A note on admissibility for unbounded bilinear control systems

Abstract

This paper studies infinite-dimensional bilinear control systems described by $y'(t)=Ay(t)+u(t)By(t)$ where $A$ generates a semigroup $(e^{tA})_{t\geq 0}$ on a Banach space $Y$ (state space), $B:D(B)(\subset Y)\rightarrow Y$ is an unbounded linear operator and $u\in L^{p}_{loc}(0,\infty)$ is a scalar control. Sufficient conditions are given for $B$ to be admissible, i.e for any $t$, the integral $\int^{t}_{0}u(s)e^{(t-s)A}By(s)ds$ should be in $Y$ and depends continuously on $u\in L^{p}(0,\infty)$, $y\in L^{q}(0,\infty;Y)$ for some appropriate positive numbers $p$, $q$. This approach enables us to obtain, through an integrated form, a unique solution for the bilinear system. The results are applied to a heat equation.