Among the major achievements of modern mathematics, which received the greatest popularity and approval in engineering circles, a special place is occupied by the mathematical theory of optimal control, created by a team of Soviet scientists led by Academician L.. C. Pontryagin . The foundations of this theory were laid out in a 1961 monograph by L....... C. PONTYAGINA, V. G. Boltyansky, R. In. Gamkrslidze, E. F. Mishchenko , awarded the Lenin Prize for 1962. In this book, the mathematical theory of optimal control is presented in the form available to an engineer who has mathematical training in the scope of a technical university. The author pays special attention to computational methods, as well as to those problems that by the time of writing the book could be solved completely. Striving for maximum simplicity of presentation, the author did not sacrifice rigor anywhere. Thus, the necessary engineer, this book will be interesting and mathematics . In the second edition, the book has been significantly revised by the author: the layout of the book has been changed, the proofs of a number of theorems have been presented in a new way, new material has been added. Thus, in fact, the reader is offered a new book ..