Null Controllability and the Algebraic Riccati Equation in Banach Spaces

Abstract

By a recent result of Priola and Zabczyk, a null controllable linear system [y'(t) = Ay(t) + Bu(t)] in a Hilbert space E is null controllable with vanishing energy if and only if it is null controllable and the only positive self-adjoint solution of the associated algebraic Riccati equation [XA + A* X - XBB* X = 0] is the trivial solution X = 0. In this paper we extend this result to Banach spaces with an elementary proof which uses only reproducing kernel Hilbert space techniques. We also show that null controllability with vanishing energy implies null controllability.