P-MOMENT EXPONENTIAL STABILITY OF DIFFERENTIAL EQUATIONS WITH RANDOM NONINSTANTANEOUS IMPULSES AND THE ERLANG DISTRIBUTION

Abstract: In some real world phenomena a process may change instantaneously at uncertain moments and act non instantaneously on finite intervals. In modeling such processes it is necessarily to combine deterministic differential equations with random variables at the moments of impulses. The presence of randomness in the jump condition changes the solutions of differential equations significantly. The study combines methods of deterministic differential equations and probability theory. In this paper we study nonlinear differential equations subject to impulses occurring at random moments. Inspired by queuing theory and the distribution for the waiting time, we study the case of Erlang distributed random variables at the moments of impulses. The p-moment exponential stability of the trivial solution is defined and Lyapunov functions are applied to obtain sufficient conditions. Some examples are given to illustrate the results.


Introduction
In some real world phenomena a process may change instantaneously at some moments.In modeling such processes one uses impulsive differential equations (see, for example, the books [5], [6], [11] and the cited references therein).In the case when the process has instantaneous changes at uncertain moments which act non instantaneously on finite intervals one combines ideas in differential equations and probability theory.When there is uncertainty in the behavior of the state of the investigated process an appropriate model is usually a stochastic differential equation where one or more of the terms in the differential equation are stochastic processes, and this usually results with the solution being a stochastic process ( [14], [15], [16], [17]).Sometimes the impulsive action starts at an random point and remains active on a finite time interval.These type of impulses are called noninstantaneous.Recently results concerning deterministic noninstantaneous impulses were obtained for differential equations ( [2], [8], [13]), delay integro-differential equations ( [9]), abstract differential equations ( [10]), and fractional differential equations ( [1], [12]).Differential equations with instantaneously acting impulses at random times were studied in [4], [18] but there are some inaccuracies there in the mixing properties of deterministic variables and random variables, and inaccuracies in the convergence of a sequence of real numbers to a random variable.
In this paper we study nonlinear differential equations subject to impulses starting abruptly at some random points and their action continue on intervals with a given finite length.We study the case of Erlang distributed random variables defining the moments of the occurrence of the impulses.The p-moment exponential stability of the solution is studied using Lyapunov functions.

Random Noninstantaneous Impulses in Differential Equations
Let T 0 ≥ 0 be a given point and the increasing sequence of positive points {T k } ∞ k=1 and the sequence of nonnegative numbers {d i } ∞ i=1 be given such that Consider the following condition: Consider the initial value problem (IVP) for the system of noninstantaneous impulsive differential equations (NIDE) with fixed points of impulses Denote the solution of NIDE (1) by x(t; T 0 , x 0 , {T k }).

H4.
For any initial value (T 0 , x 0 ) the ODE x ′ = f (t, x) with x(T 0 ) = x 0 has an unique solution x(t) = x(t; T 0 , x 0 ) defined for t ∈ [T 0 , P ) where P = ∞ if condition (H1) is satisfied and P = T + B is condition (H2) is satisfied.
In Section 5 we will need the following result for the initial value problem for a scalar linear differential inequality with noninstantaneous fixed moments of impulses: (2) Continue this process.
Let the probability space (Ω, F, P ) be given.Let {τ k } ∞ k=1 be a sequence of random variables defined on the sample space Ω.Assume ∞ k=1 τ k = ∞ with probability 1.
Remark 1.The random variables τ k will define the time between two consecutive impulsive moments of the impulsive differential equation with random impulses.
We will assume the following condition is satisfied H5.The random variables {τ k } ∞ k=1 , τ k ∈ Erlang(α k , λ) are independent with two parameters: a positive integer "shape" α k and a positive real "rate" λ.
We will recall some properties of the Erlang distribution: and the probability density function (PDF) is Proposition 2. Let condition (H5) be satisfied and the sequence of random variables {Ξ k } ∞ k=1 be such that Define the increasing sequence of random variables {ξ k } ∞ k=0 by where The random variable ξ n will be called the waiting time and it gives the arrival time of n-th impulses in the impulsive differential equation with random impulses.
Let the points t k be arbitrary values of the corresponding random variables τ k , k = 1, 2, . . . .Define the increasing sequence of points Note T k are values of the random variables ξ k .The set of all solutions x(t; T 0 , x 0 , {T k }) of NIDE (1) for any values t k of the random variables τ k , k = 1, 2, . . .generates a specific stochastic process with state space R n .We denote it by x(t; T 0 , x 0 , {τ k )} and we will say that it is a solution of the following initial value problem for differential equations with noninstantaneous random moments of impulses (RNIDE) Definition 1.The solution x(t; T 0 , x 0 , {T k }) of the IVP for the IDE with fixed points of impulses ( 1) is called a sample path solution of the IVP for the RIDE (6).Definition 2. A stochastic process x(t; T 0 , x 0 , {τ k )} is said to be a solution of the IVP for the system of RIDE (6) if for any values t k of the random variable τ k , k = 1, 2, 3, . . .and T k = T 0 + k i=1 t i , k = 1, 2, . . . the corresponding function x(t; T 0 , x 0 , {T k }) is a sample path solution of the IVP for RIDE (6).

Preliminary Results for Erlang Distributed Moments of Impulses
For any t ≥ T 0 consider the events where the random variables ξ k , k = 1, 2, . . .are defined by (4).
Proposition 3.For any t ≥ T 0 the equality Corollary 1. (Upper bound of S 0 (t)).For any t ≥ T 0 the inequality Proof.We have the following Apply the inequality Γ(a, x) ≤ x a e −x x−a−1 (see [7]) for the upper incomplete gamma function Γ(a, x) = ∞ x y a−1 e −y dy and obtain Lemma 1.Let conditions (H1),(H5) be satisfied and t ≥ T 0 .
Then the probability that there will be exactly k impulses until time t is where Proof.From the definition of S j (t) we get Let j > k.From the definition of k it follows that (9) and formula (3) we obtain Also, Remark 3. Let conditions (H1),(H5) be satisfied and α j+1 = 1, i.e. τ j+1 ∈ Exp(α).Then the formula (8) reduces to Corollary 2. Let conditions (H1),(H5) be satisfied with α i = α, i = 1, 2, . . .and t ≥ T 0 + k i=1 d i .Then the probability that there will be exactly k impulses until time t is Lemma 2. Let conditions (H2),(H5) be satisfied and t ≥ T 0 .Then the probability that there will be exactly k impulses until time t is where A j = j i=1 α i and Proof.The proof of the case t < T 0 + B is similar to the proof in Lemma 1.
Let t ≥ T 0 + B. Then for all natural number j the inequality t − T 0 − holds and the proof is similar to the one of Lemma 1.
Lemma 3. (Upper bound of S k (t)).Let condition (H5) and one of (H1) or (H2) be satisfied.Then for any natural number j we have holds where Proof.From equality (12), inequality and Remark 4 it follows that (13) is true.
Lemma 4. Let conditions (H1), (H5) hold.Then the probability the time t is immediately after the k-th random impulse but not far away than d k from it is given by where Proof.From the definition of W j (t) and the random variables Ξ j we get If j > k then similar to the proof in Lemma 1 we get From Proposition 2 and equality (3) we obtain Also, Lemma 5. Let conditions (H2),(H5) be satisfied and t ≥ T 0 .Then the probability the time t is immediately after the k-th random impulse but not far away than d k from it is given by where Lemma 6. (Upper bound of W k (t)).Let condition (H5) and one of (H1) or (H2) be satisfied.Then for any natural number j we have Proof.Using the Integral mean value Theorem and j i=1 α i ≥ 1 we obtain
Lemma 7. Let the following conditions be satisfied: 1. Condition (H5) and one of the conditions (H1) or (H2) is fulfilled.
Then the solution of the IVP for the scalar linear differential equation with random noninstantaneous moments of impulses ( 18) is and the expected value of the solution is Proof.The sample path solution of ( 18) is given by

The above equality and Definition 2 establishes (19).
From formula (19) and the independence of the random variables τ k we obtain Using the definition of the density function of the Erlang distribution and substituting (m i−1 + λ)x = s we get Substitute ( 22) and ( 23) in ( 21) and obtain (20).

p-Moment Exponential Stability for RNIDE
The main goal of the paper is to define the exponential stability of the zero solution of RNIDE (6) (with x 0 = 0) and to obtain some sufficient conditions for it.
Definition 3. Let p > 0. Then the trivial solution (x 0 = 0) of the RNIDE ( 6) is said to be p-moment exponentially stable if for any initial point x 0 ∈ R n there exist constants α, µ > 0 such that the inequality E[||x(t; T 0 , x 0 , {τ k )})|| p ] < α||x 0 || p e −µ(t−T 0 ) holds for all t ≥ T 0 , where x(t; T 0 , x 0 , {τ k )} is the solution of the IVP for the RNIDE (6).Remark 6.We note that the two-moment exponentially stability for stochastic equations is known as exponentially stability in mean square.Definition 4. Let J ⊂ R + be a given interval and ∆ ⊂ R n , 0 ∈ ∆ be a given set.We will say that the function V (t, x) : J × ∆ → R + , V (t, 0) ≡ 0 belongs to the class Λ(J, ∆) if it is continuous on J × ∆ and locally Lipschitzian with respect to its second argument.

The function
(ii) there exists a constant m : 0 < m ≤ λ such that the inequality 3. There exist positive constants D, µ : µ < λ+m and Then the trivial solution of the RNIDE ( 6) is p-moment exponentially stable.