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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

$dX\left(t\right)=\left(A\left(t\right)X\left(t\right)+F\left(X\left(t\right),\nu \left(t\right)\right)\right)dt+G\left(X\left(t\right)\right)dM\left(t\right)$

in a Hilbert space, where $A\left(t\right)$ is an unbounded linear operator, $F$ and $G$ are differentiable functions with bonded derivatives, $M$ is a continuous martingale, and $\nu \left(t\right)$ 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:
 93E20 Optimal stochastic control (systems) 49K45 Optimal stochastic control (optimality conditions) 60H15 Stochastic partial differential equations 60H10 Stochastic ordinary differential equations 93E03 General theory of stochastic systems
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