Optimal Periodic Control: A General Theory of Necessary Conditions

Does time-dependent periodic control yield better process performance than optimal steady-state control? This paper examines exhaustively the role of first order necessary conditions in answering this question. For processes described by autonomous, ordinary differential equations, a very general optimal periodic control problem (OPC) is formulated. By considering control and state functions which are constant, a finite-dimensional optimal steady-state problem (OSS) is obtained from OPC. Three solution sets are introduced: $\mathcal{S}$(OSS)—the solutions of OSS, $\mathcal{S}$(OPC)—the solutions of OPC, $\mathcal{S}$(NCOSS)—the solutions of OPC which are constant. Necessary conditions for elements of each of these sets are derived; their solution sets are denoted, respectively, by $\mathcal{S}$(NCOSS), $\mathcal{S}$(NCOPC), and $\mathcal{S}$(NCSSOPC). The relationship between these six solutions sets is a central issue. Under various hypotheses certain pair-wise inclusions of the six sets are determined a...