Introduction to optimal control theory. (English) Zbl 0624.49009

We will show several formulations of the optimal control of a dynamical system described by ordinary differential equations. Then, we will present briefly two main aspects of optimal control: Pontryagin’s maximum principle (which generalizes the calculus of variations) and Bellman’s dynamic programming. After that, an example of optimal control in pharmacology will present one of the many applications, and one of the many mathematical techniques employed in solving control problems.
Some important aspects of optimal control will not be treated here among them, the study of stochastic systems and adaptive (or self-organizing) control, and the study of systems governed by partial differential equations.


49K15 Optimality conditions for problems involving ordinary differential equations
49L20 Dynamic programming in optimal control and differential games
93C15 Control/observation systems governed by ordinary differential equations
34H05 Control problems involving ordinary differential equations
92Cxx Physiological, cellular and medical topics
Full Text: DOI


[1] Landau Y. D., Adaptive Control (1979)
[2] Sariois G., Self-Organizing Control of Stochastic Systems (1979)
[3] Tsypkin Y. Z., Adaptation and Learning in Automatic Systems (1971) · Zbl 0221.68049
[4] Lions J. C., Contrôle Optimal des Systèmes Gouvernés par des équations aux dérivées partielles, (Dunod-Gauthiers Villars (1968)
[5] Chen C. T., Rinehart and Winston (1970)
[6] Pontryagin L., The Mathematical Theory of Optimal Processes (1962) · Zbl 0112.05502
[7] DOI: 10.1007/978-3-642-93126-0
[8] Roitenberg I., Théorie du Contrôle Automatique (MIR (1974) · Zbl 0302.93001
[9] Boltianski V., Commande Optimale des systèmes discrets (MIR (1976) · Zbl 0363.49001
[10] Zadeh L. A., Linear Systems Theory (1963)
[11] Bellman R., Dynamic Programming and Modern Control Theory (1975) · Zbl 0245.49015
[12] Hestenes M. R., Calculus of Variation and Optimal Control Theory (1966) · Zbl 0173.35703
[13] Larson R. E., State Increment Dynamic Programming (1968) · Zbl 0204.47101
[14] Cherruault Y., Modélisation et Méthodes Mathématiques en Biomédecine (Masson (1977)
[15] Lootsma F. A., Numerical Methods for Non-Linear Optimization (1972) · Zbl 0265.00015
[16] DOI: 10.1142/0028
[17] Cherruault Y., Que sais-je? (1983)
[18] Gilles J. C., Théorie et Calcul des Asservissements (Dunod (1959)
[19] Barnett S., Introduction to Mathematical Control Theory (1975) · Zbl 0307.93001
[20] Tou S. T., Modern Control Theory (1963) · Zbl 0196.45603
[21] Feldbaum A., Systèmes Asservis Optimaux (MIR (1973) · Zbl 0269.49004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.