The Pontryagin Maximum Principle . Proof and Applications", written by the author's team as part of Arutyunov A. In. , Magaryl-Ilyaeva G. G. and Tikhomirov V. M. , is a unique edition that immerses the reader in the world of mathematical analysis and optimization. This work will be a real find for students, graduate students and specialists in mathematics, physics and engineering, as well as for anyone interested in control theory and process optimization. The Pontryagin maximum principle is one of the cornerstones of the theory of optimal control, which allows you to find optimal strategies in various tasks. The authors of the book not only examine in detail the theoretical aspects of this principle, but also offer the reader a deep understanding of its evidence and practical applications. In this context, the book serves not just as a textbook, but as a real guide to complex mathematical concepts that are of great importance in modern scientific research and development. Readers interested in mathematical optimization, game theory, economics and engineering will find in this book many examples and tasks that will help to consolidate the knowledge gained in practice. The book will be especially useful for those who seek to develop their skills in the analysis of complex systems and decision-making based on mathematical models. It is suitable for both senior students and professionals who want to deepen their knowledge in the field of optimal management. The topics raised in the book cover a wide range of issues, ranging from the basics of optimization theory to complex applications in real-world problems. The authors emphasize the importance of the Pontryagin maximum principle in various fields of science and technology, demonstrating how this principle can be applied to solve problems in economics, physics and engineering. The book also contains many illustrations and graphs that help visualize complex concepts and make the material more easily understood. The style of the authors is distinguished by clarity and logical presentation, which allows the reader to easily follow the course of thought and absorb complex mathematical ideas. Arutyunov, Magaryl-Ilyaev and Tikhomirov are experienced specialists in their fields, and their joint work reflects a deep understanding of the subject and the desire to convey knowledge to the reader. Each statement is supported by clear evidence and examples, which makes the book not only theoretically sound, but also practically useful. The Pontryagin Maximum Principle . Proof and Applications” is not just a textbook, but a real encyclopedia of knowledge for those who want to understand the complex issues of optimization. It will become an indispensable assistant for students, graduate students and researchers, as well as for practitioners working in the field of management and optimization. If you are looking for literature that will help you understand the principles of optimal management and their application in real life, this publication is your ideal choice. Do not miss the opportunity to immerse yourself in the world of mathematics and optimization with a book that opens new horizons and inspires further research. The Pontryagin Maximum Principle . Evidence and Applications” is your key to understanding complex systems and effective decision-making.