Optimal control of self-adjoint nonlinear operator equations in Hilbert spaces

Abstract

In this paper, we formulate and study a general optimal control problem governed by nonlinear operator equations described by unbounded self-adjoint operators in Hilbert spaces. This problem extends various particular control models studied in the literature, while it has not been considered before in such a generality. We develop an efficient way to construct a finite-dimensional subspace extension of the given self-adjoint operator that allows us to design the corresponding adjoint system and finally derive an appropriate counterpart of the Pontryagin Maximum Principle for the constrained optimal control problem under consideration by using the obtained increment formula for the cost functional and needle type variations of optimal controls.

Keywords:

• optimal control
• operator equations
• maximum principle

AMS Subject Classifications:

• 49K20
• 49K22

Acknowledgments

The authors thank the King Fahd University of Petroleum and Minerals for excellent facilities provided to support scientific research. Research of the second author was partly supported by the USA National Science Foundation under Grant DMS-1007132.