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In this article, a notion of viscosity solutions is introduced for second-order Hamilton–Jacobi–Bellman (HJB) equations with delays associated with optimal control problems for stochastic differential equations with delays. We identify the value functional of optimal control problem as the unique viscosity solution to the second-order HJB equation. [ABSTRACT FROM AUTHOR]

In this article, optimal control problems of differential equations with delays are investigated for which the associated Hamilton–Jacobi–Bellman (HJB) equations are nonlinear partial differential equations with delays. This type of HJB equation has not been previously studied and is difficult to solve because the state equations do not possess smoothing properties. We introduce a new notion of viscosity solutions and identify the value functional of the optimal control problems as the unique solution to the associated HJB equations. An analytical example is given as application. [ABSTRACT FROM AUTHOR]

In this article, a notion of viscosity solutions is introduced for first order path-dependent Hamilton–Jacobi–Bellman (PHJB) equations associated with optimal control problems for path-dependent evolution equations in Hilbert space. We identify the value functional of optimal control problems as unique viscosity solution to the associated PHJB equations without the assumption (A.4) on page 231 of Li and Yong (1995). We also show that our notion of viscosity solutions is consistent with the corresponding notion of classical solutions, and satisfies a stability property. [ABSTRACT FROM AUTHOR]

In this paper, a controlled stochastic delay heat equation with Neumann boundary-noise and boundary-control is considered. The existence and uniqueness of the mild solution for the associated Hamilton–Jacobi–Bellman equations are obtained by means of the backward stochastic differential equations, which is applied to the optimal control problem. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

We consider the optimal control for a Banach space valued stochastic delay evolution equation. The existence and uniqueness of the mild solution for the associated Hamilton-Jacobi-Bellman equations are obtained by means of backward stochastic differential equations. An application to optimal control of stochastic delay partial differential equations is also given. [ABSTRACT FROM AUTHOR]

In this article, we consider an optimal control problem in which the controlled state dynamics is governed by a stochastic evolution equation in Hilbert spaces and the cost functional has a quadratic growth. The existence and uniqueness of the optimal control are obtained by the means of an associated backward stochastic differential equations with a quadratic growth and an unbounded terminal value. As an application, an optimal control of stochastic partial differential equations with dynamical boundary conditions is also given to illustrate our results. [ABSTRACT FROM AUTHOR]