Pontryagin’s maximum principle for an optimal control system governed by an ordinary differential equation with endpoint constraints is proved under the assumption that the control domain has no linear structure. We also obtain the variational equation, adjoint equation and Hamilton system for our problem.

We study the boundary control problems for stochastic parabolic equations with Neumann boundary conditions. Imposing super-parabolic conditions, we establish the existence and uniqueness of the solution of state and adjoint equations with non-homogeneous boundary conditions by the Galerkin approximations method. We also find that, in this case, the adjoint equation (BSPDE) has two boundary conditions (one is non-homogeneous, the other is homogeneous). By these results we derive necessary optimality conditions for the control systems under convex state constraints by the convex perturbation method.

We obtain the existence and uniqueness of Lp-solutions (1 ≤ p ≤ ∞) for Fokker–Planck equations with irregular and degenerate coefficients. Moreover, as an application, we also derive the existence and uniqueness of L1-solutions for the Fokker–Planck–Boltzmann equation.

In the present paper,we study stochastic boundary control problems where the system dynamics is a controlled stochastic parabolic equation with Neumann boundary control and boundary noise. Under some assumptions,the continuity and differentiability of the value function are proved.We also define a new type of Hamilton–Jacobi–Bellman(HJB) equation and prove that the value function is a viscosity solution of this HJB equation also defineanewtypeofHamilton–Jacobi–Bellman(HJB)equationandprovethatthevalue

By means of backward stochastic differential equations, the existence and uniqueness of the mild solution are obtained for the nonlinear Kolmogorov equations associated with stochastic delay evolution equations. Applications to optimal control are also given.