## A dynamic programming approach to nonlinear boundary control problems of parabolic type.(English)Zbl 0823.49017

The authors study a Hamilton-Jacobi equation in a Hilbert space that in fact is a generalization of the dynamic programming equation related to the boundary control of a parabolic equation with Neumann boundary conditions. They manage to devise a revised definition of a viscosity solution which allows them to get existence and uniqueness results in case of an unbounded Hamiltonian.
The main difficulty is to select appropriate relaxations of the unobunded terms appearing in the Hamilton-Jacobi equation. The authors borrow some ideas from Ishii (1992) and a very recent paper of Tataru to obtain existence results when the value function of the optimal control problem is Lipschitz continuous with respect to the negative powers of the operator $$-A$$ appearing in the Hamilton-Jacobi equation and give a simplified version of their existence and uniqueness results.

### MSC:

 49L20 Dynamic programming in optimal control and differential games 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games 49J20 Existence theories for optimal control problems involving partial differential equations
Full Text: