February 25, 2008
Chen Dubi
Title: Detecting the stable subsets of upper triangular switching systems
A discrete time switching system is defined by:
\begin{eqnarray*}
x(n+1)&=&A(n)x(n), \quad A(n) \in \mathbb{A}=\{A_{1},A_{2}, \dots A_{N} \}.\\
x(0)&=&X_{0}.
\end{eqnarray*}
For the continuous time model, we have :
\begin{equation*}
x^{\prime}(t)=A(t)x(t), \quad A(t) \in \mathbb{A}=\{A_{1},A_{2}, \dots A_{N} \}.
\end{equation*}
Finding a necessary and sufficient condition for the asymptotic stability of such systems has been the interest of a large number of studies in recent years, and is generally considered a hard problem. It is a trivial observation that a necessary condition for the stability of the system is that each matrix in $\mathbb{A}$ is stable (in the proper sense) --- yet by no means is this condition a sufficient one. One specific case in which stability of each matrix forces asymptotic stability of the switching systems is when the set of matrices $\mathbb{A}$ admits a simultaneous triangularization. In the talk, we introduce an algorithm that will detect all stable initial conditions for a switching system admitting a simultaneous triangularization.