# Practical Stability and Integral Stability for Singular Differential Systems with Maxima

Authors: Junyan Bao; Peiguang Wang; Yanjun Li
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, we introduce various definitions of practical stability and integral stability for nonlinear singular differential systems with maxima and give criteria of stability for such systems via the Lyapunov method and comparison principle.


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## Body

## 1. Introduction and Preliminaries

Differential equations with maxima are a special type of differential equations that contain the maximum of the unknown function over a previous interval, of which many examples are found in the fields of application such as automatic control, population dynamics, disease control, and so on. Recently, the research interest in differential equations with maxima has increased exponentially. Some stability results for such equations can be found in the monographs [1, 2], the papers [3–9], and references cited therein.In practical applications, many problems can be described by singular system models, such as optimal control problems and constrained control problems, which can be found in the monographs of Campbell [10] and Dai [11]. Singular system is a type of dynamic system which is more complicated than the ordinary one. Owing to its complicated structure and many other factors, the study of stability for singular systems involves greater difficulty than that of nonsingular systems. Till now, various types of stability for singular systems have been investigated via Lyapunov functions. However, most previous studies focused on the singular systems described by ordinary differential equations [10–13], difference equations [14–17], and delay differential equations [18–20], and there are a few results for singular differential systems with maxima. In addition, differential equations with maxima have some different properties from the well-known differential equations and delay differential equations.The purpose of this paper is to integrate these two areas and analyze the practical stability and integral stability of nonlinear singular systems with maxima. To extend Lyapunov’s stability and support the specific needs of singular systems, we introduce the functionqt,x and obtain some different types of stability criteria by using the Lyapunov function method and comparison principle.


## 2. Practical Stability

The practical stability, being quite different from the stability in the sense of Lyapunov, is neither weaker nor stronger than the usual stability. It is significant from the perspective of engineering application (see [21–25]). In this section, by using Lyapunov functions and the comparison principle, we study some practical stability for the following singular differential systems.Consider the singular differential systems with maxima(1)Ex˙=ft,x,maxs∈t−τ,txs,fort≥t0≥0,xt0=φt,t∈−τ,0,where E∈Rn×n with rank E<n is a singular constant matrix, x∈Rn, f∈CR+×Rn×Rn,Rn, ft,0,0≡0, τ>0 is a constant, and φ∈C−τ,0,Rn.Firstly, we introduce the following notations and sets for convenience.LetTk=0,tk, where 0<tk≤+∞; qt,x∈C1J×Rn,Rm, qt,0≡0, J⊆R+. Skt0 is a set of all consistent initial functions at initial time t0. Then, for any φ∈Skt0, there exists at least one continuous solution of systems (1) in t0−τ,+∞ through t0,φ (see [20]).K=at∈CTk,R+at;is strictly increasing anda0=0K∗=at0,r∈CTk×R+,R+at0,r;is strictly increasing inrandat0,0=0Qε=x∈Rnqt,x<ε,t∈Tk,ε>0is a constantDA=x∈Rnqt,x<A,A>0is a constantΛ=VV∈CR×DA,R+andVis  locally Lipschitzian inxG⟹H means that G implies HWe denote byxt≡xt;t0,φ the solution of the initial value problems (1).Definition 1.
LetV∈Λ,t∈Tk, and we define the derivative of the function V (t; x) along the trajectory of solution of the singular systems (1) as follows:(2)D2.1+Vt,xt=limsuph⟶01hVt+h,xt+hft,xt,maxs∈−τ,0xt+s−Vt,xt.Definition 2.
Letφ∈Skt0. The singular systems (1) is said to beS1 stable with respect to qt,x,Tk if for any ε>0, and some t0∈Tk, there exists δt0,ε>0, such that(3)maxs∈−τ,0φs<δt0,ε⟹qt,xt;t0,φ<ε,fort≥t0.Definition 3.
Letφ∈Skt0. The singular systems (1) is said to be PS1 practically stable for given λ,A with 0<λ<A and some t0∈Tk, such that(4)maxs∈−τ,0φs<λ⟹qt,xt;t0,φ<A,fort≥t0,PS2 uniformly practically stable if PS1 holds for all t0∈TkPS3 practically quasistable for given λ,B,T with λ,B,T>0, and some t0∈Tk, we have(5)maxs∈−τ,0φs<λ⟹qt,xt;t0,φ<B,fort≥t0+T,PS4 uniformly practically quasistable if PS1 holds for all t0∈TkPS5 strongly practically stable if PS1 and PS3 hold simultaneouslyPS6 strongly uniformly practically stable if PS2 and PS4 hold simultaneouslyRemark 1.
If
q
t
,
x
=
x
,
t
k
=
+
∞
,
and
S
k
t
0
,
t
k
=
C
−
τ
,
0
,
R
n
, then Definitions 2 and 3 reduce to the concepts of classic Lyapunov stability.
It is well known that the comparison principle plays an important role in the development of stability theory. By the comparison principle, we can reduce the study of a given complicated differential system to that of a relatively simpler differential equation. For this purpose, we give the following lemma and definition.Lemma 1 (See [1]).
Assume that the following conditions holdA1mt∈CR+,R+,gt,u∈CR+×R+,R and for any t∈Tk such that mt>mt+s for s∈−τ,0, the inequality(6)D+mt≤gt,mt,holds, whereD+mt=limsuph⟶0+1/hmt+h−mt, gt,0≡0A2 the maximal solution rt≡rt;t0,u0 of the scalar equation(7)u˙=gt,u,ut0=u0,exists, ont0,+∞. Then, mt≤rt,t≥t0, provided maxs∈−τ,0mt0+s≤u0.Definition 4.
Comparison equation (7) is said to be PS7 practically stable if for given λ,A with 0<λ<A and some t0∈R+, we have u0<λ implies ut<A, for t≥t0PS8 uniformly practically stable if PS7 holds for all t0∈R+PS9 practically quasistable if for given λ,B,T with 0<λ<A, B>0, T>0, and some t0∈R+, we have that u0<λ implies ut<B, for t≥t0+TPS10 uniformly practically quasistable if PS9 holds for all t0∈R+Theorem 1.
Assume that the following conditions hold.A3qt,x,Tk,λ,A with 0<λ<A are givenA4 there exists a function V∈CTk×DA,R+ and V∈Λ such that(i)
for anyt>t0, Vt,xt>Vt+s,xt+s, s∈−τ,0, the inequality(8)D2.1+Vt,xt≤gt,Vt,xt,
holds, whereg∈CTk×R+,R and gt,0≡0
(ii)
b
q
t
,
x
≤
V
t
,
x
≤
a
x, where a⋅, b⋅∈K and aλ<bA
Then, equation (7) is (uniformly) practically stable with respect to aλ,bA implies that system (1) is (uniformly) practically stable with respect to qt,x,Tk,λ,A.Proof.
Assume thatut;t0,u0 is a solution of the equation (7), and is practically stable with respect to aλ,bA for given 0<λ<A. Let mt=Vt,xt, where xt is a solution of the systems (1). From the condition (i) of A4, it follows that D+mt≤gt,mt, for t≥t0. Let(9)ut0=maxs∈−τ,0Vt0+s,φs=maxs∈−τ,0mt0+s.
By Lemma1, we know that the inequality Vt,xt=mt≤rt,fort≥t0, holds, where rt is the maximal solution of comparison equation (7) existing on Tk. Assume that maxs∈−τ,0φs<λ, then, we have(10)ut0=maxs∈−τ,0Vt0+s,φs≤maxs∈−τ,0aφs=amaxs∈−τ,0φs<aλ.
Furthermore, from the condition (ii) ofA4 and Lemma 1, we get bqt,x≤Vt,x=mt≤rt<bA. Thus, maxs∈−τ,0φs<λ implies qt,x<A, t≥t0, that is, system (1) is practically stable with respect to qt,x,Tk,λ,A.
Similarly, we can prove that equation (7) is uniformly practically stable with respect to aλ,bA implies that the systems (1) is uniformly practically stable with respect to qt,x,Tk,λ,A. The proof is completed.
By Theorem1, we can obtain the following corollaries.Corollary 1.
Assume that the conditionsA3 and (ii) of (A4) hold in Theorem 1, andA5 there exists a function V∈CTk×DA,R+ and V∈Λ such that for any t>t0, Vt,xt>Vt+s,xt+s, s∈−τ,0, the inequality(11)D2.1+V≤0holds. Then, system (1) is uniformly practically stable with respect to qt,x,Tk,λ,A.The conclusion of Corollary1 can be obtained by considering the case of gt,u≡0 and u˙=0 is uniformly practically stable with respect to aλ,bA for given 0<λ<A.Corollary 2.
Assume that the conditionsA3 and (ii) of A4 hold in Theorem 1, andA6 there exists a function Vt,x∈CTk×DA,R+ and V∈Λ such that for any t>t0, Vt,xt>Vt+s,xt+s, s∈−τ,0, the inequality(12)D2.1+V≤αtFVt,x,holds, whereF∈CR+,R+ and 0<FV≤VA7 the inequalities(13)∫t0tαtdt≤M<+∞,M≤lnbAaλ,t∈t0,+∞,holdThen, system (1) is uniformly practically stable with respect to qt,x,Tk,λ,A.Proof.
By Theorem1, we only prove that the system u˙=αtFu is uniformly practically stable with respect to aλ,bA. In fact, let ut0=ut0;t0,u0=maxs∈−τ,0Vt0+s,φs. Assume that maxs∈−τ,0φs<λ, it follows from the condition (ii) of A4 that(14)ut0;t0,u0≤maxs∈−τ,0aφs=amaxs∈−τ,0φs<aλ.
Furthermore, by conditionA7, the inequality(15)ut;t0,u0≤u0exp∫t0tαsds<aλeM<bA,holds. Then, system (1) is uniformly practically stable with respect to qt,x,Tk,λ,A. The proof is completed.Corollary 3.
Assume that the conditions (A3) and (ii) of (A4) hold in Theorem 1, andA8 there exists a function V∈CTk×DA,R+ and V∈Λ such that for any t>t0, Vt,xt>Vt+s,xt+s for s∈−τ,0, the inequality(16)D2.1+V≤−αFVt,x+β,holds, in whichα and β are positive constants, 0<FV≤VA9 the inequalities(17)aλ<bA,aλ+βα≤bA,holdThen, system (1) is uniformly practically stable with respect to qt,x,Tk,λ,A.Proof.
In fact, we only need to prove that the systemu˙=−αFu+β is uniformly practically stable with respect to aλ,bA. Let ut0=ut0;t0,u0=maxs∈−τ,0Vt0+s,φs; then, we have(18)ut0;t0,u0≤maxs∈−τ,0aφs=amaxs∈−τ,0φs<aλ.
Furthermore, by conditionA8, the inequality(19)ut;t0,u0≤u0e−αt−t0+βα<aλ+βα≤bAholds. Then, system (1) is uniformly practically stable with respect to qt,x,Tk,λ,A.Theorem 2.
Assume that the conditions (A3) and (ii) of (A4) hold in Theorem 1, and (iii) Vt,x≥bqt,x,Vt0,x≤at0,x, where b⋅∈K,at0,⋅∈K∗.Then, equation (7) is (uniformly) practically stable with respect to at0,λ,bA implies that the systems (1) is (uniformly) practically stable with respect to qt,x,Tk,λ,A.Proof.
In fact, by the condition (iii) ofA4, we have(20)ut0=maxs∈−τ,0Vt0+s,φs≤maxs∈−τ,0at0,φs=at0,maxs∈−τ,0φs.
Then, we can get the result by using a method similar to Theorem1. We omit its details.Theorem 3.
Assume that the following conditions holdA10qt,x,Tk,λ,A,B,T with 0<λ<A, 0<B<A and T>0 are givenA4 there exists a function V∈CTk×DA,R+ and V∈Λ such that(i)
for anyt>t0, Vt,xt>Vt+s,xt+s, s∈−τ,0, the inequality(21)D2.1+Vt,xt≤gt,Vt,xt,
holds, whereg∈CTk×R+,R, gt,0≡0
(ii)
b
q
t
,
x
≤
V
t
,
x
≤
a
x, where a⋅, b⋅∈K and aλ<bA
Then, equation (7) is (uniformly) practically quasistable with respect to aλ,bB,T implies that system (1) is (uniformly) practically quasistable with respect to qt,x,Tk,λ,B,T.Proof.
Assume thatut;t0,u0 is a solution of equation (7) and is practically quasistable with respect to aλ,bB,T for given 0<λ<A, 0<B<A and T>0. Let mt=Vt,xt, where xt is a solution of system (1). It follows from the condition A4 that(22)D+mt≤gt,mt,fort≥t0.
Letut0=maxs∈−τ,0Vt0+s,φs=maxs∈−τ,0mt0+s. Then, by Lemma 1, we know that the inequality(23)Vt,x=mt≤rt,fort≥t0,holds, where rt is the maximal solution of comparison equation (7) existing on Tk. Assume that φs∈C−τ,0,Rn and maxs∈−τ,0φs<λ. Then, we obtain(24)ut0=maxs∈−τ,0Vt0+s,φs≤maxs∈−τ,0aφs=amaxs∈−τ,0φs<aλ.
Furthermore, by comparison equation (7) which is practically quasistable with respect to aλ,bB,T, the condition A4, and Lemma 1, we get(25)bqt,x≤Vt,x=mt≤rt<bB,fort≥t0+T.
Thus,maxs∈−τ,0φs<λ implies qt,x<B, t≥t0+T, that is, system (1) is practically quasistable with respect to qt,x,Tk,λ,B,T.
Similarly, we can prove that equation (7) is uniformly practically stable with respect to aλ,bB,T implies that the systems (1) is uniformly practically stable with respect to qt,x,Tk,λ,B,T.Theorem 4.
Assume that the conditions (A10) and (i) of (A4) hold in Theorem 3, and the condition (ii) of (A4) is replaced by(iv)
Vt,x≥bqt,x,Vt0,x≤at0,x, where b⋅∈K,at0,⋅∈K∗
Then, equation (7) is (uniformly) practically quasistable with respect to aλ,bB,T implies that the systems (1) is (uniformly) practically quasistable with respect to qt,x,Tk,λ,B,T.The proof of Theorem4 is similar to that of Theorem 3, so we omit its details.


## 3. Integral Stability

The concept of integral stability, which was introduced for ordinary differential equations by Vrhoc in 1959 [26] and Lakshmikantham in 1969 [27], enlarges the collection of dynamical properties of solutions which can be investigated by the direct Lyapunov method. The integral stability theory has been rapidly developed recently. For example, Martynyuk [28], Salvadori and Visentin [29], Soliman and Abdalla [30] obtained the integral stability criteria for nonlinear differential equations, respectively; Hristova [31] obtained the integral stability in terms of two measures for impulsive differential equations; and Sood and Srivastava [32] gave the φ0-integral stability criteria for impulsive differential equations. The main purpose of this section is to discuss the integral stability of singular differential systems with maxima and its perturbed systems.Consider singular differential system (1) and its perturbed systems(26)Ex˙=ft,x,maxs∈t−τ,txs+ht,x,maxs∈t−τ,txs,fort≥t0≥0,xt0=φt,t∈−τ,0,where h∈CR+×Rn×Rn,Rn, ht,0,0≡0.LetSpkt0 be a set of all consistent initial functions of (1) and (26) in t0,tk through t0,φ. For any φ∈Spkt0, assume that there exists a continuous solution of (1) and (26) in t0,tk through t0,φ at least.(27)Bφ,δ=ψ∈C−τ,0,Rnmaxs∈−τ,0ψs−φs<δ,φ∈C−τ,0,Rn,δ∈R+.Definition 5.
Letφ∈Spkt0. Singular system (1) is said to beIS1 equi-integrally stable on qt,x,Tk, if for given α≥0 and t0∈Tk, there exists a positive function β=βt0,α, which is continuous in t0 for each α and β∈K, such that, for every solution xt;t0,φ of the perturbed systems (26),(28)qt,x∈Dβ,t≥t0,holds, provided thatφ∈B0,α∩Spkt0 and(29)∫t0t0+Tsupqt,x≤βhs,x,maxu∈s−τ,sxuds≤α,forT>0;IS2 uniformly integrally stable on qt,x,Tk, if the β in IS1 is independent of t0IS3 equiasymptotically integrally stable on qt,x,Tk, if IS1 holds and for every ϵ>0, α≥0 and t0∈Tktk=+∞, there exist positive functions T=Tt0,α,ϵ and γ=γt0,α,ϵ, which are continuous in t0 for each α and ϵ, and for every solution xt;t0,φ of the perturbed systems (26),(30)qt,x∈Dϵ,t≥t0+T,holds, provided thatφ∈B0,α∩Spt0, and(31)∫t0+∞supqt,x≤βhs,x,maxu∈s−τ,sxuds≤γ,IS4 uniformly asymptotically integrally stable on qt,x,Tk if the T and γ in IS3 are independent of t0 and IS2 holds
Now, we consider comparison scalar differential equation (7) and its perturbed equation(32)u˙=gt,u+pt,ut0=u0,where gt,0≡0, g∈CTk×R+,R, pt∈CTk,R+.Definition 6.
Equation (7) is said to be equi-integrally stable, if for given α1≥0,t0∈Tk, there exists a positive function β1=β1t0,α1 that is continuous in t0 for each α1 and β1∈K, such that, for every solution ut;t0,u0 of the perturbed differential equation (32), the inequality(33)ut;t0,u0<β1,holds, provided that u0≤α1 and ∫t0t0+Tpsds≤α1 for every T>0.Remark 2.
Similar to Definition5, we can give the corresponding concepts of stability of equation (7).
Next, we investigate the integral stability of system (1) via the Lyapunov function method and comparison principle.Theorem 5.
Assume that the condition (i) of (A4) holds in Theorem 1, and condition (ii) of (A4) is replaced by(34)A4′bqt,x≤Vt,x,t≥t0,whereb∈K,andbr⟶+∞asr⟶+∞.Then, equation (7) which is equi-integrally stable implies that system (1) is equi-integrally stable on qt,x,Tk.Proof.
Letφ∈B0,α∩Spkt0 for every α≥0, t0∈Tk. Since Vt,x is Lipschitzian in x, we have(35)Vt,x−Vt,y≤Lx−y,L>0is a constant,maxs∈−τ,0Vt0+s,φs≤Lmaxs∈−τ,0φs≤Lα=α1.
Letxt=xt;t0,φ be any solution of (26). Thus, by the condition (i) of A4 and (29), we get(36)D21+Vt,x≤gt,Vt,x+Lht,x,maxs∈t−τ,txs.
Definingpt=Lht,x,maxs∈t−τ,txs and choosing u0=maxs∈−τ,0Vt0+s,φs, by Lemma 1, we have(37)Vt,x≤rt;t0,u0,where rt;t0,u0 is the maximal solution of (32).
If equation (7) is equi-integrally stable, then for α1≥0 and t0∈Tk, there exists a β1=β1t0,α1, which is continuous in t0 for each α1 and β1∈K, such that, for every solution ut;t0,u0 of (32), the inequality(38)ut;t0,u0<β1,t≥t0holds, whenever u0≤α1 and ∫t0t0+Tpsds≤α1 for any T>0.
By the conditionA4′, it is possible to choose a β=βt0,α satisfying(39)bβ≥β1.
It is easily shown thatβ is continuous in t0 for each α and β∈K for each t0∈Tk. Moreover, we claim that system (1) is equi-integrally stable on qt,x,Tk. In fact, if this is not true, there exists a t1>t0 such that(40)qt1,x=β,andqt,x<β,t∈t0,t1.
From (37)–(40), we have(41)β1≤bβ≤Vt1,xt1;t0,φ≤rt1;t0,u0<β1,which is a contradiction. Thus, system (1) is equi-integrally stable on qt,x,Tk.Corollary 4.
Assume that the conditions of Theorem5 hold. Then, equation (7) which is uniformly integrally stable implies that system (1) is uniformly integrally stable on {q(t; x); Tk}.The detailed proof of Corollary4 is similar to the proof in Theorem 5, so we omit it.Theorem 6.
Assume that the conditions of Theorem5 hold and tk = +∞ Then, equation (7) which is equiasymptotically integrally stable implies that system (1) is equiasymptotically integrally stable on {q(t. x), R+}.Proof.
It can be known from the proof of Theorem5 that system (1) is equi-integrally stable on qt,x,R+. Let φ∈B0,α∩Spt0,+∞ and α1=Lα, for given α≥0 and t0∈0,+∞. For given .., α1≥0 and t0∈R+, there exist γ1=γ1t0,α1,ε and T=Tt0,α1,ε, such that for every solution ut;t0,u0 of (32),(42)ut;t0,u0<bε,t≥t0+T,holds, whenever u0=maxs∈−τ,0Vt0+s,φs≤α1 and ∫t0∞psds<γ1.
Choosing a positive numberγ=γt0,α,ϵ such that Lγ=γ1, for given γ and T, system (1) satisfies IS3 of Definition 5. In fact, suppose that the conclusion is not true, then qt,x∈Dϵ, t≥t0+T cannot be satisfied when(43)φ∈B0,α∩Spt0,+∞and∫t0+∞supqt,x≤βhs,x,maxu∈s−τ,sxuds≤γ.
Lettk be a sequence such that tk≥t0+T and limk⟶+∞tk=+∞. Suppose that there is a solution xt=xt;t0,φ of system (26), such that for every k,(44)qtk,xtk;t0,φ≥ε.
By the conditionA4′ and (44), we obtain(45)bε≤bqtk,xtk;t0,φ≤Vtk,xtk;t0,φ.
Furthermore, by the equiasymptotical integral stability of equations (7), (37), and (42)–(59), we can get(46)bϵ≤Vtk,xtk;t0,φ≤rtk;t0,u0<bϵ.which is a contradiction. Thus, system (1) is equiasymptotically integrally stable on qt,x,0,+∞.Corollary 5.
Assume that the conditions of Theorem6 hold. Then, equation (7) which is uniformly asymptotically integrally stable implies that system (1) is uniformly asymptotically integrally stable on {q(t; x); Tk}.In fact, we can show that the positive numbersT and γ1 in proof of Theorem 6 are independent of t0; therefore, (44) implies that γ is independent of t0. The rest of the proof is similar to that of Theorem 6, so we omit the details here.For the comparison equation (7), if we suppose that the function gt,u is nonincreasing in u for t∈t0,+∞; then, we can get the uniform asymptotic integral stability of system (1) by the uniform asymptotic stability of the comparison equation (7). Therefore, we firstly give the following definition and Lemmas, which can be found in [27].Definition 7.
Comparison equation (7) is said to beS1 equistable, if for each ϵ>0, t0∈R+, there exists a positive function δ=δt0,ϵ, such that(47)u0≤δ⟹ut;t0,u0<ε,fort≥t0,S2 uniformly stable if δ in S1 is independent of t0S3 equiasymptotically stable, if it is equistable and for each ϵ>0, t0∈R+, there exists a positive function δ0=δ0t0, T=Tt0,ϵ, such that(48)u0≤δ0⟹ut;t0,u0<ϵ,fort≥t0+T,S4 uniformly asymptotically stable if S2 holds and the numbers δ0 and T in S3 are independent of t0Lemma 2 (see [27]).
Equation (7) is uniformly stable if and only if there exists a function a(r) ∈ K, such that(49)u0≤δ⟹ut;t0,u0≤au0,fort≥t0.Lemma 3 (see [27]).
Equation (7) is uniformly asymptotically stable if and only if there exist functions a(r) ∈ K and σr∈S, such that(50)u0≤δ⟹ut;t0,u0≤au0σt−t0,fort≥t0,where F=σr∈CR+,R+σr is monotone decreasing in r, and σr⟶0 as r⟶+∞.Theorem 7.
Assume that the conditions of Theorem6 hold, the function gt;u be nonincreasing in u for any t∈t0,+∞, and(51)Vt,xt>Vt+s,xt+s−L∫tt+shv,x,maxu∈v−τ,vxudv,fors∈−τ,0.Then, equation (7) which is uniformly asymptotically stable implies that system (1) is uniformly asymptotically integrally stable on qt,x,Tk.Proof.
Firstly, we prove system (1) is uniformly integrally stable. Because equation (7) is uniformly stable, by Lemma 2, there exists a function β1∈K, such that(52)0<u0≤α1impliesut;t0,u0<β1u0,t≥t0.
For givenα>0 and t0∈Tk, where Lα=α1, let φ∈B0,α∩Spt0. Since Vt,x is Lipschitzian in x, we have(53)maxs∈−τ,0Vt0+s,φs≤Lmaxs∈−τ,0φs≤Lα=α1.
Letxt=xt;t0,φ be any solution of (26) with φ∈B0,α∩Spt0, and(54)Vt,xt;t0,φ≡mt+pt,where pt=L∫t0ths,x,maxu∈s−τ,sxuds. According to the condition (i) of A4, we get(55)D+mt≤D3.1+Vt,xt−Lht,x,maxs∈t−τ,txs≤D2.1+Vt,xt≤gt,mt.
By (51) and Lemma 1, we have(56)mt≤rt;t0,u0,where rt;t0,u0 is the maximal solution of (7) with u0=maxs∈−τ,0mt0+s.
We choosingβ>0, such that(57)bβ>β1Lα+Lα.
In view of the conditionbr⟶+∞ as r⟶+∞, the choice of β is reasonable. It is obvious that β=βα and β∈K. At the same time, we can claim that system (1) is uniformly integrally stable. In other words, the solution of system (26) satisfies qt,x∈Dβ, t≥t0, whenever φ∈B0,α∩Spt0, and(58)∫t0t0+Tsupqt,x≤βhs,x,maxu∈s−τ,sxuds≤α,forT>0.
Suppose that this is not true; there exists at1>t0 such that(59)qt1,xt1d=βandqt,xt≤β,t∈t0,t1.
From the conditionA4′ and (54)–(58), we get(60)bβ≤Vt1,xt1≤rt1;t0,u0+pt1≤β1Lα+L∫t0t1supqt,x≤βhs,x,maxu∈s−τ,sxuds≤β1Lα+Lα<bβ.
This is a contradiction, and then system (1) is uniformly integrally stable.
Secondly, we prove that system (1) is uniformly asymptotically integrally stable. By the uniform asymptotic stability of equation (7) and Lemma 3, we have(61)ut;t0,u0≤β1u0σt−t0,t≥t0,where β1∈K and σ∈F. For given ϵ>0, α≥0 and t0∈Tk, let φ∈B0,α∩Spt0, and(62)L∫t0+∞supqt,x≤βhs,x,maxu∈s−τ,sxuds≤Lγ1<bε,where γ=minγ1,α. For any solution xt=xt;t0,φ of (26) and (56), holds whenever u0=maxs∈−τ,0Vt0+s,φs. By (54), (56), and (61), together with A4′, we can obtain the inequality(63)bqt,xt≤Vt,xt≤rt;t0,u0+L∫t0ths,x,maxu∈s−τ,sxuds≤β1Lασt−t0+L∫t0tsupqt,x≤βhs,x,maxu∈s−τ,sxuds<β1Lασt−t0+Lγ.
Sinceσ∈F, then there exists a T=Tα,ϵ, such that(64)σt−t0<bϵ−Lγβ1Lα,t≥t0+T.
Furthermore, we have(65)bqt,xt<bϵ,t≥t0+T,which implies that(66)qt,xt<ϵ,t≥t0+T,provided φ∈B0,α∩Spt0, and (62) is satisfied. Therefore, system (1) is uniformly asymptotically integrally stable.


## 4. Conclusion

This paper discussed a class of nonlinear singular differential systems with maxima. Some notions of practical stability and integral stability for such systems were introduced, and various stability criteria were obtained by using the Lyapunov method and comparison principle.



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*Source: 4792183-2020-04-14.xml*