A Characterization of Semilinear Dense Range Operators and Applications

Abstract

We characterize a broad class of semilinear dense range operators $G_H:W\to Z$ given by the following formula, $G_{H}w=Gw+H(w),w\in W$, where $Z$, $W$ are Hilbert spaces, $G\in L(W,Z)$, and $H:W\to Z$ is a suitable nonlinear operator. First, we give a necessary and sufficient condition for the linear operator $G$ to have dense range. Second, under some condition on the nonlinear term $H$, we prove the following statement: If $\overline{\text{R}\text{a}\text{n}\text{g}(G)}=Z$, then $\overline{\text{R}\text{a}\text{n}\text{g}(G_H)}=Z$ and for all $z\in Z$ there exists a sequence given by $w_{\alpha }=G^{*}(\alpha I+G{G}^{*}{)}^{-1}(z-H(w_{\alpha }))$, such that $\text{l}\text{i}\text{m}\alpha \to 0^{+}\{G{u}_{\alpha }+H(u_{\alpha })\}=z$. Finally, we apply this result to prove the approximate controllability of the following semilinear evolution equation: $z^{\prime }=Az+Bu(t)+F(t,z,u(t)),z\in Z,u\in U,t>0$, where $Z$, $U$ are Hilbert spaces, $A:D(A)\subset Z\to Z$ is the infinitesimal generator of strongly continuous compact semigroup $\{T(t){\}}_{t\ge 0}$ in $Z,B\in L(U,Z)$, the control function $u$ belongs to $L^2(0,\tau ;U)$, and $F:[0,\tau ]\times Z\times U\to Z$ is a suitable function. As a particular case we consider the controlled semilinear heat equation.