A Characterization of Semilinear Dense Range Operators and Applications

Abstract EN

We characterize a broad class of semilinear dense range operators GH:W→Z given by the following formula, GHw=Gw+H(w), w∈W, where Z, W are Hilbert spaces, G∈L(W,Z), and H:W→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 Rang(G)¯=Z, then Rang(GH)¯=Z and for all z∈Z there exists a sequence {wα∈Z:0<α≤1} given by wα=G*(αI+GG*)-1(z-H(wα)), such that limα→0+{Guα+H(uα)}=z.

Finally, we apply this result to prove the approximate controllability of the following semilinear evolution equation: z′=Az+Bu(t)+F(t,z,u(t)), z∈Z, u∈U, t>0, where Z, U are Hilbert spaces, A:D(A)⊂Z→Z is the infinitesimal generator of strongly continuous compact semigroup {T(t)}t≥0 in Z,B∈L(U,Z), the control function u belongs to L2(0,τ;U), and F:[0,τ]×Z×U→Z is a suitable function.

As a particular case we consider the controlled semilinear heat equation.