Infinite-dimensional Hamilton-Jacobi equations and Dirichlet boundary control problems of parabolic type. (English) Zbl 0880.49029

Cannarsa, Piermarco; Tessitore, Maria Elisabetta

SIAM J. Control Optimization 34, No. 6, 1831-1847 (1996).

In this paper the authors study the existence and the uniqueness of viscosity solutions to the infinite-dimensional Hamilton-Jacobi equation \[ \lambda\upsilon(x)+ \langle Ax+\Phi(x),D\upsilon(x)\rangle+ H(A^{\beta}x, D\upsilon(x))=0 \] with \[ \lambda >0\quad \text{and}\quad \beta \in (0,1). \] The study is motivated by its connections to boundary control of parabolic equations under Dirichlet boundary conditions. In this framework, the main tool for constructing optimal boundary control is represented by the Riccati equation. For problems which are not linear-quadratic the role of the Riccati equation is played by the above dynamic programming equation. The authors, following a well established scheme suitably adapted to this problem, prove that the unique viscosity solution is the corresponding value function.

Reviewer: P.Zezza (Firenze)

MSC:

49L20	Dynamic programming in optimal control and differential games
49L25	Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35G10	Initial value problems for linear higher-order PDEs
35K25	Higher-order parabolic equations
35K55	Nonlinear parabolic equations

Keywords:

Hamilton-Jacobi equation; boundary control; existence; uniqueness; viscosity solutions