Function space null controllability of linear delay systems with limited power (Q1095848)

The system under consideration has the form \[ (1)\quad \dot x(t)=L(t,x_ t)+B(t)u(t),\quad t\geq \tau \] in the standard notation for functional differential equations. Here B(t) is an n by m matrix continuous in t and the mapping \(\phi\to L(t,\phi)\) for \(\phi\) in \(W_ 2^{(1)}([-h,0],E^ n)\) satisfies appropriate conditions. The main result is the following. Assume (i) that the system (1) is ull controllable; and (ii) the system \(\dot x(t)=L(t,x_ t)\) is uniformly asymptotically stable. Then (1) is null controllable with constraints. The term null controllable here means that for each \(t_ 1>\tau\) and each \(\phi\) in \(W_ 2^{(1)}\), there exists a control u in \(L_ 2([\tau,t_ 1],E^ m)\) for which the solution \(x_ t\) is \(\phi\) at \(t=\tau\) and zero at \(t=t_ 1\). Null controllable with constraints is similarly defined but with controls restricted to lie in the set with \(L_ 2\)-norm \(\| u\| \leq 1.\) Necessary and sufficient conditions for null controllability of (1) are obtained when the operator L has a tem A(t)\(\phi\) (0), and null controllability with constraints is proved for certain differential- difference equations.