Exponential stabilizability of stochastic systems with Markovian jumping parameters (Q1304041)

Consider a linear system with uncertain coefficients described by stochastic differential equations \[ \begin{multlined} dx(t)=[A(\theta (t))+\Delta A(\theta (t))]x(t) dt+ [B(\theta (t))+\Delta B(\theta (t))]u(t) dt+ \sigma (x(t), \theta (t),t) dw(t),\\ x(0)=x_0,\end{multlined} \] where \(x(t)\in\mathbb R^n\) is a state of the system, \(u(t)\in\mathbb R^m\) is a control process, \(w(t)\) is a standard Brownian motion and \(\theta (t)\) is a continuous in time Markov process with values in \(\{1,\dots, N\}\) and given infinitesimal transition probabilities. The authors give conditions when the linear system is exponentially mean-square stabilizable, and an associated nonlinear system (when the coefficients depend also on \(x(t)\)) is exponentially stabilizable.