Quantitative analysis of jump Markovian nonlinear stochastic hybrid systems: Practical stability (Q2783613)

The author investigates stability properties of stochastic dynamical systems which have the form of an SDE whose drift and diffusion coefficients are controlled by a finite state Markov process \(\eta_t\): \( dx_t = f(t,x_t,\eta_t) dt + \sigma(t,x_t,\eta_t) dW_t, \quad x(t_0) =x_0 \in {\mathbb R}^N \). Here stability is used in the weak sense of practical stability: A dynamical system is called \textit{practically stable} (in the \(p^{th}\) mean with respect to \(\alpha\), \(\lambda\))fif \(0<\lambda<\alpha<\infty\) and if \({\mathbb E}|X(t_0)|^p < \lambda\) implies \({\mathbb E}|X(t)|^p < \alpha\) for all \(t \geq t_0\) (\(X(t)\) denoting the solution of the SDE). Sufficient conditions for practical stability are given in terms of Lyapunov type functions and stability properties of an associated system of first order ODEs.