scholarly journals An elementary derivation of Pontrayagin's maximum principle of optimal control theory

1977 ◽  
Vol 20 (2) ◽  
pp. 142-156 ◽  
Author(s):  
J. M. Blatt ◽  
J. D. Gray
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Control Theory ◽  
Optimal Control Theory ◽  
Pontryagin’S Maximum Principle ◽  
Time Variable ◽  
Pontryagin's Maximum Principle ◽  
Elementary Derivation ◽  
Mathematical Techniques

AbstractPontryagin's maximum principle is derived by elementary mathematical techniques. The conditions on the functions which enter are generally somewhat more stringent than in Pontryagin's derivation, but one (practically very awkward) condition of Pontryagin can be relaxed: continuity in the time variable can be replaced by a much weaker condition.

scholarly journals Distributed-Order Non-Local Optimal Control

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 124
Author(s):  
Faïçal Ndaïrou ◽  
Delfim F. M. Torres
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Control Theory ◽  
Optimality Condition ◽  
Optimal Control Theory ◽  
Sufficient Optimality Condition ◽  
Weak Version ◽  
Fractional Optimal Control ◽  
Non Local ◽  
Distributed Order

Distributed-order fractional non-local operators were introduced and studied by Caputo at the end of the 20th century. They generalize fractional order derivatives/integrals in the sense that such operators are defined by a weighted integral of different orders of differentiation over a certain range. The subject of distributed-order non-local derivatives is currently under strong development due to its applications in modeling some complex real world phenomena. Fractional optimal control theory deals with the optimization of a performance index functional, subject to a fractional control system. One of the most important results in classical and fractional optimal control is the Pontryagin Maximum Principle, which gives a necessary optimality condition that every solution to the optimization problem must verify. In our work, we extend the fractional optimal control theory by considering dynamical system constraints depending on distributed-order fractional derivatives. Precisely, we prove a weak version of Pontryagin’s maximum principle and a sufficient optimality condition under appropriate convexity assumptions.

scholarly journals CONSTRAINT ALGORITHM FOR EXTREMALS IN OPTIMAL CONTROL PROBLEMS

2009 ◽  
Vol 06 (07) ◽  
pp. 1221-1233 ◽  
Author(s):  
MARÍA BARBERO-LIÑÁN ◽  
MIGUEL C. MUÑOZ-LECANDA
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Optimal Control Problem ◽  
Control Theory ◽  
Control Problem ◽  
Optimal Control Theory ◽  
Optimal Control Problems ◽  
Necessary Conditions ◽  
Control Problems ◽  
Affine System

A geometric method is described to characterize the different kinds of extremals in optimal control theory. This comes from the use of a presymplectic constraint algorithm starting from the necessary conditions given by Pontryagin's Maximum Principle. The algorithm must be run twice so as to obtain suitable sets that once projected must be compared. Apart from the design of this general algorithm useful for any optimal control problem, it is shown how to classify the set of extremals and, in particular, how to characterize the strict abnormality. An example of strict abnormal extremal for a particular control-affine system is also given.

scholarly journals Optimal Control Strategies in an Alcoholism Model

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
Xun-Yang Wang ◽  
Hai-Feng Huo ◽  
Qing-Kai Kong ◽  
Wei-Xuan Shi
Keyword(s):  
Mathematical Model ◽  
Optimal Control ◽  
Maximum Principle ◽  
Numerical Simulations ◽  
Control Strategies ◽  
Pontryagin’S Maximum Principle ◽  
Social Phenomenon ◽  
Pontryagin's Maximum Principle ◽  
Existence And Stability ◽  
Objective Functional

This paper presents a deterministic SATQ-type mathematical model (including susceptible, alcoholism, treating, and quitting compartments) for the spread of alcoholism with two control strategies to gain insights into this increasingly concerned about health and social phenomenon. Some properties of the solutions to the model including positivity, existence and stability are analyzed. The optimal control strategies are derived by proposing an objective functional and using Pontryagin’s Maximum Principle. Numerical simulations are also conducted in the analytic results.

scholarly journals Pontryagin’s Maximum Principle for the Loewner Equation in Higher Dimensions

2015 ◽  
Vol 67 (4) ◽  
pp. 942-960 ◽  
Author(s):  
Oliver Roth
Keyword(s):  
Maximum Principle ◽  
Optimal Control Theory ◽  
Necessary Conditions ◽  
Higher Dimensions ◽  
Pontryagin’S Maximum Principle ◽  
Extremal Problems ◽  
Pontryagin's Maximum Principle ◽  
Loewner Equation ◽  
First Order ◽  
The Pontryagin Maximum Principle

AbstractIn this paper we develop a variational method for the Loewner equation in higher dimensions. As a result we obtain a version of Pontryagin’s maximum principle from optimal control theory for the Loewner equation in several complex variables. Based on recent work of Arosio, Bracci, and Wold, we then apply our version of the Pontryagin maximum principle to obtain first-order necessary conditions for the extremal mappings for a wide class of extremal problems over the set of normalized biholomorphic mappings on the unit ball in ℂn.

scholarly journals Optimal control with a cost of switching control

1976 ◽  
Vol 19 (3) ◽  
pp. 316-332 ◽  
Author(s):  
John M. Blatt
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Existence Theorem ◽  
Switching Control ◽  
Pontryagin’S Maximum Principle ◽  
Pontryagin's Maximum Principle ◽  
Theory Of Optimal Control ◽  
General Existence

AbstractThe Pontryagin theory of optimal control is modified by assuming a positive cost associated with switching control from one discrete value to another. The resulting new theory permits a general existence theorem. Pontryagin's maximum principle is replaced by an “indifference principle”.

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