Path-Dependent Hamilton--Jacobi Equations with Super-Quadratic Growth in the Gradient and the Vanishing Viscosity Method

We establish necessary and sufficient conditions for viability of evolution inclusions with locally monotone operators in the sense of Liu and Röckner [J. Funct. Anal., 259 (2010), pp. 2902-2922]. This allows us to prove wellposedness of lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations associated to the optimal control of evolution inclusions. Thereby, we generalize results in Bayraktar and Keller [J. Funct. Anal., 275 (2018), pp. 2096-2161] on Hamilton-Jacobi equations in infinite dimensions with monotone operators in several ways. First, we permit locally monotone operators. This extends the applicability of our theory to a wider class of equations such as Burgers' equations, reaction-diffusion equations, and 2D Navier-Stokes equations. Second,our results apply to optimal control problems with state constraints. Third, we have uniqueness of viscosity solutions. Our results on viability and lower semicontinuous solutions are new even in the case of monotone operators.

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Path-Dependent Hamilton--Jacobi Equations with Super-Quadratic Growth in the Gradient and the Vanishing Viscosity Method

Cited by 7 publications

References 38 publications

A convergence theorem for Crandall–Lions viscosity solutions to path-dependent Hamilton–Jacobi–Bellman PDEs

A convergence theorem for Crandall–Lions viscosity solutions to path-dependent Hamilton–Jacobi–Bellman PDEs

Optimal control of stochastic delay differential equations: Optimal feedback controls

Viability for locally monotone evolution inclusions and lower semicontinuous solutions of Hamilton–Jacobi–Bellman equations in infinite dimensions

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