A Characterization of Semilinear Dense Range Operators and Applications. (18th March 2013) - Leiva, H.; Merentes, N.; Sanchez, J.
Abstract : We characterize a broad class of semilinear dense range operatorsG H : W → Z given by the following formula, G H w = G w + H ( w ), w ∈ W, whereZ, W are Hilbert spaces, G ∈ L ( W, Z ), andH : W → Z is a suitable nonlinear operator. First, we give a necessary and sufficient condition for the linear operatorG to have dense range. Second, under some condition on the nonlinear termH, we prove the following statement: IfR a n g ( G ) ¯ = Z, thenR a n g ( G H ) ¯ = Z and for allz ∈ Z there exists a sequence{ w α ∈ Z : 0 < α ≤ 1 } given byw α = G * ( α I + G G * ) - 1 ( z - H ( w α ) ), 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 ∈ Z, u ∈ U, t > 0, whereZ, U are Hilbert spaces, A : D ( A ) ⊂ Z → Z is the infinitesimal generator of strongly continuous compact semigroup{ T ( t ) } t ≥ 0 inZ, B ∈ L ( U, Z ), the control functionu belongs toL 2 ( 0, τ ; U ), andF : [ 0, τ ] × Z × U → Z is a suitable function. As a particular case we consider the controlled semilinear heat equation.