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Necessary conditions for optimal control of stochastic evolution equations in Hilbert spaces. (English) Zbl 1234.93112

The paper studies a system governed by a stochastic evolution equation

dX(t)=A ( t ) X ( t ) + F ( X ( t ) , ν ( t ) )dt+G(X(t))dM(t)

in a Hilbert space, where A(t) is an unbounded linear operator, F and G are differentiable functions with bonded derivatives, M is a continuous martingale, and ν(t) is a control.

The main problem considered in the article is minimizing the cost functional over a set of admissible controls. This problem is approached through using the theory of backward stochastic differential equations for deriving a stochastic maximum principle for this control problem. In fact, the adjoint equation the derived in the paper turns out to be a backward stochastic partial differential equation and it can be dealt with by using previous results by the same author.

93E20Optimal stochastic control (systems)
49K45Optimal stochastic control (optimality conditions)
60H15Stochastic partial differential equations
60H10Stochastic ordinary differential equations
93E03General theory of stochastic systems
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