Optimal control of backward stochastic heat equation with Neumann boundary control and noise

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Abstract
This paper considers a stochastic control problem in which the dynamic system is a controlled backward stochastic heat equation with Neumann boundary control and boundary noise and the state must coincide with a given random vector at terminal time. Through defining a proper form of the mild solution for the state equation, the existence and uniqueness of the mild solution is given. As a main result, a global maximum principle for our control problem is presented. The main result is also applied to a backward linear-quadratic control problem in which an optimal control is obtained explicitly as a feedback of the solution to a forward–backward stochastic partial differential equation.

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