Optimal Periodic Control: A General Theory of Necessary Conditions

Abstract

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 and it is shown that no others can be obtained. Tests which imply that time-dependent periodic control is better than optimal steady-state control $((\mathcal{S}({\text{SSOPC}} = \emptyset ,\mathcal{S}({\text{OSS}}) \ne \emptyset )$, including these based on relaxed steady-state control, are investigated and limits to what tests exist are established. The results integrate and amplify results which have appeared in the literature. Examples provide insight which supports the theory.