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**Optimal control.**
*(English)*
Zbl 0952.49001

Control Theory is a by now classical field which received a strong impulse in the last decades, especially due to the many applications to other disciplines for which modelizations as control problems were required. Indeed the range of applications of Control Theory is very wide, covering fields as mechanics, chemistry, biology; we may say that practically every phenomenon which requires a choice of the operating strategy among admissible choices can be modelized as a control problem.

The book by R. Vinter is really well written; it goes from the very basic tools in variational analysis, as direct methods and regularity, to the most recent results in dynamic programming. Special attention is devoted to nonsmooth analysis tools; indeed this is not simply for a desire of generalization but mainly because derivatives of nonsmooth functions naturally appear in the study of solutions of control problems. Necessary conditions of optimality, Pontryagin maximum principle, Hamilton-Jacobi equation, optimal synthesis, all require a good knowledge of nonsmooth analysis theory to be correctly formulated.

The concept of viscosity solution and the method of dynamic programming are other topics which are very important in a modern presentation of Control Theory. In the book by R. Vinter the reader can find a very clear introduction to them as well as their applications to control problems.

The book is enriched by many examples: this is a very important point for a reader who wants to approach Control Theory, and makes the volume suitable not only for specialists but also for students at a Ph. D. level.

The only criticism I can make on the book is that it neglects the case of control problems governed by partial differential equations, but I understand that it has been more a choice of the author rather than an oversight.

The book by R. Vinter is really well written; it goes from the very basic tools in variational analysis, as direct methods and regularity, to the most recent results in dynamic programming. Special attention is devoted to nonsmooth analysis tools; indeed this is not simply for a desire of generalization but mainly because derivatives of nonsmooth functions naturally appear in the study of solutions of control problems. Necessary conditions of optimality, Pontryagin maximum principle, Hamilton-Jacobi equation, optimal synthesis, all require a good knowledge of nonsmooth analysis theory to be correctly formulated.

The concept of viscosity solution and the method of dynamic programming are other topics which are very important in a modern presentation of Control Theory. In the book by R. Vinter the reader can find a very clear introduction to them as well as their applications to control problems.

The book is enriched by many examples: this is a very important point for a reader who wants to approach Control Theory, and makes the volume suitable not only for specialists but also for students at a Ph. D. level.

The only criticism I can make on the book is that it neglects the case of control problems governed by partial differential equations, but I understand that it has been more a choice of the author rather than an oversight.

Reviewer: Giuseppe Buttazzo (Pisa)

### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

49-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control |