Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay

Xianlong Fu

doi:10.3934/eect.2017026

Abstract

The controllability of non-autonomous evolution systems is an important and difficult topic in control theory. In this paper, we study the approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay. The theory of linear evolution operators is used instead of \begin{document}$C_0-$\end{document} semigroup to discuss the problem. Some sufficient conditions of approximate controllability are formulated and proved here by using the resolvent operator condition. Finally, two examples are provided to illustrate the applications of the obtained results.

Highlights

In this paper, we investigate the approximate controllability of systems represented in the following semilinear non-autonomous evolution system with state-dependent delay: dx(t) dt = −A(t)x(t) + Bu(t) + F (t, xρ(t,xt)), t ∈ [0, T ] := J, (1)x0 = φ ∈ Bα, where the state variable x(·) takes values in a Hilbert space X and the control function u(·) is given in Hilbert space L2(J; U ) with U a Hilbert space
Bα ⊂ B, and B is a phase space given
The notation xt represents the function defined by xt : (−∞, 0] → X, xt(θ) = x(t + θ), and belongs to some abstract phase space Bα described axiomatically and ρ : J × Bα →

Summary

Introduction

We investigate the approximate controllability of systems represented in the following semilinear non-autonomous evolution system with state-dependent delay: dx(t) dt = −A(t)x(t) + Bu(t) + F (t, xρ(t,xt)), t ∈ [0, T ] := J,. X0 = φ ∈ Bα, where the state variable x(·) takes values in a Hilbert space X and the control function u(·) is given in Hilbert space L2(J; U ) with U a Hilbert space. B is a bounded linear operator from U into X. Bα ⊂ B, and B is a phase space given . The notation xt represents the function defined by xt : (−∞, 0] → X, xt(θ) = x(t + θ), and belongs to some abstract phase space Bα described axiomatically and ρ : J × Bα → Approximate controllability, non-autonomous evolution equation, linear evolution system, state-dependent delay, fractional power operator.

XIANLONG FU