Course title: Optimal control (Commande optimale) Teacher: J. Blot

UE 2

Major(s): ORO

18 hours (2.5 ECTS) Evaluation: Written exam

Prerequisites:

Basic static optimization, differential and integral calculus Presentation:

The main aim of this course is to introduce to the theory of the Optimal Control, mainly the viewpoint of Pontryagin in a continuous-time setting. Considering controlled dynamical systems, the problem is to find the optimal processes, i.e. the processes which are the better following a given criterion. This theory is an important part of the dynamic optimization. After a short recall on elementary facts about the dynamical systems, we give the description of the classes of problems which are considered, and after the relations between these classes, we establish the theorems, so-called principles of Pontryagin, which provide necessary conditions of optimality; the list of these conditions contains: conditions of sign, slackness condition, adjoint equation, maximum principle, transversality condition, etc. We also establish sufficient conditions which are related to these principles. To illustrate these general results, we treat several explicit examples. The fields of application of this theory are numerous: dynamical macroeconomics, sustainable development, various kinds of management problems (forests, fisheries, etc.), medical problems, physical problems, etc.

References:

UE 2

Major(s): ORO

18 hours (2.5 ECTS) Evaluation: Written exam

Prerequisites:

Basic static optimization, differential and integral calculus Presentation:

The main aim of this course is to introduce to the theory of the Optimal Control, mainly the viewpoint of Pontryagin in a continuous-time setting. Considering controlled dynamical systems, the problem is to find the optimal processes, i.e. the processes which are the better following a given criterion. This theory is an important part of the dynamic optimization. After a short recall on elementary facts about the dynamical systems, we give the description of the classes of problems which are considered, and after the relations between these classes, we establish the theorems, so-called principles of Pontryagin, which provide necessary conditions of optimality; the list of these conditions contains: conditions of sign, slackness condition, adjoint equation, maximum principle, transversality condition, etc. We also establish sufficient conditions which are related to these principles. To illustrate these general results, we treat several explicit examples. The fields of application of this theory are numerous: dynamical macroeconomics, sustainable development, various kinds of management problems (forests, fisheries, etc.), medical problems, physical problems, etc.

References:

- L. Pontryagin, V. Boltyanski, R. Gramgrelidze, E. Mitchenko, “Théorie mathématique des processus optimaux”, traduction française, MIR, Moscou, 1974.
- A. Ioffe, V. Tihomirov, “Theory of extremal problems”, Norht-Holland, Amsterdam, 1979.
- V. Alexéev, V. Tihomirov, S. Fomin, “Commande optimale”, traduction française, MIR, Moscou, 1982.
- D. Léonard, N. V. Long, “Optimal control theory and static optimization in Economics”, Cambridge University Press, Cambridge, 1992.