is a metric space.

The dynamic programming approach studies the properties of the so called value function for the problem, identifies it as a solution of the associated Hamilton-Jacobi-Bellman (HJB) equation through the dynamic programming principle, and then tries to use this PDE to construct optimal feedback controls, obtain conditions for optimality, do numerical computations, etc.. There exists an extensive literature on the subject for optimal control of ordinary differential equations, i.e. when the HJB equations are finite dimensional (see for instance the books [@BCD; @CLSW; @FR; @FS; @Lo; @Vinter; @YongZhou] and the references therein). The situation is much more complicated for optimal control of partial differential equations (PDE) or abstract evolution equations, i.e. when the HJB equations are infinite dimensional, nevertheless there is by now a large body of results on such HJB equations and the dynamic programming approach ([@B1; @B2; @B3; @B4; @BaBaJe; @BaDaP1; @BaDaP2; @BaDaP3; @BaDaP4; @BaDaPPo; @Ca1; @CaCa; @CaDaP1; @CaDaP2; @CaDiB; @CaGoSo; @CaFr1; @CaFr2; @CaTe1; @CaTe2; @CL4; @CL5; @CL6; @CL7; @DiB; @GSS; @I; @KoSo; @LiYong; @Sh; @Sri; @T1; @T2] and the references therein). Numerous notions of solutions are introduced in these works, the value functions are proved to be solutions of the dynamic programming equations, and 