  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 $\mathbb{S}^2$.
