Tests for properness in periodic control of functional differential systems

A fundamental problem in optimal periodic control may be formulated as follows: Suppose one has an optimal steady state x0 corresponding to a constant control u0. Can performance be improved by allowing for trajectories x and controls u being periodic with some common period ?? > 0? If this is the case, the problem is called proper. For systems governed by ordinary differential equations the so called ??-criterion is a second order variational test for (local) properness. It has been proposed by Bittanti, Fronza, and Guarbadassi [1] and proven by Bernstein and Gilbert [3]; the most general version can be found in Bernstein [2]. Watanabe, Nishimura and Matsubara [12] gave a variant of the ??-criterion (\´singular control test\´ or \´infinite frequency ??-criterion\´) which tests properness for sufficiently high frequencies and requires significantly less computational effort. The ??-criterion is of some relevance in chemical engineering and aircraft flight performance optimization (cp. Sincic and Bailey [9], Speyer [11] and the survey papers by Matsubara, Nishimura, Watanabe, Onogi [7] and Noldus [8]). This paper presents a generalization to functional differential systems of the ??-criterion and its "high-frequency" variant.