ABSTRACT
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