Some Recent Results on the Maximum Principle of Optimal Control Theory

1997 ◽  
pp. 351-372 ◽  
Author(s):  
H. J. Sussmann
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Control Theory ◽  
Optimal Control Theory ◽  
The Maximum Principle

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 Sub-Riemannian geometry and swimming at low Reynolds number: the Copepod case

2019 ◽  
Vol 25 ◽  
pp. 9 ◽  
Author(s):  
P. Bettiol ◽  
B. Bonnard ◽  
A. Nolot ◽  
J. Rouot
Keyword(s):  
Optimal Control ◽  
Reynolds Number ◽  
Maximum Principle ◽  
Numerical Simulations ◽  
Riemannian Geometry ◽  
Optimal Control Theory ◽  
Geometric Analysis ◽  
Optimal Solutions ◽  
The Maximum Principle ◽  
Recovery Stroke

In Takagi [Phys. Rev. E 92 (2015) 023020], based on copepod observations, Takagi proposed a model to interpret the swimming behaviour of these microorganisms using sinusoidal paddling or sequential paddling followed by a recovery stroke in unison, and compares them invoking the concept of efficiency. Our aim is to provide an interpretation of Takagi’s results in the frame of optimal control theory and sub-Riemannian geometry. The maximum principle is used to select two types of periodic control candidates as minimizers: sinusoidal up to time reparameterization and the sequential paddling, interpreted as an abnormal stroke in sub-Riemannian geometry. Geometric analysis combined with numerical simulations are decisive tools to compute the optimal solutions, refining Takagi computations. A family of simple strokes with small amplitudes emanating from a center is characterized as an invariant of SR-geometry and allows to identify the metric used by the swimmer. The notion of efficiency is discussed in detail and related with normality properties of minimizers.

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.

Application of Pontryagin’s Principle to a Minimum Weight Design Problem

1970 ◽  
Vol 92 (2) ◽  
pp. 245-250 ◽  
Author(s):  
B. M. E. de Silva
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Optimal Control Theory ◽  
Design Problem ◽  
Minimum Weight ◽  
Inequality Constraints ◽  
Minimum Weight Design ◽  
The Maximum Principle ◽  
Weight Design ◽  
And Control

A minimum weight design problem has been formulated as a general problem in optimal-control theory with the addition of state and control inequality constraints. Complete analytical solutions have been derived using the maximum principle of Pontryagin.

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