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A characterization of semilinear dense range operators and applications

Abstract:

We characterize a broad class of semilinear dense range operators G H: W → Z given by the following formula, GH w = Gw + H (w), w ε W, where Z, W are Hilbert spaces, G ε 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 R a n g (G) ̄ = Z, then R a n g (G H) ̄ = Z and for all z ε Z there exists a sequence {wα ε Z: 0 < α ≤ 1 } given by wα = G * (α I + G G *) - 1 (z - H (wα)), such that l i m α → 0 + { G u α + H (u α) } = z. Finally, we apply this result to prove the approximate controllability of the following semilinear evolution equation: z ′ = A z + B u (t) + F (t, z, u (t)), z ε Z, u ε 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 ε 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. © 2013 H. Leiva et al.