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,"

References

[1] H. Leiva, “Exact Controllability of the Suspension Bridge Model Proposed by Lazer and Mckenna,” Journal of Mathematical Analysis and Applications, Vol. 309, No. 2, 2005, pp. 404-419.

[2] E. Iturriaga and H. Leiva, “A Necessary and Sufficient Condition for the Controllability of Linear Systems in Hilbert Spaces and Applications,” IMA Journal of Mathematics, Control and Information, Vol. 25, No. 3, 2008, pp. 269-280.

[3] H. Leiva, “Exact Controllability of Semilinear Evolution Equation and Applications,” International Journal of Systems and communications, Vol. 1, No. 1, 2008, pp. 1- 12.

[4] H. Leiva and J. Uzcategui, “Exact Controllability of Semilinear Difference Equation and Application,” Journal of Difference Equations and Applications, Vol. 14, No. 7, 2008, pp. 671-679.

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

[6] R. F. Curtain and A. J. Pritchard, “Infinite Dimensional Linear Systems,” Lecture Notes in Control and Information Sciences, Vol. 8, Springer Verlag, Berlin, 1978.

[7] R. F. Curtain and H. J. Zwart, “An Introduction to Infinite Dimensional Linear Systems Theory,” Tex in Applied Mathematics, Vol. 21. Springer Verlag, New York, 1995.

[8] H. Leiva, “A Lemma on C0-Semigroups and Applications PDEs Systems” Quaestions Mathematicae, Vol. 26, No. 3, 2003, pp. 247-265.

[1] H. Leiva, “Exact Controllability of the Suspension Bridge Model Proposed by Lazer and Mckenna,” Journal of Mathematical Analysis and Applications, Vol. 309, No. 2, 2005, pp. 404-419.

[2] E. Iturriaga and H. Leiva, “A Necessary and Sufficient Condition for the Controllability of Linear Systems in Hilbert Spaces and Applications,” IMA Journal of Mathematics, Control and Information, Vol. 25, No. 3, 2008, pp. 269-280.

[3] H. Leiva, “Exact Controllability of Semilinear Evolution Equation and Applications,” International Journal of Systems and communications, Vol. 1, No. 1, 2008, pp. 1- 12.

[4] H. Leiva and J. Uzcategui, “Exact Controllability of Semilinear Difference Equation and Application,” Journal of Difference Equations and Applications, Vol. 14, No. 7, 2008, pp. 671-679.

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

[6] R. F. Curtain and A. J. Pritchard, “Infinite Dimensional Linear Systems,” Lecture Notes in Control and Information Sciences, Vol. 8, Springer Verlag, Berlin, 1978.

[7] R. F. Curtain and H. J. Zwart, “An Introduction to Infinite Dimensional Linear Systems Theory,” Tex in Applied Mathematics, Vol. 21. Springer Verlag, New York, 1995.

[8] H. Leiva, “A Lemma on C0-Semigroups and Applications PDEs Systems” Quaestions Mathematicae, Vol. 26, No. 3, 2003, pp. 247-265.