Minimal and maximal Lyapunov exponents of bilinear control systems
This paper deals with control systems of the form (1) \(\dot x=A_ 0(u_ 0)x+ \sum_{i=1}^ m u_ i A_ i(u_ 0)x\), \(x\in\mathbb{R}^ d\), where \(u_ 0: \mathbb{R}\to\Omega\) represents a locally integrable constrained control (\(\Omega\) is a compact subset of \(\mathbb{R}^ l\)) and \(u=(u_ 1,\dots,u_ m):\mathbb{R} \to\mathbb{R}^ m\) is a locally integrable control (not subject to constraints). The stabilization (or destabilization) properties of (1) are related to the minimal and maximal Lyapunov exponents \(\sup_ u \sup_{x_ 0} \lambda(x_ 0,u)\), \(\inf_ u \inf_{x_ 0} \lambda(x_ 0,u)\), where \(\lambda(x_ 0,u)=\limsup {{\log| x(t,x_ 0,u)|} \over t}\) and \(x(t,x_ 0,u)\) denotes the solution of (1) corresponding to the control \(u\) and the initial value \(x_ 0\). The paper is devoted to a careful investigation of maximal and minimal Lyapunov exponents and the related uniform notions. The authors give several characterizations of these notions, as well as of the set of initial values for which these extremal exponents are realized. The analysis is carried out by considering an optimization problem for the system induced by (1) on the projective space \(\mathbb{P}\) of \(\mathbb{R}^ d\). Some applications are included (high gain stabilization, stabilization of linear systems by time-varying feedback, the linear oscillator problem).
stabilization of linear systems by time- varying feedback
scientific article; zbMATH DE number 166900