A Comparison Principle for Hamilton-Jacobi Equations Related to Controlled Gradient Flows in Infinite Dimensions
We develop new comparison principles for viscosity solutions of Hamilton–Jacobi equations associated with controlled gradient flows in function spaces as well as the space of probability measures. Our examples are optimal control of Ginzburg–Landau and Fokker–Planck equations. They arise in limit considerations of externally forced non-equilibrium statistical mechanics models, or through the large deviation principle for interacting particle systems. Our approach is based on two key ingredients: an appropriate choice of geometric structure defining the gradient flow, and a free energy inequality resulting from such gradient flow structure. The approach allows us to handle Hamiltonians with singular state dependency in the nonlinear term, as well as Hamiltonians with a state space which does not satisfy the Radon–Nikodym property. In the case where the state space is a Hilbert space, the method simplifies existing theories by avoiding the perturbed optimization principle.
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