## On control of solution of stochastic differential equation in Hilbert space.(Ukrainian, English)Zbl 1026.60081

Teor. Jmovirn. Mat. Stat. 65, 53-59 (2001); translation in Theory Probab. Math. Stat. 65, 61-68 (2002).
Summary: Let $$H$$ be a real separable Hilbert space with the inner product $$(h_1,h_2)$$, and the norm $$\|h\|$$ and let $$H_{-}$$ be the completed Hilbert space with the norm $$\|h\|_{-}=\|W^{-1/2}h\|$$, where $$W$$ is a self-adjoint positive definite operator on $$H$$. Consider a stochastic differential equation of the form $d\xi_t=f(t,\xi,u_t(\xi)) dt+dW_t, \quad \xi_0=0, \quad t\in[0,T],$ where $$\{W_t$$, $$t\in[0,T]\}$$ is a Wiener process with values from $$H_-$$, $$u=\{u_t$$, $$t\in[0,T]\}$$ is a control function with values from the set of possible control functions $$U$$, and $$f(t,x,u)$$ is an $$H$$-valued drift function. The problem considered is to find a control $$u^{\ast}\in U$$ which minimizes $$E\int_0^TC(t,\xi) dt$$, where $$C(t,\xi)$$ is a risk function. Under some conditions on the functions $$f(t,x,u)$$ and $$C(t,\xi)$$ the authors prove the existence of the optimal control $$u^{\ast}\in U$$.

### MSC:

 60H20 Stochastic integral equations 93E20 Optimal stochastic control

### Keywords:

Wiener process; separable Hilbert space; optimal control