Existence of optimal controls for SPDE with locally monotone coefficients

Abstract

The aim of this paper is to investigate the existence of optimal controls for systems described by stochastic partial differential equations (SPDEs) with locally monotone coefficients controlled by external forces which are feedback controls. To attain our objective we adapt the argument of Lisei (2002) where the existence of optimal controls to the stochastic Navier–Stokes equation was studied. The results obtained in the present paper may be applied to demonstrate the existence of optimal controls to various types of controlled SPDEs such as: a stochastic nonlocal equation and stochastic semilinear equations which are locally monotone equations; we also apply the result to a monotone equation such as the stochastic reaction–diffusion equation and to a stochastic linear equation.

Keywords:

• Stochastic optimal control
• stochastic partial differential equation
• stochastic systems
• control problems

Acknowledgments

The authors wishes to express his gratitude to the anonymous referees for the several helpful comments, which in fact improve the aspect of the present work.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

EdsonA. Coayla-Teran  http://orcid.org/0000-0002-6329-4929

Additional information

Funding

This work was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) [grant number 480356/2010-6].