Nonlinear Optimal Control with Infinite Horizon for Distributed Parameter Systems and Stationary Hamilton–Jacobi Equations

Optimal control problems, with no discount, are studied for systems governed by nonlinear 'parabolic' state equations, using a dynamic programming approach. If the dynamics are stabilizable with respect to cost, then the fact that the value function is a generalized viscosity solution of the associated Hamilton-Jacobi equation is proved. This yields the feedback formula. Moreover, uniqueness is obtained under suitable stability assumptions.