A new general sufficient condition for almost sure stability of jump linear systems. (English) Zbl 0870.93048

Fang, Yuguang

IEEE Trans. Autom. Control 42, No. 3, 378-382 (1997).

Almost sure asymptotic stability of the discrete-time linear stochastic system \(x_{k+1}= H(z_k)x_k\) is considered, where \((z_k)\) is a finite state Markov chain, not necessarily ergodic. With the help of a Lyapunov function, a sufficient condition is derived, together with several variants. It is given in terms of the quotients “norms of the possible new states over norm of the old state”, the norms properly chosen. The condition is necessary and sufficient in the case of a one-dimensional system; and it is a necessary condition for second moment stability.

Reviewer: V.Wihstutz (Charlotte)

MSC:

93E15	Stochastic stability in control theory
93C55	Discrete-time control/observation systems

Keywords:

almost sure stability; stochastic stability; asymptotic stability; discrete-time linear stochastic system; finite state Markov chain; Lyapunov function; second moment stability