Constrained Optimal Control of A Fractionally Damped Elastic Beam

2020 ◽  
Vol 21 (3-4) ◽  
pp. 389-395
Author(s):  
Beyza Billur İskender Eroğlu ◽  
Derya Avcı ◽  
Necati Özdemir
Keyword(s):  
Optimal Control ◽  
Fractional Derivatives ◽  
Space Form ◽  
Hamiltonian Formalism ◽  
Elastic Beam ◽  
Control Law ◽  
Caputo Fractional Derivatives ◽  
Constrained Optimal Control ◽  
Pontryagin Principle ◽  
One Step

AbstractThis work presents the constrained optimal control of a fractionally damped elastic beam in which the damping characteristic is described with the Caputo fractional derivative of order 1/2. To achieve the optimal control that involves energy optimal control index with fixed endpoints, the fractionally damped elastic beam problem is first converted to a state space form of order 1/2 by using a change of coordinates. Then, the state and the costate equations are set in terms of Hamiltonian formalism and the constrained control law is acquired from Pontryagin Principle. The numerical solution of the problem is obtained with Grünwald-Letnikov approach by utilizing the link between the Riemann-Liouville and the Caputo fractional derivatives. Application of the formulations is demonstrated with an example and the illustrations are figured by MATLAB. Also, the effectiveness of the Grünwald-Letnikov approach is exhibited by comparing it with an iterative method which is one-step Adams-Bashforth-Moulton method.

scholarly journals Optimal control for obstacle problems involving time-dependent variational inequalities with Liouville–Caputo fractional derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Parinya Sa Ngiamsunthorn ◽  
Apassara Suechoei ◽  
Poom Kumam
Keyword(s):  
Optimal Control ◽  
Variational Inequalities ◽  
Obstacle Problem ◽  
Fractional Derivatives ◽  
Necessary Conditions ◽  
Existence Results ◽  
The State ◽  
Time Dependent ◽  
Obstacle Problems ◽  
Caputo Fractional Derivatives

AbstractWe consider an optimal control problem for a time-dependent obstacle variational inequality involving fractional Liouville–Caputo derivative. The obstacle is considered as the control, and the corresponding solution to the obstacle problem is regarded as the state. Our aim is to find the optimal control with the properties that the state is closed to a given target profile and the obstacle is not excessively large in terms of its norm. We prove existence results and establish necessary conditions of obstacle problems via the approximated time fractional-order partial differential equations and their adjoint problems. The result in this paper is a generalization of the obstacle problem for a parabolic variational inequalities as the Liouville–Caputo fractional derivatives were used instead of the classical derivatives.

scholarly journals Towards a combined fractional mechanics and quantization

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Agnieszka Malinowska ◽  
Delfim Torres
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Fractional Calculus ◽  
Calculus Of Variations ◽  
Fractional Derivatives ◽  
Hamiltonian Formalism ◽  
Partial Differential ◽  
Caputo Fractional Derivatives ◽  
Fractional Mechanics ◽  
Fractional Calculus Of Variations

AbstractA fractional Hamiltonian formalism is introduced for the recent combined fractional calculus of variations. The Hamilton-Jacobi partial differential equation is generalized to be applicable for systems containing combined Caputo fractional derivatives. The obtained results provide tools to carry out the quantization of nonconservative problems through combined fractional canonical equations of Hamilton type.

Analysis of Constrained Optimal Control with Related Constraints of Input Variables

2014 ◽  
Vol 39 (5) ◽  
pp. 679-689 ◽  
Author(s):  
Xiong-Lin LUO ◽  
Xiao-Long ZHOU ◽  
Shu-Bin WANG
Keyword(s):  
Optimal Control ◽  
Constrained Optimal Control ◽  
Input Variables

Solution of inverse coefficient problem for fractional anomalous diffusion equation

2012 ◽  
Vol 9 (2) ◽  
pp. 65-70
Author(s):  
E.V. Karachurina ◽  
S.Yu. Lukashchuk
Keyword(s):  
Anomalous Diffusion ◽  
Fractional Derivatives ◽  
Descent Method ◽  
Extremum Problem ◽  
Steepest Descent Method ◽  
Coefficient Problem ◽  
Caputo Fractional Derivatives ◽  
Inverse Coefficient Problem ◽  
Residual Functional ◽  
Inverse Coefficient

An inverse coefficient problem is considered for time-fractional anomalous diffusion equations with the Riemann-Liouville and Caputo fractional derivatives. A numerical algorithm is proposed for identification of anomalous diffusivity which is considered as a function of concentration. The algorithm is based on transformation of inverse coefficient problem to extremum problem for the residual functional. The steepest descent method is used for numerical solving of this extremum problem. Necessary expressions for calculating gradient of residual functional are presented. The efficiency of the proposed algorithm is illustrated by several test examples.

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