Optimal controls for diffusion in \(R^ d\)- a min-max max-min formula for the minimal cost growth rate
The infinite horizon optimal control of systems whose dynamics are described by a stochastic differential equation is studied. The expected cost over the [0,T] time interval is assumed to diverge as T tends to infinity, for every control. The following formula is established: \[ \sup_{\phi}\inf_{x}L\phi \leq \alpha \leq \inf_{\phi}\sup_{x}L\phi, \] where L is a second order operator associated with the control problem, and \(\phi\) ranges over a certain class of functions. Existence results for optimal controls are established, assuming the existence of a solution to the infinite horizon H.J.B. equation which satisfies a certain growth condition. Optimal controls are studied both for the long-run average cost and for the overtaking optimality criterion.
scientific article; zbMATH DE number 4146939
scientific article