Approximate Controllability of Distributed Systems by Distributed Controllers

Approximate controllability problem for a linear distributed control system with possibly unbounded input operator, connected in a series to another distributed system without control is investigated. An initial state of the second distributed system is considered as a control parameter. Applications to control partial equations governed by hyperbolic controller, and to control delay systems governed by hereditary controller are considered. 1. Statement of the Problem Research in control theory started for single control systems. However, many technical applications use control systems interconnected in many ways. The goal of the present paper is to establish approximate controllability conditions for a control system interconnected in a series with a second homogeneous system without control in such a way that a control function of the first control system is an output of the second one, so a control is considered as an initial state of a second system. Let X,U,Z be Hilbert spaces, and let A,C be infinitesimal generators of strongly continuous C0-semigroups SA(t) in X and SC(t) in Z correspondingly in the class C0 [5, 8]. Consider the abstract evolution control equation ẋ(t) = Ax(t) +Bu(t), x(0) = x0, u(t) = Kz(t), 0 ≤ t < +∞, (1.1) where z(t) is a mild solution of another evolution equation of the form ż(t) = Cz(t), z(0) = z0, 0 ≤ t < +∞. (1.2) Here x(t), x0 ∈ X, u(t) ∈ U , z(t), z0 ∈ Z, B : U → X is a linear possibly unbounded operator, K : Z → U is a linear possibly unbounded onto operator. Equation (1.2) is said to be a controller equation. A control u(t) is defined by u(t) = Kz(t) as an output of controller equation (1.2). 2000 Mathematics Subject Classification. 93B05.