Practical Stability and Integral Stability for Singular Differential Systems with Maxima

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. )e 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 function q(t, x) and obtain some different types of stability criteria by using the Lyapunov function method and comparison principle.


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][4][5][6][7][8][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][11][12][13], difference equations [14][15][16][17], and delay differential equations [18][19][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. e 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 function q(t, x) and obtain some different types of stability criteria by using the Lyapunov function method and comparison principle.

Practical Stability
e 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][22][23][24][25]). In this section, by using Lyapunov functions and the comparison principle, we study some practical stability for the following singular differential systems.
Firstly, we introduce the following notations and sets for convenience. Let is a set of all consistent initial functions at initial time t 0 . en, for any φ ∈ S k (t 0 ), there exists at least one continuous solution of systems (1) in [20]).
is strictly increasing and a(0) � 0} K * � a(t 0 , r) ∈ C(T k × R + , R + ) | a(t 0 , r); is strictly increasing in r and a(t 0 , 0) � 0} We denote by x(t) ≡ x(t; t 0 , φ) the solution of the initial value problems (1). Definition 1. Let V ∈ Λ, t ∈ T k , and we define the derivative of the function V (t; x) along the trajectory of solution of the singular systems (1) as follows: Definition 2. Let φ ∈ S k (t 0 ). e singular systems (1) is said to be Definition 3. Let φ ∈ S k (t 0 ). e singular systems (1) is said to be (PS 1 ) practically stable for given (λ, A) with 0 < λ < A and some t 0 ∈ T k , such that (PS 2 ) uniformly practically stable if (PS 1 ) holds for all t 0 ∈ T k (PS 3 ) practically quasistable for given (λ, B, T) with λ, B, T > 0, and some t 0 ∈ T k , we have (PS 4 ) uniformly practically quasistable if (PS 1 ) holds for all t 0 ∈ T k (PS 5 ) strongly practically stable if (PS 1 ) and (PS 3 ) hold simultaneously (PS 6 ) strongly uniformly practically stable if (PS 2 ) and (PS 4 ) hold simultaneously 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 hold
Definition 4. Comparison equation (7) is said to be (PS 7 ) practically stable if for given (λ, A) with 0 < λ < A and some holds for all t 0 ∈ R + (PS 9 ) practically quasistable if for given (λ, B, T) with 0 < λ < A, B > 0, T > 0, and some t 0 ∈ R + , we have that u 0 < λ implies u(t) < B, for t ≥ t 0 + T (PS 10 ) uniformly practically quasistable if (PS 9 ) holds for all t 0 ∈ R + Theorem 1. Assume that the following conditions hold en, equation (7) is (uniformly) practically stable with respect to (a(λ), b(A)) implies that system (1) is Proof. Assume that u(t; t 0 , u 0 ) is a solution of the equation (7), and is practically stable with respect to (a(λ), b(A)) for By Lemma 1, we know that the inequality V(t, x(t)) � m(t) ≤ r(t), for t ≥ t 0 , holds, where r(t) is the maximal solution of comparison equation (7) existing on T k . Assume that max s∈[−τ,0] ‖φ(s)‖ < λ, then, we have Furthermore, from the condition (ii) of (A 4 ) and Lemma 1, Similarly, we can prove that equation (7) is uniformly practically stable with respect to (a(λ), b(A)) implies that the systems (1) is uniformly practically stable with respect to (q(t, x), T k , λ, A). e proof is completed.
e conclusion of Corollary 1 can be obtained by considering the case of g(t, u) ≡ 0 and _ u � 0 is uniformly practically stable with respect to (a(λ), b(A)) for given 0 < λ < A.

Corollary 2.
Assume that the conditions (A 3 ) and (ii) of (A 4 ) hold in eorem 1, and hold en, system (1) is uniformly practically stable with respect to (q(t, x), T k , λ, A).

Corollary 3. Assume that the conditions (A 3 ) and (ii) of (A 4 ) hold in eorem 1, and (A 8 ) there exists a function V ∈ C(T k × D(A), R + ) and
V ∈ Λ such that for any t > t 0 , V(t, x(t)) > V(t + s, x (t + s)) for s ∈ [−τ, 0), the inequality holds, in which α and β are positive constants,

hold en, system (1) is uniformly practically stable with respect to (q(t, x), T k , λ, A).
Proof. In fact, we only need to prove that the system _ u � −αF(u) + β is uniformly practically stable with respect to Furthermore, by condition (A 8 ), the inequality holds. en, system (1) is uniformly practically stable with respect to (q(t, x), T k , λ, A).
Proof. In fact, by the condition (iii) of (A 4 ), we have en, we can get the result by using a method similar to eorem 1. We omit its details.
Similarly, we can prove that equation (7) is uniformly practically stable with respect to (a(λ), b(B), T) implies that the systems (1) is uniformly practically stable with respect to (q(t, x), T k , λ, B, T).

□ Theorem 4. Assume that the conditions (A 10 ) and (i) of (A 4 ) hold in eorem 3, and the condition (ii) of (A 4 ) is replaced by
en, equation (7) is (uniformly) practically quasistable with respect to (a(λ), b(B), T) implies that the systems (1) is (uniformly) practically quasistable with respect to (q(t, x), T k , λ, B, T). e proof of eorem 4 is similar to that of eorem 3, so we omit its details.

Integral Stability
e 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. e 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 φ 0 -integral stability criteria for impulsive differential equations. e 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 where h ∈ C(R + × R n × R n , R n ), h(t, 0, 0) ≡ 0. Let S pk (t 0 ) be a set of all consistent initial functions of (1) and (26) in [t 0 , t k ) through (t 0 , φ). For any φ ∈ S pk (t 0 ), assume that there exists a continuous solution of (1) and (26) in [t 0 , t k ) through (t 0 , φ) at least.
Remark 2. Similar to Definition 5, 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 (A 4 ) holds in eorem 1, and condition (ii) of (A 4 ) is replaced by
and b(r) ⟶ + ∞ as r ⟶ + ∞.