Galerkin approximations of nonlinear optimal control problems in Hilbert spaces

TL;DR

This paper establishes convergence theorems for Galerkin approximations of nonlinear optimal control problems in Hilbert spaces, including applications to climate models and heat equations on manifolds.

Contribution

It introduces a set of natural assumptions enabling convergence results for a broad class of nonlinear evolution equations and control strategies.

Findings

Convergence of value functions for Galerkin approximations.

Applicability to semilinear heat equations on manifolds.

Application to climate energy balance models.

Abstract

Nonlinear optimal control problems in Hilbert spaces are considered for which we derive approximation theorems for Galerkin approximations. Approximation theorems are available in the literature. The originality of our approach relies on the identification of a set of natural assumptions that allows us to deal with a broad class of nonlinear evolution equations and cost functionals for which we derive convergence of the value functions associated with the optimal control problem of the Galerkin approximations. This convergence result holds for a broad class of nonlinear control strategies as well. In particular, we show that the framework applies to the optimal control of semilinear heat equations posed on a general compact manifold without boundary. The framework is then shown to apply to geoengineering and mitigation of greenhouse gas emissions formulated for the first time in terms of optimal control of energy balance climate models posed on the sphere $\mathbb{S}^2$.