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Nonlinear optimal control problems in Hilbert spaces are considered for whichwe derive approximation theorems for Galerkin approximations. Approximationtheorems are available in the literature. The originality of our approachrelies on the identification of a set of natural assumptions that allows us todeal with a broad class of nonlinear evolution equations and cost functionalsfor which we derive convergence of the value functions associated with theoptimal control problem of the Galerkin approximations. This convergence resultholds for a broad class of nonlinear control strategies as well. In particular,we show that the framework applies to the optimal control of semilinear heatequations posed on a general compact manifold without boundary. The frameworkis then shown to apply to geoengineering and mitigation of greenhouse gasemissions formulated for the first time in terms of optimal control of energybalance climate models posed on the sphere $\mathbb{S}^2$.

Galerkin approximations of nonlinear optimal control problems in Hilbert spaces