Density and Cost in Non-linear Control

The duality between cost and flow is well known and carefully studiedin several scientific fields, including economics and fluid mechanics.The purpose of this paper is to show how this duality relates to thebasic concepts of nonlinear control theory for ordinary differentialequations. A duality theorem for optimal control is derived using theHahn-Banach theorem. This puts focus on convexity aspects in controlsynthesis and the recent theory of almost global stability. As aclosely related result, it is shown that existence of a densityfunction is both necessary and sufficient for almost global stabilityin a nonlinear system.