IJPAM: Volume 109, No. 1 (2016)

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

R. Agarwal$^1$, S. Hristova$^2$, D. O'Regan$^3$, P. Kopanov$^4$
$^1$Department of Mathematics
Texas A&M University
Kingsville, Kingsville, TX 78363, USA
$^{2,4}$Department of Applied Mathematics and Modeling
Plovdiv University
Tzar Asen 24, 4000 Plovdiv, BULGARIA
$^2$e-mail: [email protected]
$^3$School of Mathematics, Statistics and Applied Mathematics
National University of Ireland
Galway, IRELAND


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.

Received: August 1, 2016

AMS Subject Classification: 34A37, 34F05, 34K20, 37B25

Key Words and Phrases: random noninstantaneous impulses, Erlang distribution, p-moment exponential stability

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DOI: 10.12732/ijpam.v109i1.3 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 109
Issue: 1
Pages: 9 - 28


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