Stochastic fractional optimal control of quasi-integrable Hamiltonian system with fractional derivative damping

2012 ◽  
Vol 70 (2) ◽  
pp. 1459-1472 ◽  
Author(s):  
F. Hu ◽  
W. Q. Zhu ◽  
L. C. Chen
Keyword(s):  
Optimal Control ◽  
Hamiltonian System ◽  
Fractional Derivative ◽  
Integrable Hamiltonian System ◽  
Fractional Optimal Control ◽  
Fractional Derivative Damping ◽  
Quasi Integrable Hamiltonian System

Stochastic dynamics and fractional optimal control of quasi integrable Hamiltonian systems with fractional derivative damping

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Lincong Chen ◽  
Fang Hu ◽  
Weiqiu Zhu
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Hamiltonian Systems ◽  
Stochastic Dynamics ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Integrable Hamiltonian Systems ◽  
Fractional Optimal Control ◽  
Fractional Derivative Damping

AbstractIn the present survey, some progress in the stochastic dynamics and fractional optimal control of quasi integrable Hamiltonian systems with fractional derivative damping is reviewed. First, the stochastic averaging method for quasi integrable Hamiltonian systems with fractional derivative damping under various random excitations is briefly introduced. Then, the stochastic stability, stochastic bifurcation, first passage time and reliability, and stochastic fractional optimal control of the systems studied by using the stochastic averaging method are summarized. The focus is placed on the effects of fractional derivative order on the dynamics and control of the systems. Finally, some possible extensions are pointed out.

On the Diffusion Along Resonant Lines in Continuous and Discrete Dynamical Systems

2003 ◽  
Vol 17 (22n24) ◽  
pp. 3964-3976
Author(s):  
Claude Froeschlé ◽  
Elena Lega
Keyword(s):  
Dynamical Systems ◽  
Hamiltonian System ◽  
Coupling Parameter ◽  
A Priori ◽  
Integrable Hamiltonian System ◽  
Discrete Dynamical Systems ◽  
Nekhoroshev Theorem ◽  
Symplectic Map ◽  
Quasi Integrable Hamiltonian System

We detect and measure diffusion along resonances in a discrete symplectic map for different values of the coupling parameter. Qualitatively and quantitatively the results are very similar to those obtained for a quasi-integrable Hamiltonian system, i.e. in agreement with Nekhoroshev predictions, although the discrete mapping does not fulfill completely, a priori, the conditions of the Nekhoroshev theorem.

The approximate solution of non-linear time-delay fractional optimal control problems by embedding process

2018 ◽  
Vol 36 (3) ◽  
pp. 713-727 ◽  
Author(s):  
E Ziaei ◽  
M H Farahi
Keyword(s):  
Optimal Control ◽  
Time Delay ◽  
Fractional Derivative ◽  
Optimal Control Problems ◽  
Linear Time ◽  
Control Problems ◽  
Linear Dynamical Systems ◽  
Fractional Optimal Control ◽  
Non Linear ◽  
Fractional Optimal Control Problems

Abstract In this paper, a class of time-delay fractional optimal control problems (TDFOCPs) is studied. Delays may appear in state or control (or both) functions. By an embedding process and using conformable fractional derivative as a new definition of fractional derivative and integral, the class of admissible pair (state, control) is replaced by a class of positive Radon measures. The optimization problem found in measure space is then approximated by a linear programming problem (LPP). The optimal measure which is representing optimal pair is approximated by the solution of a LPP. In this paper, we have shown that the embedding method (embedding the admissible set into a subset of measures), successfully can be applied to non-linear TDFOCPs. The usefulness of the used idea in this paper is that the method is not iterative, quite straightforward and can be applied to non-linear dynamical systems.

A Numerical Scheme and an Error Analysis for a Class of Fractional Optimal Control Problems

2009 ◽  
Author(s):  
Om P. Agrawal
Keyword(s):  
Optimal Control ◽  
Error Analysis ◽  
Fractional Derivative ◽  
Performance Index ◽  
Numerical Solutions ◽  
Fractional Power ◽  
The State ◽  
Control Variables ◽  
Fractional Optimal Control ◽  
Linear Algebraic Equations

There has been a growing interest in recent years in the area of Fractional Optimal Control (FOC). In this paper, we present a formulation for a class of FOC problems, in which a performance index is defined as an integral of a quadratic function of the state and the control variables, and a dynamic constraint is defined as a Fractional Differential Equation (FDE) linear in both the state and the control variables. The fractional derivative is defined in the Caputo sense. In this formulation, the FOC problem is reduced to a Fractional Variational Problem (FVP), and the necessary differential equations for the problems are obtained using the recently developed theories for FVPs. For the numerical solutions of the problems, a direct approach is taken in which the solutions are approximated using a truncated Fractional Power Series (FPS). An error analysis is also performed. It is demonstrated that the solution converges from above in the sense that the value of the approximate performance index is always higher than the optimum performance index. An expression for the error in the performance index is also given. The application of a FPS and an optimality criterion reduces the FOC to a set of linear algebraic equations which are solved using a linear solver. It is demonstrated numerically that the solution converges as the number of terms in the series increases, and the approximate solution approaches to the analytical solution as the order of the fractional derivative approaches to an integer order derivative. Numerical results are presented to demonstrate the performance of the Formulation.

Solving a class of fractional optimal control problems by the Hamilton–Jacobi–Bellman equation

2016 ◽  
Vol 24 (9) ◽  
pp. 1741-1756 ◽  
Author(s):  
Seyed Ali Rakhshan ◽  
Sohrab Effati ◽  
Ali Vahidian Kamyad
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Bellman Equation ◽  
Optimal Control Problems ◽  
Control Problems ◽  
Fractional Optimal Control ◽  
Solution Scheme ◽  
Hamilton Jacobi Bellman Equation ◽  
Fractional Optimal Control Problems ◽  
Hamilton Jacobi Bellman

The performance index of both the state and control variables with a constrained dynamic optimization problem of a fractional order system with fixed final Time have been considered here. This paper presents a general formulation and solution scheme of a class of fractional optimal control problems. The method is based upon finding the numerical solution of the Hamilton–Jacobi–Bellman equation, corresponding to this problem, by the Legendre–Gauss collocation method. The main reason for using this technique is its efficiency and simple application. Also, in this work, we use the fractional derivative in the Riemann–Liouville sense and explain our method for a fractional derivative of order of [Formula: see text]. Numerical examples are provided to show the effectiveness of the formulation and solution scheme.

NOISY CHAOS IN A QUASI-INTEGRABLE HAMILTONIAN SYSTEM WITH TWO DOF UNDER HARMONIC AND BOUNDED NOISE EXCITATIONS

2012 ◽  
Vol 22 (05) ◽  
pp. 1250117 ◽  
Author(s):  
C. B. GAN ◽  
Y. H. WANG ◽  
S. X. YANG ◽  
H. LEI
Keyword(s):  
Hamiltonian System ◽  
Integrable Hamiltonian System ◽  
Melnikov Method ◽  
Threshold Region ◽  
Mean Square ◽  
Bounded Noise ◽  
Onset Of Chaos ◽  
Extended Approach ◽  
The Mean ◽  
Quasi Integrable Hamiltonian System

This paper presents an extended form of the high-dimensional Melnikov method for stochastically quasi-integrable Hamiltonian systems. A quasi-integrable Hamiltonian system with two degree-of-freedom (DOF) is employed to illustrate this extended approach, from which the stochastic Melnikov process is derived in detail when the harmonic and the bounded noise excitations are imposed on the system, and the mean-square criterion on the onset of chaos is then presented. It is shown that the threshold of the onset of chaos can be adjusted by changing the deterministic intensity of bounded noise, and one can find the range of the parameter related to the bandwidth of the bounded noise excitation where the chaotic motion can arise more readily by investigating the changes of the threshold region. Furthermore, some parameters are chosen to simulate the sample responses of the system according to the mean-square criterion from the extended stochastic Melnikov method, and the largest Lyapunov exponents are then calculated to identify these sample responses.

A spectral framework for the solution of fractional optimal control and variational problems involving Mittag–Leffler nonsingular kernel

2020 ◽  
pp. 107754632097481
Author(s):  
Haniye Dehestani ◽  
Yadollah Ordokhani
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Lagrange Multiplier ◽  
Caputo Fractional Derivative ◽  
Variational Problems ◽  
Computational Technique ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Optimal Control

A new fractional-order Dickson functions are introduced for solving numerically fractional optimal control and variational problems involving Mittag–Leffler nonsingular kernel. The type of fractional derivative in the proposed problems is the Atangana–Baleanu–Caputo fractional derivative. In the process of the method, we use fractional-order Dickson functions and their properties to provide an accurate computational technique for calculating operational matrices, at first. Then, with the help of operational matrices and the Lagrange multiplier method, these problems are reduced to a system of algebraic equations. At last, to demonstrate the effectiveness of the new method, we enforce the proposed algorithm for several examples.

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