# On Some Sufficiency-Type Global Stability Results for Time-Varying Dynamic Systems with State-Dependent Parameterizations

Authors: M. De la Sen
Journal: International Journal of Differential Equations
 (2019)

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## Abstract

This paper formulates sufficiency-type global stability and asymptotic stability results for, in general, nonlinear time-varying dynamic systems with state-trajectory solution-dependent parameterizations. The stability proofs are based on obtaining sufficiency-type conditions which guarantee that either the norms of the solution trajectory or alternative interval-type integrals of the matrix of dynamics of the higher-order than linear terms do not grow faster than their available supremum on the preceding time intervals. Some extensions are also given based on the use of a truncated Taylor series expansion of chosen truncation order with multiargument integral remainder for the dynamics of the differential system.


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## Body

## 1. Introduction

It is of interest to investigate explicit solution forms, if possible, and the stability and asymptotic stability properties of ordinary differential equations whose coefficients eventually depend on the solution trajectory and on its relevant derivatives with respect to time. In particular, differential equations whose coefficients depend on the solution and derivatives up till a certain order have been formally investigated, for instance, in [1, 2] and references therein. On the other hand, the stability properties of time-varying dynamic linear and nonlinear systems have been also investigated, for instance, in [3–12] and some of the references therein. In particular, Lyapunov second method for stability theory has been successfully used to address and discuss the boundedness and stability properties of the solutions of two class of third-order differential equations whose coefficients are time-varying functions which might depend on the solution and their two first-order time-derivatives.The duality which exists between ordinary and functional differential equations of orders higher than one is well known. This duality is also reflected in their alternative equivalent descriptions in terms of sets of systems of first-order differential equations. On the other hand, it is also well known the usefulness of such descriptions to find explicit closed forms of solutions and to perform the analysis of their stability properties. Note that the descriptions of ordinary and functional differential equations of ordern through sets of n first-order differential equations, eventually coupled, is suitable to formalize both the analytic solutions and, in particular, the stability properties of real-world dynamic systems of order n. This procedure might allow, to some extent, to get standard solution trajectory analytic expressions in a closed form and to obtain conditions of stability for the original differential equation.This paper states and proves sufficiency-type global Lyapunov’s stability and Lyapunov’s asymptotic stability results of a differential system of an arbitraryn-th order decomposed as a set of n first-order ordinary differential equations whose parameterization is, in general, time-varying and depends on the solution trajectory. The differential system is, in fact, the description via first-order differential equations of an ordinary, or functional, differential equation of the same order whose coefficients depend on the solution, its relevant time-derivatives, and, eventually, explicitly with time. The technical mathematical proofs are based on getting “ad hoc” mathematical expressions which are obtained analytically for upper-bounds depending on time of the supremum of the solution norm. It is assumed that such a supremum norm evolves with time at a sufficiently slow rate. It is proved that either the global stability or the global asymptotic stability holds under the stability of a matrix function which generates the fundamental matrix of a certain reference unforced system and the sufficiently smallness of the error matrix function between the dynamics of the whole differential system and the above-mentioned stability matrix.Alternative sufficiency-type global asymptotic stability conditions are got under the time integrability of the norm, or a power of the norm, of the error matrix function with sufficiently small values of the time-interval integral. Taking advantage of the fact that the decomposition of a matrix in a sum of matrices is not unique, there is a freedom in the choice of the matrix which is requested to be stable, the so-called nominal matrix of dynamics, but then the error matrix related to that of the system has to have a sufficiently small norm to achieve the stability properties.Some extended results are also given if a truncated Taylor series expansion with an integral remainder around the equilibrium point is developed. In particular, the remainder expression is obtained by respecting the compatibility of the regularity conditions, with respect to the state-trajectory solution and time, of the matrix defining the whole dynamics of the differential system and the truncation order in the series expansion chosen to define the error matrix. The first one of those matrices defines the nominal matrix of dynamics, which exhibits stability properties, while the second one defines the error matrix related to the whole matrix function associated to the differential system at hand.On the other hand, it can be pointed out that there are certain epidemic models where some of the coefficients describing the differential equations are time-varying and depending on the state defined by the subpopulations which take part of the model. In particular, a normalized SIR epidemic model (that is, with susceptible, infectious, and recovered integrated subpopulations) whose recovery rate is time-varying and state dependent is described and analyzed in [13]. In [14], an epidemic model with random screening (that is, the detected infectious are removed into a special class) is proposed with a nonlinear incidence rate which depends on the susceptible and the infectious, that is on the model state. On the other hand, in [15], a true-mass action type SEIR epidemic model (that is, with susceptible, exposed, infectious, and recovered integrated subpopulations) such that the coefficient disease transmission rate is normalized with the total population is analyzed. In such a way, such a normalized parameter becomes state-dependent, and thus time-varying, in the model. The above three types of state-dependent parameterizations are very common to some commonly used epidemic models. This feature results in their describing differential equations, or their state equations, to have state-dependent and time-varying parameterizations.The following basic notation is used through the manuscript:(1)R+=z∈R:z>0,R0+=z∈R:z≥0,Z+=z∈Z:z>0,Z0+=z∈Z:z≥0.The disjunction and conjunction logic propositions are, respectively, denoted by the symbols “∨” and “∧”:(2)n¯=1,2,…,n.L∞n0,t and, respectively, Lpn0,t are the sets of real n-vector functions which are bounded or, respectively, p-integrable on 0,t for any p∈Z+. In particular, L∞n=L∞n0,∞ and Lpn=Lpn0,∞ for any p∈Z+; clS denotes the closure of the set S.


## 2. Problem Statement

Consider then-th differential system of first-order equations.(3)z˙t=Bzt,tzt,z0=z0,where the matrix function B:R0+×Rn×R0+⟶Rn×n has piecewise continuous entries for each pair zt,t∈Rn×R0+ such that z:R0+⟶Rn is a solution of (3). Such a matrix function can be, in general nonuniquely, decomposed as(4)Bzt,t=Azt,t+A˜zt,t,∀t∈R0+.It is assumed thoroughly in the paper that the only (nonnecessarily stable) equilibrium point of (3) for the given A˜zt,t and for the case when A˜zt,t≡0 is ze=0∈Rn; ∀t∈R0+. In mathematical terms,(5)ze∈Rn:ze∈KerBze,t∨ze∈KerAze,t;∀t∈R0+⟺ze=0∈Rn,what implies that KerB0,t=KerA0,t=0⊂Rn; ∀t∈R0+, as a result. This feature introduces a further constraint on the nonunique additive decomposition (4).It is, furthermore, assumed that the matrix functionsA,A˜:R0+×Rn×R0+⟶Rn×n have piecewise continuous entries for each pair zt,t∈Rn×R0+ such that z:R0+⟶Rn is a solution of (3). To fix ideas, it is interpreted that A.,. defines the nominal dynamics and A˜.,. defines the error dynamics of (3), subject to (4).Example 1.
The differential system (3) can compactly describe a time-varying n-th ordinary differential system whose coefficients depend also on the derivatives up till n-th order and which are linear in the solution and its first n−1-th derivatives. For instance, a differential equation subject to a more general forcing term than equation (3) of [2] which, together with its second order analogous, plays an important role in the phase locked loop model realized by a T.V. system (see [2, 16, 17]), may be (in general, nonuniquely) described in the form (3) and (4), as follows:(6)zt=xt,x˙t,x¨tT,ezt,t=czt,t+A˜zt,tzt,∀t∈R0+,according, for instance, to
(a)(7)Aa=Aazt,t=010001−a10−a20−a30,A˜azt,t=000000a10a20−btgxta30−atfx˙t,which is clearly equivalent to the third-order differential equation:(8)x…t+atfx˙tx¨t+btgxtx˙t=ext,x˙t,x¨t,t−cxt,x˙t,x¨t,t,independent of any real constants ai0 for i=1,2,3. If ct,zt=atxt and A˜b=A˜bzt,t=0; ∀t∈R0+, it is possible to describe the differential system, again in alternative ways, for instance, as follows:
(b)(9)Abt=Abzt,t=010001−at−btgxt−atfx˙t,A˜b=A˜bzt,t=0;∀t∈R0+,
(c)(10)Act=Aczt,t=000000−a1t−btgxt−atfx˙t,A˜ct=A˜czt,t=000001−a2t00,∀t∈R0+,with a2t=at−a1t; ∀t∈R0+,
(d)(11)Ad=Adzt,t=000000−a10−a20+b0gx0−a30+a0fx˙0,∀t∈R0+,A˜dzt,t=000001a10−ata20+b0gx0−btgxta30+a0fx˙0−atfx˙t,∀t∈R0+.
The following further assumptions are made to be used in some of the main subsequent results.Assumption 1.
A
z
t
,
t commutes with ∫0tAzτ,τdτ; ∀t∈R0+.Assumption 2.
The matrix functionA:R0+×Rn×R0+⟶Rn×n satisfies Azt,t≤a, and it is a stability matrix for each pair zt,t∈Rn×R0+ such that z:R0+⟶Rn is a solution of (3).Assumption 3.
A
˜
z
t
,
t
≤
a
˜; ∀zt,t∈Rn×R0+ such that z:R0+⟶Rn is a solution of (3).Remark 1.
Note that the decomposition (4) is always possible and nonunique if B:R0+×Rn×R0+⟶Rn×n has piecewise continuous entries for each pair zt,t∈Rn×R0+ such that z:R0+⟶Rn is a solution of (3). On the other hand, taking advantage that the decomposition (4) is not unique, one concludes that Assumption 1 is not restrictive at all, in practice. It would suffice, for instance, to take a diagonal Azt,t and A˜zt,t=Bzt,t−Azt,t; ∀t∈R0+.
The main objective of the paper is to derive and prove sufficiency-type global stability and global asymptotic stability results of the differential system (3), subject to (4) by examining the growing rules of its norm through time through “ad hoc” derived integral inequalities.


## 3. Some Stability Results

The subsequent result relies on the existence and uniqueness of the solution of (3).Theorem 1.
If Assumption 1 holds, then then-th differential system (3) has a unique solution for each initial condition z0=z0∈Rn which is given by(12)zt=e∫0tAzτ,τdτz0+∫0te∫τtAzσ,σdσA˜zτ,τzτdτ,∀t∈R0+,where Ψz0,t,0=e∫0tAzτ,τdτ; ∀t∈R0+ is the fundamental matrix function of the n-th differential system differential system y˙t=Azt,tyt; y0=y0, which satisfies Ψz0,0,0=In; ∀z0∈Rn:(13)Ψ˙z0,t,0=Azt,te∫0tAzτ,τdτ=Azt,tΨt,0=Ψt,0Azt,t,∀t,τ∈R0+×R0+.Proof.
Note from (12) that z0=z0 and that (12) is everywhere time-differentiable on R0+ whose derivative is by using Leibnitz’s rule:(14)z˙t=Azt,te∫0tAzτ,τdτz0+e∫ttAzσ,σdσA˜zt,tzt+∫0tAzt,te∫τtAzσ,σdσA˜zτ,τzτdτ=Azt,te∫0tAzτ,τdτz0+∫0te∫τtAzσ,σdσA˜zτ,τzτdτ+e∫ttAzσ,σdσA˜zt,tzt=Azt,tzt+A˜zt,tzt,∀t∈R0+,after replacing (3) in the second identity of (14). Note that (12) is a unique solution for each z0=z0∈Rn since it is a closed formula and that it is calculated by taking the auxiliary system y˙t=Azt,tyt as the unforced system with y0=y0 and whose solution is yt=Ψz0,t,0y0=e∫0tAzτ,τdτy0; ∀t∈R0+.□
It can be pointed out that time-varying matrices, contrarily to constant matrices, do not preserve the stability under arbitrary transformations. There are specific transformations, as, for instance, the so-called Bohl transformations [5], which keep the stability properties from the original representation.Example 2.
Retaking Example 1 withet≡0 and the particular parameterization of (a) of the matrix Aa=010001−a10−a20−a30 yields that the fundamental matrix is Ψt,0=eAt; ∀t∈R0+. Assumption 1 holds trivially and the boundedness part of Assumption 2 also holds.
The subsequent result is related to the boundedness of the solution and the stability properties of (3).Theorem 2.
If Assumptions 1–3 hold, then then-th differential system (3), subject to (4), has a uniformly bounded solution for any finite initial conditions and it is globally asymptotically stable at large (i.e., in Rn) if a˜ is sufficiently small satisfying a maximum measurable guaranteed upper-bound amount which is given explicitly in the proof.Proof.
Note that, from Assumption 2,eAzt,t≤KAzt,te−∫0tρAzτ,τdτ for some bounded piecewise-continuous functions KA:R0+×Rn×R0+⟶1,∞ and ρA:R0+×Rn×R0+⟶0,+∞.
Then,(15)Ψz0,t,0=e∫0tAzτ,τdτ≤Ke−∫0tρAzτ,τdτ≤Ke−ρ0z0,tt≤Ke−ρt,∀t∈R0+,where(16)K=supt∈R0+supz0∈RnKAz0,t,0<ρ0z0,t≤∫0tρAzτ,τdτt,ρ=inft∈R0+infz0∈Rnρ0z0,t.
Note thatρ>0 since ρ0z0,t>0 for t∈R+ and, from L’Hopital rule,(17)limt⟶0+ρ0z0,t=limt⟶0+∫0tρAzτ,τdτt=ρAz0,0>0.
Then,(18)zt≤Ke−ρtz0+∫0teρτA˜zτ,τzτdτ;≤Ke−ρtz0+eρt−1ρa˜sup0≤τ≤tzτ;∀t∈R0+.
Define(19)tt=maxτ∈0,t:zτ=sup0≤σ≤τzσ,∀t∈R0+,(20)sup0≤τ≤tzτ=ztt≤Ke−ρttz0+K1−e−ρttρa˜sup0≤τ≤tzτ,∀t∈R0+.
Now, assume thata˜<ρ/Kinft≥01/1−e−ρtt≤ρ/K, then zt≤sup0≤τ<∞zτ≤ρKz0/ρ−Ka˜; ∀t∈R0+ since zt≤sup0≤τ≤tzτ≤ρKe−ρttz0/ρ+Ke−ρtt−1a˜≤ρKz0/ρ−Ka˜<+∞; ∀t∈R0+.
Then, the solution (12) is uniformly bounded for all time for any given finite initial conditions, and the differential system (3) is globally uniformly stable at large in Lyapunov’s sense as a result. It is now proved by contradiction argument that the stability is asymptotic. Assume on the contrary that there is a sequence tjj=0∞⊂R0+ such that suptk+1≤t<tk+2zt≥suptk≤t<tk+1zt; ∀tk∈tjj=0∞. Assume that the following cases can occur.Case a.
There is a subsequencetjii=0∞⊆tjj=0∞ such that suptk+1≤t<tk+2zt>suptk≤t<tk+1zt; ∀tk∈tjii=0∞. This case is not possible since then the solution is not uniformly bounded for all t∈R0+.Case b.
There is a subsequencetjii=0∞⊆tjj=0∞ such that suptk+1≤t<tk+2zt=suptk≤t<tk+1zt>0; ∀tk∈tjii=0∞. Then,(21)suptk+j+1≤τ<tk+j+2zτ=ztk+j+1+σk+j+1≤Ke−ρtk+j+1−tk+1+σk+j+1−σk+1suptk+1≤τ<tk+2zτ+K1−e−ρtk+j+1−tk+1+σk+j+1−σk+1ρa˜sup0≤τ<tk+j+2zτ,∀tk∈tjii=0∞,∀t∈R0+,where σk∈0,tk+1−tk, ∀tk∈tjii=0∞, such that suptk≤τ<tk+1zτ=ztk+σk if and only if τ=0; i.e., tk+σk is the largest time instant in tk,tk+1 where the supremum is reached within such an interval for all tk∈tjii=0∞. Now, if the subsequence tjii=0∞ is not unique then, with no loss in generality, take the one such that there is some finite tj0∈tjii=0∞ and there is no t≥tj0∈R0+ such that zt>suptk≤τ<tk+1zτ; ∀tk∈tjii=0∞ so that supτ≥tj0zτ=suptj0≤τ<tkzτ; ∀tk≥tj0∈tjj=0∞ and tj0∈tjii=0∞. Thus, one gets from (21) that(22)suptj0≤τ<tk+j+2zτ≤Ke−ρtk+j+1−tk+1+σk+j+1−σk+1suptj0≤τ<tk+j+2zτ+K1−e−ρtk+j+1−tk+1+σk+j+1−σk+1ρa˜suptj0≤τ<tk+j+2zτ;∀tk∈tjj=0∞,∀t∈R0+,(23)1−Ke−ρtk+j+1−tk+1+σk+j+1−σk+1−K1−e−ρtk+j+1−tk+1+σk+j+1−σk+1ρa˜suptj0≤τ<tk+j+2zτ≤0;∀tk∈tjj=0∞,∀t∈R0+,so that(24)limsupj⟶∞1−Ke−ρtk+j+1−tk+1+σk+j+1−σk+1−K1−e−ρtk+j+1−tk+1+σk+j+1−σk+1ρa˜suptj0≤τ<tk+j+2zτ≤0,for some tj0∈tjii=0∞; ∀tk∈tjj=0∞, ∀t∈R0+. Then, 1−Ka˜/ρsuptj0≤τ<∞zτ≤0 which implies that for any subsequence tjii=0∞⊆tjj=0∞ such that z¯=limsuptk⟶∞suptk≤t<tk+1zt; ∀tk∈tjii=0∞ it holds that z¯=0, a contradiction to the assumption of Case b. Then, Case b is not possible either. As a result, the proved global stability is also asymptotic at large.□
Some necessary, but not sufficient, conditions for global stability or global asymptotic stability can be got by simple direct inspection of the various parameterizations of the differential system (3), subject to (4), given in the particular Example 1. Such conditions are basically addressed by inspecting the (3, 3)-entry of the corresponding matrices Azt,t as discussed in the next result.Proposition 1.
Consider the third-order Example 1 of the differential system (3) subject to (4). Then, the following properties hold:(i)
Ifa10=0, bt=a20/gxt and at=a30/fx˙t with a20>0 and a30>0, then the system is not globally asymptotically stable at large, but it can be globally stable at large. The property can also hold if a10=0, bt=a20/gxt and at=a30/fx˙t do not hold for all time while A˜zt,t≤a˜; ∀zt,t∈Rn×R0+ (Assumption 3) with sufficiently small a˜.(ii)
IfAbt satisfies Assumption 1 and its third row has no entry either being identically zero or unbounded or negative for all time, then the system (3), subject to (4), can be globally asymptotically stable only if(25)limt⟶∞∫0taτfx˙τdτ=−∞.(iii)
IfAct satisfies Assumption 1 and its third row has no entry either being identically zero or negative for all time, and if A˜ct satisfies Assumption 3 with a2t being sufficiently small for all t∈R0+, then the system (3), subject to (4), can be globally asymptotically stable only if(26)limt⟶∞∫0taτfx˙τdτ=−∞.(iv)
If the entries of the third row ofAd satisfy a10>0 and, for each given initial conditions, a20>−b0gx0 and a30>−a0fx˙0 while A˜zt,t satisfies either(27)at=a10,bt=a20+b0gx0gxt,at=a30+a0fx˙0fx˙t,or if some of the above equalities fail but Assumption 3 holds, with a2t being sufficiently small for all t∈R0+, then the system (3), subject to (4), can be globally asymptotically stable.Proof.
Note that Assumption 1 and the first part of Assumption 2 on boundedness hold trivially sinceAa is constant. On the other hand, since A˜azt,t=0; ∀t∈R0+, Assumption 3 holds trivially and the stability of the differential system reduces to the stability of the constant matrix Aa. Note that the characteristic equation of Aa is ps=s3+a20s+a30=0, and since the coefficient of s2 in the characteristic polynomial is zero, then the matrix Aa is not a stability matrix and then the differential system cannot be globally asymptotically stable but can be globally stable depending on the values of a20 and a30 since they are positive. Furthermore, since deteAat=detettrAa=dete−a30t⟶0 as t⟶+∞, (at least) an eigenvalue Λ1t of the fundamental matrix of multiplicity pα<n∈Z+ vanishes exponentially with time as time tends to infinity (note that pα cannot equalize n since then Aa would be a stability matrix). Thus, the remaining eigenvalues of eAat of spectrum speAat=Λit;i∈n¯ fulfill that ∏i=1n−1Λit=otpα−1e−αt. If the given equalities fail but Assumption 3 holds, then the nonasymptotic property can still holds under extra sufficiency-type conditions from Theorem 2. Proposition 1 (i) has been proved. Propositions 1 (ii) and 1 (iii) follow in the same way provided that limt⟶∞∫0taτfx˙τdτ=−∞ which guarantees that limt⟶∞dete∫0tAbτdτ=0, respectively, limt⟶∞dete∫0tAcτdτ=0 which is a necessary condition for global asymptotic stability. Proposition 1 (iv) follows from close arguments as those invoked for proving Proposition 1 (i) if Ad is a stability matrix (note that this is possible depending on its eigenvalues while contrarily Aa can never be a stability matrix) and A˜dzt,t is either zero or with sufficiently norm for all time.Example 3.
Assume that the system of Example 1 withet≡0 is parameterized with the parameterization of (a). The constant matrix Aa is a stability matrix if its characteristic equation s3+a10s2+a20s+a30=0 has all its roots in the open complex left-hand-side. This holds from the Routh-Hurwitz criterion if and only if a10>0, a20>0, and 0<a30<a10a20. If the parameterization of (d) is used, then the modified matrix of dynamics Ad is a stability matrix if and only if(28)a10>0,a20+b0gx0>0,a10a20+b0gx0>a0fx˙0.(i)
The parameterization of (a) is useful to guarantee the global asymptotic stability of the differential system (3), subject to (4), that is if Aa is stable and, furthermore, A˜at is sufficiently small for all time in the sense that if −ρMa is the stability abscissa of Aa, i.e., the absolute value of its (stable) eigenvalue being closer to the complex imaginary axis satisfying:(29)a˜1t2+a˜2t2+a˜3t2=a10−at2+a20−btgxt2+a30−atfx˙t2<ρMa2.(ii)
The parameterization of (d) is useful ifAd is a stability matrix of stability abscissa −ρMd, and(30)a10−at2+a20+b0gx0−btgxt2+a30+a0fx˙0−atfx˙t2<ρMd2.Remark 2.
Theorem 2 proves thatsup0≤t<∞zt≤ρKz0/ρ−Ka˜ under Assumption 3 which requests the uniform boundedness of Azt,t and A˜zt,t for any pair zt,t∈Rn×R0+ such that z:R0+⟶Rn is a solution of (3). The theorem concludes the global asymptotic stability at large of (3). This suggests that the global asymptotic stability in a closed ball (rather than in the large) can be formulated for a certain closed ball of Rn containing the solution trajectory of (3) for any initial conditions in a given closed ball without invoking Assumption 3. Thus, define the closed balls B00,r0=z0∈Rn:z0≤r0 and(31)B0,ρKr0ρ−Ka˜=z∈Rn:z≤ρKr0ρ−Ka˜∧z0∈B00,r0.
Thus, Theorem 2 has the following useful corollary which does need “a priori” boundedness conditions on the norm of the supremum of the solution as invoked in Assumption 3.Corollary 1.
If Assumptions 1 and 2 hold, then then-th differential system (3), subject to (4), has a uniformly bounded solution for any initial conditions in B00,r0=z0∈Rn:z0≤r0 and it is globally asymptotically stable with the trajectory solution contained in (32)B0,ρKr0ρ−Ka˜=z∈Rn:z≤ρKr0ρ−Ka˜∧z0∈B00,r0,if a˜ is small enough such that a˜<ρ/K (see Theorem 2).□
The subsequent result is similar to Theorem 2 without assuming the boundedness ofA˜zt,t which can be expanded in series for all values of the trajectory solution at any time according to a small time-varying parameter. Also, it can grow unboundedly with time if sup0≤τ≤tzτ is strictly upper-bounded by the inverse of a small time-varying function which can vanish asymptotically.
The bounded closed domain which contains the state trajectory solution for all time may be defined depending on the relevant parameters of the differential system.Example 4.
Let the initial conditions of (3), subject to (4), satisfy z0≤r0. Then, sufficient conditions for zt≤r<+∞; ∀t∈R0+ for some r=μr0>r0 can be got from Theorem 2, equation (18), as follows:(33)zt≤Ke−ρt+1ρsup0≤t<+∞A˜tμr0≤μr0=r,∀t∈R0+,μ=rr0≥K1+1ρsup0≤t<+∞A˜tμ≥Ke−ρt+1ρsup0≤t<+∞A˜tμ,∀t∈R0+.
So,r≥K1+1/ρsup0≤t<+∞A˜tμr0=Kr0+K/ρsup0≤t<+∞A˜tr, or r≥Kρr0/ρ−Ksup0≤t<+∞A˜t if sup0≤t<+∞A˜t<ρ/K. Note that μ≥Kρ/ρ−Ksup0≤t<+∞A˜t>K≥1 as a result.
Now, assume thatr0 is given as the radius of the closed ball around zero which fixes the domain for initial conditions and r is prefixed as the radius of the suitable closed ball which guarantees that the solution remains within it for all time. Thus, ρr−Kr0≥rKsup0≤τ<+∞A˜t so that the constraint sup0≤τ<+∞A˜t≤ρμ−Kr0/rK guarantees the respective radii r0 and r=μr0 for a given ρ>0 provided that sup0≤t<+∞A˜t<ρ/K. The combination of both norm constraints leads that the first one is stronger since sup0≤τ<+∞A˜t≤ρ/Kmin1,μ−K/μ=ρ/K1−K/μ.Theorem 3.
Let Assumptions 1–2 hold and assume also that(1)
The pair of real constantsρ,K defined in the proof of Theorem 2 satisfies the constraint ρ>K/1−ε¯0(2)
There exists a functionε:0,t⟶R+; ∀t∈R0+ such that ε0t=εtsup0≤τ≤tzτ≤ε¯0<1 (note that εt is allowed to be asymptotically vanishing) and such that A˜zt,t≤∑i=0∞εitsup0≤τ≤tizτ
Then, then-th differential system (3), subject to (4), has a uniformly bounded solution for any finite initial conditions, and it is globally asymptotically stable at large (i.e., in Rn).Proof.
Note thatεt=ε0t/sup0≤τ<∞zτ; ∀t∈R0+ for some ε0t∈0,ε¯0 and some 0≤ε¯0≤1; ∀t∈R0+ implies that(34)A˜zt,t≤∑i=0∞εitsup0≤τ≤tizτ=∑i=0∞ε0it=11−ε0=11−εtsup0≤τ≤tzτ;∀t∈R0+,with(35)sup0≤τ≤tzτ≤ε¯0εt<1εt.
Equation (20) is modified as follows:(36)sup0≤τ≤tzτ=ztt≤Ke−ρttz0+K1−e−ρttρ11−εtsup0≤τ<∞zτsup0≤τ≤tzτ,∀t∈R0+,leading to(37)1−ε¯0sup0≤τ≤tzτ≤1−εtsup0≤τ<∞zτsup0≤τ≤tzτ=1−εtsup0≤τ<∞zτztt≤K1−εtsup0≤τ<∞zτe−ρttz0+K1−e−ρttρsup0≤τ≤tzτ,∀t∈R0+,or(38)1−ε¯0−Kρsup0≤τ≤tzτ≤1−ε¯0−K1−e−ρttρsup0≤τ≤tzτ≤K1−εtsup0≤τ<∞zτe−ρttz0,∀t∈R0+,so that(39)1−ε¯0−Kρsup0≤τ≤tzτ≤ρKρ1−ε¯0−K1−εtsup0≤τ<∞zτe−ρttz0,∀t∈R0+,and since ρ>K/1−ε¯0≥K/1−ε¯0+Kεte−ρttz0 and sup0≤τ≤tzτ=ε0t/εt≤ε¯0/εt<1/εt; ∀t∈R0+, one has(40)sup0≤τ≤tzτ≤ρKρ1−ε¯0−K1−εtsup0≤τ≤tzτe−ρttz0,∀t∈R0+,(41)sup0≤τ≤tzτ≤ρKe−ρttz0ρ1−ε¯0+Kεte−ρttz0−K<+∞,∀t∈R0+,which proves the global uniform stability at large. The asymptotic stability at large is proved by contradiction arguments by the construction of the appropriate solution sequences as in Theorem 2.Corollary 2.
Theorem 2 still holds ifA˜zt,t≤∑i=0kεitsup0≤τ≤tizτ; ∀t∈R0+ for some given positive integer k with ε0t=εtsup0≤τ≤tzτ≤ε¯0<1; ∀t∈R0+.Proof.
In this case, one has(42)A˜zt,t≤∑i=0kεitsup0≤τ≤tizτ=1−εktsup0≤τ≤tzτ1−εtsup0≤τ≤tzτ≤11−εtsup0≤τ≤tzτ;∀t∈R0+.
The proof follows by using the above constraint.□
The subsequent result does not invoke Assumption 3. Instead, an upper-bound maximum growing time-interval condition on the time integral of the norm ofA˜zt,t is used to address sufficiency-type conditions for the global stability and asymptotic stability of (3), subject to (4).Theorem 4.
If Assumptions 1 and 2 hold and, furthermore,(43)∫tt+TA˜zτ,τ4dτ≤c1T+c2,∀t∈R0+,∀T∈R+,for some sufficiently small c1,c2∈R0+, then the differential system (3), subject to (4), is globally asymptotically Lyapunov’s stable at large.Proof.
One gets from (18) in the proof of Theorem 1 and Hölder’s inequality that(44)zt+T≤Ke−ρTzt+∫tt+TKe−ρt+T−τA˜zτ,τzτdτ≤Ke−ρTzt+K∫tt+Te−2ρt+T−τdτ1/2∫tt+TA˜zτ,τzτ2dτ1/2≤Ke−ρTzt+K∫tt+Te−2ρt−τdτ1/2∫tt+TA˜zτ,τ4dτ1/4∫tt+Tzτ4dτ1/4≤Ke−ρTzt+K1−e−2ρT2ρ∫tt+TA˜zτ,τ4dτ1/4∫tt+Tzτ4dτ1/4≤Ke−ρTzt+K1−e−2ρT2ρc1T+c21/4supt≤τ≤t+TzτT1/4≤Ke−ρTzt+K1−e−2ρT2ρc11/4T1/4+c21/4T1/4supt≤τ≤t+Tzτ,∀t∈R0+,and for some tt∈t,t+T defined as tt=maxτ∈t,t+T:zτ=sup0≤σ≤Tzt+σ; ∀t∈R0+,(45)supt≤τ≤t+Tzτ=ztt≤Ke−ρtt−tzt+K1−e−2ρT2ρc11/4T1/4+c21/4T1/4supt≤τ≤t+Tzτ,∀t∈R0+.
Sincec1 and c2 are small enough such that 1>K1−e−2ρT/2ρc11/4T1/4+c21/4T1/4, one has(46)supt≤τ≤t+Tzτ≤Ke−ρtt−tzt1−K1−e−2ρT/2ρc11/4T1/4+c21/4T1/4,∀t∈R0+.
Assume thatlimt⟶∞zt=+∞. Since T is arbitrary in (43), there are nonunique strictly increasing real sequences kT0∞⊂R0+ such that zk+jkT>zkT; for some sufficiently large finite jk=jkT∈Z+, ∀k∈Z0+. Then, for any given finite T>0 and k∈Z0+, sufficiently large values of jk∈Z0+ and sufficiently small related constants c1=c1T,c2T, one has(47)zkT<zk+jjT≤Ke−ρjkTzkT1−K1−e−2ρjkT/2ρc11/4jkT1/4+c21/4jkT1/4<zkT,∀t∈R0+,because of the dominance of the second right-hand-side numerator related to the denominator for sufficiently large jk and sufficiently small c1,c2 since T is finite. Hence, a contradiction for some sufficiently large finite T>0. Thus, the differential system (3), subject to (4), is globally stable at large as a result. On the other hand, consider three particular cases concerned with the differential system (3), subject to (4).Case a.
It is globally stable at large for sufficiently small constantsc1,c2. Thus, no further proof is needed.Case b.
Its solution is unbounded for sufficiently small constantsc1andc2. In this case, it is unstable what contradicts its already proved global Lyapunov’s stability at large. So, this case is impossible.Case c.
Its solution is bounded but oscillatory. Thus, there is a time interval0,ta, of finite or zero measure, with ta=taz0≥0 such that sup0≤t≤tazt=sup0≤t<+∞zt so that there is a subsequence of time instants tkj0∞⊂R0+ defined by tkj=ta+kjT+τkj, with kj∈Z0+ and τkj∈0,T; ∀j∈Z0+, such that ztkj=sup0≤t≤tazt so that one gets from the first inequality of (44) that(48)sup0≤t≤tazt=ztkj≤Ke−ρkjT+τkjzta+sup0≤t≤tazt∫0kjT+τkjKe−ρkjT+τkj−τA˜zτ+ta,τ+tadτ≤Ke−ρkjT+τkjzta+Kρ1−e−ρkjT+τkjc1kjT+τkj+c2sup0≤t≤tazt,and then one gets the following contradiction for some sufficiently large finite T>0:(49)sup0≤t≤tazt≤Ke−ρkjT+τkjzta1−K/ρ1−e−ρkjT+τkjc1kjT+τkj+c2<zta,∀t∈R0+,by defining the subsequences kjj=0∞⊂Z0+, εkjj=0∞⊂R+ and mkjj=0∞⊂R+ with εkj=e−ρkjT+τkj (note that εkj⟶0) such that(50)0<εkj<min1,1Kzta,kj=−1Tlnεkjρ+τkj∈Z+,mkj=c1kjT+c2<1K−εkjztaρ1−εkj,and note also that it suffices to see that the contradiction holds for the finite first element k0∈kjj=0∞ and sufficiently small c1,c2 related to ρ/K. Thus, the solution to (3), subject to (4), for any finite initial conditions cannot be bounded and oscillatory. Thus, one concludes that the differential system is globally asymptotically Lyapunov’s stable at large if c1 and c2 are sufficiently small.
The particular 4-th power inA˜zτ,τ in the integral (43) assumed in Theorem 4 is not crucial to the proof except for the “amount of smallness” needed for the constants c1 and c2 to guarantee the theorem. In this context, note the subsequent result.Corollary 3.
Assume that the inequality (43) in Theorem 4 is replaced with(51)∫tt+TA˜zτ,τ2dτ≤d1T+d2,∀t∈R0+,∀T∈R,for some sufficiently small d1,d2∈R0+, or with(52)∫tt+TA˜zτ,τdτ≤f1T+f2,∀t∈R0+,∀T∈R+,for some sufficiently small f1,f2∈R0+. Then, Theorem 4 still holds under the same given remaining assumptions.Outline of Proof.
The relevant inequalities of (44) are modified as follows if (43) is replaced with (51):(53)zt+T≤Ke−ρTzt+∫tt+TKe−ρt+T−τA˜zτ,τzτdτ≤Ke−ρTzt+K∫tt+TA˜zτ,τ2dτ1/2∫tt+Tzτ2dτ1/2≤Ke−ρTzt+Kd1T+d21/2supt≤τ≤t+TzτT1/2≤Ke−ρTzt+Kd1T+d2Tsupt≤τ≤t+Tzτ,∀t∈R0+,and the rest of the proof follows “mutatis-mutandis” to that of Theorem 4. In the same way, if (43) is replaced with (52), then the modified set of inequalities holds:(54)zt+T≤Ke−ρTzt+K∫tt+TA˜zτ,τ2dτ1/2∫tt+Tzτ2dτ1/2≤Ke−ρTzt+Kf1T+f2supt≤τ≤t+TzτT1/2≤Ke−ρTzt+Kf1T3/2+f2Tsupt≤τ≤t+Tzτ;∀t∈R0+,and the stability result is again proved.□Remark 3.
Note that the global stability of the various proved results guarantees that the solution of the differential system and all its time-derivatives up till n-th order are bounded for all time for any given finite initial conditions. On the other hand, the global asymptotic stability guarantees in addition the asymptotic convergence of the above functions to zero.Theorem 5.
Assume the following particular differential system (3) and (4) which is also forced with an external function(55)z˙t=Btzt+ωt,z0=z0,∀t∈R0+,where the matrix function B:×R0+⟶Rn×n has piecewise continuous entries and it is nonuniquely decomposed as Bt=At+A˜t; ∀t∈R0+, and assume also that the amended Assumptions 1–3 hold referred just to time, instead to the pair zt,t, in view of (55). Then, the following properties hold:(i)
The differential system is globally stable if the unforced system is globally asymptotically stable andω∈L∞n or ω∈Lpn for any p∈Z+ if a˜ is small enough.(ii)
Assume, in addition, thatω is of exponential order −ρω<0 with ρω being larger than the average stability abscissa of At. Then, the differential system is globally asymptotically stable at large.Proof.
In this case, (12) is replaced as follows:(56)zt=zunft+zft,∀t∈R0+,where the unforced and forced solutions are(57)zunft=e∫0tAτdτz0+∫0te∫τtAσdσA˜τzτdτ,zft=∫0te∫τtAσdσωτdτ,∀t∈R0+.
Note that closely to the arguments used in the proof of Theorem 2 thate∫0tAτdτ≤Ke−∫0tρAτdτ≤Ke−ρ0z0,tt≤Ke−ρt; ∀t∈R0+, where K=supt∈R0+supz0∈RnKAz0,t; 0<ρ≤infτ,t>τ∈R0+∫τtρAτdτ/t−τ. Thus, the following additive terms have to be added for the obtained unforced upper-bounds of the unforced sup0≤τ≤tzunfτ; ∀t∈R0+ in the various former given results applied to this particular differential system:(a)
K∫0te−ρt−τωτdτ≤Ksup0≤τ<∞ωτ1/ρ=KM∞/ρ; ∀t∈R0+if ω∈L∞n and ω∞=M∞. Then, sup0≤τ≤tzτ≤sup0≤τ≤tzunfτ+KM∞/ρ.(b)
K∫0te−ρt−τωτdτ≤K∫0tωτdτ=KM1; ∀t∈R0+ if ω∈L1n and ω1=M1. Then, sup0≤τ≤tzτ≤sup0≤τ≤tzunfτ+KM1.(c)
K∫0te−ρt−τωτdτ≤K∫0te−ρpt−τ1/p∫0tωτp1/pdτ=K/pρ1/pMp; ∀t∈R0+ if ω∈L1n and ω1=M1. Then, sup0≤τ≤tzτ≤sup0≤τ≤tzunfτ+KM1; if ω∈Lpn and ωp=Mp for any p∈Z+. Then, sup0≤τ≤tzτ≤sup0≤τ≤tzunfτ+K/pρ1/pMp; ∀t∈R0+. Note that since ρ>K≥1 for the results on asymptotic stability, the upper-bound K/ρM1 of (c) might improve, in general, that of (b), i.e., KM1 if ω∈L1n. One concludes that, under any of the conditions obtained for the unforced differential system to be globally asymptotically stable, the forced one remains globally stable with the corresponding above new given norm upper-bounds for all time. Property (i) has been proved. Property (ii) follows since if ω is of negative exponential order −ρω with ρω>ρ then zft⟶0 exponentially as t⟶∞ since(58)zft=∫0te∫τtAσdσωτdτ≤K∫0te−∫τtρAσdσK1e−ρωτdτ≤KK1e−ρt∫0teρ−ρωτ≤KK1e−ρteρ−ρωt−1ρ−ρω,∀t∈R0+.Remark 4.
The upper-bounds of Theorem 5 for the norm of the whole solution of (55) may be improved if the forcing function belongs jointly to several normed function spaces on 0,∞ by using the minimum of the corresponding finite norms. For instance, one has(59)sup0≤τ≤tzτ≤sup0≤τ≤tzunfτ+Mω1,∀t∈R0+,where Mω1=K/ρminM∞/ρ,M1/ρ if ω∈L∞n∩L1n. Since L∞n∩L1n⊃L2n, then Mω1 may be replaced with Mω2=K/ρminM∞/ρ,M1/ρ,M2/2 in (59) since ω∈L∞n∩L1n∩L2n.
The so-called property of global ultimate boundedness guarantees also the global Lyapunov’s (nonasymptotic) stability property since the smooth solution of (3) and (4) has to be necessarily continuous for all time so that it cannot have finite escape times (i.e., right or left finite-time discontinuity points to ±∞), as a result. An explicit related result follows.Theorem 6.
Under Assumptions 1–2, the following properties hold for the differential system (3), subject to (4):(i)
Assume thatlimsupt⟶∞A˜zt,tzt≤M∞ for some finite real constant M∞ and any given finite initial conditions. Then, the system is globally stable at large with ultimate boundedness satisfying the asymptotic finiteness norm constraint limsupt⟶∞zt≤K/ρM∞ for any given finite initial conditions and ρ being defined in Theorem 2.(ii)
Assume thatlimsupt⟶∞∫0tA˜zτ,τzτpdτ≤Mp for some real p>0 and finite real constant Mp and any given finite initial conditions. Then, the system is globally stable at large with ultimate boundedness satisfying the asymptotic finiteness norm constraint limsupt⟶∞zt≤KMp/ρp1/p for any given finite initial conditions with ρ being defined in Theorem 2.Proof.
Since the unforced part of the solution converges asymptotically to zero, Property (i) follows since(60)limsupt⟶∞zt≤limsupt⟶∞∫0tKe−ρt−τA˜zτ,τzτdτ≤KρM∞,and Property (ii) follows since limsupt⟶∞∫0te−ρt−τdτp≤1/ρp so that(61)limsupt⟶∞zt≤Klimsupt⟶∞∫0te−ρt−τA˜zτ,τzτdτ≤Klimsupt⟶∞∫0te−ρt−τdτp1/p×limsupt⟶∞∫0tA˜zτ,τzτdτ1/p≤KMpρp1/p.Example 5.
Assume that, for some finite and positive real constantszm, θ, and q≥1 and some bounded function K˜:R0+⟶R+, the solution of (3), subject to (4), satisfies(62)A˜zt,t≤K˜t,ifzt≤zm,K˜tztq+θ,ifzt≤zm,A˜zt,tzt≤A˜zt,tzt≤K˜tzm,ifzt>zm,K˜tztztq+θ≤K˜tztq−1+θzt,ifzt>zm,then for some positive finite real constant M∞, limsupt⟶∞A˜zt,tzt≤M∞ so that the condition of Property (i) of Theorem 6 holds. Thus, Theorem 6 (i) holds under Assumptions 1–2.


## 4. Some Remarks on the Problem Point of View under Then Taylor’s Series Remainder

A brief remind of well-known results on Taylor’s series expansion follow to be then used for the problem at hand in this paper. Consider a real open intervalI=a,b and a real function fx which is continuous on clI=a,b, m-th continuously differentiable on I, and such that its m-th derivative is absolutely continuous on clI and its m+1-th derivative exists and it is absolutely continuous on clI. Then, the Taylor’s series expansion formula with truncation and integral remainder gives for any x0,x∈I that(63)fx=∑j=0mfjx0j!x−x0j+1m!∫x0xfm+1ξx−ξmdξ=∑j=0mfjx0j!x−x0j+1m!∫011−ξmfm+1ξdξ,where the last right-hand-side additive term is the integral remainder. Note that if fmξ is just continuous, rather than absolutely continuous, on x0,x, then one gets from the mean value theorem,(64)Rmx=1m+1!fm+1ξx−ξm+1=1m!fm+1ςx−ςmx−x0,for some ξ,ς∈x0,x, the first expression of (64) being the Lagrange remainder form while the second one is the Cauchy remainder form.The formula (63), subject to (64), be easily extended to the differential system (3), subject to (4), by expanding z˙t in Taylor’s series about the equilibrium point, that is, by expanding the real vector function Bzt,tzt for each fixed t∈R0+ with the assignations x0⟶ze=0∈Rn, x⟶zt for each t∈R0+, in particular, x⟶z0 for t=0. Then, assume that all the components of Bzt,t, of rows Bizt,t for i=1,2,…,n, are continuously differentiable with respect to all the components of zt=z1t,z2t,…,zntT up till some order m and that the m+1-th higher-order differentials of all the entries of Bzt,t exist at ze=0∈Rn, in some open subset of Rn containing z0 and the unique equilibrium point ze=0∈Rn. Thus, we have(65)z˙t=0+∇zTBij0,tztze=0zt+R1z,t=B0,t.0︸0+∇zTBij0,tze=0zt+∑j=2m∑k=1n1j!∑i1=1j⋯∑ij=1j∂jB1kzt,tzk∂zi1⋯∂zijze=0∏ℓ=1jziℓ⋮⋮1j!∑i1=1j⋯∑ij=1j∂jBnkzt,tzk∂zi1⋯∂zijze=0∏ℓ=1jziℓ+Rmzt,t,∀t∈R0+,where the ∇Nabla-operator stands for the gradient, i.e., ∇zTBij0,tze=0=∂Bij0,t/∂zjij; ∀t∈R0+. A useful simplified notation for (65) is(66)z˙t=∇zTBij0,tze=0zt+∑2≤α≤m1α!DαB1kz,tz0zα⋮⋮∑2≤α≤m1α!DαBnkz,tz0zα×idm−1+Rmzt,t,∀t∈R0+,where the discrete binary indicator function id:Z0+⟶0,1 is defined as idm−1=0 if m=0,1 and idm−1=1 if m≥2, and the remainder real vector of the series expansion in differential and integral forms becomes(67)Rmzt,t=∑k=1n1m+1!∑i1=1m+1⋯∑i1=1m+1∂jB1kzt,tzkt∂zi1⋯∂zijze=0∏ℓ=1m+1ziℓ⋮⋮1m+1!∑i1=1m+1⋯∑ij=1m+1∂jBnkzt,tzkt∂zi1⋯∂zijze=0∏ℓ=1m+1ziℓ,(68)=∑α=m+1m+1α!zα∫011−tmDαB1ktztdt⋮⋮∑α=m+1m+1α!zα∫011−tmDαBnktztdt,∀t∈R0+,where Dαf=∂αf/∂x1α1...∂xnαn, where α=∑i=1nαi, α!=∏i=1nαi!, and zα=∏i=1nzi−zei=∏i=1nzi since ze=0 so that ze+tz−z3=tzt in (68) after using a variable change s⟶ze+tz−ze to convert the integral from ze=0 to z into one from t=0 to t=1. Note that, if (12), subject to (67), equivalently to (68), is used to describe (3), subject to (4), with(69)Azt,t=∇zTBij0,tze=0,A˜zt,tzt=∑2≤α≤m1α!DαB1kz,tz0zα⋮⋮∑2≤α≤m1α!DαBnkz,tz0zα×idm−1+Rmzt,t,∀t∈R0+,then Assumptions 1–2 hold provided that the unique equilibrium point ze=0 is stable and Azt,t=Aze,t=A0,t=A; ∀t∈R0+ is a constant (i.e., independent of time) stability matrix.Theorem 2 holds under the following simplified “ad hoc” form.Theorem 7.
Assume that then-th differential system (3), subject to (4), is described by (69), subject to (68) with Azt,t=A; ∀t∈R0+ being a stability matrix, such that the unique equilibrium point ze=0 is locally asymptotically stable, and that Assumption 3 holds with a˜ being sufficiently small related to the absolute value of the stability abscissa of A. Then, the system is globally asymptotically stable at large.



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