Random relaxed controls and partially observed stochastic systems (Q1314873)

The basic problem of optimal stochastic controls is to find a control \(u\) which satisfies \[ j(u)= \inf\{ j(v):\;v \text{ is an ordinary control}\}, \tag{1} \] where \(j(u)\) is the cost of \(u\). In general an ordinary control \(u\) which satisfies (1) does not exist. The authors introduce a new class of generalized controls called random relaxed controls. They show that under quite general conditions an optimal random relaxed control \(u\) exists and satisfies \(j(u)= \inf\{ j(v)\): \(v\) is an ordinary control\(\}\). They also show that the optimal admissible control can be approximated arbitrarily well by very simple, ordinary controls.