Dual effect free stochastic controls

Abstract

In stochastic optimal control, a key issue is the fact that "solutions" are searched for in terms of "feedback" over available information and, as a consequence, a major potential difficulty is the fact that present control may affect future available information. This is known as the "dual effect" of control.Given a minimal framework (that is, an observation mapping from the product of a control set and of a random set towards an observation set), we define open-loop lack of dual effect as the property that the information provided by observations under open-loop control laws is fixed, whatever the open-loop control. Our main result consists in characterizing the maximal set of closed-loop control laws for which the information provided by observations closed with such a feedback remains also fixed.We then address the multi-agent case. To obtain a comparable result, we are led to generalize the precedence and memory-communication binary relations introduced by Ho and Chu for the LQG problem, and to assume that the precedence relation is compatible with the memory-communication relation.When the precedence relation induces an acyclic graph, we prove that, when open-loop lack of dual effect holds, the maximal set of closed-loop control laws for which the information provided by observations closed with such a feedback remains fixed is the set of feedbacks measurable with respect to this fixed information. We end by studying the dual effect for discrete time stochastic input-output systems with dynamic information structure, for which the same result holds.