Viscosity Solutions of Hamilton-Jacobi Equations in Infinite Dimensions. VII. The HJB Equation Is Not Always Satisfied

Let A = −Δ with domain H10(Ω)∩H2(Ω) where Ω is open, smooth, and bounded. Run the state equation dXt/dt + AXt = αt with the control α and the initial value X0 = x in L2(Ω) to determine Xt. The values αt of the control are constrained to lie in a fixed bounded subset A of H−1(Ω). Given a running cost f, the value function u(x) = infα∫∞0e−tf(Xt) dt is expected to satisfy the HJB equation of dynamic programming, [formula]. The delicate situation in which the boundedness of A in H−1 is coupled with uniform continuity of f on H1(Ω) is examined. An example is given in which f(x) = β(||x||H1(Ω)) where β is smooth and compactly supported and does not vanish identically and A is the unit ball, but u ≡ 0. On the positive side, it is proved that the value function may still be characterized as the maximal subsolution of the HJB equation. Part of the issue is to formulate the correct "viscosity" solution notions in this case. Conditions are given on f for which u is indeed a solution of the HJB equation, and it is shown that the value function is the uique solution when there is a solution. Various generalizations of the special cases above are presented.