Stability of systems with state-dependent and random delays. (English) Zbl 1010.34078

Verriest, Erik J.

IMA J. Math. Control Inf. 19, No. 1-2, 103-114 (2002).

The author focusses on dynamical systems with a delay or time lag varying in time. By Lyapunov-Krasovskii theory, sufficient conditions for the global asymptotic stability are derived. In the first part, the case of stochastic delay differential equations with Brownian motion as driving process and a randomly varying time delay independent of Brownian motion (but Markovian) is considered. Secondly, linear deterministic systems, but with nonlinear state-dependent delay, are investigated, and an oscillator system with friction and state-dependent delay is studied in more detail. One is lead to consider a so-called delay algebraic Riccati equation in order to derive stability.

Reviewer: Markus Reiß (Berlin)

MSC:

34K50	Stochastic functional-differential equations
34K20	Stability theory of functional-differential equations

Keywords:

Lyapunov functional; Riccati equation; Brownian motion