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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.

MSC:
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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