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This research monograph deals with optimal periodic control problems for systems governed by ordinary and functional differential equations of retarded type. Particular attention is given to the problem of local properness, i.e. whether system performance can be improved by introducing periodic motions. Using either Ekeland's Variational Principle or optimization theory in Banach spaces, necessary optimality conditions are proved. In particular, complete proofs of second-order conditions are included and the result is used for various versions of the optimal periodic control problem. Furthermore a scenario for local properness (related to Hopf bifurcation) is drawn up, giving hints as to where to look for optimal periodic solutions. The book provides mathematically rigorous proofs for results which are potentially of importance in chemical engineering and aerospace engineering.
Annotation. This research monograph deals with optimal periodic control problems for systems governed by ordinary and functional differential equations of retarded type. Particular attention is given to the problem of local properness, i.e. whether system performance can be improved by introducing periodic motions. Using either Ekeland's Variational Principle or optimization theory in Banach spaces, necessary optimality conditions are proved. In particular, complete proofs of second-order conditions are included and the result is used for various versions of the optimal periodic control problem. Furthermore a scenario for local properness (related to Hopf bifurcation) is drawn up, giving hints as to where to look for optimal periodic solutions. The book provides mathematically rigorous proofs for results which are potentially of importance in chemical engineering and aerospace engineering.
"Existing conditions for locally optimizing a periodic process with free period are corrected, extended, and clarified using the strong properties of the Hamiltonian structure of the Optimal Periodic Control (OPC) Problem and the relationships of the second variation that result from the problem's periodicity constraints. A neighboring optimum feedback controller is determined which regulates all perturbations back to the locally minimizing periodic trajectory. This periodic regulator extends many of the concepts of the time invariant regulator. An illustrative problem is constructed to demonstrate the basic characteristics of this class of problem. A relatively comprehensive numerical investigation is conducted identifying a multiplicity of extremal solutions which are shown to form one-parameter families of solutions to the OPC. Bifurcation points, which define the critical periodic solutions common to intersecting families, are computed. Extremal solutions, satisfying all first order conditions, are tested for local sufficiency conditions by verifying the existence of the Ricatti' variable over one full period. The neighboring optimal feedback control law for a locally minimizing solution is tested by demonstrating the limit cycle behavior that results from perturbations to the initial conditions in a closed loop application. An asymptotic series expansion is derived for the illustrative problem using a perturbation technique. An extremal solution in the form of a Fourier series is obtained that accurately predicts the optimal period, the locally minimizing periodic solution, and the associated values of the Hamiltonian and the cost of the principal family."--Abstract, report documentation p.