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J. L. Lions, “Optimal Control of Systems Governed by Partial Differential Equations,” Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen Band 170, 1971.

has been cited by the following article:

  • TITLE: A Characterization of Semilinear Surjective Operators and Applications to Control Problems

    AUTHORS: Edgar Iturriaga, Hugo Leiva

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

    JOURNAL NAME: Applied Mathematics, Vol.1 No.4, October 29, 2010

    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.