Existence of optimal controls for stochastic partial differential equations with fully local monotone coefficients
Gaofeng Zong
TL;DR
This work proves the existence of stochastic optimal feedback controls for controlled stochastic partial differential equations with fully local monotone coefficients. It combines Faedo-Galerkin approximations, uniform energy estimates, and tightness arguments to establish well-posedness and strong convergence of approximate solutions, even in nonconvex settings. From minimizing sequences of controls, it extracts a limit control Φ^* whose induced state X_{Φ^*} achieves the minimum of the cost J, via lower semicontinuity and Fatou's lemma. The results extend prior locally monotone frameworks to fully monotone coefficients and apply to a wide range of stochastic PDEs, including quasilinear and Navier–Stokes-type systems, with concrete exemplars such as a controlled quasilinear SPDE. Overall, the paper provides a rigorous pathway to existence of optimal feedback controls in complex stochastic evolution equations with local monotone structure, enabling applications across physics, engineering, and finance.
Abstract
This paper deals with a stochastic optimal feedback control problem for the controlled stochastic partial differential equations. More precisely, we establish the existence of stochastic optimal feedback control for the controlled stochastic partial differential equations with fully monotone coefficients by a minimizing sequence for the control problem. Using the Faedo-Galerkin approximations, the uniform estimates and the tightness in some appropriate space for the Faedo-Galerkin approximating solution can be obtain to prove the well-posedness of the controlled stochastic partial differential equations with fully monotone coefficients. The results obtained in the present paper may be applied to various types of controlled stochastic partial differential equations, such as the controlled stochastic convection diffusion equation.