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Regularity of solutions of a second order Hamilton-Jacobi equation and application to a control problem. (English) Zbl 0842.49021
A certain Hamilton-Jacobi equation on a Hilbert space with initial conditions is thoroughly studied and existence and regularity results are established. Then the results are applied to solve a certain stochastic optimal control problem. The solution to the Hamilton-Jacobi equation turns out to be the value function of the optimal control problem. To solve the Hamilton-Jacobi equation, a linearized version is solved first. This can be done by using known techniques. The resulting semigroup is then used to write an integral version of the Hamilton-Jacobi equation and then its solution follows by a fixed point argument. The results are of interest in functional analysis, probability, and stochastic optimal control.

49L99 Hamilton-Jacobi theories
49N60 Regularity of solutions in optimal control
70H20 Hamilton-Jacobi equations in mechanics
93E20 Optimal stochastic control
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