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We prove existence of optimal control for the deterministic and stochasticshallow lake problem without any restrictions on the parameter space and weestablish a generalization of the Arrhenius Law in the case of noise-dependentpotentials, which naturally arise in control theory problems. We also prove aresult about convergence of the derivatives in the viscosity approximation ofthe value function and use this result to derive the Arrhenius Law for theshallow lake problem.

Optimal control, viscosity approximation and Arrhenius Law for the shallow lake problem