Introduction.

X u(0) = x, where W is a cylindrical Wiener process on i f , A generates a semigroup on H and the control u belongs to a com pact control space V . Our objective is to find a control u minimizing the following cost functional:

J{u) = e { j f 1 b {X u(t), u(t))dt + / i (X “( l ) ) | . (2)

Before passing to formal presentation of the problem we must notice that even in the sim­ plest cases the optimal control does not need to exist. Consider the following deterministic equation on H = R:

X '( t ) = u(t), X ( 0 ) = x with the cost functional ^

J{u) = j ( x 2(t) — u2( t f jd t , where u(t) E V = [—1,1]. Putting un(t) = (—l)[nil we compute that J ( u n) = — 1 —► —1 as n —> oo. Since J(u) > — 1 for any control u we derive that infw J(u) = — 1. But for any