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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 \[ dX(t) = \big(A(t)X(t) + F(X(t),\nu(t))\big)dt + G(X(t))dM(t) \] 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 \(\nu(t)\) 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:

93E20 Optimal stochastic control
49K45 Optimality conditions for problems involving randomness
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93E03 Stochastic systems in control theory (general)
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