Global asymptotic stabilization of bilinear control systems with periodic coefficients

Abstract

MSC: 34D23, 34H15, 93D15c V. A. Zaitsev GLOBAL ASYMPTOTIC STABILIZATION OF BILINEAR CONTROL SYSTEMS WITH PERIODIC COEFFICIENTS 1Sufficient conditions for uniform global asymptotic stabilization of the origin are obtained for bilinear control systems with periodic coefficients. The proof is based on the use of the Krasovsky theorem on global asymptotic stability of the origin for periodic systems. The stabilizing control function is feedback control constructed as the quadratic form of the phase variables and depends on time periodically.

Uniform Global Asymptotic Stabilization of Semilinear Periodic Discrete-Time Systems

“…Implication (1 ⇒ 2) in Theorems 2 and 3 is a converse Lyapunov theorem. A similar statement is known for linear periodic continuous-time systems; see the proof, e.g., in [13,Theorem 6]. This proof uses the reducibility of a periodic system (by a periodic Lyapunov-Floquet transformation to a time-invariant system).…”

“…The matrices of systems ( 13) and ( 3) are related by the equality B(t) = L(t + 1)A(t)L −1 (t). For system (13), denote by Θ(t, τ ) the transition matrix and by Σ(t) = Θ(t + ω, t) the monodromy matrix. Note that Σ(t + ω) = Σ(t).…”

“…Then K(t + ω) = K(t) and det K(t) = 0, t ∈ Z. Let us make the transformation z(t) = K(t)y(t) for system (13). We obtain the system…”