Density and Cost in Non-linear Control

The duality between cost and flow is well known and carefully studied in several scientific fields, including economics and fluid mechanics. The purpose of this paper is to show how this duality relates to the basic concepts of nonlinear control theory for ordinary differential equations. A duality theorem for optimal control is derived using the Hahn-Banach theorem. This puts focus on convexity aspects in control synthesis and the recent theory of almost global stability. As a closely related result, it is shown that existence of a density function is both necessary and sufficient for almost global stability in a non-linear system.

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