Calculus of variations and optimal control. 23rd Brazilian mathematics colloquium (Q2748480)

This textbook gives an introduction to the Calculus of Variations, Optimal Control Theory, and its Applications. It is well suited to advanced undergraduate students or first-year graduate students in the fields of Science or Engineering. More advanced material, fruit of the research interests of the author, is also included: optimal control problems with infinite horizon; optimal impulsive control problems. Although many good references on the subject are available, some of them included at the end of the book in the Bibliography, texts in the Portuguese language are a rarity, and the present book is a very good contribution to fill the gap. The text is organized in four chapters and one appendix. In Chapter 1 the problems of the calculus of variations are introduced. Special attention is given to classical necessary optimality conditions, and to sufficient conditions under convexity assumptions. Three classical problems of the calculus of variations are analyzed: geodesics in the plane; geodesics in the sphere; and the Brachistochrone problem. In Chapter 2 the problems of optimal control are presented, and the connection with the calculus of variations is established. In particular, the necessary optimality conditions for both families of problems are compared. Problems of the calculus of variations with isoperimetric and Lagrange constraints are studied. Under additional hypotheses of convexity sufficient conditions are proved for problems without constraints on the values of the control variables. Several concrete examples are analyzed, and bang-bang extremals naturally obtained. The weak maximum principle is proved using the Lagrange multiplier theorem. In Chapter 3 the Pontryagin maximum principle is formulated for several classes of optimal control problems: with finite horizon, infinite horizon, and impulsive optimal control problems, that is, for problems which admit discontinuous trajectories. The third chapter ends with the discussion of the applicability of the Pontryagin maximum principle in applications, and several examples, from different fields, are analyzed. The complete proof of the Pontryagin maximum principle is given in Chapter 4 and Appendix. The text is self-contained and the proofs are given. All chapters finish with a list of proposed exercises for the student.