二阶Hamilton-Jacobi-Bellman(HJB)方程已广泛应用于随机时滞方程最优控制问题的研究。在前期研究中，申请者得到了随机时滞发展方程对应的二阶HJB方程温和解的存在唯一性。本项目基于随机分析和最优控制理论，定性研究Hilbert空间中二阶HJB方程的粘性解及其在随机时滞方程最优控制问题中的应用。主要研究随机时滞微分方程，随机时滞发展方程和边界受控的随机时滞热方程等三类受控方程。针对随机时滞方程化成随机发展方程后，无界微分算子仅生成强连续(非压缩)半群，所以拟直接构建最优控制问题所对应的二阶HJB方程，结合动态规划原理得到上述方程最优控制问题的值函数是相应的二阶HJB方程的粘性解；拟通过定义一个非负算子克服时滞所带来的困难，从而将粘性解理论用于研究相应的二阶HJB方程粘性解的唯一性。本方法为随机时滞方程最优控制的研究提供了新的途径。

Second order Hamilton-Jacobi-Bellman (HJB) equations have been widely applied to optimal control problems for stochastic delay equations. With our earlier contributions in the existence and uniqueness of the mild solution for the second order HJB equations associated with stochastic delay evolution equations, this project qualitatively studies the viscosity solutions of the second order HJB equations in Hilbert spaces and its applications in optimal control problems for stochastic delay equations on the base of stochastic analysis and optimal control theories. We mainly study the following three types of controlled equations: stochastic delay differential equations, stochastic delay evolution equations and stochastic delay heat equations with boundary control. The unbounded differential operator only generates a strongly continuous (not contraction) semigroup when stochastic delay equations be reformulated as stochastic evolution equations, therefore, we intend to construct the second order HJB equations associated with the optimal control problems directly, and combined with the dynamic programming principle we show the value functions for the optimal control problems of the above equations are the viscosity solutions of the associated second order HJB equations; by defining a non-negative operator, we expect to overcome the difficulties brought by the delay, then the theories of viscosity solutions can be used to study the uniqueness of the viscosity solution for the corresponding second order HJB equations. It provides a new way to the study of optimal control problems for stochastic delay equations.