Abstract
In this paper we characterize a broad class of semilinear surjective operators given by the following formula where Z are Hilbert spaces, and is a suitable nonlinear function. First, we give a necessary and sufficient condition for the linear operator to be surjective. Second, we prove the following statement: If and is a Lipschitz function with a Lipschitz constant small enough, then and for all the equation admits the following solution .We use these results to prove the exact controllability of the following semilinear evolution equation , , where , are Hilbert spaces, is the infinitesimal generator of strongly continuous semigroup in the control function belong to and is a suitable function. As a particular case we consider the semilinear damped wave equation, the model of vibrating plate equation, the integrodifferential wave equation with Delay, etc.
Highlights
In this paper we characterize a broad class of semilinear surjective operators GH V Z given by the following formula
We apply our results to prove the exact controllability of the following semilinear evolution equation
Under some conditions on F, we prove that the controllability of the linear system (1.3) is preserved by the semilinear system (1.2)
Summary
In this paper we characterize a broad class of semilinear surjective operators GH V Z given by the following formula. The results are so general that can be apply to those control systems governed by evolutions equations like the one studied in [1,2,3] and [4]. We present here a variational approach to find solutions of the semilinear equation Gw H (w) z which is motivated by the one used to prove the interior controllability for some control system governed by PDE’s, see [5]. These results can be used to motivate the study of semilinear range dense operator in order to characterize the approximate controllability of evolution equations
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