- Open Access
Exponential stability for differential equations with random impulses at random times
© Agarwal et al.; licensee Springer. 2013
- Received: 1 October 2013
- Accepted: 21 November 2013
- Published: 20 December 2013
Impulsive differential equations with impulses occurring at random times arise in the modeling of real world phenomena in which the state of the investigated process changes instantaneously at uncertain moments. The investigation of these differential equations uses ideas in the qualitative theory of differential equations and probability theory. In this paper differential equations with randomly occurring impulses are considered and the p-moment exponential stability of the solutions is studied.
- impulsive differential equations
- random moments of impulses
- p-moment exponential stability
Impulsive differential equations are studied extensively in the literature. Many authors consider impulsive differential equations with determined impulsive moments (see, for example, the monographs [1–4] and the references cited therein). However, in some real world phenomena the investigated process changes instantaneously at uncertain moments. When modeling such processes, it is necessary to use random variables in jump conditions and impulsive differential equations with random impulses occurring at random moments. The presence of randomness in the jump condition changes the behavior of solutions of differential equations significantly. In the case of impulses occurring at random moments, the solution is a stochastic process.
In the literature a number of results have been obtained for stochastic differential equations with jumps [5–7]. Also, some results on the qualitative properties of equations with random impulses have been obtained [8–10]. In the monograph , impulsive differential equations with fixed impulses and random amplitude of jumps were studied.
In this paper we study nonlinear differential equations subject to random impulses occurring at random moments. Randomness is introduced both through the time between impulses, which is distributed exponentially, and through the amount of impulses. The p-moment exponential stability of the solution is studied by employing appropriate generalized Lyapunov’s functions. In the literature many authors study the stability of impulsive systems with deterministic moments of impulses (see [1, 2, 4] and the references cited therein), the exponential stability of impulsive delay differential equations with deterministic moments of impulses [12–15] and the p-moment stability of stochastic differential equations with or without impulses [6, 7, 16, 17]. The behavior of solutions of stochastic equations is totally different from the behavior of ordinary differential equations, so the study of stability properties of differential equations with impulses occurring at random moments is important.
Let the probability space be given. Let be a sequence of independent exponentially distributed random variables with a parameter that are defined on the sample space Ω. We will call the random variables waiting times since they will define the time between two consecutive impulses of the considered impulsive differential equation.
Define the sequence of random variables such that and , , where is a fixed point.
We note that is an increasing sequence of random variables.
Assume with probability 1.
where , , and .
The solution of the impulsive differential equation with fixed moments of impulses (1) depends not only on the initial condition but on the moments of impulses , , i.e., the solution depends on the initially chosen arbitrary values of the random variables , . We denote the solution of initial value problem (1) by . We will assume that .
Remark 1 We note that the length of the interval of the continuity of the solutions of the initial value problem for impulsive differential equation with fixed moments of impulses (1) is equal to , that is, a value of the random variable , called the waiting time.
For any values of the random variables , the solution of the initial value problem for the impulsive equation with fixed points of impulses (1) will be called a sample path solution of RIDE (2).
We note that for any fixed point t and any element , there exists a natural number k such that and for , or for any fixed point t, there exists a natural number k such that and for .
We will prove the following result for the stochastic processes .
Lemma 2.1 Let be independent exponentially distributed random variables (IED) with a parameter λ and .
we have (4). □
Proof The result follows immediately from the definition of the event , Lemma 2.1 and the fact that . □
Now we will illustrate some differences between impulsive differential equations with deterministic moments of impulses and differential equations with randomly occurring impulses.
Example 1 (Ordinary differential equation)
The solution of (6) is stable but is not approaching 0.
Example 2 (Impulsive differential equations with fixed points of impulses)
The solution of IVP (7) is for .
The solution is a piecewise continuous function.
The behavior of depends significantly on the amplitude of jumps.
If , then is approaching 0 (different from the corresponding ordinary differential equation considered in Example 1).
Example 3 (Impulsive differential equation with random points of impulses)
Let the sequence of independent exponentially distributed random variables , (waiting time) be given. Define (moments of impulses).
Let, for any , the point be an arbitrary value of the random variable . Define the increasing sequence of points , , that are values of the random variables .
The solution of (9) is for .
It depends not only on but on the moments of impulses , i.e., on the initially chosen arbitrary values of the random variables , .
The set of all solutions of IVP (9) for any values of the random variables generates a stochastic process with state space . We will say it is a solution of the impulsive differential equations with random moments of impulses (8) and it is for .
The solution is a stochastic process.
If , then is approaching 0 (compare with the impulsive differential equation with fixed moments of impulses considered in Example 2).
We will say that conditions (H) are satisfied if
(H1) For any initial values , , the initial value problem , has a unique solution defined for .
(H2) and for , .
(H3) The random variables are independent exponentially distributed random variables with a parameter λ and , .
Definition 1 A stochastic process with an uncountable state space is said to be a solution of the initial value problem for the system of equations with randomly occurring impulses (2), , if for any sample values of the random variables , , correspondingly, the process satisfies the initial value problem for the impulsive equation with fixed points of impulses (1), where the moments of impulses are defined by , .
Remark 2 We note that if condition (H1) is satisfied, then the sample path solution of the initial value problem for the impulsive equation with impulses at random moments (2) exists for all , provided that the times between two consecutive impulses are such that .
Remark 3 We note that if the values are values of the random variables , , correspondingly, then the value is a value of the random variable , .
Definition 2 We will say that the stochastic processes and satisfy the inequality for if the state space of is .
Definition 3 Let . Then the trivial solution of the impulsive differential equation with random impulses (2) is said to be p-moment exponentially stable if for any initial data and , there exist constants such that for all , where is the solution of the initial value problem for the impulsive differential equation with random impulses (2).
Remark 4 We note that the two-moment exponential stability for stochastic equations is known as exponential stability in mean square.
In this section we will use Lyapunov functions to obtain sufficient conditions for the p-moment exponential stability of the trivial solution of the nonlinear impulsive random system at random moments (2).
where , , .
Condition (H3) is satisfied.
The functions ().
This solution generates a continuous stochastic process that is defined by (11). It is a solution of the initial value problem for the linear impulsive differential equation with random moments of impulses (10).
In the special case when the expected values of all amplitudes of jumps are the same, we obtain the following result.
In the special case when all amplitudes of jumps are deterministic, we obtain the following result.
Condition (H3) is satisfied.
The functions , ().
Then the state space of the solution of the linear impulsive differential inequalities with random moments of impulses (16) is .
We prove that any solution of inequalities (17) is a piecewise continuous function that is nonnegative. Assume the contrary.
Suppose that there exists a point such that . Then there exists a point such that and . That contradicts the first inequality of (17). Therefore on .
According to the second inequality of (17), it follows that if , then .
Using induction, assume that on . If there exists a point such that , then, as in the proof above, we obtain a contradiction.
Since the above proof does not depend on the points of jumps , it follows that for any points , the solution of (17) will be nonnegative.
Therefore the stochastic process , generated by all nonnegative functions , will have as a state space. □
We will use Lyapunov functions in order to obtain sufficient conditions for the p-moment exponential stability of the trivial solution of systems of nonlinear impulsive differential equations with impulses occurring at random moments.
where , .
Remark 5 The function V does not depend on t but its derivative D V depends on t because of the function f.
Conditions (H) are satisfied.
- 2.The function and there exist positive constants a, b such that
- (ii)for any , the inequality
holds, where and ;
- (iii)there exist constants and C such that , , such that(19)
Then the trivial solution of the impulsive differential equations with random moments of impulses (2) is p-moment exponentially stable.
Consider , . The stochastic process satisfies (16) with and . According to Lemma 3.2, the state space of is , i.e., the inequality , holds.
Inequality (24) proves the p-moment exponential stability. □
Then the inequality holds.
Example 4 (Exponential stability of IDE with random moments of impulses)
where a, b are constants such that , .
Consider the Lyapunov function .
Since , condition 2(i) of Theorem 3.1 is satisfied for .
Therefore the conditions of Theorem 3.1 are satisfied for , , and .
where , and .
Therefore, the presence of impulses at random time changes the behavior of the solution, i.e., the solution of IDE with random impulses is exponentially stable in mean square.
Research was partially supported by Fund Scientific Research MU13FMI002, Plovdiv University.
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