Optimal Control TheoryThis book is an introduction to the mathematical theory of optimal control of processes governed by ordinary differential eq- tions. It is intended for students and professionals in mathematics and in areas of application who want a broad, yet relatively deep, concise and coherent introduction to the subject and to its relati- ship with applications. In order to accommodate a range of mathema- cal interests and backgrounds among readers, the material is arranged so that the more advanced mathematical sections can be omitted wi- out loss of continuity. For readers primarily interested in appli- tions a recommended minimum course consists of Chapter I, the sections of Chapters II, III, and IV so recommended in the introductory sec tions of those chapters, and all of Chapter V. The introductory sec tion of each chapter should further guide the individual reader toward material that is of interest to him. A reader who has had a good course in advanced calculus should be able to understand the defini tions and statements of the theorems and should be able to follow a substantial portion of the mathematical development. The entire book can be read by someone familiar with the basic aspects of Lebesque integration and functional analysis. For the reader who wishes to find out more about applications we recommend references [2], [13], [33], [35], and [50], of the Bibliography at the end of the book. |
Contents
Examples of Control Problems | 1 |
Formulation of the Control Problem | 14 |
Existence Theorems with Convexity Assumptions | 39 |
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admissible pair admissible trajectories assume Assumption 4.1 attainable set Bolza calculus of variations Chapter compact interval compact set constraint sets control constraints control problem convex set Corollary definition denote the set differential equations dx dt En+1 end conditions end point equi-absolutely continuous EXERCISE existence theorem finite fixed follows formulation func function defined function f given Hence holds hypotheses of Theorem inequality initial point integrable L₁ Lagrange problem Lemma let f lower semicontinuous mapping matrix maximum principle meas measurable function minimizing sequence obtain optimal control optimal pair orthogonal proof of Theorem relaxed problem relaxed trajectory right hand side satisfy Section solution statement t₁ t₂ Theorem 4.1 Theorem III 5.1 tion to,xo torty toxo trajectory corresponding transversality condition unique upper semicontinuous vector weak Cesari property zero хо