Abstract
In the theory control systems, there are many various qualitative control problems that can be considered. In our previous work, we have analyzed the approximate controllability and observability of the nonautonomous Riesz-spectral systems including the nonautonomous Sturm-Liouville systems. As a continuation of the work, we are concerned with the analysis of stability, stabilizability, detectability, exact null controllability, and complete stabilizability of linear non-autonomous control systems in Banach spaces. The used analysis is a quasisemigroup approach. In this paper, the stability is identified by uniform exponential stability of the associated C0-quasisemigroup. The results show that, in the linear nonautonomous control systems, there are equivalences among internal stability, stabizability, detectability, and input-output stability. Moreover, in the systems, exact null controllability implies complete stabilizability.
Highlights
In this paper we focus on linear nonautonomous control system ẋ (t) = A (t) x (t) + B (t) u (t), x (0) = x0, t ≥ 0 (1)
Where x(t) ∈ X is the state, u(t) ∈ U is the control, and X and U are complex Hilbert spaces of the state and control, respectively; A(t) is a densely defined operator in X with domain D(A(t)) = D, independent of t; and B(t) : U → X is a bounded operator such that B(⋅) ∈ L∞(R+, Ls(U, X)), where Ls(V, W) and L∞(Ω, W) denote the space of bounded operators from V to W equipped with strong operator topology and the space of bounded measurable functions from Ω to W provided with essential supremum norm, respectively
Rabah et al [10] prove that exact null controllability implies complete stabilizability for neutral type linear systems in Hilbert spaces
Summary
For non-autonomous control systems of the finite-dimensional spaces, Ikeda et al [7] proved that if the system is null controllable, it is completely stabilizable. Phat and Kiet [8] investigated relationship between stability and exact null controllability extending the Lyapunov equation in Banach spaces. Rabah et al [10] prove that exact null controllability implies complete stabilizability for neutral type linear systems in Hilbert spaces. In the linear nonautonomous systems in Hilbert spaces, Niamsup and Phat [12] have proved that exact null controllability implies the complete stabilizability. Until now there is no research which investigates the qualitative control problems of the nonautonomous control systems implementing C0-quasisemigroup theory.
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