Lyapunov exponents and robust stabilization

Abstract

In recent years there has been considerable interest in the analysis and synthesis of time-varying control systems. Here we present some open problems in this area. Let K = ℝ, ℂ and consider a family of bilinear systems

$$ \begin{array}{*{20}{c}} {\dot x\left( t \right) = \left( {{A_0} + \sum\limits_{i = 1}^m {{v_i}\left( t \right){A_i}} } \right)x\left( t \right)} \\ {x\left( 0 \right) = {x^0} \in {K^n}} \\ {v\left( t \right) = \left( {{v_1}\left( t \right),...,vm\left( t \right)} \right)' \in Va.a.t \geqslant 0} \end{array} $$

((18.1))

where A ₀,…,A _m ∈ K ^n×n and V ⊂ K ^m is a convex, compact set with 0 ∈ int V. The set of admissible control functions V is the set of measurable functions v: ℝ → V. The solution determined by an initial condition x ⁰ and a control v ∈ V is denoted by ϕ(·;x ⁰,v).