉∈𝑋⟹𝑇(𝑡,𝜃)𝜉(2.24) is bounded, forall𝜉∈𝑋, forall𝑡(>𝜃)∈𝐑₊,𝜃∈𝐑₀₊⟹‖𝑇(𝑡,𝜃)‖≤𝐶_𝑇,some𝐑∋𝐶_𝑇≥1,∀𝑡(>𝜃)∈𝐑₊,𝜃∈𝐑₀₊(2.25) (from the uniform boundedness principle). Now, note that the operator Γ∶Dom(Γ)≡𝑋→𝐿^𝑝(𝐑₊,𝑋) is closed and then bounded from the closed graph theorem, so that the proof of Property (ii) is complete.

Remark 2.4. Stabilization by impulsive controls may be combined with the design of regular stabilization controllers or used as the sole stabilization tool. Some advantages related to the use of impulsive control for stabilization of stabilizable systems arise in the cases when the classical regular controller are of high design and maintenance costs.

3. Stability

The global asymptotic stability of the controlled system is now investigated. Firstly, a conservative stability result follows from Theorem 2.3 (2.16)–(2.21), which does not take into account possible compensations of the impulsive controls for stabilization purposes.

Theorem 3.1. Assume that the sequence Imp is infinite, ‖Ψ_𝑠(𝑡,𝜏)‖≤𝑘_Ψ𝑒^{−𝜌𝜓(𝑡−𝜏)}, forall𝑡≥𝜏+𝑡₀, some finite 𝑡₀>0,some 𝐑₊∋𝑘_Ψ>0, and some 𝜌_𝜓∈𝐑₊ as follow: 𝑘_Ψ1+sup^{𝑡𝑘+𝑖𝑝≤𝜏≤𝑡𝑘+(𝑖+1)𝑝}‖‖∑^𝑞𝑖=1∑^𝑁𝑗=1𝜆_𝑖𝑗(𝜏)𝐴^∗𝑖𝑗(𝜏)‖‖₂𝜌_Ψ+‖‖‖‖‖_{𝑡𝑗∈Imp𝑡𝑘+𝑖𝑝,𝑡𝑘+(𝑖+1)𝑝𝑁}_𝑗=1𝜆_0𝑗𝑡_𝑗𝐵_0𝑗𝑡_𝑗𝐾^0𝑗𝑡_𝑗𝑒^{−𝜌Ψ(𝑡𝑘+(𝑗+1)𝑝−𝑡𝑗)}‖‖‖‖‖₂⎞⎟⎟⎠≤1,(3.1) for some 𝑝∈𝐙₊, some finite 𝑘∈𝐙₀₊, some subsequence {𝑡_{𝑘+𝑖𝑝}}∈Imp, forall𝑖∈𝐙₀₊. Thus, the closed-loop system (2.8)-(2.9) is globally stable. If the above inequality is strict, then the system is globally asymptotically stable. Also, if the sequence Imp is finite, then the results are valid 𝑘_Ψ(1+sup_{𝑡𝑘≤𝜏<∞}‖∑^𝑞𝑖=1∑^𝑁𝑗=1𝜆_𝑖𝑗(𝜏)𝐴^∗𝑖𝑗(𝜏)‖₂/𝜌_Ψ)≤1(<1)with 𝑡_𝑘 being the last element of the finite sequence Imp

Now, a general stability result follows, which proves that (in general, nonasymptotic) global stability is achievable by some sequence of impulsive controls generated from appropriate impulsive controller gains.

Theorem 3.2. There is a sequence of impulsive time instants Imp∶={𝑡_𝑖∈𝐑₀₊} such that the closed-loop system (2.6)–(2.7) is globally stable for any function of initial conditions 𝜑∈BPC⁽⁰⁾([−ℎ,0],𝐑^𝑛) for some sequence of impulsive controller gains 𝐾^0𝑗∶𝐑₀₊→𝐑^𝑛×𝑚, forall𝑗∈𝑁,forall𝑖∈𝑞∪{0}.

Proof. The basic equation to build the stability proof is 𝑥(𝑡⁺)−𝑥(𝑡)=∑^𝑁𝑗=1𝜆_0𝑗(𝑡)𝐵_0𝑗(𝑡)𝐾^0𝑗(𝑡)𝑥(𝑡), forall𝑡∈Imp and any sequence of impulsive time instants Imp. Consider prefixed real constants 𝐾_𝑖∈𝐑₊(𝑖∈4) fulfilling 𝐾₁≤𝐾₃−𝜀₁ and 𝐾₄≤𝐾₂−𝜀₂ with 𝜀₁∈(0,𝐾₃)∩𝐑₀₊ and 𝜀₂∈(0,𝐾₂)∩𝐑₀₊such that 𝑥_𝑘(0)∈[𝐾₃,𝐾₄]⊂[𝐾₁+𝜀₁,𝐾₂−𝜀₂]⊂[𝐾₁,𝐾₂], for all 𝑘∈𝑛. The proof of global stability is now made by complete induction. Assume that some finite or infinite 𝑡∈𝐑₊ exists such that 𝑥_𝑘(𝜏)∈[𝐾₁,𝐾₂]; forall𝜏∈[0,𝑡), but 𝑥_𝑘(𝑡)∈((−∞,𝐾₃)∪(𝐾₄,∞))∩[𝐾₁,𝐾₂]for some 𝑘∈𝑛, some 𝐾₃∈𝐑 with an existing (perhaps empty) partial sequence of impulsive time instants Imp[0,𝑡) until time 𝑡. Such a time 𝑡 always exists from the boundedness and almost everywhere continuity of the state-trajectory solution. Then, 𝑡∈Imp so that Imp[0,𝑡]=Imp[0,𝑡)∪{𝑡} is fixed as the first impulsive time instant and −∞<𝐾₃≤𝑥_𝑘𝑡⁺=𝛿(𝑘,ℓ)+^𝑁_𝑚𝑗=1_𝑛𝑖=1_ℓ=1𝜆_0𝑗(𝑡)𝐵_{0𝑗𝑘𝑖}(𝑡)𝐾^{0𝑗𝑖ℓ}(𝑡)𝑥_ℓ(𝑡)≤𝐾₄<∞,(3.2) where the entry notation 𝑀=(𝑀_𝑖𝑗)for a matrix 𝑀 is used, provided that the impulsive controller gain 𝐾^{0𝑗𝑖𝑘}(𝑡) is chosen so that the following constraint holds: 𝐾₃−∑^𝑁𝑗=1∑^{𝑚𝑖(≠𝑘)=1}∑^{𝑛ℓ(≠𝑘)=1}𝜆_0𝑗(𝑡)𝐵_{0𝑗𝑘𝑖}(𝑡)𝐾^{0𝑗𝑖ℓ}(𝑡)𝑥_ℓ(𝑡)−𝑥_𝑘(𝑡)≤𝐾_{0𝑗𝑘𝑘}1+∑^𝑁𝑗=1𝜆_0𝑗(𝑡)𝐵_{0𝑗𝑘𝑖}(𝑡)𝐾^{0𝑗𝑖𝑘}(𝑡)𝑥_𝑘(𝑡)(𝑡)≤𝐾⁴−∑^𝑁𝑗=1∑^{𝑚𝑖(≠𝑘)=1}∑^{𝑛ℓ(≠𝑘)=1}𝜆_0𝑗(𝑡)𝐵_{0𝑗𝑘𝑖}(𝑡)𝐾^{0𝑗𝑖ℓ}(𝑡)𝑥_ℓ(𝑡)−𝑥_𝑘(𝑡)1+∑^𝑁𝑗=1𝜆_0𝑗(𝑡)𝐵_{0𝑗𝑘𝑖}(𝑡)𝐾^{0𝑗𝑘𝑘}(𝑡)𝑥_𝑘(𝑡).(3.3)
Note by direct inspection of (3.3) that such a controller gain always exists. As a result, 𝑥_𝑘(𝑡⁺)∈[𝐾₃,𝐾₄]⊂[𝐾₁,𝐾₂], forall𝑘∈𝑛. By continuity of the state-trajectory solution, there exists a finite 𝑇(𝑡,𝐾^)∈𝐑₊ such that 𝑥_𝑘(𝜏)∈[𝐾₃−𝐾^,𝐾₄+𝐾^]⊂[𝐾₁,𝐾₂] for any prefixed 𝐾^∈𝐑₊, forall𝜏∈[𝑡,𝑡+𝑇(𝑡,𝐾^)), forall𝑘∈𝑛 provided that𝐾₃−𝐾₁≤𝐾^≤𝐾₂−𝐾₄. Since 𝑥_𝑘(𝑡+𝑇(𝑡,𝐾^))∈((−∞,𝐾₃)∪(𝐾₄,∞))∩[𝐾₁,𝐾₂]then 𝑥_𝑘(𝜏)⊂[𝐾₁,𝐾₂],forall𝜏∈[0,𝑡+𝑇(𝑡)), forall𝑘∈𝑛. Also, 𝑥_𝑘(𝜏)⊂[𝐾₁,𝐾₂] for all𝜏∈[0,𝑡+𝑇(𝑡)], forall𝑘∈𝑛 if an impulsive controller gain is chosen at time 𝑡+𝑇(𝑡) by replacing 𝑡→𝑡+𝑇(𝑡) in (3.3) and Imp[0,𝑡+𝑇(𝑡)]=Imp[0,𝑡+𝑇(𝑡))∪{𝑡+𝑇(𝑡)} with Imp[0,𝑡+𝑇(𝑡))=Imp[0,𝑡]. It has been proven that 𝑥_𝑘(𝜏)∈[𝐾₁,𝐾₂], forall𝜏∈[0,𝑡) for any given 𝑡∈𝐑₀₊, forall𝑘∈𝑛 then 𝑥_𝑘(𝜏)∈[𝐾₁,𝐾₂], forall𝜏∈[0,𝑡+𝑇(𝑡)], for some 𝑇(𝑡)∈𝐑₊,and for all 𝑘∈𝑛 so that the result holds by complete induction for forall𝑡∈𝐑₀₊ with a bounded sequence of impulsive controller gains at some appropriate sequence of impulsive time instants Imp:={𝑡_𝑖∈𝐑₀₊}.

Remark 3.3. Note that Theorem 3.2 holds irrespective of the values of the regular controller gain functions 𝐾_𝑖𝑗∶𝐑₀₊→𝐑^𝑚×𝑛 for some appropriate sequence of impulsive controller gains 𝐾^0𝑗∶𝐑₀₊→𝐑^𝑛×𝑚, forall𝑗∈𝑁,forall𝑖∈𝑞∪{0}. The reason is that the stabilization mechanism consists of decreasing the absolute values of the state components as much as necessary at its right limits at the impulsive time instants for any values of their respective left-hand-side limits and values at previous values at the intervals between consecutive impulsive time instants.

The subsequent result establishes that the stabilization is achievable with the stabilizing impulsive controller gains being chosen arbitrarily except at some subsequence of the impulsive time instants.

Theorem 3.4. The closed-loop system (2.6)–(2.7) is globally stable for any 𝜑∈BPC⁽⁰⁾([−ℎ,0],𝐑^𝑛) and any given set of regular controller gain functions 𝐾_𝑖𝑗∶𝐑₀₊→𝐑^𝑛×𝑚 if the sequence of impulsive time instants Imp∶={𝑡_𝑖∈𝐑₀₊} is chosen so that (1)the sequence of impulsive controller gains 𝐾^0𝑗∶𝐑₀₊→𝐑^𝑛×𝑚, forall𝑗∈𝑁;forall𝑖∈𝑞∪{0} is chosen appropriately for some subsequence of impulsive time instants Imp^∗∶={𝑡^∗𝑘}⊂Impsatisfying 𝑡^∗𝑘+1−𝑡^∗𝑘≤𝑇^∗(𝑡^∗𝑘)<∞, for each two consecutive 𝑡^∗𝑘,𝑡^∗𝑘+1∈Imp^∗(2)such a sequence of impulsive controller gains is chosen arbitrarily for the sequence Imp⧵Imp^∗.

Proof. Consider the following Lyapunov functional candidate 𝑉∶𝐑₀₊×𝐑^𝑛→𝐑₀₊, [17]: 𝑉𝑡,𝑥_𝑡∶=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+^𝑞_𝑖=1^{𝑡𝑡−ℎ}_𝑖(𝑡)𝑥^𝑇(𝜏)𝑆_𝑖(𝜏)𝑥(𝜏)𝑑𝜏,(3.4) where 𝐑^𝑛×𝑛∋𝑃=𝑃^𝑇≻0 and 𝑆_𝑖∈BPC⁽⁰⁾(𝐑₀₊,𝐑^𝑛×𝑛) fulfils 𝑆_𝑖(𝑡)≻0, forall𝑡∈𝐑₀₊, forall𝑖∈𝑞. One gets by taking time-derivatives in (3.4) using (2.6) as follow: 𝑉̇𝑡,𝑥_𝑡∶=2𝑥^𝑇(𝑡)𝑃^𝑁_𝑗=1𝜆_0𝑗(𝑡)𝐵_0𝑗(𝑡)𝐾^0𝑗(𝑡)𝛿(0)+^𝑞_𝑁𝑖=0_𝑗=1+𝜆_𝑖𝑗(𝑡)𝐴^∗𝑖𝑗(𝑡)𝑥𝑡−ℎ_𝑖(𝑡)𝑥(𝑡)_𝑞_𝑖=1𝑥^𝑇=𝑥̂_𝑇(𝑡)𝑆_𝑖(𝑡)𝑥(𝑡)−1−ℎ̇_𝑖(𝑡)𝑥^𝑇𝑡−ℎ_𝑖(𝑡)𝑆_𝑖𝑡−ℎ_𝑖(𝑡)𝑥^𝑇𝑡−ℎ_𝑖(𝑡)(3.5)(𝑡)𝑄(𝑡)𝑥̂(𝑡)=−𝑥̂^𝑇(𝑡)𝑄_𝑑(𝑡)+𝑄_𝑜𝑑(𝑡)𝑥̂(𝑡),(3.6) where 𝑥̂𝑄(𝑡)(𝑡=)∶=𝑥^𝑇B(l𝑡o)c𝑥k^𝑇m𝑡a−triℎx₁(𝑄𝑡)_𝑖𝑗⋯𝑥^𝑇𝑡−ℎ_𝑞(𝑡)^𝑇,(𝑡)∶𝑖,𝑗∈𝑞+1,(3.7) with 𝑄₁₁(𝑡)=^𝑁_𝑗=1+𝑃𝜆_0𝑗(_𝑁𝑡)𝐴^∗0𝑗(𝑡)+𝐵_0𝑗(𝑡)𝐾^0𝑗(𝑡)𝛿(0)^𝑇𝑃_𝑗=1𝜆_0𝑗(𝑡)𝐴^∗0𝑗(𝑡)+𝐵_0𝑗(𝑡)𝐾^0𝑗(𝑡)𝛿(0)+^𝑞_𝑖=1𝑄𝑆_𝑖(𝑡)_1,𝑖+1(𝑡)=𝑄^𝑇𝑖+1,1(𝑡)∶=^𝑁_𝑗=1𝑄𝜆_𝑖𝑗(𝑡)𝑃𝐴^∗𝑖𝑗(𝑡),∀𝑖∈𝑞,_𝑖𝑖𝑄(𝑡)∶=−1−ℎ̇_𝑖(𝑡)𝑆_𝑖𝑡−ℎ_𝑖(𝑡),𝑄_𝑖𝑗(𝑡)=0,∀𝑖,𝑗(≠𝑖)∈𝑞+1⧵{1},_𝑑𝑄(𝑡)=Blockdiag−𝑄₁₁(𝑡)−𝑄₂₂(𝑡)−𝑄_{𝑞+1,𝑞+1}(𝑡)_𝑜𝑑,(𝑡)=−𝑄(𝑡)+𝑄_𝑑(𝑡)=⎡⎢⎢⎢⎢⎢⎣0−𝑄¹²−𝑄_𝑇12(𝑡)⋯−𝑄_1,𝑞+1(𝑡)−𝑄⋮_𝑇𝑞+1,1(𝑡)⋮0−𝑄⋱₂₃(𝑡)⋯−𝑄⋮_2,𝑞+1(𝑡)(𝑡)−𝑄^𝑇𝑞+1,2(𝑡)(𝑡)−𝑄^{𝑇𝑞+1,𝑞}(𝑡)(𝑡)⋯0⎤⎥⎥⎥⎥⎥⎦,(3.8) so that the following cases arise:
(1) if 𝑡∉𝐷,then 𝑄₁₁(𝑡)=^𝑁_𝑗=1𝜆_0𝑗(𝑡)𝐴^∗𝑇0𝑗(𝑡)𝑃+𝑃^𝑁_𝑗=1𝜆_0𝑗(𝑡)𝐴^∗0𝑗(𝑡)+^𝑞_𝑖=1𝑄𝑆_𝑖(𝑡),_1,𝑖+1(𝑡)=𝑄^𝑇𝑖+1,1(𝑡)∶=^𝑁_𝑗=1𝑄𝜆_𝑖𝑗(𝑡)𝑃𝐴^∗𝑖𝑗(𝑡),∀𝑖∈𝑞,_𝑖𝑖(𝑡)∶=−1−ℎ̇_𝑖(𝑡)𝑆_𝑖𝑡−ℎ_𝑖(𝑡),𝑄_𝑖𝑗(𝑡)=0,∀𝑖,𝑗(≠𝑖)∈𝑞+1⧵{1},(3.9)
(2) if 𝑡∈𝐷⧵Imp, then (3.8) still holds to the left of any 𝑡∈𝐑₀₊. Similar equations as (3.9) stand for 𝑡⁺ by replacing 𝑡→𝑡⁺ in all the matrix functions entries which become modified only if the time instant 𝑡 is a discontinuity point of the corresponding matrix function entry,
(3) if 𝑡∈Imp, then the left-hand-side limit of 𝑄(𝑡) is defined with block matrices as follow: 𝑄₁₁(𝑡)=^𝑁_𝑗=1+𝑃𝜆_0𝑗𝑁(𝑡)𝐴^∗0𝑗(𝑡)+𝐵_0𝑗(𝑡)𝐾^0𝑗(𝑡)𝛿(0)^𝑇𝑃_𝑗=1𝜆_0𝑗(𝑡)𝐴^∗0𝑗(𝑡)+𝐵_0𝑗(𝑡)𝐾^0𝑗(𝑡)𝛿(0)+^𝑞_𝑖=1𝑄𝑆_𝑖(𝑡),_1,𝑖+1(𝑡)=𝑄^𝑇𝑖+1,1(𝑡)∶=^𝑁_𝑗=1𝑄𝜆_𝑖𝑗(𝑡)𝑃𝐴^∗𝑖𝑗(𝑡),∀𝑖∈𝑞,_𝑖𝑖(𝑡)∶=−1−ℎ̇_𝑖(𝑡)𝑆_𝑖𝑡−ℎ_𝑖(𝑡),𝑄_𝑖𝑗(𝑡)=0,∀𝑖,𝑗(≠𝑖)∈𝑞+1⧵{1},(3.10) and the right-hand-side limits are defined with block matrices as follow: 𝑄₁₁𝑡⁺=^𝑁_𝑗=1+𝑃𝜆_0𝑗𝑁(𝑡)𝐴^∗0𝑗𝑡⁺+𝐵_0𝑗(𝑡)𝐾^0𝑗(𝑡)^𝑇𝑃_𝑗=1𝜆_0𝑗(𝑡)𝐴^∗0𝑗𝑡⁺+^𝑁_𝑗=1𝐵_0𝑗(𝑡)𝐾^0𝑗(𝑡)+^𝑞_𝑖=1𝑄_1,𝑖+1𝑆_𝑖𝑡⁺𝑄𝑡⁺=𝑄^𝑇𝑖+1,1𝑡⁺∶=^𝑁_𝑗=1𝜆_𝑖𝑗(𝑡)𝑃𝐴^∗𝑖𝑗𝑡⁺,∀𝑖∈𝑞,_𝑖𝑖𝑡⁺∶=−1−ℎ̇_𝑖(𝑡)𝑆_𝑖𝑡−ℎ_𝑖(𝑡)⁺,𝑄_𝑖𝑗(𝑡)=0,∀𝑖,𝑗(≠𝑖)∈𝑞+1⧵{1},(3.11) since from Assumption 2.1, the scalar functions 𝜆_𝑖𝑗(𝑡) and the matrix functions 𝐵_0𝑗(𝑡), forall𝑖∈𝑞∪{0},forall𝑗∈𝑁 cannot be discontinuous at the sequence Imp. As in (3.11), a matrix function entry at 𝑡⁺ is more distinct than its left-hand-side limit at 𝑡 only if it has a discontinuity at the time instant 𝑡. Thus, 𝑉̇𝑡⁺,𝑥_𝑡+−𝑉̇𝑡𝑉,𝑥̇_𝑡𝑡=𝑥̂^𝑇(𝑡)𝑄𝑡⁺−𝑄(𝑡)𝑥̂(𝑡),𝑉𝑡⁺,𝑥_𝑡−𝑉𝑡,𝑥_𝑡=0,∀𝑡∉Imp,₊,𝑥_𝑡+−𝑉̇𝑡,𝑥_𝑡=0,∀𝑡∉𝐷since𝑄𝑡⁺=𝑄(𝑡).(3.12) Furthermore, in view of (3.5), 𝑉̇𝑡⁺,𝑥_𝑡+−𝑉̇𝑡,𝑥_𝑡=𝑥̂^𝑇𝑡⁺𝑄𝑡⁺𝑥̂𝑡⁺−𝑥̂^𝑇(𝑡)𝑄(𝑡)𝑥̂(𝑡),∀𝑡∈Imp.(3.13) If, in addition, 𝑡∉𝐷_𝑝, that is, if 𝑡∈Imp∩𝐷_𝑝, and since 𝑄(𝑡⁺)=𝑄(𝑡), (3.13) becomes 𝑉̇𝑡⁺,𝑥_𝑡+−𝑉̇𝑡,𝑥_𝑡=𝑥̂^𝑇𝑡⁺𝑄(𝑡)𝑥̂𝑡⁺−𝑥̂^𝑇(𝑡)𝑄(𝑡)𝑥̂(𝑡),(3.14) Furthermore, 𝑉𝑡⁺,𝑥_𝑡+−𝑉𝑡,𝑥_𝑡=×^𝑡+_𝑡𝑥_𝑇𝑉̇𝜏,𝑥_𝜏𝑑𝜏(𝑡)⎛⎜⎜⎝^𝑁_𝑗=1𝜆_0𝑗(𝑡)𝐵_0𝑗(𝑡)𝐾^0𝑗(𝑡)^𝑇𝑃^𝑁_𝑗=1+2𝑃_𝑁𝜆_0𝑗(𝑡)𝐵_0𝑗(𝑡)𝐾^0𝑗(𝑡)_𝑗=1𝜆_0𝑗(𝑡)𝐵_0𝑗(𝑡)𝐾^0𝑗(𝑡)𝑥(𝑡),∀𝑡∈Imp.(3.15) since 𝑥(𝑡⁺)−𝑥(𝑡)=∑^𝑁𝑗=1𝜆_0𝑗(𝑡)𝐵_0𝑗(𝑡)𝐾^0𝑗(𝑡)𝑥(𝑡) from (2.4) (which results to be zero for 𝑡∉Imp). Now, for any 𝑘∈𝐙₀₊and some 𝑝_𝑘∈𝐙₊, consider a sequence of consecutive impulsive time instants Imp(𝑡_𝑘,𝑡_{𝑘+𝑝𝑘})∶={𝑡_𝑘,𝑡_𝑘+1,…,𝑡_{𝑘+𝑝𝑘}}⊂Imp, so that, 𝑉𝑡=^+𝑘+𝑝_𝑘,_{𝑡+𝑘𝑘+𝑝𝑡+𝑘}𝑥_𝑡^{+𝑘𝑘+𝑝}−𝑉𝑡^+𝑘,𝑥_𝑡^+𝑘=𝑉̇𝜏,𝑥_𝜏𝑑𝜏_𝑝𝑘_𝑖=1^{𝑡−𝑘+𝑖}_𝑡^{+𝑘+𝑖−1}𝑉̇𝜏,𝑥_𝜏𝑑𝜏+2𝑥^𝑇𝑡_𝑘+𝑖𝑃^𝑁_𝑗=1+𝑥_𝑇𝜆_0𝑗𝑡_𝑘+𝑖𝐵_0𝑗𝑡_𝑘+𝑖𝐾^0𝑗𝑡_𝑘+𝑖𝑥𝑡_𝑘+𝑖×𝑃𝑡_{𝑘+𝑖𝑁}^𝑁_𝑗=1𝜆_0𝑗𝑡_𝑘+𝑖𝐾^𝑇0𝑗𝑡_𝑘+𝑖𝐵^𝑇0𝑗𝑡_𝑘+𝑖_𝑗=1𝜆_0𝑗𝑡_𝑘+𝑖𝐵_0𝑗𝑡_𝑘+𝑖𝐾^0𝑗𝑡_𝑘+𝑖𝑥𝑡_𝑘+𝑖(3.16)≤−^𝑝𝑘_𝑖=1^{𝑡𝑘+𝑖}_𝑡^{+𝑘+𝑖−1}+𝑥_𝑇𝛼(𝜏)||𝑥^𝑇(𝜏)𝑄_𝑑(𝜏)𝑥(𝜏)||𝑑𝜏+^𝑡𝑘+1_{𝑡+𝑘+𝑖−1}𝛽(𝜏)||𝑥^𝑇(𝜏)𝑄_𝑜𝑑(𝜏)𝑥(𝜏)||𝑑𝜏×𝑃𝑡_{𝑘+𝑖𝑁}^𝑁_𝑗=1𝜆_0𝑗𝑡_𝑘+𝑖𝐾^𝑇0𝑗𝑡_𝑘+𝑖𝐵^𝑇0𝑗𝑡_𝑘+𝑖_𝑗=1−2𝜇𝑡_𝑘+𝑖𝜆_0𝑗𝑡_𝑘+𝑖𝐵_0𝑗𝑡_𝑘+𝑖𝐾^0𝑗𝑡_𝑘+𝑖𝑥𝑡_𝑘+𝑖|||||𝑥^𝑇𝑡_𝑘+𝑖𝑃^𝑁_𝑗=1𝜆_0𝑗𝑡_𝑘+𝑖𝐵_0𝑗𝑡_𝑘+𝑖𝐾^0𝑗𝑡_𝑘+𝑖𝑥𝑡_𝑘+𝑖|||||⎤⎥⎥⎦(3.17) by using the binary indicator functions as follow:(a)𝛼∶𝐑₀₊→{1,−1} defined by 𝛼(𝑡)=1 if 𝑥^𝑇(𝑡)𝑄_𝑑(𝑡)𝑥(𝑡)>0 and 𝛼(𝑡)=−1 otherwise, forall𝑡∈𝐑₀₊,(b)𝛽∶𝐑₀₊→{1,−1} defined by 𝛽(𝑡)=1 if 𝑥^𝑇(𝑡)𝑄_𝑜𝑑(𝑡)𝑥(𝑡)>0 and 𝛽(𝑡)=−1 otherwise, forall𝑡∈𝐑₀₊,(c)𝜇∶Imp→{1,−1} defined by,𝜇(𝑡_𝑘)=1 if 𝑥^𝑇(𝑡^+𝑗)𝑃(∑^𝑁𝑗=1𝜆_0𝑗(𝑡_𝑗)𝐵_0𝑗(𝑡_𝑗)𝐾^0𝑗(𝑡_𝑗))𝑥(𝑡^+𝑗)>0 and 𝜇(𝑡_𝑘)=−1 otherwise; forall𝑡_𝑘∈Imp. Equation (3.17) is less than or equal to zero, which implies that 𝑉(𝑡^+𝑘+𝑝_𝑘,𝑥_𝑡^{+𝑘𝑘+𝑝})≤𝑉(𝑡^+𝑘,𝑥_𝑡^+𝑘) if ^𝑝𝑘_𝑖=1^{𝑡𝑘+𝑖}_𝑡^{+𝑘+𝑖−1}+𝑥_𝑇𝛼(𝜏)||𝑥^𝑇(𝜏)𝑄_𝑑(𝜏)𝑥(𝜏)||𝑑𝜏+^{𝑡𝑘+𝑖}_{𝑡+𝑘+𝑖−1}𝛽(𝜏)||𝑥^𝑇(𝜏)𝑄_𝑜𝑑(𝜏)𝑥(𝜏)||𝑑𝜏×𝑃𝑡_{𝑘+𝑖𝑁}^𝑁_𝑗=1𝜆_0𝑗𝑡_𝑘+𝑖𝐾^𝑇0𝑗𝑡_𝑘+𝑖𝐵^𝑇0𝑗𝑡_𝑘+𝑖_𝑗=1−2𝜇𝑡𝜆_{0𝑗𝑘+𝑖}𝑡_𝑘+𝑖𝐵_0𝑗𝑡_𝑘+𝑖𝐾^0𝑗𝑡_𝑘+𝑖𝑥𝑡_𝑘+𝑖|||||𝑥^𝑇𝑡_𝑘+1𝑃^𝑁_𝑗=1𝜆_0𝑗𝑡_𝑘+𝑖𝐵_0𝑗𝑡_𝑘+𝑖𝐾^0𝑗𝑡_𝑘+𝑖𝑥𝑡_𝑘+𝑖|||||⎤⎥⎥⎦≥0,(3.18) which holds with an existing Imp∋𝑡_𝑘^∗=𝑡_{𝑘+𝑝𝑘}∈[𝑡_𝑘,𝑡_{𝑘+𝑝𝑘}] for each𝑡_𝑘+𝑖∈Imp(forall𝑖∈𝑝_𝑘∪{0}) with impulsive control gains 𝐾^0𝑗(𝑡_𝑘^∗)=Λ_0𝑗(𝑡_𝑘^∗)𝐵^𝑇0𝑗(𝑡_𝑘^∗)𝑃 of the jth parameterization of the polytopic system, where 𝐑^𝑛×𝑛∋Λ_0𝑗(𝑡_𝑘^∗)=Λ^𝑇0𝑗(𝑡_𝑘^∗), forall𝑗∈𝑁 if ^𝑁_𝑗=1⎛⎜⎜⎝𝜆_0𝑗𝑡_𝑘^∗𝜆_maxΛ_0𝑗𝑡_𝑘^∗‖‖𝑃𝐵_0𝑗𝑡_𝑘^∗‖‖₂×⎛⎜⎜⎝1−𝜆_0𝑗𝑡_𝑘^∗𝜆^2minΛ_0𝑗𝑡_𝑘^∗𝜆_min𝐵^𝑇0𝑗𝑡_𝑘^∗𝑃²𝐵_0𝑗𝑡_𝑘^∗≤1𝜆_maxΛ_0𝑗𝑡_𝑘^∗⎞⎟⎟⎠⎞⎟⎟⎠‖‖𝑥𝑡_𝑘∗‖‖²²⎛⎜⎜⎝^𝑘∗_𝑖=1^{𝑡𝑘+𝑖}_𝑡^{+𝑘+𝑖−1}+𝛼(𝜏)||𝑥^𝑇(𝜏)𝑄_𝑑(𝜏)𝑥(𝜏)||𝑑𝜏+^{𝑡−𝑘+𝑖}_{𝑡+𝑘+𝑖−1}𝛽(𝜏)||𝑥^𝑇(𝜏)𝑄_𝑜𝑑(𝜏)𝑥(𝜏)||𝑑𝜏_𝑘∗−1_𝑖=1𝑥^𝑇𝑡_𝑘+𝑖^𝑁_𝑗=1×𝑃_𝑁𝜆_0𝑗𝑡_𝑘+𝑖𝐾^𝑇0𝑗𝑡_𝑘+𝑖𝐵^𝑇0𝑗𝑡_𝑘+𝑖_𝑗=1−2|||||𝑥_𝑇𝜆_0𝑗𝑡_𝑘+𝑖𝐵_0𝑗𝑡_𝑘+𝑖𝐾^0𝑗𝑡_𝑘+𝑖𝑥𝑡_𝑘+𝑖𝑡_𝑘+𝑖^𝑁_𝑗=1𝜆_0𝑗𝑡_𝑘+𝑖𝑃𝐵_0𝑗𝑡_𝑘+𝑖𝐾^0𝑗𝑡_𝑘+𝑖𝑥𝑡_𝑘+𝑖|||||⎤⎥⎥⎦⎞⎟⎟⎠,(3.19) where 𝑡_𝑘^∗∶=max_{𝑖∈𝑝𝑘}(𝑡_𝑘+𝑖∈Imp∶𝑥^𝑇(𝑡_𝑘+𝑖)(∑^𝑁𝑗=1𝜆_0𝑗(𝑡_𝑘+𝑖)𝑃𝐵_{0𝑗(𝑡𝑘+𝑖)}𝐾^0𝑗(𝑡_𝑘+𝑖))𝑥(𝑡_𝑘+𝑖)≠0)
The existence of 𝑡_𝑘^∗∈[𝑡_𝑘,𝑡_{𝑘+𝑝𝑘}] has been proven for time instants 𝑡_𝑘+𝑖∈Imp(forall𝑖∈𝑝_𝑘∪{0}), and some 𝑝_𝑘∈𝐙₀₊ such that 𝑥^𝑇(𝑡^∗𝑘)(∑^𝑁𝑗=1𝜆_0𝑗(𝑡^∗𝑘)𝑃𝐵_0𝑗(𝑡^∗𝑘)𝐾^0𝑗(𝑡^∗𝑘))𝑥(𝑡^∗𝑘)≠0 if𝑥(𝑡^∗𝑘)≠0 for appropriate impulsive controller gains 𝐾^0𝑗(𝑡_𝑘),forall𝑗∈^⟵𝑁. In particular, if Λ_0𝑗(𝑡_𝑘^∗)=𝜈(𝑡_𝑘^∗)𝐼_𝑚≠0 with 𝜈(𝑡_𝑘^∗)∈𝐑⧵{0} being a scalar common for the impulses injected at all the parameterizations of the polytopic system, then the condition in (3.19) becomes in particular,𝜈𝑡_𝑘^∗≤1𝜌𝑡_𝑘^∗‖‖𝑥𝑡_𝑘^∗‖‖²²×⎛⎜⎜⎝^𝑘∗_𝑖=1^{𝑡𝑘+𝑖}_𝑡^{+𝑘+𝑖−1}+𝛼(𝜏)||𝑥^𝑇(𝜏)𝑄_𝑑(𝜏)𝑥(𝜏)||𝑑𝜏+^{𝑡𝑘+𝑖}_{𝑡+𝑘+𝑖−1}𝛽(𝜏)||𝑥^𝑇(𝜏)𝑄_𝑜𝑑(𝜏)𝑥(𝜏)||𝑑𝜏_𝑘∗−1_𝑖=1𝑥^𝑇𝑡_𝑘+𝑖^𝑁_𝑗=1×𝑃_𝑁𝜆_0𝑗𝑡_𝑘+𝑖𝐾^𝑇0𝑗𝑡_𝑘+𝑖𝐵^𝑇0𝑗𝑡_𝑘+𝑖_𝑗=1−2|||||𝑥_𝑇𝜆_0𝑗𝑡_𝑘+𝑖𝐵_0𝑗𝑡_𝑘+𝑖𝐾^0𝑗𝑡_𝑘+𝑖𝑥𝑡_𝑘+𝑖𝜌𝑡_𝑘∗𝑡_𝑘+𝑖^𝑁_𝑗=1𝜆_0𝑗𝑡_𝑘+𝑖𝑃𝐵_0𝑗𝑡_𝑘+𝑖𝐾^0𝑗𝑡_𝑘+𝑖𝑥𝑡_𝑘+𝑖|||||⎤⎥⎥⎦⎞⎟⎟⎠,:=^𝑁_𝑗=1⎛⎜⎜⎝𝜆_0𝑗𝑡_𝑘^∗𝜆_maxΛ_0𝑗𝑡_𝑘^∗‖‖𝑃𝐵_0𝑗𝑡_𝑘^∗‖‖₂×⎛⎜⎜⎝1−𝜆_0𝑗𝑡_𝑘^∗𝜆^2minΛ_0𝑗𝑡_𝑘^∗𝜆_min𝐵^𝑇0𝑗𝑡_𝑘^∗𝑃²𝐵_0𝑗𝑡_𝑘^∗𝜆_maxΛ_0𝑗𝑡_𝑘^∗⎞⎟⎟⎠⎞⎟⎟⎠.(3.21) It follows by simple inspection that 𝜈(𝑡_𝑘^∗) may be chosen to satisfy (3.21) and, furthermore, −∞≤𝜈(𝑡_𝑘^∗)≤∞ for some 𝑡_𝑘^∗∈Imp∩[𝑡_𝑘,𝑡_{𝑘+𝑝𝑘}]. It is now proven by contradiction that the sequences {‖𝑥(𝑡_𝑘)‖_{𝑡𝑘∈Imp}}and {‖𝑥(𝑡^+𝑘)‖_{𝑡𝑘∈Imp}} are both bounded. Assume that 𝑆_𝜈∶={𝜈(𝑡_𝑘^∗)}_{𝑡𝑘∗∈Imp∩[𝑡𝑘,𝑡𝑘𝑘+𝑝]} is an unbounded sequence. Then, there is an infinite subsequence 𝑆^𝜈⊂𝑆_𝜈 such that 𝑆^𝜈∋𝜈(𝑡_𝑘^∗)→±∞ as 𝑡_𝑘^∗→∞, forall𝑡_𝑘∈Imp. From the definition of the Lyapunov function candidate (3.4) and the guaranteed property 𝑉(𝑡^+𝑘_∗,𝑥_𝑡^+𝑘∗)≤𝑉(𝑡^+𝑘,𝑥_𝑡^+𝑘)≤𝑉(𝑡⁺¹,𝑥_𝑡⁺¹)<∞, (from (3.17), if (3.21) holds), it follows that ‖𝑥(𝑡^+𝑘_∗)‖≤𝑀^+𝜑𝑘_∗<∞ for any positive finite constant 𝑀_𝜑 depending on the bounded function of initial conditions of the system (3.4). Also, ‖𝑥(𝑡_𝑘∗)‖≤𝑀_𝜑𝑘∗<∞ since the discontinuities at the state trajectory solution caused by impulses are second-class finite jump-type discontinuities. Then, the sequences {‖𝑥(𝑡^∗𝑘)‖}_𝑡^∗𝑘_∈Imp∗and {‖𝑥(𝑡^𝑘∗+)‖}_{𝑡∗𝑘∈Imp∗} are bounded by positive real constants, 𝑀_𝜑=max_{𝑡𝑘∈Imp}(max_𝑡^∗𝑘_{∈[𝑡𝑘,𝑡𝑘𝑘+𝑝]}𝑀_𝜑𝑘∗) and 𝑀^+𝜑=max_{𝑡𝑘∈Imp}(max_𝑡^∗𝑘_{∈[𝑡𝑘,𝑡𝑘𝑘+𝑝]}𝑀^+𝜑𝑘_∗), respectively. This implies that (3.21) may be fulfiled with −∞<𝜈(𝑡_𝑘^∗)<∞, also, since (1)the state-trajectory solution of the closed-loop system is continuous and almost everywhere time differentiable except at second-class discontinuity points on a set of zero measure, and(2)the state-trajectory solution of the closed-loop system is bounded on the subsequence Imp^∗. Thus, it cannot beunbounded on Imp⧵Imp^∗ since, otherwise, it could not be an almost everywhere smooth state-trajectory solution.As a result, it exist 𝐶=𝐶(𝑇^∗,Imp)∈𝐑₊ and 𝐶⁺=𝐶⁺(𝑇^∗,Imp)∈𝐑₊ such that ‖𝑥(𝑡_𝑘)‖≤𝐶𝑀_𝜑 and ‖𝑥(𝑡^+𝑘)‖≤𝐶⁺𝑀^+𝜑, forall𝑡_𝑘∈Imp and the candidate (3.4) is a Lyapunov functional. The result has been proven.

The proof of the global asymptotic stability of the system requires to extend Theorem 3.4 by guaranteeing that the state-trajectory solution converges asymptotically to zero as time tends to infinity. This requires also stabilizability-type conditions on the nonimpulsive part of the closed-loop solution. The following result holds.

Theorem 3.5. The closed-loop system (2.6)–(2.7) is globally asymptotically stable for any 𝜑∈BPC⁽⁰⁾([−ℎ,0],𝐑^𝑛) and a given sequence of impulsive time instants if the regular controller gain functions 𝐾_𝑖𝑗∶𝐑₀₊→𝐑^𝑛×𝑚 and the sequence of impulsive controller gains 𝐾^0𝑗∶𝐑₀₊→𝐑^𝑛×𝑚, forall𝑗∈𝑁,forall𝑖∈𝑞∪{0} are chosen so that the following matrix inequalities hold for some 𝐑^𝑛×𝑛∋𝑃=𝑃^𝑇≻0 and 𝑆_𝑖∈BPC⁽⁰⁾(𝐑₀₊,𝐑^𝑛×𝑛) which fulfils 𝑆_𝑖(𝑡)≻0, forall𝑡∈𝐑₀₊, forall𝑖∈𝑞 as follow: 𝑄𝑄₁₁(𝑡)≺0,𝑄₂₂(𝑡)−𝑄^𝑇12(𝑡)𝑄⁻¹¹¹(𝑡)𝑄^𝑇12(𝑡)≺0,∀𝑡∈𝐑₀₊⧵D,(3.22)₁₁𝑄𝑡⁺≺0,𝑄₂₂𝑡⁺−𝑄^𝑇12𝑡⁺𝑄⁻¹¹¹𝑡⁺𝑄^𝑇12𝑡⁺≺0,∀𝑡∈𝐷⧵Imp,₁₁𝑡⁺≺0,𝑄₂₂𝑡𝑄⁺𝑡−𝑄^𝑇12𝑡⁺𝑄⁻¹¹¹(𝑡⁺)𝑄^𝑇12𝑡⁺≺0,∀𝑡∈𝐷⧵Imp,(3.23)₊−𝑄(𝑡)≼0,∀𝑡∈Imp,(3.24) where 𝑄₁₂(𝑡)=𝑄^𝑇21(𝑡)=^𝑁_𝑗=1𝜆_1𝑗(𝑡)𝑃𝐴^∗1𝑗(𝑡)^𝑁_𝑗=1𝜆_2𝑗(𝑡)𝑃𝐴^∗1𝑗(𝑡)⋯^𝑁_𝑗=1𝜆_𝑞𝑗(𝑡)𝑃𝐴^∗1𝑗(𝑡),(3.25)𝑄₂₂(𝑡):=−Blockdiag×𝑆1₂₂−ℎ̇₁(𝑡)𝑆₁₁𝑡−ℎ₁(𝑡)1−ℎ̇₂(𝑡)𝑡−ℎ₂(𝑡)⋯1−ℎ̇_𝑞(𝑡)𝑆_𝑞𝑞𝑡−ℎ_𝑞(𝑡).(3.26)

Proof. From (3.6) and (3.12), the system is globally asymptotically stable if the Lyapunov functional candidate (3.4) is in fact a Lyapunov functional which is guaranteed if(a)𝑉̇(𝑡,𝑥_𝑡)<0, forall𝑡∈𝐑₀₊ such that 𝑥_𝑡≠0 what holds if and only if 𝑄(𝑡)≺0, forall𝑡∈𝐑₀₊(b)𝑉(𝑡⁺,𝑥_𝑡+)≤𝑉(𝑡,𝑥_𝑡), forall𝑡∈Imp what holds if and only if 𝑄(𝑡⁺)≺𝑄(𝑡), forall𝑡∈𝐑₀₊.The first condition holds from (3.9) via Schur’s complement if (3.22)-(3.23) hold. The second condition holds if (3.24) holds.

Remark 3.6. Theorem 3.5 can be tested directly from (3.9)–(3.11) with direct algebraic tests. However, it is very restrictive since it does not provide with conditions guaranteeing a cooperative achievement of global asymptotic stability among the non-impulsive and impulsive parts. Note that necessary conditions for the fulfilment of Theorem 3.5 are from (3.9)–(3.11): 𝑄₁₁(𝑡)≺0, 𝑄₁₁(𝑡⁺)≺0 (i.e., the Lyapunov matrix inequality holds for forall𝑡∈𝐑₀₊⧵𝐷, and for the left and right limits of all 𝑡∈𝐷), and 𝑄₁₁(𝑡⁺)≺𝑄₁₁(𝑡)≺0, forall𝑡∈Imp.

Remark 3.7. Note that (3.22)-(3.23) imply that 𝑄₂₂(𝑡)≺𝑄^𝑇12(𝑡)𝑄⁻¹¹¹(𝑡)𝑄^𝑇12(𝑡)≼0,∀𝑡∈𝐑₀₊⧵𝐷,(3.27)𝑄₂₂(𝑡)≺𝑄^𝑇12(𝑡)𝑄⁻¹¹¹(𝑡)𝑄^𝑇12(𝑡)≼0,𝑄₂₂𝑡⁺≺𝑄^𝑇12𝑡⁺𝑄⁻¹¹¹𝑡⁺𝑄^𝑇12𝑡⁺≼0,(3.28)forall𝑡∈𝐷⧵Imp since 𝑄^𝑇12(𝑡)𝑄⁻¹¹¹(𝑡)𝑄^𝑇12(𝑡) is symmetric and𝑄⁻¹¹¹(𝑡)≺0and 𝑄₂₂(𝑡)≺𝑄^𝑇12(𝑡)𝑄⁻¹¹¹(𝑡)𝑄^𝑇12(𝑡)≼0,𝑄₂₂𝑡⁺≺𝑄^𝑇12𝑡⁺𝑄⁻¹¹¹𝑡⁺𝑄^𝑇12𝑡⁺≼0,∀𝑡∈𝐷⧵Imp,(3.29) As a result, sup_𝑡∈𝐑⁺⁰[max_𝑖∈𝑞(max(ℎ̇_𝑖(𝑡),ℎ̇_𝑖(𝑡⁺)))]<1is a necessary condition for Theorem 3.5 to hold.

Concerning Theorem 3.5, (3.22)-(3.23), note that isolated bounded discontinuities in 𝑉̇(𝑡,𝑥_𝑡) do not affect to maintain 𝑉(𝑡,𝑥_𝑡) as a positive strictly monotonically decreasing functional on 𝐑₀₊. Therefore, Theorem 3.5 can be relaxed by removing a set of zero measure of 𝐑₀₊⧵𝐷(and then of 𝐑₀₊) to evaluate (3.22)-(3.23) and also bounded discontinuities at the sequence Imp from (3.24). The resulting modified stability result follows.

Corollary 3.8. The closed-loop system (2.6)–(2.7) is globally asymptotically stable for any 𝜑∈BPC⁽⁰⁾([−ℎ,0],𝐑^𝑛) and a given sequence of impulsive time instants if the regular controller gain functions 𝐾_𝑖𝑗∶𝐑₀₊→𝐑^𝑛×𝑚 and the sequence of impulsive controller gains 𝐾^0𝑗∶𝐑₀₊→𝐑^𝑛×𝑚, forall𝑗∈𝑁,forall𝑖∈𝑞∪{0} are chosen so that the following matrix inequalities hold for some 𝐑^𝑛×𝑛∋𝑃=𝑃^𝑇≻0 and 𝑆_𝑖∈BPC⁽⁰⁾(𝐑₀₊,𝐑^𝑛×𝑛) which fulfils 𝑆_𝑖(𝑡)≻0, forall𝑡∈𝐑₀₊, forall𝑖∈𝑞 as follow: ^𝑁_𝑗=1𝜆_0𝑗(𝑡)𝐴^∗𝑇_0𝑗(𝑡)𝑃+𝑃^𝑁_𝑗=1𝜆_0𝑗(𝑡)𝐴^∗0𝑗(𝑡)+^𝑞_𝑖=1𝑆_𝑖(𝑡)≺0,(3.30) almost everywhere in 𝐑₀₊, Blockdiag1−×ℎ𝑆̇₁(₂₂𝑡)𝑆₁₁𝑡−ℎ₁(𝑡)1−ℎ̇₂(𝑡)≻𝑡−ℎ₂(𝑡)⋯1−ℎ̇_𝑞(𝑡)𝑆_𝑞𝑞𝑡−ℎ_𝑞(𝑡)_𝑁_𝑗=1𝜆_1𝑗(𝑡)𝑃𝐴^∗1𝑗(𝑡)^𝑁_𝑗=1𝜆_2𝑗(𝑡)𝑃𝐴^∗1𝑗(𝑡)⋯^𝑁_𝑗=1𝜆_𝑞𝑗(𝑡)𝑃𝐴^∗1𝑗(𝑡)^𝑇×−^𝑁_𝐽=1𝜆_0𝑗(𝑡)𝐴^∗𝑇_0𝑗(𝑡)𝑃−𝑃^𝑁_𝐽=1𝜆_0𝑗(𝑡)𝐴^∗0𝑗(𝑡)−^𝑞_𝑖=1𝑆_𝑖(𝑡)⁻¹×^𝑁_𝑗=1𝜆_1𝑗(𝑡)𝑃𝐴^∗1𝑗(𝑡)^𝑁_𝑗=1𝜆_2𝑗(𝑡)𝑃𝐴^∗1𝑗(𝑡)⋯^𝑁_𝑗=1𝜆_𝑞𝑗(𝑡)𝑃𝐴^∗1𝑗(𝑡),(3.31) almost everywhere in𝐑₀₊, and 𝑄=⎡⎢⎢⎢⎣𝑡⁺−𝑄(𝑡)_𝑁_𝑗=1𝜆_0𝑗(𝑡)𝐾^𝑇0𝑗(𝑡)𝐵^𝑇0𝑗(𝑡)𝑃+𝑃^𝑁_𝑗=1𝜆0_0𝑗(𝑡)𝐵_0𝑗(𝑡)𝐾^0𝑗(𝑡)∀𝜆𝑡_0𝑗∈(𝑡I)m𝐵p_0𝑗,(𝑡)𝐾^0𝑗(𝑡)00⎤⎥⎥⎥⎦≼0,(3.32) where 𝑄_𝑖𝑗(𝑡⁺)=𝑄_𝑖𝑗(𝑡) if 𝑄_𝑖𝑗(𝑡) is not impulsive and 𝑄_𝑖𝑗(𝑡⁺)=𝑄_𝑖𝑗(𝑡⁺), otherwise.

Proof. Equations (3.30)-(3.31) follow from Theorem 3.5 by expanding 𝑄₁₁(𝑡)≺0,𝑄₂₂(𝑡)−𝑄^𝑇12(𝑡)𝑄⁻¹¹¹(𝑡)𝑄^𝑇12(𝑡)≺0 from (3.8)-(3.9) and (3.25)-(3.26) on 𝐑₀₊excepting time instants of bounded isolated discontinuities. Equation (3.32) follow from (3.24), forall𝑡∈Imp also excluding bounded discontinuities at the time-derivative of the Lyapunov functional since they are irrelevant for analysis since they do not generate bounded jumps at the Lyapunov functional.

Corollary 3.8 holds in terms of more restrictive but it is easier to test conditions given in the subsequent result.

Corollary 3.9. Corollary 3.8 holds if ^𝑁_𝑗=1𝜆_0𝑗(𝑡)𝐴^∗𝑇_0𝑗(𝑡)𝑃+𝑃^𝑁_𝑗=1𝜆_0𝑗(𝑡)𝐴^∗0𝑗(𝑡)+^𝑞_𝑖=1𝑆_𝑖(𝑡)≺−𝑞(𝑡)𝐼_𝑛,(3.33) almost everywhere in 𝐑₀₊ for some 𝑞∈BPC⁽⁰⁾(𝐑₀₊,𝐑₊) which satisfies 𝑞(𝑡)>min⎛⎜⎜⎜⎝𝑞₀,‖‖∑^𝑁𝑗=1𝜆_1𝑗(𝑡)𝑃𝐴^∗1𝑗(𝑡)∑^𝑁𝑗=1𝜆_2𝑗(𝑡)𝑃𝐴^∗1𝑗(𝑡)⋯∑^𝑁𝑗=1𝜆_𝑞𝑗(𝑡)𝑃𝐴^∗1𝑗(𝑡)‖‖²²min_𝑖∈𝑞𝜆_min𝑆_𝑖𝑖𝑡−ℎ_𝑖(𝑡),⎞⎟⎟⎟⎠(3.34) provided that max_𝑖∈𝑞ℎ̇_𝑖(𝑡)<min⎛⎜⎜⎜⎝𝛾,1−‖‖∑^𝑁𝑗=1𝜆_1𝑗(𝑡)𝑃𝐴^∗1𝑗(𝑡)∑^𝑁𝑗=1𝜆_2𝑗(𝑡)𝑃𝐴^∗1𝑗(𝑡)⋯∑^𝑁𝑗=1𝜆_𝑞𝑗(𝑡)𝑃𝐴^∗1𝑗(𝑡)‖‖²²𝑞(𝑡)min_𝑖∈𝑞𝜆_min𝑆_𝑖𝑖𝑡−ℎ_𝑖(𝑡)⎞⎟⎟⎟⎠,(3.35) almost everywhere in 𝐑₀₊, and (3.32) holds forall𝑡∈Imp.

The following result states that stabilization is achievable under impulsive control impulses which respect a maximum separation time interval and exceed an upper bound of the maximum delay provided that it is bounded.

Corollary 3.10. Assume that (1)all the delays are uniformly bounded for all time,(2)¬∃𝑡∈𝐑₀₊∶𝜆_0𝑗(𝑡)𝐵_0𝑗(𝑡)≠0forall𝑗∈𝑁 and fix a real constant ∞>𝑇≥𝑇(≥sup_{𝑡∈𝐑0+}ℎ(𝑡)). Fix a real constant ∞>𝑇≥𝑇(≥sup_{𝑡∈𝐑0+}ℎ(𝑡)). Thus, there is always a globally stabilizing impulsive control law by appropriate design of one of the impulsive controller gains and choice of the interval sequences of impulsive instants as follow:𝐾^0ℓ∶𝐑₀₊⟶𝐑^𝑚×𝑛,ℓ∈𝑁,Imp(𝑗𝑇,(𝑗+1)𝑇](3.36) for each time interval (𝑗𝑇,(𝑗+1)𝑇], forallj(≥j₀)∈𝐙₀₊ and some given arbitrary finite 𝑗₀∈𝐙₀₊.

Proof. One has from (3.4) that Δ𝑉𝑡,𝑥_𝑡=𝑉𝑡⁺,𝑥_𝑡+−𝑉𝑡,𝑥_𝑡=𝑥^𝑇𝑡⁺𝑃𝑥𝑡⁺−𝑥^𝑇(𝑡)𝑃𝑥(𝑡),∀𝑡∈𝐑₀₊,(3.37) which equalizes zero at 𝑡∉Imp, since ^𝑞_𝑖=1^𝑡+_{𝑡−ℎ𝑖(𝑡)}⁺𝑥^𝑇(𝜏)𝑆_𝑖(𝜏)𝑥(𝜏)𝑑𝜏=^𝑞_𝑖=1^{𝑡𝑡−ℎ}_{𝑖(𝑡)}𝑥^𝑇(𝜏)𝑆_𝑖(𝜏)𝑥(𝜏)𝑑𝜏,∀𝑡∈𝐑₀₊,(3.38) since the discontinuities of the state vector at 𝑡∈Imp are bounded. Thus, one has for any arbitrary 𝑇∈𝐑₊that 𝑉+𝑡⁺+𝑇−𝑉𝑡⁺≤−^{𝑡𝑡+𝑇}𝑥̂^𝑇(𝜏)𝑄(𝜏)𝑥̂(𝜏)𝑑𝜏_{𝑡𝑖∈Imp](𝑡,𝑡+𝑇}≤−_{𝑡𝑡+𝑇}𝑥^𝑇𝑡^+𝑖𝑃𝑥𝑡^+𝑖−𝑥^𝑇𝑡_𝑖𝑃𝑥𝑡_𝑖+𝑥̂^𝑇(𝜏)𝑄(𝜏)𝑥̂(𝜏)𝑑𝜏_{𝑡𝑖∈Imp](𝑡,𝑡+𝑇}𝑥^𝑇×𝑃𝑡_𝑖𝐼_𝑛𝐼_𝑛+^𝑁_𝑗=1𝜆_0𝑗𝑡_𝑖𝐾^𝑇0𝑗𝑡_𝑖𝐵_0𝑗𝑡_𝑖+^𝑁_𝑗=1𝜆_0𝑗𝑡_𝑖𝐵_0𝑗𝑡_𝑖𝐾^0𝑗∀𝑡𝑡∈_𝑖𝐑−_𝟎+𝑃𝑥𝑡_𝑖,.(3.39) Define 𝑡^∗=𝑡^∗(𝑡,𝑇)∀=𝑡_𝑖𝑡_𝑖∈Imp(𝑡,𝑡+𝑇]∶𝑥𝑡^∗≠0∧Imp𝑡^∗,𝑡+𝑇≠∅⟹𝑥𝑡^∗𝑖=0,∈Imp𝑡^∗,𝑡+𝑇∈(𝑡,𝑡+𝑇]∩𝐑₀₊(3.40) as the last impulsive sampling instant in (𝑡,𝑡+𝑇], where the state vector is nonzero. Thus, 𝑉(𝑡⁺+𝑇)≤𝑉(𝑡⁺) if 𝑥^𝑇𝑡^∗×𝑃𝐼𝐼_𝑛𝑛+^𝑁_𝑗=1𝜆_0𝑗𝑡^∗𝐾^𝑇0𝑗𝑡^∗𝐵^𝑇0𝑗𝑡^∗+^𝑁_𝑗=1≤_{𝑡𝑖+𝑇}𝜆_0𝑗𝑡^∗𝐵_0𝑗𝑡^∗𝐾^0𝑗𝑡^∗−𝑃𝑥𝑡^∗−𝑥̂^𝑇(𝜏)𝑄(𝜏)𝑥̂(𝜏)𝑑𝜏_{𝑡𝑖∈Imp(𝑡,𝑡∗](𝑡,𝑇)}𝑥^𝑇×𝑃𝐼𝑡_𝑖𝑛𝐼_𝑛+^𝑁_𝑗=1𝜆_0𝑗𝑡_𝑖𝐵^𝑇0𝑗𝑡_𝑖+^𝑁_𝑗=1𝜆_0𝑗𝑡_𝑖𝐵_0𝑗𝑡_𝑖𝐾^0𝑗𝑡_𝑖−𝑃𝑥𝑡_𝑖(3.41) from (3.6). Since the interval (𝑡,𝑡+𝑇) is finite, it follows that the Lyapunov functional candidate is bounded on the interval, provided that it is bounded at a single point. The result follows by applying the above upper-bounding constraint recursively for 𝑡=𝑗𝑇,forall𝑗≥𝑗₀and appropriate choice of the impulsive sequence Imp(𝑗𝑇,(𝑗+1)𝑇]since the state vector cannot be identically zero on (𝑗𝑇,(𝑗+1)𝑇] for ∞>𝑇≥𝑇(≥sup_{𝑡∈𝐑0+}ℎ(𝑡))except for the trivial state-trajectory solution.

Remark 3.11. Corollary 3.10 may be directly reformulated under weaker (but easier to deal with) conditions by using 𝑉̇𝑡,𝑥_𝑡≤≤𝑥−̂^𝑇(𝑡𝜆)𝑄_min(𝑡)𝑥̂(𝑡)=−𝑥̂^𝑇(𝑡)𝑄_𝑑(𝑡)+𝑄_𝑜𝑑(𝑡)𝑥̂(𝑡)Δ𝑉𝑡,𝑥_𝑡𝑄_𝑑(𝑡)−𝜆_max𝑄^𝑇0𝑑(𝑡)𝑄_0𝑑(𝑡)‖𝑥̂(𝑡)‖²²,(3.42):==𝑥𝑉_𝑇𝑡⁺,𝑥_𝑡+−𝑉𝑡,𝑥_𝑡(𝑡)^𝑁_𝑗=1𝜆_0𝑗(𝑡)𝐾^𝑇0𝑗(𝑡)𝐵^𝑇0𝑗(𝑡)𝑃^𝑁_𝑗=1+2_𝑁𝜆_0𝑗(𝑡)𝐵_0𝑗(𝑡)𝐾^0𝑗(𝑡)_𝑗=1𝜆_0𝑗(𝑡)𝐾^𝑇0𝑗(𝑡)𝐵^𝑇0𝑗(𝑡)𝑃𝑥(𝑡).(3.43)

4. Examples

4.1. Example for Scalar Systems

𝑥̇(𝑡)=𝑎𝑥(𝑡)+𝑎₀(𝑡)𝑥(𝑡−ℎ)+∑_{𝑡𝑘∈Imp(0,𝑡)}𝐾(𝑡_𝑘)𝑥(𝑡_𝑘)𝛿(𝑡−𝑡_𝑘)for some constant delay ℎ≥0. Its solution satisfies for 𝑇_𝑘=𝑡_𝑘+1−𝑡_𝑘, forall𝜃∈[0,𝑇_𝑘] with 𝑈(𝑡) being the unit step (Heaviside) function,𝑥𝑡^+𝑘+𝜃=𝑒^𝑎𝜃𝑥𝑡^+𝑘+^𝜃0×s𝑒up^−𝑎𝜏𝑎₀𝑡_{𝑘𝑡𝜏∈𝑘𝑡−ℎ,max𝑘+1−ℎ,𝑡𝑘}+𝜏𝑥𝑡_𝑘+𝜏−ℎ𝑑𝜏+𝐾𝑡_𝑘+1𝑈𝜃−𝑇_𝑘𝑥𝑡_𝑘+1≤||1+𝐾𝑡_𝑘+1||𝑥(𝜏)||𝑈𝜃−𝑇_𝑘||||||𝑒^𝑎𝜃𝑥𝑡^+𝑘+^𝜃0𝑒^−𝑎𝜏𝑎₀∀𝜃𝑡_𝑘∈+𝜏0,𝑥𝑇_𝑘𝑡_𝑘+𝜏−ℎ𝑑𝜏||||,⟹sup_{𝑡𝜃∈𝑘,𝑡𝑘+1}max_{𝜃∈0,𝑇𝑘}||𝑥(𝜏)||≤max_{𝜃∈0,𝑇𝑘}||1+𝐾𝑡_𝑘+1𝑈𝜃−𝑇_𝑘|||||||𝑒_𝑎𝜃||𝑥𝑡^+𝑘||+||||𝑒^𝑎𝜃−1×max_{𝜏∈0,𝑇𝑘}𝑎||||⟹sup_{𝑡𝜏∈𝑘,𝑡𝑘+1}||𝑎₀𝑡_𝑘+𝜏||sup_{𝑡𝜏∈𝑘𝑡−ℎ,max𝑘+1−ℎ,𝑡𝑘}||𝑥(𝜏)|||||||,∀𝜃∈0,𝑇_𝑘||𝑥(𝜏)||≤m×axmax1,_{𝜃∈0,𝑇𝑘}max_{𝜃∈0,𝑇𝑘}||1+𝐾𝑡_𝑘+1𝑈𝜃−𝑇_𝑘||𝑒^𝑎𝜃||𝑥𝑡^+𝑘||+||||𝑒^𝑎𝜃−1𝑎||||max_{𝜏∈0,𝑇𝑘}×maxsup_{𝑡𝜏∈𝑘−ℎ,𝑡𝑘+1−ℎ}||𝑎₀𝑡_𝑘+𝜏||||𝑥(𝜏)||,sup_{𝑡𝜏∈𝑘+1−ℎ,𝑡𝑘}||𝑥(𝜏)||𝑈𝑡_𝑘−𝑡_𝑘+1−ℎ.(4.1)

Note that ||𝑥𝑡_𝑘+1||≤𝐶𝑡_𝑘+1𝜏∈𝑡_𝑘sup||𝑥𝑡−ℎ,max𝑡_𝑘+1−ℎ,𝑡_𝑘||𝑥(𝜏)_+𝑘+1||,||≤𝐶𝑡^+𝑘+1𝜏∈𝑡_𝑘sup−ℎ,max𝑡_𝑘+1−ℎ,𝑡_𝑘||𝑥(𝜏)||,(4.2) with𝐶𝑡^+𝑘+1=||1+𝐾𝑡_𝑘+1||𝐶𝑡_𝑘+1,𝐶𝑡_𝑘+1=𝑒^𝑎𝑇𝑘+||||𝑒^𝑎𝑇𝑘−1𝑎||||max_{𝜏∈0,𝑇𝑘}||𝑎₀𝑡_𝑘+𝜏||,(4.3) and also||||𝑥𝑥𝑡𝑡_𝑘+1+𝜃||≤𝐶_𝜃𝑡_𝑘+1sup_{𝑡𝜏∈𝑘𝑡−ℎ,max𝑘+1−ℎ,𝑡𝑘}||𝑥(𝜏)_+𝑘+1||,+𝜃||≤𝐶_𝜃𝑡^+𝑘+1sup_{𝑡𝜏∈𝑘𝑡−ℎ,max𝑘+1−ℎ,𝑡𝑘}||𝑥(𝜏)||,(4.4) for all θ ∈[0,𝑇_𝑘+1], so that ||||𝑥𝑥𝑡𝑡_𝑘+1+𝜃||≤max_𝑖∈𝑝𝐶_𝜃𝑡_𝑘+𝑖sup_{𝑡𝜏∈𝑘𝑡−ℎ,max𝑘+∑𝑝𝑖=1𝑇𝑘+𝑖−ℎ,𝑡𝑘}||𝑥(𝜏)_+𝑘+1||+𝜃||≤max_𝑖∈𝑝𝐶_𝜃𝑡^+𝑘+𝑖sup_{𝑡𝜏∈𝑘𝑡−ℎ,max𝑘+∑𝑝𝑖=1𝑇𝑘+𝑖−ℎ,𝑡𝑘}||𝑥(𝜏)||(4.5)forall𝜃∈[0,∑^𝑝𝑖=1𝑇_𝑘+𝑖] and any finite 𝑝∈𝐙₊. Thus, thereexistΩ_𝜃(𝑡_𝑘+1−ℎ,𝑡_𝑘)∈[1,∞)∩𝐑₊, which might be computed with direct simple calculations via (4.3), which equalizes max_{𝑖∈𝑝𝑘+1}(𝐶_𝜃(𝑡^+𝑘+𝑖)) with 𝑝_𝑘+1being a positive integer accounting for a subsequence of consecutive impulsive time instants{𝑡_𝑘+𝑖∶𝑖∈𝑝_𝑘+1}. Thus, it follows from (4.5) that max≤Ω_𝜃sup_{𝑡𝜏∈𝑘−ℎ,𝑡𝑘+1−ℎ}||𝑥(𝜏)||,sup_{𝑡𝜏∈𝑘+1−ℎ,𝑡𝑘}||𝑥(𝜏)||𝑈𝑡_𝑘−𝑡_𝑘+1−ℎ𝑡_𝑘+1−ℎ,𝑡_𝑘sup_{𝑡𝜏∈𝑘−ℎ,𝑡𝑘+1−ℎ}||𝑥(𝜏)||.(4.6) It follows directly from (4.6) into (4.1) and complete induction that if max_{𝜃∈0,𝑡𝑘+1−𝑡𝑘}||1+𝐾𝑡_𝑘+1𝑈𝜃−𝑇_𝑘||max_{𝜃∈0,𝑡𝑘+1−𝑡𝑘𝑒𝑎𝜃}+||||𝑒^𝑎𝜃−1𝑎||||max_{𝜏∈0,𝑡𝑘+1−𝑡𝑘}||𝑎₀𝑡_𝑘+𝜏||Ω_𝜃𝑡_𝑘+1−ℎ,𝑡_𝑘≤1,(4.7)forall𝑡_𝑘∈Imp(𝑡₀,∞) for some finite 𝑡₀∈𝐑₀₊then the system is globally uniformly stable for any admissible function of initial conditions 𝜑∈BPC⁽⁰⁾([−ℎ,0],𝐑) withsup_{𝑡∈𝐑0+}||𝑥(𝑡)||≤sup_{𝑡∈−ℎ,𝑡0,}||𝑥(𝑡)||≤𝐾_𝑥<∞,(4.8) with 𝑥(𝑡)=𝜑(𝑡),forall𝑡∈[−ℎ,0]. If the inequality in (4.7) is strict, then the system is globally asymptotically stable for any 𝜑∈BPC⁽⁰⁾([−ℎ,0],𝐑).

Note that if, furthermore, 𝑡_𝑘+1>𝑡_𝑘+ℎ, forall𝑡_𝑘,𝑡_𝑘+1∈Imp, then Ω_𝜃(𝑡_𝑘+1−ℎ,𝑡_𝑘)=1,forall𝑡_𝑘,𝑡_𝑘+1∈Imp so that (4.7) holds if the subsequent constraints hold for some real constant 𝛾∈(0,1]∩𝐑₀₊ as follow:max_{𝜃∈0,𝑡𝑘+1−𝑡𝑘}𝑒^𝑎𝜃+||||𝑒^𝑎𝜃−1𝑎||||max_{𝜏∈0,𝑡𝑘+1−𝑡𝑘}max_{𝜃∈0,𝑡𝑘+1−𝑡𝑘}||𝑎₀𝑡_𝑘+𝜏||≤𝛾,||1+𝐾𝑡_𝑘+1𝑈𝜃−𝑇_𝑘||≤𝛾⁻¹,(4.9) which may be fulfiled without requiring neither 𝑎≤0 (global stability of the auxiliary system with no delayed dynamics) nor 𝑎+|𝑎₀(𝑡)|≤0(global stability independent of the delay size) by using appropriate impulses of appropriate signs so that the above inequalities hold. A similar consideration applies for global asymptotic stability one of the inequalities in (4.9) being well posed and strict without requiring neither 𝑎<0(global asymptotic stability of the auxiliary system with no delayed dynamics) nor 𝑎+|𝑎₀(𝑡)|<0 (global asymptotic stability independent of the delay size). Note also that these above results are particular results of Theorem 3.1 for a scalar system (2.1)-(2.4) with a single parameterization with the non-impulsive controller being identically zero and the control parameter b being unity. If the scalar dynamic system is of polytopic type 3d by𝑥̇(𝑡)=^𝑁_𝑗=1+_𝑞𝜆_0𝑗(𝑡)⎛⎜⎜⎝𝑎_0𝑗(𝑡)𝑥(𝑡)+_{𝑡𝑘∈Imp(0,𝑡)}𝐾_0𝑗𝑡_𝑘𝑥𝑡_𝑘𝛿𝑡−𝑡_𝑘⎞⎟⎟⎠_𝑁𝑖=1_𝑗=1=𝑎𝑥(𝑡)+𝜆_𝑖𝑗𝑁(𝑡)𝑎_𝑖𝑗(𝑡)𝑥𝑡−ℎ_𝑖(𝑡)_𝑗=1_{𝑡𝑘∈Imp(0,𝑡)}+_𝑞𝜆_0𝑗𝑡_𝑘𝐾_0𝑗𝑡_𝑘𝑥𝑡_𝑘𝛿𝑡−𝑡_𝑘_𝑁𝑖=0_𝑗=1𝜆_𝑖𝑗(𝑡)𝑎_𝑖𝑗(𝑡)𝑥𝑡−ℎ_𝑖(𝑡),(4.10) provided that ∑^𝑁𝑗=1𝜆_0𝑗(𝑡)=1, 𝜆_0𝑗(𝑡)∈𝐑₀₊, 𝑎_0𝑗(𝑡)=𝑎_0𝑗(𝑡)−𝑎, 𝑎_𝑖𝑗(𝑡)=𝑎_𝑖𝑗(𝑡), forall𝑖∈𝑞,forall𝑗∈𝑁,forall𝑡∈𝐑₀₊ and any arbitrary constant 𝑎∈𝐑 so that,𝑥𝑡^+𝑘+𝜃=𝑒^𝑎𝜃𝑥𝑡^+𝑘++^𝜃0𝑞_𝑁𝑖=0_𝑗=1𝜆_𝑖𝑗𝑡_𝑘+𝜏𝑒^−𝑎𝜏𝑎_𝑖𝑗𝑡_𝑘+𝜏𝑥𝑡_𝑘+𝜏−ℎ_𝑖(𝜏)𝑑𝜏_𝑁_𝑗=1𝜆_0𝑗𝑡_𝑘+1𝐾_0𝑗𝑡_𝑘+1𝑈𝜃−𝑇_𝑘𝑥𝑡_𝑘+1,∀𝜃∈0,𝑇_𝑘,(4.11)

Thus, the first inequality of (4.1) becomes,sup_{𝑡𝜃∈𝑘,𝑡𝑘+1}||𝑥(𝜏)||≤max_{𝜃∈0,𝑇𝑘}|||||1+_𝑁_𝑗=1×max_{𝜃∈0,𝑇𝑘}𝜆_0𝑗𝑡_𝑘+1𝐾_0𝑗𝑡_𝑘+1𝑈𝜃−𝑇_𝑘||||||||||⎛⎜⎜⎝𝑒_𝑎𝜃||𝑥𝑡^+𝑘||+||||𝑒^𝑎𝜃−1𝑎||||max_{𝜏∈0,𝑇𝑘}|||||_𝑞_𝑁𝑖=0_𝑗=1×⎛⎜⎜⎝sup_{𝑡𝜏∈𝑘−𝑡ℎ(𝑡),max𝑘+1−ℎ(𝑡),𝑡𝑘}𝜆_𝑖𝑗𝑡_𝑘+𝜏𝑎_𝑖𝑗𝑡_𝑘+𝜏|||||||𝑥(𝜏)||⎞⎟⎟⎠⎞⎟⎟⎠||||||,(4.12)forall𝜃∈[0,𝑇_𝑘], where ℎ(𝑡)=max_𝑖∈𝑞sup(ℎ_𝑖(𝑡)). Thus, (4.7) is modified as follows.max_{𝜃∈0,𝑡𝑘+1−𝑡𝑘}|||||1+_𝑁_𝑗=1×max_{𝜃∈0,𝑡𝑘+1−𝑡𝑘𝑒𝑎𝜃}𝜆_0𝑗𝑡_𝑘+1𝐾_0𝑗𝑡_𝑘+1𝑈𝜃−𝑇_𝑘|||||+||||𝑒^𝑎𝜃−1𝑎||||max_{𝜏∈0,𝑡𝑘+1−𝑡𝑘}|||||_𝑞_𝑁𝑖=0_𝑗=1×Ω_𝜃𝜆_𝑖𝑗𝑡_𝑘+𝜏𝑎_𝑖𝑗𝑡_𝑘+𝜏|||||𝑡_𝑘+1−ℎ,𝑡_𝑘≤1,(4.13) which guarantees global stability from Theorem 3.1 and if the above inequality is strict, then global asymptotic stability is guaranteed.

Example 4.1. This example refers to the stability of the impulsive closed-loop system (2.8), subject to (2.7) and (2.9), by application of Corollary 3.10 to Theorems 3.4-3.5 and Remark 3.11. Assume that the non impulsive controller gains𝐾_𝑖𝑗(𝑡)are identically zero for all time so that𝐴^∗𝑖𝑗(𝑡)=𝐴_𝑖𝑗(𝑡), forall𝑖∈𝑞∪{0},forall𝑗∈𝑁and (3.7)–(3.12) are stated for this particular case. Then, the system is controlled by the impulsive controller gains which are nonzero only at set of zero measure defined by all the sequence of impulsive time instants. Note from Remark 3.11 that Δ𝑉(𝑡,𝑥_𝑡)=0if 𝑡∉Imp. Note also that if there only one Imp∋𝑡_𝑖∈(𝑡,𝑡+𝑇] at which 𝑥(𝑡_𝑖)≠0 so that for the controller gain choice 𝐾^0𝑗(𝑡_𝑖)=𝜈₀(𝑡_𝑖)𝐼_𝑛 then 𝑉≤𝑡−⁺+_{𝑡𝑡+𝑇+}𝑇,𝑥_𝑡+−𝑉𝑡,𝑥_𝑡𝜆^min𝑄_𝑑+𝜆(𝜏)−𝜆_max𝑄^𝑇0𝑑(𝜏)𝑄_0𝑑(𝜏)‖𝑥̂(𝜏)‖²²𝑑𝜏_min^𝑁_𝑗=1𝜆_0𝑗𝑡_𝑖𝐾^𝑇0𝑗𝑡_𝑖𝐵^𝑇0𝑗𝑡_𝑖𝑃^𝑁_𝑗=1−2‖‖‖‖_𝑁𝜆_0𝑗𝑡_𝑖𝐵_0𝑗𝑡_𝑖𝐾^0𝑗𝑡_𝑖_𝑗=1𝜆_0𝑗𝑡_𝑖𝐾^𝑇0𝑗𝑡_𝑖𝐵^𝑇0𝑗𝑡_𝑖𝑃‖‖‖‖₂⎤⎥⎥⎦‖‖𝑥𝑡_𝑖‖‖²²≤−^{𝑡𝑡+𝑇}₊−+𝜆𝜆_max𝑄^𝑇0𝑑(𝜏)𝑄_0𝑑(𝜏)𝜆_min𝑄_𝑑(𝜏)‖𝑥̂(𝜏)‖²²𝑑𝜏_min^𝑁_𝑗=1𝜆_0𝑗𝑡_𝑖𝐵^𝑇0𝑗𝑡_𝑖𝑃^𝑁_𝑗=1−2‖‖‖‖_𝑁𝜆_0𝑗𝑡_𝑖𝐵_0𝑗𝑡_𝑖𝜈₀𝑡_𝑖_𝑗=1𝜆_0𝑗𝑡_𝑖𝐵^𝑇0𝑗𝑡_𝑖𝑃‖‖‖‖₂⎤⎥⎥⎦𝜈₀𝑡_𝑖‖‖𝑥𝑡_𝑖‖‖²²≤0(4.14) if 𝜈₀𝑡_𝑖∈⎡⎢⎢⎢⎣0,𝑏𝑡^𝑖+𝑏²𝑡_𝑖+4𝑎𝑡_𝑖𝑐𝑡_𝑖2𝑎𝑡_𝑖⎤⎥⎥⎥⎦,𝜆^min𝑄_𝑑(𝑡)≥𝜆_max𝑄^𝑇0𝑑(𝑡)𝑄_0𝑑(𝑡),∀𝑡∈𝐑(₀₊4,.15) where 𝑎𝑡_𝑖∶=𝜆_min^𝑁_𝑗=1𝜆_0𝑗𝑡_𝑖𝐵^𝑇0𝑗𝑡_𝑖𝑃^𝑁_𝑗=1𝑏𝑡_𝑖𝜆_0𝑗𝑡_𝑖𝐵_0𝑗𝑡_𝑖‖‖𝑥𝑡_𝑖‖‖²²,𝑐𝑡_𝑖∶=2‖‖‖‖^𝑁_𝑗=1𝜆_0𝑗𝑡_𝑖𝐵^𝑇0𝑗𝑡_𝑖𝑃‖‖‖‖₂‖‖𝑥𝑡_𝑖‖‖²²,∶=^{𝑡𝑡+𝑇}₊𝜆^min𝑄_𝑑(𝜏)−𝜆_max𝑄^𝑇0𝑑(𝜏)𝑄_0𝑑(𝜏)‖𝑥̂(𝜏)‖²²𝑑𝜏,(4.16) which can always be fulfilled with 𝜈_0𝑖(𝑡_𝑖)=0(i.e., zero impulsive controller of the given class of impulsive controllers) since the right-hand equation (4.16) holds, which is a condition of global stability of the impulse-free system. If (4.16) is replaced with 𝜈₀𝑡_𝑖∈⎛⎜⎜⎜⎝0,𝑏𝑡^𝑖+𝑏²𝑡_𝑖−4𝑎𝑡_𝑖𝑐𝑡_𝑖2𝑎𝑡_𝑖⎞⎟⎟⎟⎠,𝜆^min𝑄_𝑑(𝑡)>𝜆_max𝑄^𝑇0𝑑(𝑡)∀𝑄𝑡∈_0𝑑(𝐑𝑡)₀₊,,(4.17) then global asymptotic stability is guaranteed. However, assume that 𝜆_min(𝑄_𝑑(𝑡))<𝜆_max(𝑄^𝑇0𝑑(𝑡)𝑄_0𝑑(𝑡))⇒𝑐(𝑡_𝑖)≤0 (except possibly on a set of zero measure) implying 𝑐(𝑡_𝑖)≤0. Then, global stability is not guaranteed without impulsive controls since the candidate is not a Lyapunov functional. However, the choice 𝜈₀(𝑡_𝑖)∈[0,(𝑏(𝑡_𝑖)+(𝑏²(𝑡_𝑖)−4𝑎(𝑡_𝑖)|𝑐(𝑡_𝑖)|)/2𝑎(𝑡_𝑖)] and a sufficiently small 𝑇(𝑡)∈𝐑₊ containing each impulsive time instant ensuring that ||𝑐𝑡_𝑖||≤𝑇(𝑡)max_{[]𝜏∈𝑡,𝑡+𝑇(𝑡)}||||𝜆_min𝑄_𝑑(𝜏)−𝜆_max𝑄^𝑇0𝑑(𝜏)𝑄_0𝑑(𝜏)‖𝑥̂(𝜏)‖²²||||≤𝑏²𝑡_𝑖4𝑎𝑡_𝑖(4.18) also guarantees global stability even although the impulsive-free system is not stable. If𝜈₀(𝑡_𝑖)∈(0,(𝑏(𝑡_𝑖)+(𝑏²(𝑡_𝑖)−4𝑎(𝑡_𝑖)|𝑐(𝑡_𝑖)|)/2𝑎(𝑡_𝑖))then global asymptotic stability is guaranteed provided that 𝜆_min(𝑄_𝑑(𝑡))>𝜆_max(𝑄^𝑇0𝑑(𝑡)𝑄_0𝑑(𝑡))⇒(𝑐(𝑡_𝑖)>0) on a connected subset of 𝐑₀₊of infinite measure in order to guarantee the global asymptotic convergence to zero of the state-trajectory solution. That means that asymptotic stability is guaranteed under the last conditions for finite time intervals but, after some finite time, the conditions (4.17) are fulfilled. Note that it has not been assumed that the polytope of vertices 𝐴^∗𝑖𝑗(𝑡)=𝐴_𝑖𝑗(𝑡),forall𝑖∈𝑞∪{0},forall𝑗∈𝑁 is a stability matrix at any time. The example is very easily extendable to the case of simultaneous control under a standard control and an impulsive one so that 𝐴^∗𝑖𝑗(𝑡)=𝐴_𝑖𝑗(𝑡)+𝐵_𝑖𝑗𝐾_𝑖𝑗, forall𝑖∈𝑞∪{0},forall𝑗∈𝑁.

Example 4.2. An automatic steering device was designed by Minorsky for the battleship New Mexico in 1962, [32]. There is a direction indicating instrument tracking the current direction of motion and there is also an instrument defining the suitable reference motion. Another problem solved by Minorsky for ships is that of the stabilization of the rolling by the activated tanks method in which ballast water is pumped from a position to another one by means of a propeller pump controlled by electronic instrumentation. The second-order delayed resulting dynamics for rolling control of the ship has the following standard form: 𝑦̈(𝑡)+𝛼𝑦̇(𝑡)+𝛽𝑦̇(𝑡−ℎ)+𝜔²⁰𝑦(𝑡)=𝑢₀(𝑡),(4.19) where the various parameters are positive, where the last left-hand side term is related to stiffness, 𝛼 is the standard dumping coefficient excluding delay effects, and 𝛽 is the dumping coefficient produced by pumping which has a delay when the dump becomes overworked (in not overworked normal operation points, the delay ℎ=0 and the dumping coefficient is 𝛼+𝛽). If the open-loop control action is modified using feedback to improve the original dynamics as follows: 𝑢₀(𝑡)⟶𝑢(𝑡)=𝑢₀(𝑡)+𝑘_𝛼𝑦̇(𝑡)+𝑘_𝛽𝑦̇(𝑡−ℎ)+𝑘_𝜔𝑦(𝑡)(4.20) then, the resulting closed-loop differential equation becomes, 𝑦̈(𝑡)+𝛼−𝑘_𝛼𝑦̇(𝑡)+𝛽−𝑘_𝛽𝑦̇(𝑡−ℎ)+𝜔²⁰−𝑘_𝜔𝑦(𝑡)=𝑢₀(𝑡),(4.21) which can be also described in the state-space form (1) through two first-order differential equations by the state variables 𝑥₁(𝑡)=𝑦(𝑡), 𝑥₂(𝑡)=𝑦(̇𝑡) as 𝑥̇¹𝑥̇₂(𝑡)(𝑡)=𝑘−0𝜔²⁰𝑘_𝛼1_𝜔−𝛼𝑥¹𝑥(𝑡)₂(𝑡)+00𝑘_𝛽0−𝛽𝑥¹𝑥(𝑡−ℎ)₂(𝑡−ℎ)+01𝑢⁰(𝑡).(4.22) The above system is positive if and only if 𝑘_𝜔≥𝜔²⁰ and 𝑘_𝛽≥𝛽 irrespective of the value of (𝑘_𝛼−𝛼) since 𝐴₀∶=[^𝑘01_𝜔−𝜔²⁰_{𝑘𝛼−𝛼}] (the system delay-free matrix) is a Metzler matrix, the control vector 𝑏>0and the delayed matrix of dynamics 𝐴₁∶=[^000𝑘_𝛽−𝛽]>0. A complete discussion about positivity is found in [32]. The fundamental matrix of the above system is Ψ(𝑡,0)=𝑒^𝐴0𝑡𝐼+^0𝑡−ℎ𝑒^−𝐴0𝜏𝐴₁Ψ(𝜏−ℎ,0)𝑈(𝑡−ℎ)𝑑𝜏,(4.23) where 𝑈(𝑡)=1(𝑡)is the unit step (Heaviside) function. In Minorsky’s problem 𝑢₀(𝑡)≡𝑎sin𝜔𝑡 which is not a positive control for all time. Now, consider the stability problem rather than the positivity one under a polytopic parameterization numbered by “1” and “2” one being stable while the other being unstable. Consider the case where switches occur between both vertices of the polytope. The polytope model is adopted to deal wit the uncertainty in the parameter (𝑘_𝜔−𝜔²⁰) which is known to be close zero, but its sign is unknown if, for instance, it is slightly time varying around zero.
(1) Assume that the uncontrolled parameterization 1 is stable independent of the delay under the following constraints: 𝑘_𝜔1<𝜔²⁰¹,𝑘_𝛼1<𝛼₁,‖‖𝐴₁₁‖‖₂<12||𝜆^max𝐴₀₁+𝐴^𝑇01||,(4.24) where 𝐴₀₁=[^𝑘01_𝜔1−𝜔²⁰¹_{𝑘𝛼1−𝛼1}],𝐴₁₁=[^000𝑘_{𝛽1−𝛽1}]. The two first constraints ensure that 𝐴₀₁ is a stability matrix while the third one ensures stability independent of the delay of the uncontrolled system or under any control guaranteeing that the modified closed-loop matrices 𝐴_𝑖1 (𝑖=1,2) satisfiy similar stability constraints.
(2) Assume that the uncontrolled parameterization 2 is unstable under the following constraints: 𝑘_𝜔2>𝜔²⁰²,𝑘_𝛼2<𝛼₂,‖‖𝐴₁₂‖‖₂<12||𝜆^max𝐴₀₂+𝐴^𝑇02||,(4.25) where𝐴₀₂=[^𝑘01_𝜔2−𝜔²⁰²_{𝑘𝛼2−𝛼2}], 𝐴₁₂=[^000𝑘_{𝛽2−𝛽2}]. The two first constraints ensure that 𝐴₀₁ is a stability matrix while the third one ensures stability independent of the delay of the uncontrolled system or under any control guaranteeing that the modified closed-loop matrices 𝐴_𝑖1 (𝑖=1,2) satisfy similar stability constraints. There are several possibilities to stabilize the system by choosing to generate impulsive controls at certain switching time instants in between parameterizations. Two of them are the following.
(1) Stabilizing Law 1 via Impulse-Free Switching between Parameterizations with Minimum Residence Time at the Stable Parameterization 1
Choose 𝑢₀≡0. Let Imp≡Ξ∶={𝑡_𝑖∈𝐑₀₊}_{𝑖∈𝐙0+}be the sequence of switching time instants in-between the parameterizations 1 and 2 and vice-versa. Prefix a designer’s choice of indexing index integer 𝑖̂∈𝐙₀₊which might be sufficiently large but finite. Thus, for any 𝐙₀₊∋𝑖(even)≥𝑖̂ the active 2-parameterization is unstable on [𝑡_𝑖,𝑡_𝑖+1) with switching to parameterization 1 at t =𝑡_𝑖+1. Proceed as follows. Choose 𝑡_𝑖+2>𝑡_𝑖+1 with sufficiently large residence time interval 𝑇_𝑖+1∶=𝑡_𝑖+2−𝑡_𝑖 at the active stable parameterization 1 so that the subsequent stability constraint holds ‖‖Ψ𝑡_𝑖+2,𝑡_𝑖+1‖‖₂‖‖Ψ𝑡_𝑖+1,𝑡_𝑖‖‖₂≤𝜎𝑡_𝑖+2,𝑡_𝑖+1≤1,(4.26) with the prefixed real sequence Θ∶={𝜎(𝑡_𝑖+2,𝑡_𝑖+1)≤1}_𝑖(≥𝑖̂_)∈𝐙0+ for any 𝑡_𝑖+1>𝑡_𝑖. The above switching law between parameterizations generates a stable polytopic system with switches at the polytope vertices. This simple law has to direct immediate extensions. (a) The use of an impulsive-free stabilizing control law which makes the parameterization 1 stable with a greater stability degree that its associate open-loop counterpart. (b) To guarantee the stability constraint by considering strips including some finite number of consecutive switches in-between parameterizations 1-2 by guaranteeing a sufficiently large residence time at the current stable active parameterization 1. Note that if the sequence Θ has infinitely many members strictly less than one, the global exponential stability of the polytopic system with switches in between vertices is guaranteed.(2) Stability Might Be Achieved with Impulsive Controls at Switching Time Instants for a Switching Sequence Ξ Indexed for 𝐙₀₊∋i(Even)≥î as Above
Proceed as follows. (1) Choose 𝑡_𝑖+2∈Ξ at time instants such that 𝑥₁(𝑡_𝑖+2)≠0for each triple of switching time instants which does not respect the stability constraint (i.e., if ‖Ψ(𝑡_𝑖+2,𝑡_𝑖+1)‖₂‖Ψ(𝑡_𝑖+1,𝑡_𝑖)‖₂>1). (2) Define a sequence of real numbers {𝜀(𝑡_𝑖+2)}_𝑖(≥𝑖̂_)∈𝐙0+ defined by 𝜀(𝑡_𝑖+2)∶=|𝜀(𝑡_𝑖+2)|sgn𝑥₁(𝑡_𝑖+2) if 𝑥₂(𝑡_𝑖+2)≠0 and 𝜀(𝑡_𝑖+2)=0 if 𝑥₂(𝑡_𝑖+2)=0 such that 𝜀(𝑡_𝑖) is zero if and only if 𝑥₂(𝑡_𝑖+2)=0. (3) Generate an impulsive control𝑢₀(𝑡_𝑖+2)=𝐾(𝑡_𝑖+2)𝛿(𝑡−𝑡_𝑖+2) with controller sequence{𝐾(𝑡_𝑖+2)}_𝑖(≥𝑖̂_)∈𝐙0+defined as 𝐾𝑡_𝑖+2=⎧⎪⎨⎪⎩𝜀𝑡^𝑖+2sgn𝑥₁𝑡_𝑖+2−𝑥₂𝑡_𝑖+2𝑥₂𝑡_𝑖+2,if𝑥²−1if𝑥₂𝑡_𝑖+2≠0,𝑥𝑡_𝑖+2=0,₁𝑡^+𝑖+2=𝑥₁𝑡_𝑖+2,𝑥₂𝑡^+𝑖+2=1+𝐾𝑡_𝑖+2𝑥₂𝑡_𝑖+2=𝜀𝑡_𝑖+2.(4.27) Now, note by taking into account from the companion form of the state-space realization that 𝑥₂(𝑡)=𝑥̇₁(𝑡), 𝑘_𝜔2>𝜔²⁰², and 𝑘_𝛼2<𝛼₂, it follows for 𝛿>0 from the mean value theorem for integrals of continuous integrands that, 𝑥̇₁𝑡^+𝑖+2=⎧⎪⎨⎪⎩𝑥²𝑥𝑡^+𝑖+2=1+𝐾𝑡_𝑖+2𝑥₂𝑡_𝑖+2=𝜀𝑡_𝑖+2∶=||𝜀𝑡_𝑖+2||sgn𝑥₁𝑡_𝑖+2if𝑥₂𝑡_𝑖+2≠0,₂𝑡^+𝑖+2=0if𝑥₂𝑡_𝑖+2=0(4.28) Since 𝑥₂(𝑡) is Lipschitz-continuous, then for any given 𝜀₀∈𝐑₊, it exists 𝛿^∗=𝛿^∗(𝑡_𝑖+2,𝜀₀)∈𝐑₊ being a monotone increasing function of the argument 𝜎 such that using the mean value theorem for integrals of continuous bounded integrands, one has forall𝜎∈[0,𝜎^∗); forall𝜎^∗∈𝐑₊ as follow:𝑥𝑥̇₁₁𝑡_𝑖+2+𝜎=𝑥₂𝑡^+𝑖+2+𝜎∈𝜀𝑡_𝑖+2−𝜀₀,𝜀𝑡_𝑖+2+𝜀₀⟹||𝑥₂𝑡_𝑖+2+𝜎||≤||𝜀𝑡_𝑖+2||+𝜀₀,𝑡_𝑖+2+𝜎=≤𝑥₁(1𝑡−_𝑖+2𝜎)+||𝜀^{𝑡𝑖+2𝑡+𝜎}_𝑖+2𝑡_𝑖+2𝑥̇₁(𝜏)𝑑𝜏=𝑥₁𝑡_𝑖+2+𝜎𝑥₂(𝜁)⟹||𝑥₁𝑡_𝑖+2+𝜎||||+𝜀₀,(4.29) for some 𝐑₊∋𝜁∈(𝑡_𝑖+2,𝑡_𝑖+2+𝛿). Thus, for sufficiently small |𝜀(𝑡_𝑖+2)| and𝜀₀, one has ‖‖𝑥𝑡_𝑖+2+𝜎‖‖₂≤1+(1−𝜎)²||𝜀𝑡_𝑖+2||+𝜀₀,(4.30) so that there is a close time instant 𝑡=𝑡_𝑖+2+𝜎 to 𝑡_𝑖+2(for a sufficiently small 𝜎∈𝐑₊) such that ‖𝑥(𝑡_𝑖+2+𝜎)‖₂ is arbitrarily small by choosing a sufficiently small |𝜀(𝑡_𝑖+2)| in the sequence and a sufficiently small 𝜀₀. Thus stabilization is achievable via impulsive controls proceeding in this way at the unstable parameterization when necessary through the above technique.

Acknowledgments

The author thanks to the Spanish Ministry of Education by its support of this work through Grant DPI2009-07197 and to the Basque Government by its support through Grants IT378-10 and SAIOTEK SPE07UN04. He is also grateful to the reviewers by their useful comments.