Necessary Conditions in Mathematical Programming and Optimal Control Theory

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.

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The Derivation of Structural Properties of LM-g Splines by Necessary Condition of Optimal Control

Abstract and Applied Analysis ◽

10.1155/2014/641435 ◽

2014 ◽

Vol 2014 ◽

pp. 1-5

Author(s):

Xiongwei Liu ◽

Xinjian Zhang ◽

Lizhi Cheng

Keyword(s):

Optimal Control ◽

Control Theory ◽

Structural Properties ◽

Optimal Control Theory ◽

State Constraints ◽

Necessary Conditions ◽

Necessary Condition ◽

Interpolating Splines ◽

The Relationship ◽

Optimization And Optimal Control

The structural properties of LM-g splines are investigated by optimization and optimal control theory. The continuity and structure of LM-g splines are derived by using a class of necessary conditions with state constraints of optimal control and the relationship between LM-g interpolating splines and the corresponding L-g interpolating splines. This work provides a new method for further exploration of LM-g interpolating splines and its applications in the optimal control.

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Higher order necessary conditions in optimal control theory

Ordinary and Partial Differential Equations - Lecture Notes in Mathematics ◽

10.1007/bfb0091379 ◽

1980 ◽

pp. 151-164 ◽

Cited By ~ 5

Author(s):

H. W. Knobloch

Keyword(s):

Optimal Control ◽

Control Theory ◽

Optimal Control Theory ◽

Necessary Conditions ◽

Higher Order

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Optimal Control Theory

10.1093/oso/9780198811084.003.0004 ◽

2018 ◽

Author(s):

John E. Prussing

Keyword(s):

Optimal Control ◽

Control Theory ◽

Optimal Control Theory ◽

Necessary Conditions ◽

Minimum Principle ◽

Minimum Cost ◽

Initial Time ◽

Pontryagin Minimum Principle ◽

Final Time ◽

Terminal Constraints

Optimal Control Theory is reviewed in detail. We consider a dynamic system that operates between a specified initial time and a final time which may be specified or unspecified. Necessary conditions for a minimum cost functional are derived. Terminal constraints are considered. Pontryagin Minimum Principle is discussed.

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Maximizing Distance of the Golf Drive: An Optimal Control Study

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.3426951 ◽

1975 ◽

Vol 97 (4) ◽

pp. 362-367 ◽

Cited By ~ 25

Author(s):

M. A. Lampsa

Keyword(s):

Mathematical Model ◽

Optimal Control ◽

Control Theory ◽

Optimal Control Theory ◽

Variational Formulation ◽

Necessary Conditions ◽

Model Parameter ◽

Steepest Ascent ◽

Parameter Values ◽

Control Study

Optimal control theory is used to search for the optimal control torques necessary to maximize distance of the golf drive. In the method, a mathematical model of a generalized golf swing is first developed. Film of the author’s swing serves to verify the model and to supply parameter values, constraints, and actual torques. The variational formulation of optimal control theory is utilized to establish necessary conditions for optimal control, in which constraint violations are discouraged by inclusion of penalty functions. Finally, the method of steepest ascent is used to compute optimal control torques. Also, comparison of optimal and actual torques is made, and the sensitivity of the results to small changes in model parameter values is investigated.

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