A Characterization of Semilinear Surjective Operators and Applications to Control Problems

Abstract

In this paper we characterize a broad class of semilinear surjective operators given by the following formulawhere 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.

In this paper we characterize a broad class of semilinear surjective operators given by the following formulawhere 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.

Keywords

Semilinear Surjective Operators, Evolution Equations, Controllability, Damped Wave Equation

Semilinear Surjective Operators, Evolution Equations, Controllability, Damped Wave Equation

Cite this paper

nullE. Iturriaga and H. Leiva, "A Characterization of Semilinear Surjective Operators and Applications to Control Problems,"*Applied Mathematics*, Vol. 1 No. 4, 2010, pp. 265-273. doi: 10.4236/am.2010.14033.

nullE. Iturriaga and H. Leiva, "A Characterization of Semilinear Surjective Operators and Applications to Control Problems,"

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