Minimal and Maximal Lyapunov Exponents of Bilinear Control Systems

Abstract

For a bilinear control system with bounded and unbounded controls x ̇ = A 0(u 0) x + [GRAPHICS] u iA i(u 0) x in R d, where u 0( t) ∈ Ω ⊂ R l compact, u i ( t) ∈ R for i = 1, ..., m, the extremal exponential growth rates of the solutions x(·, x 0, u) are analyzed: If λ( x 0, u) = lim sup t→∞ (1/ t) log | x( t, x 0, u)|, then K = sup u ∈ U sup x 0 ≠ 0 λ( x 0, u) and K * = inf u ∈ U inf x 0 ≠ 0 λ( x 0, u) are the maximal (and minimal, respectively) Lyapunov exponents of the system. This paper gives several characterizations of these rates, together with the corresponding uniform concepts (with respect to the initial value or the control). We describe the situations, in which K = +∞ and K * = −∞, and characterize the sets of initial values, from which K and K * can actually be realized. The techniques are applied to high gain stabilizalion. and the example of the linear oscillator with parameter controlled restoring force is treated in detail. Finally we indicate how the results can be used for feedback stabilization of linear systems, when the feedback is allowed to be time varying, but restricted to certain types of (bounded) gain matrices.