Dynamic-programming equations of stochastic optimal-control in Banach-space

In this paper the authors study the dynamic programming equation (Bellman equation) in Banach space, arising from optimal control problem for stochastic systems driven by a cylindrical Brownian motion, a generalization of white noise in infinite dimensions. A direct method which was introduced by Da Prato is further developed to prove the existence of mild solutions for the semilinear parabolic equations.