Abstract
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.
Highlights
In some real world phenomena a process may change instantaneously at some moments
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
Summary
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). Sometimes the impulsive action starts at an random point and remains active on a finite time interval. These type of impulses are called noninstantaneous. 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. The p-moment exponential stability of the solution is studied using Lyapunov functions
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