Remarks on risk sensitive adaptive control of Markov processes

The study of adaptive control problems with multiplicative cost functionals, which appear in particular in risk sensitive control problems requires a different approach than that for risk neutral (average cost per unit time) problems. Assuming that the unknown parameter is selected from a known compact set, it is proved that there is a finite class of nearly optimal control functions for a parametrized family of risk sensitive control problems. Such a class is used to test the value of the cost functional over a finite time horizon, and the control function where the cost is largest is chosen to form an adaptive control. It is shown that this procedure is nearly optimal where the quality of this approximation depends on the choice of the nearly optimal controls and the length of the testing intervals