Robust stability of nonlinear piecewise deterministic systems under matching conditions (Q679407)

The authors consider an uncertain piecewise deterministic nonlinear system described by \[ \dot x(t)= A(x(t),\eta(t),t)+ \delta A(x(t),\eta(t),a,t)+ [B(x(t),\eta(t),t)+ \delta B(x(t),\eta(t),a,t)]u(t),\tag{1} \] where \(x(t)\in\mathbb{R}^n\), and \(u(t)\in\mathbb{R}^m\) are the state and control vector, respectively; \(A(x,\eta,t)\), \(\delta A(x,\eta,a,t)\), \(B(x,\eta,t)\) and \(\delta B(x,\eta,a,t)\) are matrices of appropriate dimensions; \(\eta(t)\) is a continuous discrete-state Markov process taking values in a finite set \(G=\{1,2,\dots,s\}\) with transition probability given by \[ P\{\eta(t+\delta t)=\beta|\eta(t)=\alpha\}= \begin{cases} q_{\alpha\beta}\delta t+ o(\delta t) &\text{if }\alpha\neq\beta\\ 1+q_{\alpha\alpha}\delta t+ o(\delta t) &\text{if }\alpha=\beta\end{cases} \] where \(q_{\alpha\beta}\) is the transition probability rate from state \(\alpha\) to state \(\beta\). The authors give sufficient conditions for the robust stability of system (1). An example is presented to illustrate the obtained results.