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

Volume 2010, Article ID 693958,33pages doi:10.1155/2010/693958

Research Article

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

M. de la Sen

Institute of Research and Development of Processes IIDP, Faculty of Science and Technology, University of the Basque Country, Campus of Leioa (Bizkaia), Aptdo, 644 Bilbao, Spain

Correspondence should be addressed to M. de la Sen,manuel.delasen@ehu.es Received 18 December 2009; Accepted 23 June 2010

Academic Editor: Oleg V. Gendelman

Copyrightq2010 M. de la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper investigates the stability properties of a class of dynamic linear systems possessing several linear time-invariant parameterizationsor configurationswhich 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 thatq1polytopic parameterizations are considered for a system withqdelays 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 Li´enard-type equations with Markovian switching is investigated in4,5. 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

1In 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.

2In20, the problem of theN-buffer switched flow networks is discussed based on a theorem on positive topological entropy.

3In 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.

4In 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.

5In 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 reducedorderthen less complexwhile maintaining closed-loop stability.

6In 25, the stabilization of switched dynamic systems is discussed through topologic considerations via graph theory.

7The stability of different kinds of switched systems subject to delays has been investigated in11–13,17,26–28.

8The stability switch and Hopf bifurcation for a diffusive prey-predator system is discussed in6in the presence of delay.

9A general theory with discussed examples concerning dynamic switched systems is provided in3.

10Some 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,29–40and 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 2discusses 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 3discusses 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 ofSection 2without 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 inSection 4.

Notation 1.1. Z, R, and C are the sets of integer, real, and complex numbers, respectively.

Zand Rdenote the positive subsets of Z, respectively, and Cdenotes the subset of C of complex numbers with positive real part, andn: {1,2, . . . , n} ⊂Z, for alln ∈ 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.

Z₀: Z∪ {0}, R₀ : R∪ {0}, C₀: C∪ {0}

Z0−: Z−∪ {0}, R0− : R−∪ {0}, C0−: C−∪ {0} 1.1 Given some linear space X usually R or C, then CⁱR₀, X denotes the set of functions of classCⁱ. Also, BPCⁱR₀, Xand PCⁱR₀, Xdenote the set of functions in Cⁱ⁻¹R0, X which, furthermore, possess bounded piecewise continuous or, respectively, piecewise continuousith derivative onX.

LXdenotes the set of linear operators fromX to X. In particular, the linear space denoted byXdenotes the state space of the dynamic system with controls in the linear space U.

I_ndenotes thenth 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 matricesM, respectively. The notationsMD, M≺D, M D,andMDstand correspondingly for M−D 0,M−D ≺0,M−D 0,and M−D0,and Superscript “T” stands for transposition of matrices and vectors.

λ_maxMandλ_minMstand for the maximum and minimum eigenvalues of a definite square real matrixM mij.

A finite or infinite strictly ordered sequence of impulsive time instants is defined by Imp : {ti ∈R₀ :t_i1 > t_i}, where an impulsive controlutiδt−t_ioccurs withδ ·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 dimensionsnandmand being subject toqtime-varying point delays:

xt ˙ q

i 0

N j 1

λ_ijt

A_ijtxt−h_it B_ijtuijt

λ^Ttxt ut, 2.1

where the incommensurate time-varying delays are h₀t 0 for allt ∈ R₀, h_i ∈ PC¹R0,R0, for alli∈q: {1,2, . . . , q}i.e., the delays are continuous time differentiable of bounded time derivative, and

λ^Tt

λ01tλ02t· · ·λ0Nt· · ·λq1tλq2t· · ·λqNt , x^Tt

x^Tt

A^T₀₁t· · ·A^T_0Nt

· · ·x^T

t−h_qt

A^T_q1t· · ·A^T_qNt , u^Tt

u^T₀₁tB^T₀₁t· · ·u^T_0NtB_0N^T t· · ·u^T_q1tB_q1^Tt· · ·u^T_qNtB^T_qNt

2.2

are vector functions from R₀to R^q1N, R^q1Nnand R^q1Nm, respectively, and

ix : R0∪−h,0 → X ⊂ Rⁿ is the state vector, which is almost everywhere time differentiable on R₀ satisfying 2.1, subject to bounded piecewise continuous initial con- ditions on−h,0,that is,x ϕ ∈ BPC⁰−h,0,Rⁿ, whereh h0 max_i∈qsuphi0 ≤ h : max_i∈q sup_t∈R0ht, andu_ij : R0 → U ⊂ R^mare the control vectors for alli ∈ q ∪ {0}for allj ∈ NandAij ∈ BPC⁰R0,R^n×nandBij ∈BPC⁰R0,R^n×mparameterize the dynamic system.

ii λij ∈ BPC⁰R0,R0, subject to the constraint q i 0

_N

j 0λijt ∈ c1, c2 ⊂ R,for allt∈R0with∞> 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 2.1related 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 system

xt ˙ ^q

i 0

N j 1

λijtx˙ijt 2.3

formed with subsystems of the form ˙xijt Aijtxt−hit Bijtuijt. The controls u_ij: R₀ → U⊂R^mare generated from the state-feedback impulsive controller as follow:

u_ijt K_ijtxt−h_it ∀i∈q

1,2, . . . , q ; ∀t∈R0,fori 0, t /∈Imp, u0jt

K0jt K_0jt

xt fori 0, t∈Imp, 2.4

where the strictly ordered Imp: {ti ∈ R0 : t_i1 > ti, 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 functionf : R → R,

ft _∞

−∞fτδt−τdτ _t

t

fτδt−τdτ lim

ε→0

_tε

t−εfτδt−τdτ, 2.5

whereδtis the Dirac distribution at timet 0 with the following notational convention being used:gt limε→0g tε/gt limε→0g t−εeither ift ∈ Imp or ifg is bounded having left and right limits at a discontinuity point t ∈ R0, andgt gtif 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, ImpT1, T₂ {t∈Imp :t∈T1, T₂}and ImpT1, T₂ {t∈Imp :t∈T1, T₂}. Note that Imp Imp0,∞. The regular and impulsive controller gain matrices are, respectively, Kij ∈ BPC⁰R0,R^m×n and K_ij : Imp → R^m×n being a discrete sequence of bounded matrices. Note that, ifK_0jtis discontinuous at the time instantt,thenK_0jt/K_0jteven ift /∈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 law2.4into the open-loop system equation2.1leads to the closed-loop functional dynamic system as follows:

xt ˙ _N

j 1λ0jt

A^∗_0jt B0jtK_0jtδ0

xt q i 1

_N

j 1λijtA^∗_ijtxt−hit, xt

I_nN

j 1λ_0jtB0jtK_0j t

xt; 2.6

for allt∈R0withK_0j t 0; for allt /∈Imp, where

A^∗_ijt Aijt BijtKijt, ∀i∈q∪ {0} ∀j∈N, 2.7 Equation2.6becomes

xt ˙ q

i 0

N j 1

λ_ijtA^∗_ijtxt−h_it, 2.8

for all t /∈Imp and also at the left limits for all t ∈ Imp, and xt − xt _N

j 1λ_0jtB0jtK_0j txt, which is zero ift /∈Imp, with

xt˙ ^N

j 1

λ0jtA^∗_0jtxt q

i 1

N j 1

λijtA^∗_ijtxt−hit 2.9

for the right limits of allt∈Imp. DefineD: Imp∪D_p, where

D_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 ofAij ∈ BPC⁰R0,R^n×n,Bij ∈ BPC⁰R0,R^n×m, λ_ij ∈BPC⁰R₀,R₀, andK_ij ∈BPC⁰R₀,R^m×nfor alli∈q∪ {0}, for allj∈Nwhich are in the respective setsDAij,DBij,Dλij,andDKij. The following technical assumptions are made.

Assumption 2.1. there existυ∈Rsuch thatt_k1−tk≥υ,for alltk, t_k1> tk∈Imp.

Assumption 2.2.

j∈ND_B0j∪

j∈ND_λij∩Imp ∅.

Assumption 2.1implies 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 2.2is needed for all the functionsλ0jB0j_k ∈ BPC⁰R₀,R0for allj ∈ N, for allk ∈ n and for all ∈ m, build with the entries B_0j ∈ BPC⁰R0,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-Lindelofftheorem, there is a unique solution for any vector function of initial conditions ϕ ∈ BPC⁰−h,0,Rⁿ and x ∈ BPC¹R,Rⁿ. The state-trajectory solution of the closed-loop system 2.8-2.9 for initial conditionsϕ ∈ BPC⁰−h,0,Rⁿis given by

xt Ψt

⎡

⎣Ψ⁻¹0x0 q

i 1

N j 1

_t

0

Ψ⁻¹τλijτA^∗_ijτxτ−h_iτdτ

tk∈Imp0,t

N j 1

λ_0jtkΨ⁻¹tkB0jtkK_0jtkxtk

⎤

⎦

⎡

⎣Ψst, t0xt0

q i 1

N j 1

_t

0

λ_ijτΨst, τA^∗_ijτxτ−h_iτdτ

tk∈Impt0,t

N j 1

λ_0jtkΨst, tkB0jtkK_0j tkxtk

⎤

⎦,

2.11

subject toxt ϕt,for allt∈−h,0, where

1 Ψt ∈ C⁰R0,R^n×n is an almost everywhere differentiable matrix function on R being time differentiable on the non connected real set

ti∈Impt_i1 − ti with unnecessarily continuous time derivatives which satisfies ˙Ψt _N

j 1λ_0jtA^∗_0jtΨton RwithΨ0 I_n. IfA_ij,B_ij,λ_ij, andK_ij for alli∈q∪ {0}, for allj ∈ N are everywhere continuous on R, then Ψt ∈ C¹R0,R^n×n, Ψs·,·: R²₀ → R^n×n asΨst, τ ΨtΨ⁻¹τfor allt≥τ, and

2Impt0, t: {tk ∈R0 :t0 ≤tk∈Imp< t} ⊂Imp is the strictly ordered sequence of impulsive time instants with input impulses occurred ont0, tfor anyt0 ∈R. Also, Impt0, t: {tk ∈Imp :t₀ < t_k< t} ⊂Imp; Impt0, t: {tk ∈Imp :t₀< t_k ≤ t} ⊂Imp are defined in a closed way.

The solution2.11is identically defined by

xt Zt

Z⁻¹0x0 ₀

−hZ⁻¹τϕτdτ

tk∈Imp0,t

N j 1

λ0jtkZ⁻¹tkB0jtkK_0j tkxtk

⎤

⎦,

2.12

where Zt ∈ C⁰R0,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, Zt 0 for allt ∈ R−. Defining the matrix function Z_s·,· : R²₀ → R^n×n as

Zst, τ ZtZ⁻¹τfor allt≥τ, one has from2.12fort∈tk, t_k1for any two consecutive givent_k, t_k1 ∈Imp as follow:

xt Z_st, tkx t_k

q

i 1

₀

−hi

Z_st, tkτϕtkτdτ

⎛

⎝^N

j 1

Z_st, tkλ0jtkB0jt_k1x t_k1

K_0jt_k1

⎞

⎠,

2.13

which becomes fort t_k1as follow:

x

t_k1 ⎛

⎝I_n^N

j 1

Z_st_k1, t_kB0jt_k1K_0jt_k1

⎞

⎠xt_k1

Zstk1, tkx t_k

q

i 1

₀

−hi

Zst, tkτϕtkτdτ

^N

j 1

Zstk1, tkλ0jtkB0jtk1xtk1K_0j tk1δt, tk1,

2.14

whereδt, t_k1 1 ift t_k1 and zero otherwise is the Kronecker delta. In view of2.12, the state-trajectory solution can be defined by the impulsive evolution operator{Tt, tk : t∈tk, t_k1,for allt_k∈Imp}, associated with{Zt:t∈R0}whereT·,·:{tk, t_k1:t_k∈ Imp∪{0}} → LX, which is represented byxt Tt, tkxt_k; for allt∈tk, t_k1,for alltk∈ Imp so that:

xt Tt, tkxt_k,

x t_k1

T t_k1, t_k

x_t

k

⎛

⎝I_n^N

j 1

λ_0jt_k1B0jt_k1K_0j t_k1

⎞

⎠Tt_k1, t_kxt_k,

2.15

for allt ∈ tk, t_k1,for alltk ∈ Imp, wherext and xt denote the strings of state solution trajectory and{xτ :τ ∈t−h, t}and{xτ:τ ∈t−h, t}, respectively. The subsequent result follows directly for the state-trajectory solution from2.11for any initial conditions ϕ∈BPC⁰−h,0,Rⁿ.

Theorem 2.3. The following properties hold.

iThe state-trajectory solution satisfies the following equations on any intervalζ, t⊂R0for anyϕ∈BPC⁰−h,0,Rⁿ:

x

t_k1 ⎛

⎝In^N

j 1

Ψstk1, t_k1λ0jtk1B0jtk1K_0j tk1

⎞

⎠xtk1 2.16

Ψstk1, ζxζ _tk1

ζ

Ψstk1, τ

⎛

⎝^q

i 1

N j 1

λijτA^∗_ijτxτ−hiτ

⎞

⎠dτ

ti∈Impζ,tk1

N j 1

λ0jtiΨstk1, tiB0jtiK_0j tixti

2.17

⎛

⎝I_n^N

j 1

Z_st_k1, t_k1B0jt_k1K_0j t_k1

⎞

⎠xt_k1 2.18

Z_st_k1, ζxζ q

i 1

₀

−hi

Z_st_k1, ζτxζτdτ

ti∈Impζ,tk1

N j 1

λ_0jtiZstk1, t_iB0jtixtiK_0j ti

2.19

T t_k1, ζ

x_ζ 2.20

ti,ti1∈Impζ,tk1

⎡

⎣

⎛

⎝In^N

j 1

λ0jti1B0jti1K_0jti1

⎞

⎠Tti1, ti

⎤

⎦xζ, 2.21

for allt_k1> ζ ∈ Imp, for allζ ∈ R0 withTtk1, t_k1 Z_stk1, t_k1 Ψstk1, t_k1 I_n. Equations2.17and 2.19are also valid by replacing t_k1 → t, for allt ∈ tk1, t_k2if t_k2 ∈ Imp and for allt ∈ t_k1,∞ if t_k1,∞∩ Imp ∅, that is, if the sequence of impulsive time instants is finite with the last impulsive time instant beingt_k1. Equation2.21has to be modified by replacingt_k1 → tand then by premultiplying it byTt, tk1.

iiAssume that

ti,ti1∈Impζ,tk1

⎡

⎣

⎛

⎝I_n^N

j 1

λ_0jti1B0jti1K_0j ti1

⎞

⎠Tti1, t_i

⎤

⎦

≤M_T ≤1 2.22

⎛

⎝In^N

j 1

λ0jtB0jtK_0j t

⎞

⎠T t, tcimp

≤MT≤1, 2.23

for allt_k1> ζ∈Imp, for allζ∈R0, and for allt≥t_cimp provided thatc_imp: card Imp0,∞<

∞,thenΓx_L^p_R,X ≤C_Γ, whereΓ: DomΓ≡X → L^pR, Xis defined byΓxt Tt, θx.

for allx∈X.

Proof. iIt follows directly for the state-trajectory solution from2.11,2.14, and2.15for any time intervalζ−h, ζof initial conditionsϕ∈BPC⁰−h,0,Rⁿ.

iiThe first part follows from the definition of the impulsive evolution operator. If, in addition,MT <1, then it follows from the following given constraints:

∃lim

t→ ∞Tt, θξ 0, ∀t> θ∈R, θ∈R0, ∀ξ∈X ⇒Tt, θξ 2.24 is bounded, for allξ∈X, for allt> θ∈R, θ∈R₀

⇒ Tt, θ ≤C_T ,some RC_T≥1,∀t> θ∈R, θ∈R₀ 2.25 from the uniform boundedness principle. Now, note that the operatorΓ: DomΓ ≡X → L^pR, Xis closed and then bounded from the closed graph theorem, so that the proof of Propertyiiis 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 fromTheorem 2.32.16–2.21, 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,Ψst, τ ≤k_Ψe^{−ρψt−τ}, for allt≥τt₀, some finitet0>0, some R k_Ψ>0, and someρψ ∈Ras follow:

k_Ψ

⎛

⎜⎝1sup_t

kip≤τ≤tki1p

q i 1

_N

j 1λ_ijτA^∗_ijτ

2

ρ_Ψ

tj∈Imp^tkip,tki1p N

j 1

λ_0j t_j

B_0j t_j

K_0j t_j

e^{−ρΨtkj1p−tj} 2

⎞

⎟⎠≤1,

3.1

for some p ∈ Z, some finite k ∈ Z0, some subsequence {t_kip} ∈ Imp, for alli ∈ Z0. Thus, the closed-loop system2.8-2.9is 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_Ψ1sup_tk≤τ<∞q

i 1

_N

j 1λijτA^∗_ijτ

2/ρ_Ψ ≤ 1< 1with tk being the last element of the finite sequenceImp

Now, a general stability result follows, which proves thatin 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 : {ti∈R0}such that the closed- loop system2.6–2.7is globally stable for any function of initial conditionsϕ∈BPC⁰−h,0,Rⁿ for some sequence of impulsive controller gainsK_0j : R0 → R^n×m, for allj∈N,for alli∈q∪ {0}.

Proof. The basic equation to build the stability proof is xt − xt _N

j 1λ0jtB0jtK_0j txt, for allt ∈ Imp and any sequence of impulsive time instants Imp. Consider prefixed real constants Ki ∈ R i ∈ 4 fulfilling K1 ≤ K3 − ε1

and K4 ≤ K2 − ε2 with ε1 ∈ 0, K3 ∩ R0 and ε2 ∈ 0, K2 ∩ R0 such that x_k0 ∈ 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 infinitet∈Rexists such thatxkτ ∈K1, K2; for allτ ∈0, t,butxkt ∈−∞, K3∪K4,∞∩K1, K2for somek ∈ n, some K₃ ∈ R with an existingperhaps emptypartial sequence of impulsive time instants Imp0, 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 Imp0, t Imp0, t∪ {t}is fixed as the first impulsive time instant and

−∞< K3≤xkt

⎛

⎝δk, ^N

j 1

m i 1

n 1

λ0jtB0jkitK_0jit

⎞

⎠xt≤K4<∞, 3.2

where the entry notationM Mijfor a matrixMis used, provided that the impulsive controller gainK_0jiktis chosen so that the following constraint holds:

K3−_N

j 1_m

i/k 1_n

/k 1λ0jtB0jkitK_0ji t

xt−xkt

1_N

j 1λ_0jtB0jkitK_0jikt x_kt

≤K_0jkk t≤ K₄−_N

j 1_m

i/k 1_n

/k 1λ_0jtB0jkitK_0jit

xt−x_kt

1_N

j 1λ_0jtB0jkitK_0jkkt

x_kt .

3.3

Note by direct inspection of3.3that such a controller gain always exists. As a result, x_kt∈K3, K₄⊂K1, K₂, for allk∈n. By continuity of the state-trajectory solution, there exists a finiteTt, K∈Rsuch thatxkτ∈K3−K, K4K⊂K1, K2for any prefixed K∈R, for allτ ∈t, tTt, K, for allk∈nprovided thatK3−K1 ≤K≤K2−K4. Since x_ktTt, K∈−∞, K3∪K4,∞∩K1, K₂thenx_kτ⊂K1, K₂,for all τ∈0, tTt, for allk ∈ n. Also, xkτ ⊂ K1, K2 for al τ ∈ 0, tTt, for allk ∈ nif an impulsive controller gain is chosen at timetTtby replacingt → t Ttin3.3and Imp0, t Tt Imp0, tTt∪ {tTt}with Imp0, tTt Imp0, t. It has been proven that xkτ∈K1, K2, for allτ ∈0, tfor any givent∈R0, for all k ∈nthenxkτ∈K1, K2, for allτ ∈ 0, tTt, for some Tt ∈ R, and for all k ∈ nso that the result holds by complete induction for for allt∈R₀with a bounded sequence of impulsive controller gains at some appropriate sequence of impulsive time instants Imp: {ti∈R0}.

Remark 3.3. Note thatTheorem 3.2holds irrespective of the values of the regular controller gain functions K_ij : R₀ → R^m×n for some appropriate sequence of impulsive controller gainsK_0j : R0 → R^n×m, for allj ∈N,for alli∈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 system2.6–2.7is globally stable for anyϕ∈BPC⁰−h,0,Rⁿ and any given set of regular controller gain functionsK_ij : R₀ → R^n×mif the sequence of impulsive time instants Imp : {ti∈R0}is chosen so that

1the sequence of impulsive controller gainsK_0j : R0 → R^n×m, for allj ∈N; for alli∈ q∪ {0}is chosen appropriately for some subsequence of impulsive time instants Imp^∗ : {t^∗_k} ⊂Imp satisfyingt^∗_k1−t^∗_k≤T^∗t^∗_k<∞, for each two consecutivet^∗_k, t^∗_k1∈Imp^∗ 2such a sequence of impulsive controller gains is chosen arbitrarily for the sequence Imp\

Imp^∗.

Proof. Consider the following Lyapunov functional candidateV : R₀×Rⁿ → R₀,17:

Vt, xt: x^TtP xt q

i 1

_t

t−hitx^TτSiτxτdτ, 3.4

where R^n×nP P^T0 andSi∈BPC⁰R₀,R^n×nfulfilsSit0, for allt∈R0, for alli∈ q. One gets by taking time-derivatives in3.4using2.6as follow:

V˙t, xt: 2x^TtP

⎡

⎣^N

j 1

λ_0jtB0jtK_0j tδ0 q

i 0

N j 1

λ_ijtA^∗_ijtxt−h_it

⎤

⎦xt

q

i 1

x^TtSitxt−

1−h˙_it

x^Tt−h_itSit−h_itx^Tt−h_it 3.5

x^TtQtxt −x^TtQdt Q_odtxt, 3.6

where

xt

x^Ttx^Tt−h1t· · ·x^T

t−hqt^T , Qt: Block matrix

Q_ijt:i, j∈q1 ,

3.7

with

Q11t

⎛

⎝^N

j 1

λ0jt

A^∗_0jt B0jtK_0jtδ0⎞

⎠

T

P

P

⎛

⎝^N

j 1

λ_0jt

A^∗_0jt B_0jtK_0jtδ0⎞

⎠ q

i 1

S_it

Q_1,i1t Q^T_i1,1t: ^N

j 1

λijtP A^∗_ijt, ∀i∈q,

Qiit: −

1−h˙it

Sit−hit, Qijt 0, ∀i, j/i∈q1\ {1}, Qdt Block diag

−Q11t−Q22t−Q_q1,q1t ,

Q_odt −Qt Q_dt

⎡

⎢⎢

⎢⎢

⎣

0 −Q12t · · · −Q_1,q1t

−Q^T₁₂t 0 −Q23t· · · −Q2,q1t

... ... . .. ...

−Q^T_q1,1t −Q_q1,2^T tt −Q^T_q1,qtt· · · 0

⎤

⎥⎥

⎥⎥

⎦,

3.8

so that the following cases arise:

1ift /∈D, then

Q₁₁t

⎛

⎝^N

j 1

λ_0jtA^∗T_0jt

⎞

⎠PP

⎛

⎝^N

j 1

λ_0jtA^∗_0jt

⎞

⎠ q

i 1

S_it,

Q_1,i1t Q^T_i1,1t: ^N

j 1

λ_ijtP A^∗_ijt, ∀i∈q,

Qiit: −

1−h˙it

Sit−hit, Qijt 0, ∀i, j/i∈q1\ {1},

3.9

2if t ∈ D\Imp, then 3.8still holds to the left of anyt ∈ R0. Similar equations as3.9stand fort 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,

3ift ∈ Imp, then the left-hand-side limit ofQtis defined with block matrices as follow:

Q11t

⎛

⎝^N

j 1

λ0jt

A^∗_0jt B0jtK_0jtδ0⎞

⎠

T

P

P

⎛

⎝^N

j 1

λ_0jt

A^∗_0jt B_0jtK_0j tδ0⎞

⎠ q

i 1

S_it,

Q_1,i1t Q^T_i1,1t: ^N

j 1

λ_ijtP A^∗_ijt, ∀i∈q,

Qiit: −

1−h˙it

Sit−hit, Qijt 0, ∀i, j/i∈q1\ {1},

3.10

and the right-hand-side limits are defined with block matrices as follow:

Q₁₁t

⎛

⎝^N

j 1

λ_0jt

A^∗_0jt B_0jtK_0jt⎞

⎠

T

P

P

⎛

⎝^N

j 1

λ_0jt

⎛

⎝A^∗_0jt ^N

j 1

B_0jtK_0jt

⎞

⎠

⎞

⎠ q

i 1

S_it

Q_1,i1t Q^T_i1,1t: ^N

j 1

λijtP A^∗_ijt, ∀i∈q,

Qiit: −

1−h˙it

Sit−hit, Qijt 0, ∀i, j/i∈q1\ {1},

3.11

since from Assumption 2.1, the scalar functions λ_ijt and the matrix functions B_0jt, for alli ∈ q∪ {0},for allj ∈ N cannot be discontinuous at the sequence Imp. As in3.11, a matrix function entry att is more distinct than its left-hand-side limit attonly if it has a discontinuity at the time instantt. Thus,

V˙t, xt−V˙t, xt x^TtQt−Qtxt, Vt, xt−Vt, xt 0, ∀t /∈Imp,

V˙t, xt−V˙t, xt 0, ∀t /∈DsinceQt Qt. 3.12

Furthermore, in view of3.5,

V˙t, x_t−V˙t, xt x^TtQtxt −x^TtQtxt, ∀t∈Imp. 3.13

If, in addition,t /∈Dp, that is, ift∈Imp∩Dp, and sinceQt Qt,3.13becomes

V˙t, xt−V˙t, xt x^TtQtxt −x^TtQtxt, 3.14