Step-guided strategies for control problems with uncertain dynamics

We study here a rather general class of nonlinear control systems with uncertain dynamics involving perturbating variables, without assuming any statistical information on them. Contrary to majority of publications on uncertain control problems, we study here the value function and seek for optimal (-optimal) control policies. Our four theorems, covering all possible situations that may happen in the context of relationships between the influence of the control parameters on the system versus the perturbating parameters, give both upper and lower bounds for the value function in terms of solutions of simplified unperturbed control problems. They also provide constructive procedures for finding strategies that, under additional assumptions, appear to be -optimal control policies in the class of so-called step guided strategies introduced by us in a very recent paper.