Necessary and sufficient condition for robust stability and stabilizability of continuous-time linear systems with Markovian jumps
The goal of this paper is to derive a necessary and sufficient condition for mean-square quadratic stability and stabilizability for continuous-time linear stochastic systems with Markovian jumps. The systems are described by the following differential equation, \[ \dot x(t)= [A(r(t))+\Delta A(r(t),t)] x(t)+ [B(r(t))+\Delta B(r(t),t)] u(t),\quad x(0)= x_0,\;r(0)= r_0, \] where \(x(t)\in \mathbb{R}^n\) is the state, \(u(t)\in\mathbb{R}^m\) is the control input, \(\Delta A\) and \(\Delta B\) are system uncertainties, the term \(r(t)\) is a continuous-time stochastic Markovian process with finite discrete state space. The criterion of mean-square quadratic robust stability is established for the autonomous system with \(u= 0\), and the criterion of quadratic stabilizability is established in the case of a feedback control of the form \(u(t)= K(r(t)) x(t)\). A quadratic guaranteed cost control problem with the same \(u(t)\) and \(K= (K(1),\dots, K(N))\) is also investigated.
scientific article; zbMATH DE number 1239677