Optimal control problem and maximum principle for fractional order cooperative systems

G. M. Bahaa

Kybernetika (2019)

  • Volume: 55, Issue: 2, page 337-358
  • ISSN: 0023-5954

Abstract

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In this paper, by using the classical control theory, the optimal control problem for fractional order cooperative system governed by Schrödinger operator is considered. The fractional time derivative is considered in a Riemann-Liouville and Caputo senses. The maximum principle for this system is discussed. We first study by using the Lax-Milgram Theorem, the existence and the uniqueness of the solution of the fractional differential system in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a (FOCP) is considered as a function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE). Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Caputo derivative, we obtain an optimality system for the optimal control. Some examples are analyzed in details.

How to cite

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Bahaa, G. M.. "Optimal control problem and maximum principle for fractional order cooperative systems." Kybernetika 55.2 (2019): 337-358. <http://eudml.org/doc/294568>.

@article{Bahaa2019,
abstract = {In this paper, by using the classical control theory, the optimal control problem for fractional order cooperative system governed by Schrödinger operator is considered. The fractional time derivative is considered in a Riemann-Liouville and Caputo senses. The maximum principle for this system is discussed. We first study by using the Lax-Milgram Theorem, the existence and the uniqueness of the solution of the fractional differential system in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a (FOCP) is considered as a function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE). Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Caputo derivative, we obtain an optimality system for the optimal control. Some examples are analyzed in details.},
author = {Bahaa, G. M.},
journal = {Kybernetika},
keywords = {fractional optimal control; cooperative systems; ; Schrodinger operator; maximum principle; existence of solution; boundary control; optimality conditions; fractional Caputo derivatives; Riemann–Liouville derivatives},
language = {eng},
number = {2},
pages = {337-358},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Optimal control problem and maximum principle for fractional order cooperative systems},
url = {http://eudml.org/doc/294568},
volume = {55},
year = {2019},
}

TY - JOUR
AU - Bahaa, G. M.
TI - Optimal control problem and maximum principle for fractional order cooperative systems
JO - Kybernetika
PY - 2019
PB - Institute of Information Theory and Automation AS CR
VL - 55
IS - 2
SP - 337
EP - 358
AB - In this paper, by using the classical control theory, the optimal control problem for fractional order cooperative system governed by Schrödinger operator is considered. The fractional time derivative is considered in a Riemann-Liouville and Caputo senses. The maximum principle for this system is discussed. We first study by using the Lax-Milgram Theorem, the existence and the uniqueness of the solution of the fractional differential system in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a (FOCP) is considered as a function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE). Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Caputo derivative, we obtain an optimality system for the optimal control. Some examples are analyzed in details.
LA - eng
KW - fractional optimal control; cooperative systems; ; Schrodinger operator; maximum principle; existence of solution; boundary control; optimality conditions; fractional Caputo derivatives; Riemann–Liouville derivatives
UR - http://eudml.org/doc/294568
ER -

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