1 Introduction

Impulsive differential equations have been one object of numerous investigations during the last few years because of their wide applications in applied sciences^{[1, 2, 3, 4, 5]}.

Differential equations with maxima may be used for the mathematical simulation of processes which are characterized by a dependence at each moment of time on the maximum value over a past time interval. Examples of such processes are described in the theory of automatic control by Popov^[6]. The necessity for the analysis of the maximum function and its impulsive changes in control theory,pharmacokinetics,economics,etc. leads to the study of impulsive differential equations with supremums. In investigation of the properties of such systems,namely dynamics,qualitative behaviors,and possibilities of control though mathematical models,one may turn to the fields where similar problems can be found. Examples include classical mechanics, solid mechanics,discrete mechanics,continuum mechanics,fluid mechanics,nonsmooth mechanics and so on. The properties of the supremum function make an impulsive differential equation with supremums strongly non-linear. Such kind of impulsive systems can be applied in engineering design problems^[7]. Therefore,it requires an independent study for its qualitative theory.

The stability analysis of nonlinear systems with disturbance input or/and hereditary information is a very important problem in control theory and engineering^{[8, 9, 10]}. External distur-bances from input and time-delay lead to the loss of stability of an otherwise stable system, where impulses can be considered as a control.

Due to the current dynamics in the latest advances in impulsive differential equations with supremums,a considerable amount of research related to equations with fixed moments of impulsive effects appeared^{[11, 12, 13, 14, 15]}. Difficulties that have arisen in the investigation of impulsive differential equations with variable impulsive perturbations include the phenomena “beating” of the solutions,bifurcation,and loss of the property of autonomy. Wider applications of this type of equations require the formulation of effective criteria for the stability of their solutions. However,to the best of our knowledge,there is seldom work so far dedicated to the investigation of the qualitative properties for impulsive differential equations with supremums and variable impulsive perturbations. Therefore,it is important to study the behavior of such models and the effect of impulses at the attractivity. Such equations are suitable for modeling a number of engineering processes,such as power and voltage of the ionic current at electrolysis. An accumulation of insulating film on the electrodes is observed in this technology. Empirically,it is found that the current breaks film randomly,which means that the current power changes rapidly.

In the present paper,problems related to the global stability of the solutions to impulsive differential equations with “supremum” and variable impulsive perturbations are considered. By employing a class of piecewise continuous functions which are generalization of the classical Lyapunov’s functions coupled with the Razumikhin technique,sufficient conditions are obtained. Some examples are also discussed to illustrate the effectiveness of our results.

2 Preliminary notes and definitions

Let r = const.≥0,R₊ = [0,∞),and Rⁿ be the n-dimensional Euclidean linear space equipped with the norm

Let x = x(s),satisfying We denote

Consider the following system of impulsive differential equations with “supremum”:

where

Let τ₀(x) ≡ t₀ for x ∈ R_n. We introduce the following conditions:

H₁ τ_k ∈ C[R_n,(t₀,∞)] (k = 1,2,· · · ).

H₂ t₀ < τ1(x) < τ2(x) < · · · (x ∈ R_n).

H₃ τ_k(x) → ∞ as k → ∞ uniformly on x ∈ R_n.

Assuming that conditions H₁,H₂,and H₃ are fulfilled,we introduce the following notations:

i.e.,σ_k (k = 1,2,· · · ) are hypersurfaces with the equations t = τ_k(x(t)).

Let J ⊆ R. Define the following class of functions:

PC[J,R_n] = {σ : J → Rⁿ : σ(t) is a continuous function with points of discontinuity t ∈ J at which σ(t − 0) and σ(t + 0) exist and σ(t − 0) = σ(t).

LetΨ₀ ∈ PC[[−r,0],R_n]. Denote by x(t) = x(t; t₀,Ψ₀) the solution to system (1),satisfying

and by J⁺(t₀,Ψ₀) the maximal interval of the type [t₀,β),at which the solution x(t; t₀,Ψ₀) is defined.

Introduce the following notations:

(i) ||Φ|| = |Φ(t − t₀)| is the norm of the function Φ ∈ PC[[−r,0],R_n];

(ii) K = {a ∈ C[R₊,R₊] : a(r) is strictly increasing and a(0) = 0}.

Introduce the following conditions:

H₄ f ∈ C[[t₀,∞) × Rⁿ × R_n,R_n].

H₅ f(t,0,0) = 0,t ∈ [t₀,∞).

H₆ The function f is Lipschitz continuous with respect to its second and third arguments in [t₀,∞) × Rⁿ × R_n,uniformly on t ∈ [t₀,∞).

H₇ There exists a constant P≥0 such that

H₈ I_k ∈ C[R_n,R_n],k = 1,2,· · · .

H₉ I_k(0) = 0,k = 1,2,· · · .

H₁₀ The integral curves of system (1) meet successively each one of the hypersurfaces σ₁,σ₂,· · · exactly once.

The condition H₁₀ guarantees the absence of the phenomenon “beating” of the solutions to system (1),i.e.,a phenomenon when a given integral curve meets more than one time or infinitely many times and the same hypersurface σ_k. Efficient sufficient conditions which guarantee the absence of “beating” of the solutions to such systems are proven by Bainov and Dishliev^[16].

Let t₁,t₂ ,· · · (t₀ < t₁ < t₂ < · · · ) be the moments in which the integral curve (t,x(t; t₀,Ψ₀)) of problem (1),(2) meets the hypersurfaces σ_k (k = 1,2,· · · ).

We will note that[16−17] if the conditions H₁-H₄,H₆-H₈,and H₁₀ are met,then t_k → ∞ as k → ∞ and J⁺(t₀,Ψ₀) = [t₀,∞).

Definition 1 We say that the solutions to system (1) are uniformly bounded if

We will use the following definitions for the global stability of the zero solution to system (1).

Definition 2 The zero solution x(t) ≡ 0 to system (1) is said to be

(a) stable if

(b) uniformly stable if the number δ in (a) is independent of t₀ ∈ R₊;

(c) globally equi-attractive if

(d) uniformly globally attractive if the number in (c) is independent of t₀ ∈ R₊;

(e) globally equi-asymptotically stable if it is stable and globally equi-attractive;

(f) uniformly globally asymptotically stable if it is uniformly stable,uniformly globally at- tractive and the solutions to system (1) are uniformly bounded;

(g) globally exponentially stable if

Introduce the set

In further considerations,we shall use the class V₀ of piecewise continuous auxiliary functions V : [t₀,∞) × Rⁿ → R₊,which are analogues of Lyapunov’s functions^[18].

Definition 3 A function V : [t₀,∞) × Rⁿ → R₊ belongs to class V₀ if

(I) V is continuous in G and locally Lipschitz continuous with respect to its second argument on each of the sets G_k (k = 1,2,· · · ).

(II) For each k = 1,2,· · · and (t₀* ,x₀* ) ∈ σ_k,there exist the finite limits

and the equality V (t₀* − 0,x₀* ) = V (t₀* ,x₀* ).

Definition 4 Given a function V ∈ V₀. For t≥t₀,t≠τ_k(Φ(t)) (k = 1,2,· · · ),and Φ ∈ PC[[t−r,t],R_n],the upper right-hand derivative of V with respect to system (1) is defined by

Note that in Definition 4,D+V (t,Φ(t)) is a functional whereas V is a function. This special feature is a source of difficulties in the application of the second method of Lyapunov for differential equations of the form (1). The speed in (1) depends on the previous history as well, i.e.,it depends on the point x(t + s),which is usually hard to find. In order to find a positive definite function V ∈ V₀ such that D+V (t,Φ(t))≤0,the value Φ(t) has a dominant role. Using simple considerations,Razumikhin^[19] proved that the derivative D+V (t,Φ(t)) might be estimated only by the elements of minimal subsets of the integral curves of the investigated system when the following condition

holds. The condition (4) is called the Razumikhin condition,and the corresponding technique is known as Razumikhin technique^{[20, 21]}.

Introduce that the set PC¹[J,R_n] = {σ ∈ PC[J,R_n] : σ(t) } is continuously differentiable everywhere except some points t_k at which σ˙ (t_k − 0) and σ˙ (t_k + 0) exist and σ˙ (t_k − 0) = σ˙ (t_k) (k = 1,2,· · · ).

We remark that,if V ∈ V₀ and V ∈ PC¹[[t₀,∞) × R_n,R₊],then (3) reduces to

In the proof of the main results,we shall use the following lemma.

Lemma 1 Assume that

(i) conditions H₁-H₄,H₆-H₈,and H₁₀ hold;

(ii) the function g : [t₀,∞) × R₊ → R₊ is continuous in each of the sets (t_k−1,t_k] × R₊ (k = 1,2,· · · ),and g(t,0) = 0 for t≥t₀;

(iii) B_k ∈ C[R₊,R₊],B_k(0) = 0,and k(u) = u + B_k(u)≥0 (k = 1,2,· · · ) are non- decreasing with respect to u;

(iv) the maximal solution u⁺(t; t₀,u0) to the scalar problem

is defined in the interval [t₀,∞);

(v) the function V ∈ V₀ satisfies

and the inequality D+V (t,Φ(t))≤g(t,V (t,Φ(t))) is valid for any t ∈ [t₀,∞),t≠t_k (k = 1,2,· · · ) and any function Φ ∈ PC[[t − r,t],R_n] for which (4) is true.

Then,

Proof It follows that the solution x = x(t; t₀,Ψ₀) to problem (1),(2) and J⁺(t₀,Ψ₀) = [t₀,∞) satisfy

The maximal solution u⁺(t; t₀,u₀) to (5) is defined by

where rk(t; t_k,u⁺ _k ) is the maximal solution to the system without impulses in the interval (t_k,t_k+1] (k = 0,1,2,· · · ),for which Let t ∈ (t₀,t₁]. Then,from the corresponding comparison theorem for the continuous case^[22], it follows that V (t,x(t; t₀,Ψ₀))≤u⁺(t; t₀,u0),i.e.,the inequality (6) is valid for t ∈ (t₀,t₁].

Suppose that (6) is satisfied for t ∈ (t_k−1,t_k] (k≥1). Then,using condition (v) of Lemma 1 and the fact that the function k is non-decreasing,we obtain

We apply again the comparison theorem for the continuous case in the interval (t_k,t_k+1],and obtain

i.e.,the inequality (6) is valid for t ∈ (t_k,t_k+1].

In the case when g(t,u) = 0 for (t,u) ∈ [t₀,∞)×R₊ and k(u) = u for u ∈ R₊ (k = 1,2,· · · ), we deduce the following corollary from Lemma 1.

Corollary 1 Assume that

(i) conditions H₁-H₄,H₆-H₈,and H₁₀ hold;

(ii) the function V ∈ V₀ satisfies

and the inequality D+V (t,Φ(t))≤0 is valid for any t ∈ [t₀,∞),t≠τ_k(Φ(t)) (k = 1,2,· · · ) and any function Φ ∈ PC[[t − r,t],R_n] for which (4) is true.

Then,

3 Main results

In this section,we shall present the main results on the global stability of the zero solution to system (1). In the proofs of our main theorems in this section,we shall use piecewise continuous Lyapunov functions V : [t₀,∞) × Rⁿ → R₊ (V ∈ V₀) for which the following condition is true:

H₁₁ V (t,0) = 0,t≥t₀.

Theorem 1 Assume that

(i) conditions H₁-H₁₁ hold;

(ii) there exists a function V ∈ V₀ such that

and is valid for any t ∈ [t₀,∞),t≠τ_k(Φ(t)) (k = 1,2,· · · ) and any function Φ ∈ PC[[t − r,t],R_n] that satisfies (4).

Then,the zero solution to system (1) is globally equi-asymptotically stable.

Proof Letε> 0. From the properties of the function V ,it follows that there exists a constant δ = δ(t₀,ε)≥0 such that if x ∈ Rⁿ : |x| < δ,then V (t₀ + 0,x) < a(ε).

Let

be the solution to problem (1),(2). It follows from H₁-H₁₀ that J⁺(t₀,Ψ₀) = [t₀,∞).

Since all conditions of Corollary 1 are met,

Since |Ψ₀(0)|≤||Ψ₀|| < δ,V (t₀ + 0,Ψ₀(0)) < a(ε).

From (7),(8),and the last inequality,there follows a(|x(t; t₀,Ψ₀)|)≤V (t,x(t; t₀,Ψ₀))≤V (

which implies that |x(t; t₀,Ψ₀)| <ε for t≥t₀. This implies that the zero solution to system (1) is stable.

Now,we shall prove that it is globally equi-attractive.

Let α = const.≥0 andΨ₀ ∈ PC[[−r,0],R_n] : ||Ψ₀|| < α.

From conditions (8) and (9),it follows that for t≥t₀,the following inequality is valid:

Let

Then,for t≥t₀ + ,from (11),it follows that V (t,x(t; t₀,Ψ₀)) < a(ε). From the last inequality and (7),we have |x(t; t₀,Ψ₀)| <ε,which means that the zero solution to system (1) is globally equi-attractive.

Theorem 2 Assume that

(i) condition (i) of Theorem 1 holds;

(ii) there exists a function V ∈ V₀ such that (8) holds and

where a,b ∈ K,h : [t₀,∞) → [1,∞),and for c ∈ K,g : [t₀,∞) → (0,∞) and the inequality is valid for any t ∈ [t₀,∞),t≠τ_k(Φ(t)) (k = 1,2,· · · ) and any function Φ ∈ PC[[t − r,t],R_n] that satisfies (4);

(iii) for each sufficiently small value of η≥0,

Then,the zero solution to system (1) is globally equi-asymptotically stable.

Proof We can prove the stability of the zero solution to system (1) by the analogous arguments,as in the proof of Theorem 1.

Now,we shall prove that the zero solution to system (1) is globally equi-attractive.

Let α≥0 be arbitrary,ε≥0 be given,and η = . Let the number = (t₀,α,ε)≥0 be chosen so that

This is possible in view of condition (iii) of Theorem 2.

Let

be the solution to problem (1),(2). If we assume that for any t ∈ [t₀,t₀ + γ ],the following inequality holds: Then,by (14) and (15),it follows that From the above inequalities and (12) and (14) for t = t₀ + γ ,we obtain which contradicts (12). The obtained contradiction shows that there exists t* ∈ [t₀,t₀ + γ ] satisfying Then,for t≥t* (hence for any t≥t₀ + γ as well),the following inequality is valid: Therefore,|x(t; t₀,Ψ₀)| <ε for t≥t₀ + γ ,i.e.,the zero solution to system (1) is globally equi-attractive.

Theorem 3 Assume that

(i) condition (i) of Theorem 1 holds;

(ii) there exists a function V ∈ V₀ such that (8) holds,and

where a(u) → ∞ as u → ∞,and is valid for any t ∈ [t₀,∞),t≠τ_k(Φ(t)) (k = 1,2,· · · ) and any function Φ ∈ PC[[t − r,t],R_n] that satisfies (4).

Then,the zero solution to system (1) is uniformly globally asymptotically stable.

Proof First,we shall show that the zero solution to system (1) is uniformly stable.

For an arbitraryε≥0,choose the positive number δ = δ(ε) so that b(δ) < a(ε).

Let

be the solution to problem (1),(2). Then,by (16),(17),and (8),for any t ∈ J⁺(t₀,Ψ₀),the following inequality is valid:

Since J⁺(t₀,Ψ₀) = [t₀,∞),|x(t; t₀,Ψ₀)| <ε for t≥t₀. Thus,it is proved that the zero solution to system (1) is uniformly stable.

Now,we shall prove that the solutions to system (1) are uniformly bounded.

Let α≥0 andΨ₀ ∈ PC[[−r,0],R_n] : ||'0|| < α. For the function a ∈ K,we have a(u) → ∞ as u → ∞. Therefore,we can choose β = β(α)≥0 so that a(β)≥b(α). Since the conditions of Corollary 1 are met,V (t,x(t; t₀,Ψ₀))≤V (t₀ + 0,Ψ₀(0)) (t ∈ [t₀,∞)).

From the above inequality,(16),and (17),we have

Therefore,|x(t; t₀,Ψ₀)| < β for t≥t₀. This implies that the solutions to system (1) are uniformly bounded.

Finally,we shall prove that the zero solution to system (1) is uniformly globally attractive.

Let α≥0 be arbitrary andε>0 be given. Let the number η = η(ε)>0 be chosen so that b(η)>a(ε). Letγ = γ(α,ε)>0 satisfy γ> .

Let

be the solution to problem (1),(2). If we assume that the inequality |x(t; t₀,Ψ₀)|≥η holds for any t ∈ [t₀,t₀+γ ]. Then,by (17) and (8),it follows that which contradicts (16). The obtained contradiction shows that there exists t* ∈ [t₀,t₀ + γ ] such that |x(t*; t₀,Ψ₀)| < η. Then,for t > t* (hence for any t > t₀ + γ as well),the following inequality is valid:

Corollary 2 If in Theorem 3,(17) is replaced by

where t ∈ [t₀,∞),t≠τ_k(Φ(t)) (k = 1,2,· · · ),and the function Φ ∈ PC[[t − r,t],R_n] satisfies (4),then the zero solution to system (1) is uniformly globally asymptotically stable.

This follows immediately from Theorem 3. However,the proof can be carried out with the fact that V (t,x(t; t₀,Ψ₀))≤V (t₀ + 0,Ψ₀(0)) exp(−c(t − t₀)) for t≥t₀,which is obtained from (8) and (18).

Theorem 4 Assume that

(i) condition (i) of Theorem 1 holds;

(ii) there exists a function V ∈ V₀ such that (8) and (18) hold. For any α≥0,there existsγ(α)≥0 such that

Then,the zero solution to system (1) is globally exponentially stable.

Proof Let α≥0 be arbitrary. Let

be the solution to problem (1),(2). From (8) and (18),we have From the above inequality and (19),we obtain which implies that the zero solution to system (1) is globally exponentially stable.

4 Examples

Example 1 Let x ∈ R and r = const.≥0.

Consider the following impulsive equation:

where

We have τ_k ∈ C[R,(0,∞)] (k = 1,2,· · · ),τ_k(x) → ∞ as k → ∞ uniformly on x ∈ R,and 0 < τ₁(x) < τ₂(x) < · · · ,x ∈ R.

LetΨ₀ ∈ C[[−r,0],R] and define the function V (t,x) = 1 2x2.

If there exists a constant c≥0 such that |β(t)| + g(t)≤α(t) − c for t≥0,then for t≥0 and for any Φ ∈ PC[[t − r,t],R] satisfying V (t + s,Φ(t + s)) < V (t,Φ(t)) (s ∈ [−r,0)),we have

Thus,all conditions of Theorem 1 are satisfied,and the zero solution to (20) is globally equi-asymptotically stable.

Example 2 Consider the following equation:

where

LetΨ ∈ PC[[0,t],R],t≥0,and x(t) = x(t; 0,Ψ) be the solution to (21),satisfying the initial condition x(s) =Ψ(s) (s ∈ [0,t]).

Define the function V (t,x) = |x|. If

for t≥0 and c≥0,then for any t≥0,t≠τ_k(Φ(t)) (k = 1,2,· · · ) and any Φ ∈ PC[[t − r,t],R] satisfying V (s,Φ(s)) < V (t,Φ(t)),s ∈ [0,t),we have

For t = τ_k(x(t)) (k = 1,2,· · · ),we obtain

Thus,all conditions of Theorem 3 are satisfied,and the zero solution to (21) is uniformly globally asymptotically stable.

5 Conclusions

In this paper,by using a suitable piecewise continuous Lyapunov function and the Razumikhin technique,sufficient conditions for the global stability of the solutions to impulsive differential equations with supremums and variable impulsive perturbations are obtained. Since time-delays and impulses can affect the dynamical behaviors of the system,it is necessary to investigate both supremums over a past time interval and impulsive effects on the stabilization of the model. These play an important role in the design and applications of stable impulsive differential equations with supremums. By means of appropriate impulsive perturbations,we can control the stability properties of the model. The technique can be extended to study other impulsive delayed systems.