# About Total Stability of a Class of Nonlinear Dynamic Systems Eventually Subject to Discrete Internal Delays

Authors: Manuel De la Sen
Journal: International Journal of Differential Equations
 (2021)

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## Abstract

This paper studies and investigates total stability results of a class of dynamic systems within a prescribed closed ball of the state space around the origin. The class of systems under study includes unstructured nonlinearities subject to multiple higher-order Lipschitz-type conditions which influence the dynamics and which can be eventually interpreted as unstructured perturbations. The results are also extended to the case of presence of multiple internal (i.e., in the state) point discrete delays. Some stability extensions are also discussed for the case when the systems are subject to forcing efforts by using links between the controllability and stabilizability concepts from control theory and the existence of stabilizing linear controls. The results are based on the ad hoc use of Gronwall’s inequality.


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## Body

## 1. Introduction

The study of the stability properties is of major interest in dynamic systems since it allows to investigate the existence and nature of the equilibrium points, their respective domains of attraction, and the eventual existence of oscillations dependent on the initial conditions or asymptotic oscillations independent on the initial conditions (limit cycles). Typical techniques of investigation of stability are those based on local and global Lyapunov stability methods and Bendixson-type theorems for investigation of the oscillatory and asymptotic oscillatory behaviors (see, for instance, [1–16] and references therein). There are also frequency-type methods of stability analysis, such as, Routh–Hurwitz criterion, Jury criterion, amplitude and phase Bode plots, Nyquist criterion, root locus, Tsypkin locus, and describing function analysis, which are very popular in the time-invariant case and also in systems including some specific types of separate nonlinearities in actuators or sensors such as, for instance, ban-bang (i.e., relay) or saturating devices (see, for instance, [4–6]). Also, the well-known Gronwall inequality in its various versions can be adapted to stability studies in the time-domain (see, for instance, [1, 2]).On the other hand, it can be noticed that delays are present in a natural way in the real life and in many real processes such as, for instance, in some biological equations as, for instance, in epidemic models, in some generalizations of the Beverton–Holt equation, in prey-predator problems, or in the sunflower equation and in population growing and diffusion problems. They are also present in many engineering diffusion problems, information package transmission, queuing storages, teleoperation systems, chemical processes, and robotic systems. The study of stability of time-delay systems has received much attention in the last decades. For instance, a nonautonomous functional differential equation of the third order is considered in [7] with multiple deviating arguments. Asymptotic stability results and uniform boundedness of the solution results are obtained based on the ad hoc use of the Lyapunov–Krasovskii functional approach in the above paper. Other stability results for third-order differential equations are proved in [9, 10]. Also, some sufficiency-type conditions of global stability and asymptotic stability results for time-varying systems with state-dependent parameterizations are obtained in [8]. On the other hand, other studies of stability and robustness of dynamic systems have been performed in [11–14] and related studies for the presence of oscillatory behaviors have been performed in [15, 16] and references therein. The study of stabilization of time-delay dynamic systems and/or integrodifferential equations has been also addressed to more general classes of equations. For instance, the exponential decay of a first-order linear Volterra equation is investigated in [17] while the exponential stability of controlled nonlinear systems subject to time-varying delays has been discussed in [18]. It can be noticed that Volterra-type equations can be interpreted as being subject to a distributed delay on their previous evolution history. On the other hand, the solvability and optimal controls of a noninstantaneous impulsive neutral stochastic integer-differential equation forced by Brownian motion are addressed in [19]. It can be noticed that neutral differential equations are a class of retarded equations where the highest-order time-derivative is affected by delays contrarily to the case of nonneutral ones.This paper gives some results on total stability of a class of nonlinear time-varying systems subject to combined multiple higher-order-type Lipschitz conditions. The stability results are of global type within some prescribed closed ball around the origin and are subject to conditions of exponential stability of the nonlinearity-free system counterpart and to maximum bounds of the various Lipschitz constants. Some applications are given for the control case. The above results are included in Section2. On the other hand, Section 3 extends the results of Section 2 for the case of presence of multiple punctual constant delays which appear jointly in the linear dynamics of the system and in the nonlinear one which is also subject to combined multiple higher-order-type Lipschitz conditions.


## 2. Total Stability of Nonlinear Differential Systems with Forcing Terms Subject to Multiple Combined Higher-Order Lipschitz Conditions

The following result is two-fold since it relies on the uniqueness of the solution exhibiting, furthermore, global stability in a bounded region around the origin for a class of nonlinear time-varying differential systems under a set of Lipschitz-extended power-type perturbations of the state. A combined analysis supported by the Banach Contraction Principle and Bellman–Gronwall Lemma [1–3] is used to get the next main result.Theorem 1.
Consider the following differential system:(1)x˙t=Ft,x,subject to initial conditions xt0=x0 where F:t0,∞×Rn⟶Rn is given by(2)Ft,x=Atxt+∑i=1pfit,x+∑i=1qgit,x,where A:t0,∞⟶Rn×n has bounded piecewise continuous entries and fi,gj:t0,∞×Rn⟶Rn; i=1,2,…,p; j=1,2,…,q are locally integrable functions of t on t0,∞ in the closed ball Br=x∈Rn:x≤r, where x denotes some vector norm, and suppose that the following two assumptions hold:(A1)fit,0=0 and fit,x1−fit,x2≤βix1−x2i; i=1,2,…,p for all x1,x2∈Br(A2)git,x1≤γiri+1 and git,x1−git,x2≤γix1−x2i; i=1,2,…,q for all x1,x2∈Br(A3) The auxiliary unforced differential system(3)z˙t=Atzt,zt0=z0,is exponentially stable ont0,∞ of stability abscissa −α<0; that is, there exist real constants α>0 and K≥1 such that the fundamental matrix Ψ:t0,∞⟶Rn, which is the mild solution of ψ˙t,t0=Atψt,t0 for t≥t0 with ψt0,t0=In (In being the n-th identity matrix) satisfies Ψt1,t2≤Ke−αt2−t1 for any t1 and t2 such that t2≥t1≥t0.(A4) The constantsβi and γj, i=1,2,…,p and j=1,2,…,q, satisfy(4)∑i=1pβiri−1+∑i=1qγiri−1≤αK,for some givenr>0.Then, the following properties hold:(i)
supt0≤t≤∞xt≤r if x0≤r/K so that differential systems (1) and (2) are globally Lyapunov stable in the closed ball of radius r centred at the origin.(5)iilimsupt⟶∞xt≤Krα∑i=1pβiri−1+∑i=1qγiri−1≤r,for any given finitex0.(iii)
Assume that A4 is restricted to a strict inequality, that is,(6)A5∑i=1pβiri−1+∑i=1qγiri−1<αK,for some givenr>0. Then, there is a unique globally Lyapunov stable solution of (1) and (2) in the closed ball of radius r centred at the origin for t≥t0 under Assumptions A1, A2, A3, and A5 satisfying(7)xt≤Ke−α−K∑i=1qβiri−1t−t0x0+K∑i=1qγiriα−K∑i=1qβiri−11−e−α−K∑i=1qβiri−1t−t0≤r,∀t≥t0.Proof.
The solution of (1) and (2) for t≥t0 is given by(8)xt=Ψt,t0x0+∑i=1p∫t0tΨt,τfiτ,xτdτ+∑i=1q∫t0tΨt,τgiτ,xτdτ,and, by using Assumptions A1 to A3, one gets(9)xt≤Ke−αt−t0x0+K∑i=1pβiri+∑i=1qγiri∫t0te−αt−τdτ,=Ke−αt−t0x0+K∑i=1pβiri−1+∑i=1qγiri−1rα1−e−αt−t0,=Ke−αt−t0x0−rα∑i=1pβiri−1+∑i=1qγiri−1+Krα∑i=1pβiri−1+∑i=1qγiri−1≤r,t≥t0,provided that A4 holds subject to(10)Kx0≤Krα−Krα∑i=1pβiri−1+∑i=1qγiri−1+r=r,which is got from (9) for t=t0. Then, Property (i) is proved. On the other hand, since α>0, one gets from A4 and (9) that (5) holds and Property (ii) is proved. Now, let Cnt0,∞ be the Banach space of continuous, bounded n-vector real functions y:t0,∞⟶Rn equipped with the norm yt0,∞=supt0≤t<∞xt. Let T an operator on Cnt0,∞ represent solution (1) which is defined pointwise via(11)Txt=Ψt,t0x0+∑i=1p∫t0tΨt,τfiτ,xτdτ+∑i=1q∫t0tΨt,τgiτ,xτdτ,for t≥t0. Then, if x0≤r/K, one has from (9) that(12)Txt≤Ke−αt−t0x0+Krα∑i=1pβiri−1+∑i=1qγiri−1≤r,and one has then from (8) for any solutions x1,x2∈Br on t0,∞ under the same initial condition x0 that(13)x1t−x2t=∑i=1p∫t0tΨt,τfiτ,x1τ−fiτ,x2τdτ+∑i=1q∫t0tΨt,τgiτ,x1τ−giτ,x2τdτ,which leads to(14)Tx1t−Tx2tt0,∞≤Krα∑i=1pβiri−1+∑i=1qγiri−1x1t−x2tt0,∞,so that T:Cnt0,∞⟶Cnt0,∞ is a contraction on the closed subset Br=y∈Cnt0,∞:y≤r of Cnt0,∞ if A5 holds. This implies from the contraction mapping theorem that solution (8) is unique. From (11) and Assumptions A–A3, one gets(15)xt≤Ke−αt−t0x0+∑i=1pβiri−1∫t0te−αt−τxτdτ+∑i=1qγiri∫t0te−αt−τdτ,t≥t0.
Also, one has from (15) that eαtxt≤ξt, t≥t0 from Bellman–Gronwall Lemma, where ξ:t0,∞⟶R is the solution of(16)ξ˙t=K∑i=1pβiri−1ξt+∑i=1qγirieαt,ξt0=Keαt0x0,for t≥t0, which is(17)ξt=eK∑i=1qβiri−1t−t0ξt0+K∑i=1qγiriα−K∑i=1qβiri−11−e−α−K∑i=1qβiri−1t−t0eαt,t≥t0,which implies (7) and proves Proposition 3.
Some direct corollaries are as follows.Corollary 1.
Consider differential system (1) with(18)Ft,x=Btxt+∑i=1pfit,x+∑i=1qgit,x,with Bt=At+A˜t. Then, Theorem 1 holds under the same Assumptions A1–A5 with the replacement(19)β1⟶β1′=β1+supt≥t0Bt−At=β1+supt≥t0supz=1Bt−Atzz,where the vector-induced matrix norm X=supz=1Xz/z is used.Corollary 2.
Consider the following controlled differential system:(20)x˙t=Ft,x+ut,x,subject to initial conditions xt0=x0, where u:t0,∞⟶Rm is a linear feedback control defined by ut,x=Ktxt, where K:t0,∞×Rn⟶Rm is a bounded piecewise continuous matrix control gain and Ft,x is defined by (18) such that z˙t=Bt+Ktzt is exponentially stable with stability abscissa −α<0. Then, Theorem 1 holds with At=Bt+Kt.Corollary 3.
Assume that in Corollary2, u:t0,∞×Rn⟶L∞ent0,∞, where L∞ent0,∞ is the set of essentially bounded real n-vector functions in t on t0,∞ for all x∈Rn and let us define u¯=supx∈Rn∫t0∞uτ,xdτ. Then, Theorem 1 still holds by replacing (4) by(21)∑i=1pβiri−1+∑i=1qγiri−1+u¯r≤αK,and (5) by its corresponding strict inequality.Proof.
It follows directly by replacing (8) and (9) by(22)xt=Ψt,t0x0+∫t0tΨt,τ∑i=1pfiτ,xτ+∑i=1q∫t0tgiτ,xτ+uτ,xτdτ,t≥t0,xt≤Ke−αt−t0x0−rα∑i=1pβiri−1+∑i=1qγiri−1+Krα∑i=1pβiri−1+∑i=1qγiri−1+u¯r≤r,t≥t0,and following similar arguments to those used to prove Theorem 1.
The following uniform-type result within a bounded ball of the solution of (1) and (2) follows from Theorem 1:Corollary 4.
Assume that A4 is modified by replacing (4) with(23)A4′max1≤i≤pβi+max1≤i≤qγi≤αKMR,where(24)MR=pMRp+qMRq,(25)MRp=max1,R,…,Rp−1,MRq=max1,R,…,Rq−1.
Then, the following properties hold under Assumptions A1–A3 and A4:(i)
Ifx0≤r/K then supt0≤t≤∞xt≤r for any given r∈BR=z∈R+:z≤R so that differential system (1) and (2) is globally Lyapunov stable in BR.(ii)
Assume that A5 is modified by replacing (5) with(26)A5′max1≤i≤pβi+max1≤i≤qγi<αKMR.
Then, there is a unique solution of (1) and (2) for t≥t0 under Assumptions A1, A2, A3, and A5 satisfying(27)xt≤Ke−α−KpMRpmax1≤i≤pβit−t0x0+KqMRqmax1≤i≤qγiα−KpMRpmax1≤i≤pβi1−e−α−KpMRpt−t0≤r,t≥t0.Outline of Proof. The proof follows directly from Theorem 1 by noting that (23) implies that(28)α≥KMRmax1≤i≤pβi+max1≤i≤qγi≥KmaxpMRpmax1≤i≤pβi,qMRqmax1≤i≤qγiand (24) implies that(29)α>KMRmax1≤i≤pβi+max1≤i≤qγi≥KmaxpMRpmax1≤i≤pβi,qMRqmax1≤i≤qγi,which makes (27) to be well-posed.


## 3. Total Stability of Nonlinear Differential Systems with Linear Constant Punctual Delays and Forcing Terms Subject to Combined Higher-Order Lipschitz Conditions

Now, consider the following unforced linear time-invariant differential system subject to a finite set ofϑ, in general, incommensurate constant point delays 0<h1<h2<⋯<hϑ=h. These delays are commensurate in the particular case that hi=ih, i=1,2,…,ϑ (see, for instance, [3, 7] and some references therein). In the sequel, fix the initial time instant to t0=0 for simplicity of the presentation of the subsequent results:(30)x˙t=A0xt+∑i=1ϑAixt−hi,with Ai∈Rn×n, i=0,1,…,ϑ which is subject to any bounded piecewise continuous function of initial conditions φ:t−h,0⟶Rn with φ0=x0 whose unique solution becomes to be(31)xt=eA0tx0+∑i=1ϑ∫0te−A0τAixτ−hidτ=eA0tx0+∑i=1ϑ∫0hie−A0τAiφτ−hidτ+∑i=1ϑ∫hite−A0τAixτ−hidτ=eA0tx0+∑i=1ϑ∫0hie−A0τAiφτ−hidτ+e−A0hi∑i=1ϑ∫0t−hie−A0τAixτdτ,where eA0t≤Ke−αt for all t≥0 and some real constants α and K≥1 so that(32)xt≤Ke−αtx0+1α∑i=1ϑAieαhi−1sup−h≤τ≤0φτ+Ke−αt∑i=1ϑeαhiAi∫0t−hieατxτdτ≤Ke−αt1+1α∑i=1ϑAieαhi−1φ¯0+Kα∑i=1ϑ1−e−αt−hiAisup0≤τ≤t−hxτ,where φ¯0=sup−h≤τ≤0φτ.Above differential system (30) can be rewritten equivalently as follows after introducing the zero delay h0=0 and the matrix A=∑i=0ϑAi:(33)x˙t=Axt+∑i=1ϑAixt−hi−xt.Note from (30) that the auxiliary differential system x˙t=A0xt describes the time-delay system as hi⟶∞i=1,2,…,ϑ, which can be interpreted alternatively as the delay-free one when the contributions of delayed dynamics are zeroed described by Ai=0; i=1,2,…,ϑ. Also, note from (33) that the auxiliary differential system x˙t=Axt describes the time-delay system as hi=0i=1,2,…,ϑ. It follows that the solution may be equivalently defined by(34)xt=eAtx0+∑i=1ϑ∫0hie−AτAiφτ−hi−φτdτ+∑i=1ϑ∫hite−AτAixτ−hi−xτdτ,where eAt≤K¯e−α¯t for all t≥0 and some real constants α¯ and K¯≥1 so that(35)xt≤Ke−α¯tx0+2α¯∑i=1ϑAieα¯hi−1sup−h≤τ≤0φτ+2K¯e−αt∑i=1ϑeαhiAi∫0t−hixτeατdτ≤Ke−α¯t1+2α¯∑i=1ϑAieα¯hi−1φ¯0+2K¯α¯∑i=1ϑ1−e−αt−hiAisup0≤τ≤t−hxτ,where φ¯0=sup−h≤τ≤0φτ.The following auxiliary result holds.Proposition 1.
The following properties hold:(i)
LetA0 be a stability matrix such that eA0t≤Ke−αt for all t≥0 and some real constants α>0 and K≥1 (with K being norm-dependent) and fix φ¯0=λr. If(36)0≤λ≤λM=α+K∑i=1ϑAieαhi−1α+∑i=1ϑAieαhi−1K,thenxt≤r; ∀t≥0, and thus, the system is globally Lyapunov stable.(ii)
LetA=∑i=0ϑAi be a stability matrix such that eAt≤K¯e−α¯t for all t≥0 and some real constants α¯>0 and K¯≥1 (with K¯ being norm-dependent) and fix φ¯0=λr. If(37)0≤λ¯≤λ¯M=α¯+2K¯∑i=1ϑAieα¯hi−1α¯+2∑i=1ϑAieα¯hi−1K¯,thenxt≤r; ∀t≥0, and this the system is globally Lyapunov stable.Proof.
Note thatλ≤1 so that sup−h≤τ≤0xτ=sup−h≤τ≤0φτ≤r and there exist some t0>0 such that sup−h≤τ≤t0xτ≤r by continuity of the solution. Now, proceed by complete induction by using (32). Assume that there is a first time instant t1>0∈t−i+1h,t−ih for some i∈Z∈0,maxintz:z≤t1/h−1 such that xt1>r. However, this gives a contradiction since then sup−h≤τ<t1xτ−h>r and then there is some t2∈0,t1−h which is the first time instant such that xt2>r. As a result, φ¯0≤α+K∑i=1ϑAieαhi−1/αK+K∑i=1ϑAieαhi−1r implies that supt≥−hxt≤r which proves Property (i). The proof of Property (ii) is similar to that of Property (i) by using (35) instead of (32).Remark 1.
Note thatλM and λ¯M are strictly decreasing functions of any delay on the positive real axis. This is seen easily by considering only one delay h=h1 and define vh=Aieαhi−1 leading to λM=λMv=αK−1+v/α+v. Assume that h′>h, so that vh′>vh then λMv′<λMv since, equivalently, 0<K−1+1αv′h−vh. A similar conclusion follows for λ¯M. This concern is that we can expect from intuition; that is, the increase in the delay sizes translates into a more strict sufficiently type constraint on the “smallness” constraint on the initial conditions to guarantee that the solution is kept under a certain prescribed closed ball through time.
Two sufficiently type conditions of global asymptotic Lyapunov stability at exponential rate of (30) are obtained in the next result.Proposition 2.
The following properties hold:(i)
Assume thatA0 is a stability matrix such that eA0t≤Ke−αt for all t≥0 and some real constants α>0 and K≥1. Then, differential system (30) is exponentially stable (i.e., for any given bounded piecewise continuous function of initial conditions) if α>K∑i=1ϑeαhiAi which is guaranteed for any delays 0<hi<hi+1; i=1,2,…,ϑ−1 satisfying hϑ<1/αlnα/Kϑ∑i=1ϑAi.(ii)
Assume thatA=∑i=0ϑAi is a stability matrix such that eAt≤K¯e−α¯t for all t≥0 and some real constants α¯>0 and K≥1 so that the delay-free version of (30) is exponentially stable. Then, differential system (30) is exponentially stable if α¯>2K¯∑i=1ϑeα¯hiAi which is guaranteed for any delays 0<hi<hi+1; i=1,2,…,ϑ−1 satisfying hϑ<1/α¯lnα¯/2K¯ϑ∑i=1ϑAi.Proof.
From the first inequality in (32), it follows that(38)vt=xteαt≤Kx0+1α∑i=1ϑAieαhi−1sup−h≤τ≤0φτ+K∑i=1ϑeαhiAi∫0tvτdτ,∀t≥0,so that, from Gronwall Lemma [1–3],(39)vt≤Kx0+1α∑i=1ϑAieαhi−1sup−h≤τ≤0φτeK∑i=1ϑeαhiAit,∀t≥0,and then(40)xt≤K1+1α∑i=1ϑAieαhi−1sup−h≤τ≤0φτe−α−K∑i=1ϑeαhiAit,∀t≥0.
On the other hand, one obtains from the first inequality in (35) and Gronwall Lemma that(41)xt≤K¯1+2α¯∑i=1ϑAieα¯hi−1sup−h≤τ≤0φτe−α¯−2K¯∑i=1ϑeα¯hiAit,∀t≥0.
Property (i) follows from (40) if A0 is stability matrix while Property (ii) follows from (41) if A is a stability matrix.
The extensions of Propositions1 and 2 to the time-varying linear case are direct as follows in the subsequent result:Proposition 3.
The following properties hold if differential system (30) is time-varying:(i)
Assume that(A6)
Ai:0,∞⟶Rn×n, i=0.1,…,ϑ have piecewise continuous and bounded entries.(A7)
the unforced systemz˙t=A0tzt is exponentially stable, that is, its fundamental matrix of Ψ0t,0, such that Ψ˙0t,0=AtΨ0t,0 with Ψ00,0=In, satisfies Ψ0t,0≤Ke−αt for all t≥0 with real constants K≥1 and α>0.Thus, ifφ¯0=sup−h≤τ≤0φτ≤α+K∑i=1ϑsupt≥0Aiteαhi−1/α+∑i=1ϑsupt≥0Aiteαhi−1Kr then xt≤r; ∀t>0. Also, if α>K∑i=1ϑeαhiAi, then differential system (30) is globally asymptotically stable, with the solution trajectory being constrained to the ball xt≤r; ∀t>0, and xt⟶0 at exponential rate −α−K∑i=1ϑeαhiAi as t⟶∞.(ii)
Assume that Assumption A6 holds and, furthermore,(A8)
the unforced systemz˙t=Atzt is exponentially stable, that is, its fundamental matrix of Ψt,0, such that Ψ˙t,0=AtΨt,0 with Ψ0,0=In, satisfies Ψt,0≤K¯e−α¯t for all t≥0 with real constants K¯≥1 and α¯>0.
Thus, ifφ¯0=sup−h≤τ≤0φτ≤α¯+2K¯∑i=1ϑsupt≥0Aiteα¯hi−1/α¯+2∑i=1ϑsupt≥0Aiteα¯hi−1K¯r, then xt≤r; ∀t>0. Also, if α>K¯∑i=1ϑeα¯hiAi, then differential system (30) is exponentially stable, with the solution trajectory being constrained to the ball xt≤r; ∀t>0, so that xt⟶0 at exponential rate −α¯−K¯∑i=1ϑeαhiAi as t⟶∞.
The following main total stability result gives sufficiency-type global stability conditions of the delayed system in the presence of nonlinearities which generalize those considered in Theorem1.Theorem 2.
Consider the following differential system:(42)x˙t=Ft,xt,xt−h1,xt−h2,…,xt−hϑ,with Ai∈Rn×ni=0,1,…,ϑ which is subject to any bounded piecewise continuous function of initial conditions φ:t−h,0⟶Rn with φ0=x0 where F:0,∞×Rn⟶Rn is given by(43)Ft,x=A0txt+∑i=1ϑAitxt−hi+∑i=1pfit,xt,xt−h1,…,xt−hϑ+∑i=1qgit,xt,xt−h1,…,xt−hϑ,where Ai:t0,∞⟶Rn×ni=1,2,…,ϑ have piecewise continuous entries and fi,gj:t0,∞×Rϑ+1n⟶Rn; i=1,2,…,p; j=1,2,…,q are locally integrable functions of t on 0,∞ in the closed ball Br=x∈Rn:x≤r. Suppose that Assumption A6 and that furthermore the following assumptions hold for some given r>0:(44)A9fit,0,…,0=0,fit,x1,x2,…,xϑ+1−fit,y1,y2,…,yϑ+1≤∑j=0ϑβijkxj−yki,i=1,2,…,p,for allxj,yk∈Br,A10git,x1,x2,…,xϑ+1≤∑j=0ϑ∑k=0ϑγijkri+1,git,x1,x2,…,xϑ+1−git,y1,y2,…,yϑ+1≤∑j=0ϑ∑k=0ϑγijkxj−yki,i=1,2,…,q,for allxj,yk∈Br,(45)A11α>maxα1,α2,which is guaranteed if α>α1+α2, where(46)α1>K∑i=1ϑsupt≥0Aiteαhi−1,α2>K∑i=1p∑j=0ϑ∑k=0ϑβijkri−1+∑i=1q∑j=0ϑ∑k=0ϑγijkri−1,(47)φ¯0=sup−h≤τ≤0φτ≤α−K∑i=1p∑j=0ϑ∑k=0ϑβijkri−1+∑i=1q∑j=0ϑ∑k=0ϑγijkri−1α+∑i=1ϑsupt≥0Aiteαhi−1Kr.
Then, the following properties hold:(i)
There is a unique globally Lyapunov stable solution of (42) and (43) satisfying(48)xt≤K1+1α∑i=1ϑAieαhi−1sup−h≤τ≤0φτe−α−K∑i=1ϑeαhisupτ≥0Aiτ+∑i=1p∑j=0ϑ∑k=0ϑβijkri−1t+K∑i=1q∑j=0ϑ∑k=0ϑγijkriα−K∑i=1p∑j=0ϑ∑k=0ϑβijkri−11−e−α−K∑i=1ϑeαhisupτ≥0Aiτ+∑i=1p∑j=0ϑ∑k=0ϑβijkri−1t≤r,∀t≥0.(ii)
Assume thatα, α1, and α2 and φ¯0 in (45)–(47) are replaced with similar equations under the replacements α1⟶α¯1=2K¯∑i=1ϑsupt≥0Aiteαhi−1, α2⟶α¯2 (obtained by replacing K⟶K¯), Assumption A1 being modified with the replacement α⟶α¯>maxα¯1,α¯2 and (47) being modified with the replacements α⟶α¯, K⟶K¯, and supt≥0Ait⟶2supt≥0Ait in the denominator of (47) and to define. Then, there is a unique globally Lyapunov stable solution of (42) and (43) satisfying(49)xt≤K¯1+2α¯∑i=1ϑAieα¯hi−1sup−h≤τ≤0φτe−α¯−2K¯∑i=1ϑeα¯hisupτ≥0Aiτ+∑i=1p∑j=0ϑ∑k=0ϑβijkri−1t+K¯∑i=1q∑j=0ϑ∑k=0ϑγijkriα¯−K¯∑i=1p∑j=0ϑ∑k=0ϑβijkri−11−e−α¯−2K¯∑i=1ϑeα¯hisupτ≥0Aiτ+∑i=1p∑j=0ϑ∑k=0ϑβijkri−1t≤r,∀t≥0.Proof.
It follows directly by combining Propositions2 and 3 for differential system (42) and (43) with a similar technique of proof of Theorem 1.Remarks 2.
(1)
It is convenient now to summarize at a first glance which are the basic ideas behind the above results concerning the studied time-delay system.(2)
Note that Proposition2 gives two results of global asymptotic Lyapunov stability in the large (i.e., for any bounced piecewise continuous function of initial conditions). In this way, the solution trajectory converges asymptotically to the zero equilibrium point at exponential rate. The result is independent of the delay size type if the delays are small enough provided that either the auxiliary system with delay-free dynamics of that with zero delay is exponentially stable.(3)
Note also that Proposition1 gives two results of global Lyapunov stability independent of the delay sizes for a certain range of values of the delays with the trajectory solution being constrained within a certain closed ball around the origin for t>0 sufficiently small initial conditions provided that one of the above auxiliary unforced systems is exponentially stable.Proposition3 states results on global stability and global asymptotic stability with the trajectory being constrained to as ball around the origin if one of the two mentioned auxiliary systems is exponentially stable.(4)
Theorem2 gives two alternative results of solution uniqueness and global stability within a ball around the origin, based on either the stability of the matrix A0, corresponding to the system without delayed dynamics, or on the stability of the delay-free system of matrix dynamics A, in the case that systems (42) and (43) become forced under forcing functions with eventual delays which satisfy general nonlinear constraints based on extending those invoked in Theorem 1 for the delay-free system. Both results are supported by the constraints that the stability of the matrices A0 and A guarantee, respectively, the exponential stability of their respective unforced systems independent of the delay sizes within a range of values with a maximum bound.
In particular, Assumption A11 has a direct interpretation as follows: firstly,α>α1 guarantees that the exponential stability of the unforced differential system z˙t=A0zt, which describes the case of absence of delayed dynamics, under any finite bounded conditions z0=z0, guarantees that of system (30) for any bounded piecewise continuous function of initial conditions involving point delays for a certain maximum allowable size of the maximum delay. Secondly, if furthermore, in (46), α2 is small enough to satisfy α2<α−α1, for given forcing functions f⋅, g⋅ subject to Assumptions A9 and A10, i.e., such that Assumption A11, is also satisfied; then, the forcing system remains globally Lyapunov stable within the closed ball of radius r centred at the origin and satisfies (48). The alternative constraints invoked to get (49) are interpreted in a similar way based on the assumption of exponential stability under any finite initial conditions of the unforced delay-free differential system z˙t=Azt, which describes the zero-delay case, that is, the case when hi=0 for i=1,2,…,ϑ.
For the following discussion, recall thatF,G is a controllable pair with F∈Rn×n and G∈Rm×n if and only if rankG,FG,…,Fn−1G=n, equivalently, if and only if ranksIn−F,G=n for any s∈C which is not an eigenvalue of F (the Popov–Belevitch–Hautus controllability test). If F,G is controllable, then the spectrum of F+GK can be prefixed arbitrarily through the choice of K∈Rm×n [4–6].
On the other hand,F,G is a stabilizable pair if and only if ranksIn−F,G=n for any s∈C with Res≥0 which is not an eigenvalue of F (the Popov–Belevitch–Hautus stabilizability test). If F,G is stabilizable, then the spectrum of F+GK can be stabilized (but it cannot be either arbitrarily assigned or prefixed subject to a prefixed stability abscissa) through the choice of K∈Rm×n so that, for some K∈Rm×n, all the eigenvalues of F+GK can be allocated in the open complex left-hand-side plane Res<0, [4, 5].
It turns out that ifF,G is controllable, then it is stabilizable, but the converse is not true.
The subsequent results rely on the stabilization of (42) and (43) though a linear feedback control of the form ut=K0xt.Corollary 5.
Assume thatB0∈Rn×m, with m≤n, are constant matrices and that the pair A0,B0 is controllable and let differential system (42) be defined by(50)Ft,x=A0xt+B0K0xt+∑i=1ϑAitxt−hi+∑i=1pfit,xt,xt−h1,…,xt−hϑ+∑i=1qgit,xt,xt−h1,…,xt−hϑ,under the assumptions on the initial conditions and matrices of dynamics of Theorem 2 and Assumptions A9–A10 for any set of ϑ point delays 0<h1<h2<⋯<hϑ≤h<+∞ with any associated matrices of dynamics supτ≥0max1≤i≤ϑAiτ≤a<+∞ and any nonnegative real constants βijk≤β<+∞, γℓjk≤γ<+∞; i=1,2,…,p; ℓ=1,2,…,q; j,k=1,2,…,ϑ. Then, for any given quadruple h,a,β,γ of nonnegative real constants, there exists a matrix K0∈Rm×n such that differential systems (42) and (43) are globally Lyapunov stable with its unforced s version being exponentially stable.Proof.
SinceA0,B0 is controllable, the spectrum (and then the stability abscissa) of any given matrix Λ0∈Rn×n can be fixed by some existing K0∈Rm×n which satisfies the matrix equality A0+B0K0=Λ0. So, for any given triple a,β,γ, it is possible to fix a stability abscissa −αΛ0<0 of some Λ0 which satisfies Assumption A11, and then Theorem 2 (i) holds.
Under stabilizability conditions, rather than controllability conditions, of the pairA0,B0, a weaker result than Corollary 5 is now given since the stability abscissa of A0+B0K0 cannot be arbitrarily prefixed.Corollary 6.
Assume the assumptions of Corollary5 except for that the pair A0,B0 is stabilizable rather than controllable. Then, a quintuple h,a,β,γ,K0 can be fixed such that differential systems (42) and (43) are globally Lyapunov stable with its unforced s version being exponentially stable.
The subsequent result relies on the stabilization of a linear time-delay system with a unique delayh1 though a linear feedback control of the form ut=K0xt+K1xt−h. It is found that the exponential stability is achievable for large delay ranges if A0,B and A1,B are, respectively, stabilizable and controllable.Corollary 7.
Consider the following delayed unforced system:(51)x˙t=Axt+A1xt−h1+But,with a control law ut=K0xt+K1xt−h and assume that B∈Rn×m, with m≤n and some matrix feedback control gains K0,K1∈Rm×n. Assume that the pairs A0,B and A1,B are, respectively, stabilizable and controllable. Then, for any existing stabilizing matrix gain K0 of the pair A0,B, there exists a matrix gain K1 such that the above system is exponentially stable for h1∈0,h¯ and some h¯=h¯K1>0 for each stabilizing control gain K0 of A0,B and which depends on the control gain K1. The stabilizing delay maximum bound h¯ can be made arbitrarily large depending on the choice of such a control gain K1.Proof.
SinceA0,B is stabilizable, there is K0 such that A0+B0K0 is a stability matrix of some nonprefixed abscissa −α0<0. Now, from Proposition 2, the above system is exponentially stable if h1<1/α0lnα0/KA1+B1K1 with K≥1 being norm-dependent. However, since A1,B is controllable, the whole spectrum of A1+B1K is freely assignable by the choice of K1; then, the spectral radius ρA1+B1K1 of A1+B1K1, which is arbitrarily close to some matrix norm, can be chosen as small as possible so that the exponential stability holds for arbitrarily large delay according to the following formula:(52)0≤h1<h¯=1α0lnα0KρA1+B1K1+ε=1α0lnα0KA1+B1K1.Corollary 8.
Assume thatA0:0,∞⟶Rn×n is bounded piecewise continuous with A0t=A0∗+A˜0t and that all the remaining assumptions of Corollary 5 hold with the modifications that A0∗,B0 is controllable and supt≥0max1≤i≤ϑAit,A˜0t≤a<+∞ and supt≥0max1≤i≤ϑAit≤a<+∞. Then, for any given quadruple h,a,β,γ of nonnegative real constants, there exists a matrix K0∈Rm×n such that differential systems (42) and (43) are globally Lyapunov stable with its unforced version being exponentially stable.Proof.
It is identical to that of Corollary5 by adding a new (zero) delay h0=0 with matrix of dynamics A˜0t and replacing the lower-bound of α1 in (45) with the corresponding summation from i=0 to i=ϑ, i.e., by incorporating the extra norm associated with supt≥0A˜0t.Remark 3.
It turns out that a similar result to Corollary8 can be directly formulated by considering that A0∗,B0 is stabilizable rather than controllable.Remark 4.
Extensions ofCorollaries 5 to 7 and the direct result got as indicated in Remark 3 are direct from Theorem 2 (ii) for the cases of either controllability or stabilizability of the pair A,B for a given B∈Rm×n and A=∑i=0ϑAi∈Rn×n describing the delay-free auxiliary system. If At is time-varying then, the problem can be stated by extending Theorem 2 (ii) with At=A∗+A˜t, A∗,B being controllable/stabilizable and A˜t being a disturbance matrix of zero delay (see Corollary 7).
It can be of interest to give some ideas about how the above results can be applied when switching through time between alternative active configurations of a time-delay system are performed. For this purpose, consider a nonempty set ofΩ distinct configurations S of (42) and (43) such that Sj=Aji,fjk,gjℓ:i∈ϑ¯∪0,k∈p¯,q∈ℓ¯, with n¯=1,2,…,n, which admit switching between them through time given by a switching law σ:0,∞⟶Ω¯ which is a piecewise continuous function. T=tii=0χ, with t0=0, is either a set or an infinite sequence (i.e., χ≤∞) of switching instants between configurations; that is, σti=σti+≠σti− and σt=σti=j∈Ω¯; ∀t∈ti,ti+1, and it is said that j is the active configuration on ti,ti+1. Assume that S=Ss∪Ss¯ such thatthe configurations ofSs satisfy Theorem 2 so that each corresponding unforced version of (42) and (43) is exponentially stable, so that it has a negative stability abscissathe configurations ofSs¯ are Ss¯=Scs∪Sus, where Scs have critically stable unforced versions so that each corresponding unforced version of (42) and (43) has a zero stability abscissa while the configurations of Sus are unstable so that each corresponding unforced version of (42) and (43) has a positive stability abscissaTheorem 3.
Assume thatS=Ss∪Ss¯ with Ss≠∅ and that there is at least one S∗=Si∗∈Ss with a pair K∗,−α∗ with α∗>0 satisfying α∗/K∗>R for some given finite real constant R>0. Then, for any given admissible function of initial conditions satisfying φ¯0≤R, there always exists switching laws σ:0,∞⟶Ω¯ with a finite number of switches and switching time instants t0,t1,…,tk such that σtk=i∗ and xt≤ri∗; ∀t≥tk.Remark 5.
Note that, in particular,t1=ti∗ is such a switching law of only one or two switching time instants, the initial one at t0=0, which can be at a distinct active configuration of Si∗ if S≠Si∗ or directly at Si∗, necessarily if S=Si∗, and the second one at t1=ti∗ which is only feasible if S≠Si∗.



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