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International Journal of Differential Equations
Volume 2011, Article ID 703189, 13 pages
http://dx.doi.org/10.1155/2011/703189
Research Article

Practical Stability in terms of Two Measures for Impulsive Differential Equations with “Supremum”

Plovdiv University "Paisii Hiledarski", 4000 Plovdiv, Bulgaria

Received 19 May 2011; Accepted 6 August 2011

Academic Editor: Jianshe Yu

Copyright © 2011 S. G. Hristova and A. Georgiev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The object of investigations is a system of impulsive differential equations with “supremum.” These equations are not widely studied yet, and at the same time they are adequate mathematical model of many real world processes in which the present state depends significantly on its maximal value on a past time interval. Practical stability for a nonlinear system of impulsive differential equations with “supremum” is defined and studied. It is applied Razumikhin method with piecewise continuous scalar Lyapunov functions and comparison results for scalar impulsive differential equations. To unify a variety of stability concepts and to offer a general framework for the investigation of the stability theory, the notion of stability in terms of two measures has been applied to both the given system and the comparison scalar equation. An example illustrates the usefulness of the obtained sufficient conditions.

1. Introduction

One of the main qualitative problem in the theory of differential equations is stability. In some real world situations, it is desired that the state of a system may be mathematically unstable; however, the system may oscillate sufficiently close to the desired state, so that its performance is deemed acceptable. For example, an aircraft may oscillate around a mathematically, unstable path, yet its performance may be acceptable. In this case, it is appropriate to be used the so-called practical stability, which is defined and studied for various types of differential equations in [18]. To unify a variety of stability concepts and to offer a general framework for the investigation of the stability theory, the notion of stability in terms of two measures has been proved to be very powerful (see, e.g., [9] and cited therein references). One of the very useful methods of investigation of stability of solutions is Lyapunov-Razumikhin method combined with a comparison method. In fact, application of Lyapunov functions allows us to consider a scalar equation and to study stability properties of its solution. Sometimes, its solution is not stable in Lyapunov sense, applying regular norm. It requires using two measures not only for the given equation but also for the comparison equation.

In the last few decades, great attention has been paid to automatic control systems and their applications to computational mathematics and modeling. Many problems in control theory correspond to the maximal deviation of the regulated quantity (see [10]). Such kind of problems could be adequately modeled by differential equations that contain the maxima operator. Note that various conditions for stability for differential equations with “maxima” are obtained in [1, 1114].

In the paper, practical stability for a nonlinear system of impulsive differential equations involving the maximal value of the unknown function over a past time interval is studied. It is applied Razumikhin method with piecewise continuous scalar Lyapunov functions and comparison results for scalar impulsive differential equations. In this paper, differently than the existing up-to-date results, two different measures are applied to both the given system and the comparison scalar equation. Sufficient conditions for practical stability in the entire space (Theorem 2.10) as well as in a ball (Theorem 2.11) are obtained. The case of a ball requires more desired conditions than the global case. The main accent of the paper is consideration of (i)a new type of functional differential equations which contain the supremum of the unknown function over a past time interval; (ii)the practical stability, which is more stronger than the stability; (iii) two pairs of different measures for both the comparison scalar equation and the given system.

An example illustrates the usefulness of the obtained sufficient conditions.

2. Main Results

Let 𝑛 be 𝑛-dimensional Euclidean space with a norm , and +=[0,).

Let {𝜏𝑘}1 be a sequence of fixed points in + such that 𝜏𝑘+1>𝜏𝑘 and lim𝑘𝜏𝑘=. Let 𝑟>0 be a fixed constant.

Consider the system of nonlinear impulsive differential equations with “supremum”: 𝑥=𝑓𝑡,𝑥(𝑡),sup[]𝑠𝑡𝑟,𝑡𝑥(𝑠)for𝑡𝑡0,𝑡𝜏𝑘,𝑥𝜏𝑘+0=𝐼𝑘𝑥𝜏𝑘0for𝑘=1,2,,(2.1) with initial condition 𝑥𝑡(𝑡)=𝜙(𝑡)for𝑡0𝑟,𝑡0,(2.2) where 𝑥𝑛, 𝑓+×𝑛×𝑛𝑛, 𝐼𝑘𝑛𝑛, 𝑘=1,2,3,,  𝑡0+, and 𝜙[𝑡0𝑟,𝑡0]𝑛.

Note that for 𝑥[𝑡𝑟,𝑡]𝑛,𝑥=(𝑥1,𝑥2,,𝑥𝑛) we denote sup[]𝑠𝑡𝑟,𝑡𝑥(𝑠)=sup[]𝑠𝑡𝑟,𝑡𝑥1(𝑠),sup[]𝑠𝑡𝑟,𝑡𝑥2(𝑠),,sup[]𝑠𝑡𝑟,𝑡𝑥𝑛(𝑠).(2.3)

Denote by PC(𝑋,𝑌)(𝑋,𝑌𝑛) the set of all functions 𝑢𝑋𝑌 which are piecewise continuous in 𝑋 with points of discontinuity of the first kind at the points 𝜏𝑘𝑋 and which are continuous from the left at the points 𝜏𝑘𝑋, 𝑢(𝜏𝑘)=𝑢(𝜏𝑘0).

In our further investigations, we will assume that for any initial function 𝜙PC([𝑡0𝑟,𝑡0],𝑛) the solution of the initial value problem for system of impulsive differential equations with “supremum” (2.1) and (2.2) exists on [𝑡0𝑟,). Note that differential equations with maxima as well as impulsive differential equations with supremum are not well studied yet. Although these equations are delayed functional differential equations, not all results known in the literature for delay differential equations are applied to them. The main reason is the presence of maximum function which is very nonlinear one. Only some partial results for existence are obtained in the continuous case (see [15, 16]).

Let 𝑋+. Denote by 𝑍(𝑋) the set of all integers 𝑘 such that 𝜏𝑘𝑋.

We will define the following sets of measures:[Γ=𝐶𝑟,)×𝑛,+min𝑥𝑛[,(𝑡,𝑥)=0foreach𝑡𝑟,)Γ=𝐶,+min𝑢,(𝑢)=0Γ=𝐶+×,+min𝑢(𝑡,𝑢)=0foreach𝑡+,𝑡,𝑢1𝑡,𝑢2||𝑢for1||||𝑢2||,𝑡+.(2.4)

Remark 2.1. Note that any norm in 𝑛 is a function from Γ and any norm in is from both classes Γ and Γ. For example, the function (𝑡,𝑢)=𝑒𝑡Γ|𝑢|.

Let 0Γ, 𝑡0+, and 𝜙PC([𝑡0𝑟,𝑡0],𝑛). We will use the following notation: 𝐻0𝑡0,𝜙=sup𝑡𝑠0𝑟,𝑡00(𝑠,𝜙(𝑠)).(2.5)

Let 𝜌>0 be a fixed number and Γ. Define [𝑆(,𝜌)={(𝑡,𝑥)𝑟,)×𝑛(𝑡,𝑥)<𝜌},𝑆[(,𝜌)={(𝑡,𝑥)𝑟,)×𝑛(𝑡,𝑥)𝜌}.(2.6)

We will introduce the definition of a practical stability for impulsive differential equations with “supremum,” based on the ideas of stability in terms of two measures considered in [8, 9].

Definition 2.2. Let the functions ,0Γ and the positive constants 𝜆,𝐴 be given. The system of impulsive differential equations with “supremum” (2.1) is said to be(S1)practically stable with respect to (𝜆,𝐴) in terms of both measures (0,) if there exists 𝑡00 such that for any 𝜙PC([𝑡0𝑟,𝑡0],𝑛) inequality 𝐻0(𝑡0,𝜙)<𝜆 implies (𝑡,𝑥(𝑡;𝑡0,𝜙))<𝐴 for 𝑡𝑡0, where the function 𝐻0 is defined by (2.5), and 𝑥(𝑡;𝑡0,𝜙) is a solution of (2.1) and (2.2);(S2)uniformly practically stable with respect to (𝜆,𝐴) in terms of both measures (0,) if (S1) is satisfied for all 𝑡0+.

Remark 2.3. In the case 𝑟=0 and 0(𝑡,𝑥)=(𝑡,𝑥)=𝑥, Definition 2.2 reduces to a definition of practical stability of impulsive differential equations, studied in [4].
In the case 𝑟=0, 𝐼𝑘(𝑥)𝑥, 𝑘=1,2,, and 0(𝑡,𝑥)=(𝑡,𝑥)=𝑥, the above-given definitions reduce to definitions for the corresponding types of practical stability of ordinary differential equations, given in the books [4, 6].

In our further investigations, we will use the initial value problem for the comparison scalar impulsive differential equation: 𝑢=𝑔(𝑡,𝑢),𝑡𝑡0,𝑡𝜏𝑘,𝑢𝜏𝑘+0=𝜉𝑘𝑢𝜏𝑘𝑢𝑡,𝑘=1,2,,(2.7)0=𝑢0,(2.8) where 𝑢,𝑢0, 𝑔+×,𝜉𝑘, and 𝑘=1,2,.

In our further investigations, we will assume that for any initial point (𝑡0,𝑢0)+× the solution of scalar impulsive equation (2.7) exists on [𝑡0,), 𝑡00. For some existence results, see the book of Bainov and Simeonov [17].

Note that the definition for practical stability in terms of two measures for scalar impulsive differential equation (2.7) is given by the following definition.

Definition 2.4. Let the functions Γ,0Γ and the positive constants 𝜆,𝐴 be given. The scalar impulsive differential equation (2.7) is said to be(S3)practically stable with respect to (𝜆,𝐴) in terms of both measures (0,) if there exists 𝑡00 such that for any 𝑢0 the inequality 0(𝑢0)<𝜆 implies (𝑡,𝑢(𝑡;𝑡0,𝑢0))<𝐴 for 𝑡𝑡0, where 𝑢(𝑡;𝑡0,𝑢0) is a solution of (2.7) and (2.8);(S4)uniformly practically stable with respect to (𝜆,𝐴) in terms of both measures (0,) if (S4) is satisfied for all 𝑡0+.

Remark 2.5. In the case (𝑡,𝑥)=𝑥 and 0(𝑥)=𝑥, Definition 2.4 reduces to a definition for practical stability of zero solution of differential equations (see [6]).

We will give an example of a scalar differential equation which is not practically stable, but at the same time it is practically stable in terms of two measures.

Example 2.6. Consider scalar differential equation 𝑢=𝑢. The zero solution of this equation is not practically stable. At the same time, if we choose measures 0(𝑢)=|𝑢| and (𝑡,𝑢)=𝑒𝑡|𝑢|, then the same scalar equation is uniformly practically stable in terms of two measures.

We will study the connection between practical stability in terms of two measures of the scalar impulsive differential equation (2.7) and the corresponding practical stability in terms of two measures for the system of impulsive differential equations with “supremum” (2.1).

Introduce the following notations: 𝐺𝑘=[𝜏𝑡𝑟,)𝑡𝑘,𝜏𝑘+1,𝑘=1,2,,𝒢=𝑘=1𝐺𝑘.(2.9)

We will introduce the class Λ of piecewise continuous Lyapunov functions which will be used to investigate the practical stability of impulsive differential equations with “supremum.”

Definition 2.7. We will say that the function 𝑉(𝑡,𝑥)Δ×Ω+, Δ[𝑟,), Ω𝑛, 0Ω, belongs to class Λ if(1)𝑉(𝑡,𝑥) is a continuous function in (Δ𝒢)×Ω and 𝑉(𝑡,0)0 for 𝑡Δ,(2)for every 𝑘𝑍(Δ) and 𝑥Ω, there exist the finite limits 𝑉𝜏𝑘𝜏,𝑥=𝑉𝑘0,𝑥=lim𝑡𝜏𝑘𝑉𝜏(𝑡,𝑥),𝑉𝑘+0,𝑥=lim𝑡𝜏𝑘𝑉(𝑡,𝑥),(2.10)(3)function 𝑉(𝑡,𝑥) is Lipschitz with respect to its second argument in the set Δ×Ω.

Let 𝑉(𝑡,𝑥)Δ×Ω+, 𝑉Λ. For any 𝑡Δ𝒢 and any function 𝜓PC([𝑡𝑟,𝑡],Ω), we will define a derivative of the function 𝑉 along a trajectory of the solution of (2.1) as follows: 𝐷(2.1)𝑉(𝑡,𝜓)=lim𝜖01sup𝜖𝑉𝑡+𝜖,𝜓(𝑡)+𝜖𝑓𝑡,𝜓(𝑡),sup[]𝑠𝑟,0.𝜓(𝑡+𝑠)𝑉(𝑡,𝜓(𝑡))(2.11)

Consider the following sets: 𝐾=𝑎𝐶+,+,𝑎(𝑟)isstrictlyincreasingand𝑎(0)=0𝒦=𝑎𝐶+,+.𝑎(𝑟)isstrictlyincreasingand𝑎(𝑠)𝑠,𝑎(0)=0(2.12)

In the further investigations, we will use the following comparison result.

Lemma 2.8 (Hristova [12]). Let the following conditions be fulfilled. (1)The functions 𝑓PC([𝑡0,𝑇]×Ω×Ω,𝑛) and 𝐼𝑘𝐶(Ω,Ω) for 𝑘𝑍([𝑡0,𝑇)), where Ω𝑛, and 𝑡0,𝑇0𝑡0<𝑇< are constants.(2)The function 𝜙PC([𝑡0𝑟,𝑡0],Ω).(3)The initial value problem (2.1) and (2.2) has a solution 𝑥(𝑡)=𝑥(𝑡;𝑡0,𝜙), such that 𝑥(𝑡)Ω on [𝑡0𝑟,𝑇].(4)The functions 𝑔PC([𝑡0,𝑇]×+,+), 𝑔(𝑡,0)0 for 𝑡[𝑡0,𝑇] and 𝜉𝑘𝒦, 𝑘𝑍((𝑡0,𝑇)).(5)For any initial point 𝑢0+, the initial value problem for the scalar impulsive differential equation (2.7) has a maximal solution 𝑢(𝑡)=𝑢(𝑡;𝑡0,𝑢0), which is defined for 𝑡[𝑡0,𝑇].(6)The function 𝑉[𝑡0𝑟,𝑇]×Ω+, 𝑉Λ is such that(i)for any number 𝑡[𝑡0,𝑇]𝑡𝜏𝑘,𝑘𝑍((𝑡0,𝑇)) and any function 𝜓PC([𝑡𝑟,𝑡],Ω) such that 𝑉(𝑡,𝜓(𝑡))𝑉(𝑡+𝑠,𝜓(𝑡+𝑠)) for 𝑠[𝑟,0), the inequality 𝐷(2.1)𝑉(𝑡,𝜓(𝑡))𝑔(𝑡,𝑉(𝑡,𝜓(𝑡)))(2.13) holds.(ii)𝑉(𝜏𝑘+0,𝐼𝑘(𝑥))𝜉𝑘(𝑉(𝜏𝑘,𝑥)),𝑘𝑍((𝑡0,𝑇)),𝑥Ω.
Then, the inequality sup𝑠[𝑟,0]𝑉(𝑡0+𝑠,𝜙(𝑡0+𝑠))𝑢0 implies the inequality 𝑉(𝑡,𝑥(𝑡))𝑢(𝑡) for 𝑡[𝑡0,𝑇].

Remark 2.9. Lemma 2.8 is valid when 𝑇=, that is, for 𝑡[𝑡0,).

We will obtain sufficient conditions for practical stability in terms of two measures for impulsive differential equations with “supremum.” We will use Lyapunov functions from class Λ. The proof is based on Razumikhin method and a comparison method employing scalar impulsive differential equations.

In the case when the Lyapunov function satisfies globally the desired conditions, we obtain the following result.

Theorem 2.10. Let the following conditions be fulfilled. (1)The function 𝑓 PC [+×𝑛×𝑛,𝑛] and 𝑓(𝑡,0,0)0.(2)The functions 𝐼𝑘𝐶(𝑛,𝑛) and 𝐼𝑘(0)=0 for 𝑘𝑍(+).(3)The functions 0,Γ and 0Γ,Γ.(4)There exists a function 𝑉(𝑡,𝑥)[𝑟,)×𝑛+ with 𝑉Λ such that(i)𝑏((𝑡,𝑥))(𝑡,𝑉(𝑡,𝑥))and0(𝑉(𝑡,𝑥))𝑎(0(𝑡,𝑥))for(𝑡,𝑥)[𝑟,)×𝑛, where 𝑎,𝑏𝐾;(ii)for any number 𝑡+𝑡𝜏𝑘,𝑘𝑍(+) and any function 𝜓PC([𝑡𝑟,𝑡],𝑛) such that 𝑉(𝑡,𝜓(𝑡))>𝑉(𝑡+𝑠,𝜓(𝑡+𝑠)) for 𝑠[𝑟,0), the inequality 𝐷(2.1)𝑉(𝑡,𝜓(𝑡))𝑔(𝑡,𝑉(𝑡,𝜓(𝑡)))(2.14) holds, where 𝑔PC(+×,+) and 𝑔(𝑡,0)0;(iii)𝑉(𝜏𝑘+0,𝐼𝑘(𝑥))𝜉𝑘(𝑉(𝜏𝑘,𝑥)), for 𝑥𝑛,𝑘𝑍(+), where 𝜉𝑘𝒦.Then, the (uniform) practical stability with respect to (𝑎(𝜆),𝑏(𝐴)) in terms of both measures (0,) of scalar impulsive differential equation (2.7) implies (uniform) practical stability with respect to (𝜆,𝐴) in terms of both measures (0,) of system of impulsive differential equations with “supremum” (2.1) where the positive constants 𝜆,𝐴 are given.

Proof. Let scalar impulsive differential equation (2.7) be practically stable in both measures (0,) with respect to (𝑎(𝜆),𝑏(𝐴)). Therefore, there exists a point 𝑡00 such that 0(𝑢0)<𝑎(𝜆) implies 𝑡,𝑢𝑡;𝑡0,𝑢0<𝑏(𝐴)for𝑡𝑡0,(2.15) where 𝑢(𝑡;𝑡0,𝑢0) is a solution of (2.7) and (2.8).
Choose a function 𝜙 PC ([𝑡0𝑟,𝑡0],𝑛) such that𝐻0𝑡0,𝜙<𝜆,(2.16) and let 𝑥(𝑡;𝑡0,𝜙) be a solution of (2.1) with initial condition (2.2).
Let 𝑢0=sup𝑠[𝑟,0]𝑉(𝑡0+𝑠,𝜙(𝑡0+𝑠)). From Lemma 2.8 for Δ=[𝑟,) and Ω=𝑛, it follows the validity of the inequality𝑉𝑡,𝑥𝑡;𝑡0,𝜙𝑢𝑡;𝑡0,𝑢0for𝑡𝑡0.(2.17) From condition 4(i), we obtain for all 𝑠[𝑟,0]0𝑉𝑡0𝑡+𝑠,𝜙0+𝑠𝑎0𝑡0𝑡+𝑠,𝜙0+𝑠<𝑎(𝜆).(2.18) From inequalities (2.18) and 0(sup𝑠[𝑟,0]𝑉(𝑡0+𝑠,𝜙(𝑡0+𝑠)))sup𝑠[𝑟,0]0(𝑉(𝑡0+𝑠,𝜙(𝑡0+𝑠))), we obtain 0𝑢0<𝑎(𝜆).(2.19) From condition 4(i), and inequalities (2.15), (2.17), and (2.19), we get for𝑡𝑡0𝑏𝑡,𝑥𝑡;𝑡0,𝜙𝑡,𝑉𝑡,𝑥𝑡;𝑡0,𝜙𝑡,𝑢𝑡;𝑡0,𝑢0<𝑏(𝐴)(2.20) or 𝑡,𝑥𝑡;𝑡0,𝜙<𝐴.(2.21)

In the case when Lyapunov function does not satisfy globally the conditions 4(ii) and 4(iii) of Theorem 2.10, we obtain the following sufficient conditions.

Theorem 2.11. Let the following conditions be fulfilled. (1)The conditions (1) and (2) of Theorem 2.10 are satisfied.(2)The functions 0,Γ, 0Γ,Γ; there exist positive constants 𝜆,𝐴 and a function Ψ𝐾, Ψ(𝑥)𝑥 such that (𝑡,𝑥)Ψ(0(𝑡,𝑥)) for (𝑡,𝑥)𝑆(0,𝜆), and (𝜏𝑘,𝑥)<𝐴 implies (𝜏𝑘,𝐼𝑘(𝑥))<𝐴 for 𝑥𝑛, 𝑘𝑍(+).(3)There exists a function 𝑉(𝑡,𝑥)𝑆(,𝐴)+ with 𝑉Λ such that(i)𝑏((𝑡,𝑥))(𝑡,𝑉(𝑡,𝑥))and0(𝑉(𝑡,𝑥))𝑎(0(𝑡,𝑥))for(𝑡,𝑥)𝑆(,𝐴), where 𝑎,𝑏𝐾;(ii)for any number 𝑡+𝑡𝜏𝑘,𝑘𝑍(+) and any function 𝜓PC([𝑡𝑟,𝑡],𝑛)(𝑡,𝜓(𝑡))𝑆(,𝐴) such that 𝑉(𝑡,𝜓(𝑡))>𝑉(𝑡+𝑠,𝜓(𝑡+𝑠)) for 𝑠[𝑟,0), the inequality 𝐷(2.1)𝑉(𝑡,𝜓(𝑡))𝑔(𝑡,𝑉(𝑡,𝜓(𝑡)))(2.22) holds, where 𝑔PC(+×,+) and 𝑔(𝑡,0)0;(iii)𝑉(𝜏𝑘+0,𝐼𝑘(𝑥))𝜉𝑘(𝑉(𝜏𝑘,𝑥))for(𝜏𝑘,𝑥)𝑆(,𝐴),𝑘𝑍(+), where 𝜉𝑘𝒦.
Then, (uniform) practical stability with respect to (𝑎(𝜆),𝑏(𝐴)) in terms of both measures (0,) of scalar impulsive differential equation (2.7) implies (uniform) practical stability with respect to (𝜆,𝐴) in terms of both measures (0,) of system of impulsive differential equations with “supremum” (2.1).

Proof. Let scalar impulsive differential equation (2.7) be practically stable with respect to (𝑎(𝜆),𝑏(𝐴)) in terms of both measures (0,). Therefore, there exists a point 𝑡00 such that 0(𝑢0)<𝑎(𝜆) implies 𝑡,𝑢𝑡;𝑡0,𝑢0<𝑏(𝐴)for𝑡𝑡0,(2.23) where 𝑢(𝑡;𝑡0,𝑢0) is a solution of (2.7) and (2.8).
Choose a function 𝜙 PC ([𝑡0𝑟,𝑡0],𝑛) such that𝐻0𝑡0,𝜙<𝜆,(2.24) and let 𝑥(𝑡;𝑡0,𝜙) be a solution of (2.1) with initial condition (2.2).
We will prove that𝑡,𝑥𝑡;𝑡0,𝜙<𝐴(2.25) holds for 𝑡𝑡0.
From inclusion (𝑡,𝜙(𝑡))𝑆(0,𝜆) for 𝑡[𝑡0𝑟,𝑡0] and conditions (2) and 3(i), it follows that(𝑠,𝜙(𝑠))Ψ0𝐻(𝑠,𝜙(𝑠))Ψ0𝑡0𝑡,𝜙<Ψ(𝜆)𝜆<𝐴,𝑠0𝑟,𝑡0,(2.26) that is, inequality (2.25) holds on [𝑡0𝑟,𝑡0].
Assume that (2.25) does not hold for 𝑡>𝑡0.
Consider the following two cases.
Case 1. Let there exists a point 𝑡>𝑡0, 𝑡𝜏𝑘,𝑘𝑍((𝑡0,)) such that 𝑡𝑡,𝑥;𝑡0,𝜙=𝐴,𝑡,𝑥𝑡;𝑡0𝑡,𝜙<𝐴for𝑡0𝑟,𝑡.(2.27)
Let 𝑢0=sup𝑠[𝑟,0]𝑉(𝑡0+𝑠,𝜙(𝑡0+𝑠)). From Lemma 2.8 for the function 𝑉(𝑡,𝑥) defined on the set {(𝑡,𝑥)[𝑡0,𝑡]×𝑛(𝑡,𝑥)𝐴}, it follows the validity of the inequality𝑉𝑡,𝑥𝑡;𝑡0,𝜙𝑢𝑡;𝑡0,𝑢0𝑡for𝑡0,𝑡.(2.28) From condition 3(i), we obtain 0𝑉𝑡0𝑡+𝑠,𝜙0+𝑠𝑎0𝑡0𝑡+𝑠,𝜙0[]+𝑠<𝑎(𝜆),𝑠𝑟,0(2.29) or 0𝑢0<𝑎(𝜆).(2.30) From inequalities (2.23), (2.28), and (2.30), the choice of the point 𝑡, and condition 3(i), we get 𝑏𝑡(𝐴)=𝑏𝑡,𝑥;𝑡0,𝜙𝑡𝑡,𝑉𝑡,𝑥;𝑡0,𝜙𝑡,𝑢𝑡;𝑡0,𝑢0<𝑏(𝐴).(2.31) The obtained contradiction proves the validity of (2.25) for 𝑡>𝑡0.
Case 2. Let there exists a number 𝑘𝑍((𝑡0,)) such that (𝑡,𝑥(𝑡;𝑡0,𝜙))<𝐴 for 𝑡[𝑡0𝑟,𝜏𝑘) and (𝜏𝑘,𝑥(𝜏𝑘;𝑡0,𝜙))=𝐴. Then, as in Case 1 for 𝑡=𝜏𝑘, we obtain a contradiction. The obtained contradictions prove the validity of (2.25) for 𝑡>𝑡0.

Remark 2.12. Note that if in condition (2) inequality (𝜏𝑘,𝑥)<𝐴 implies (𝜏𝑘,𝐼𝑘(𝑥))𝐴 for 𝑥𝑛, 𝑘𝑍(+), then the claim of Theorem 2.11 holds if functions 𝜉𝒦,𝜉(𝑥)>𝑥, and 𝑘𝑍(+).

Corollary 2.13. Let the following conditions be fulfilled.(1)Conditions (1) and (2) of Theorem 2.10 are satisfied.(2)The functions 0,Γ, and there exists a positive constant 𝐴 such that (𝜏𝑘,𝑥)<𝐴 implies (𝜏𝑘,𝐼𝑘(𝑥))<𝐴 for 𝑥𝑛, 𝑘𝑍(+).(3)There exists a function 𝑉(𝑡,𝑥)[𝑟,)×𝑛+ with 𝑉Λ such that(i)𝑏((𝑡,𝑥))𝑉(𝑡,𝑥)𝑎(0(𝑡,𝑥))for(𝑡,𝑥)𝑆(,𝐴) where 𝑎,𝑏𝐾; (ii)for any number 𝑡+𝑡𝜏𝑘,𝑘𝑍(+) and any function 𝜓PC([𝑡𝑟,𝑡],𝑛)(𝑡,𝜓(𝑡))𝑆(,𝐴) such that 𝑉(𝑡,𝜓(𝑡))>𝑉(𝑡+𝑠,𝜓(𝑡+𝑠)) for 𝑠[𝑟,0), the inequality 𝐷(2.1)𝑉(𝑡,𝜓(𝑡))0(2.32) holds;(iii)𝑉(𝜏𝑘+0,𝐼𝑘(𝑥))𝑉(𝜏𝑘,𝑥)for(𝜏𝑘,𝑥)𝑆(,𝐴),𝑘𝑍(+).
Then, the system of impulsive differential equations with “supremum” (2.1) is uniformly practically stable with respect to (𝜆,𝐴) in terms of both measures (0,).

Proof. The proof of Corollary 2.13 follows from the one of Theorem 2.10 for 𝑔(𝑡,𝑥)0 and 𝜉(𝑥)𝑥. In this case, the scalar equation (2.7) is uniformly practically stable in terms of measures 0(𝑢)=|𝑢|, and (𝑡,𝑢)=|𝑢|.

3. Applications

Consider the following system of impulsive differential equations with “supremum”: 𝑥𝑥(𝑡)=𝑦(𝑡)2(𝑡)+𝑦2(𝑡)sin2𝑡+sup[]𝑠𝑡𝑟,𝑡𝑦𝑥(𝑠),𝑥(𝑡)=𝑥(𝑡)2(𝑡)+𝑦2(𝑡)sin2𝑡+sup[]𝑠𝑡𝑟,𝑡𝑦(𝑠),𝑡𝑡0,𝑡𝑘,𝑥(𝑘+0)=𝑎𝑥(𝑘),𝑦(𝑘+0)=𝑏𝑦(𝑘),(3.1) with initial conditions 𝑥(𝑡)=𝜙1𝑡𝑡0,𝑦(𝑡)=𝜙2𝑡𝑡0𝑡for𝑡0𝑟,𝑡0,(3.2) where 𝑥,𝑦, 𝑟>0 is small enough constant, 𝑡00, and 𝑎,𝑏(1,2). Without loss of generality, we could assume that 𝑡0<1.

Let 0(𝑡,𝑥,𝑦)=|𝑥|+2|𝑦|, (𝑡,𝑥,𝑦)=𝑒3𝑡(𝑥2+𝑦2), 0(𝑢)=|𝑢|, and (𝑡,𝑢)=𝑒3𝑡|𝑢|. Consider 𝑉2+, 𝑉(𝑥,𝑦)=𝑥2+2𝑦2. It is easy to check condition 3(i) of Theorem 2.10 for functions 𝑎(𝑠)=𝑠2𝐾 and 𝑏(𝑠)=𝑠𝐾.

Let 𝑡+,𝑡𝑘,𝑘=1,2, and 𝜓PC([𝑡𝑟,𝑡],2),𝜓=(𝜓1,𝜓2) be such that 𝜓21(𝑡)+2𝜓22(𝑡)>𝜓21(𝑡+𝑠)+2𝜓22[(𝑡+𝑠)for𝑠𝑟,0)(3.3) or 𝑉(𝜓1(𝑡),𝜓2(𝑡))>𝑉(𝜓1(𝑡+𝑠),𝜓2(𝑡+𝑠)).

Let 𝑖=1,2. If there exists a point 𝜂[𝑡𝑟,𝑡] such that sup𝑠[𝑡𝑟,𝑡]𝜓𝑖(𝑠)=𝜓𝑖(𝜂), then (sup𝑠[𝑡𝑟,𝑡]𝜓𝑖(𝑠))2=(𝜓𝑖(𝜂))2sup𝑠[𝑡𝑟,𝑡](𝜓21(𝑠)+2𝜓22(𝑠))=𝜓21(𝑡)+2𝜓22(𝑡).

The above inequality is also true if sup𝑠[𝑡𝑟,𝑡]𝜓𝑖(𝑠)>𝜓𝑖(𝜂) for all 𝜂[𝑟,𝑡]; that is, there exists 𝑘(𝑡𝑟,𝑡) such that sup𝑠[𝑡𝑟,𝑡]𝜓𝑖(𝑠)=𝜓𝑖(𝑘+0).

Then, for 𝑖=1,2, we obtain 𝜓𝑖(𝑡)sup[]𝑠𝑡𝑟,𝑡𝜓𝑖||𝜓(𝑠)𝑖||||||(𝑡)sup[]𝑠𝑡𝑟,𝑡𝜓𝑖||||=(𝑠)𝜓𝑖(𝑡)2sup[]𝑠𝑡𝑟,𝑡𝜓𝑖(𝑠)2𝜓21(𝑡)+2𝜓22𝜓(𝑡)=𝑉1(𝑡),𝜓2.(𝑡)(3.4) Therefore, if inequality (3.3) is fulfilled, then we have 𝐷(3.1)𝑉𝜓1(𝑡),𝜓2=𝜓(𝑡)1(𝑡)sup[]𝑠𝑡𝑟,𝑡𝜓1(𝑠)+2𝜓2(𝑡)sup[]𝑠𝑡𝑟,𝑡𝜓2𝜓(𝑠)3𝑉1(𝑡),𝜓2.(𝑡)(3.5) For any 𝑘, we obtain 𝑉𝑎(𝑎𝑥,𝑏𝑦)=2𝑥2+2𝑏2𝑦2𝑐2𝑥2+2𝑦2=𝑐2𝑉(𝑥,𝑦),(3.6) where 𝑐=max(𝑎,𝑏)>1.

Now, consider the initial value problem for the scalar comparison impulsive differential equation 𝑢=3𝑢for𝑡𝑘,𝑢(𝑘+0)=𝑐2𝑡𝑢(𝑘),𝑢0=𝑢0.(3.7)

The solution of the above initial value problem for impulsive differential equation is 𝑢(𝑡)=(𝑘𝑖=1(𝑐21))𝑢0𝑒3(𝑡𝑡0) for 𝑡[𝑘,𝑘+1),𝑘=1,2,. Let numbers 0<𝜆<𝐴 be given and |𝑢0|<𝜆2. Then, (𝑡,𝑢(𝑡))=𝑒3𝑡|𝑢|=𝑘𝑖=1𝑐2||𝑢10||𝑒3𝑡0||𝑢0||<𝜆2𝐴,(3.8) that is, scalar comparison equation is uniformly practically stable in terms of measures (0,). Therefore, according to Theorem 2.10, the system of impulsive differential equations with “supremum” (3.1) is uniformly practically stable in terms of two measures, that is, the inequality sup𝑠[𝑟,0](|𝜙1(𝑠)|+2|𝜙2(𝑠)|)<𝜆 implies 𝑒3𝑡(𝑥2(𝑡;𝑡0,𝜙)+2𝑦2(𝑡;𝑡0,𝜙))<𝐴 for 𝑡𝑡0.

4. Conclusions

Two types of sufficient conditions for practical stability of impulsive differential equations with “supremum” are obtained. The global case and the case on a ball are considered. Both types of results are based on the application of Lyapunov piecewise continuous functions and comparison results for scalar impulsive differential equations. To unify a variety of stability concepts and to offer a general framework for the investigation of the stability theory, the notion of stability in terms of two measures has been applied to both the given system and the comparison scalar equation. Note that in studying stability of differential equations everywhere in the literature two different measures are applied only to the given equation. Also, in the particular case of identity impulsive functions, that is, 𝐼𝑘(𝑥)𝑥,𝑘=1,2,, the obtained results reduce to results for practical stability of differential equations with “maxima” which are also new ones. The obtained results are generalizations of the results for practical stability of ordinary differential equations (see [4, 6]), results for impulsive differential equations (see [4]), results for differential difference equations, and results for impulsive differential difference equations (see [2, 5]).

Acknowledgment

The research was partially supported by NI11FMI004/30.05.2011, Fund “Scientific Research,” Plovdiv University.

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