Numerical solutions of an optimal control problem governed by a ginzburg-landau model in superconductivity

1998 ◽  
Vol 19 (7-8) ◽  
pp. 737-757 ◽  
Author(s):  
Zhiming Chen ◽  
K.-H. Hoffmann
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Numerical Solutions ◽  
Ginzburg Landau ◽  
Landau Model

Solving the Problem of UAV Air Combat Game Based on Differential Variational Inequality and D-Gap Function

2014 ◽  
Vol 1042 ◽  
pp. 172-177
Author(s):  
Guang Yan Xu ◽  
Ping Li ◽  
Biao Zhou
Keyword(s):  
Optimal Control ◽  
Variational Inequality ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Differential Game ◽  
Numerical Solutions ◽  
Game Problem ◽  
Gap Function ◽  
Differential Variational Inequality ◽  
Air Combat

The strategy of unmanned aerial vehicle air combat can be described as a differential game problem. The analytical solutions for the general differential game problem are usually difficult to obtain. In most cases, we can only get its numerical solutions. In this paper, a Nash differential game problem is converted to the corresponding differential variational inequality problem, and then converted into optimal control problem via D-gap function. The nonlinear continuous optimal control problem is obtained, which is easy to get numerical solutions. Compared with other conversion methods, the specific solving process of this method is more simple, so it has certain validity and feasibility.

scholarly journals Solving Optimal Control Problem of Monodomain Model Using Hybrid Conjugate Gradient Methods

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Kin Wei Ng ◽  
Ahmad Rohanin
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Conjugate Gradient ◽  
Optimization Problem ◽  
Numerical Solutions ◽  
Gradient Methods ◽  
Parabolic Partial Differential Equation ◽  
Conjugate Gradient Methods ◽  
Monodomain Model

We present the numerical solutions for the PDE-constrained optimization problem arising in cardiac electrophysiology, that is, the optimal control problem of monodomain model. The optimal control problem of monodomain model is a nonlinear optimization problem that is constrained by the monodomain model. The monodomain model consists of a parabolic partial differential equation coupled to a system of nonlinear ordinary differential equations, which has been widely used for simulating cardiac electrical activity. Our control objective is to dampen the excitation wavefront using optimal applied extracellular current. Two hybrid conjugate gradient methods are employed for computing the optimal applied extracellular current, namely, the Hestenes-Stiefel-Dai-Yuan (HS-DY) method and the Liu-Storey-Conjugate-Descent (LS-CD) method. Our experiment results show that the excitation wavefronts are successfully dampened out when these methods are used. Our experiment results also show that the hybrid conjugate gradient methods are superior to the classical conjugate gradient methods when Armijo line search is used.

A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems

2016 ◽  
Vol 23 (1) ◽  
pp. 16-30 ◽  
Author(s):  
SS Ezz-Eldien ◽  
EH Doha ◽  
D Baleanu ◽  
AH Bhrawy
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Fractional Derivative ◽  
Control Problem ◽  
Performance Index ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Fractional Optimal Control Problem ◽  
Fractional Optimal Control ◽  
Orthonormal Polynomials

The numerical solution of a fractional optimal control problem having a quadratic performance index is proposed and analyzed. The performance index of the fractional optimal control problem is considered as a function of both the state and the control variables. The dynamic constraint is expressed as a fractional differential equation that includes an integer derivative in addition to the fractional derivative. The order of the fractional derivative is taken as less than one and described in the Caputo sense. Based on the shifted Legendre orthonormal polynomials, we employ the operational matrix of fractional derivatives, the Legendre–Gauss quadrature formula and the Lagrange multiplier method for reducing such a problem into a problem consisting of solving a system of algebraic equations. The convergence of the proposed method is analyzed. For confirming the validity and accuracy of the proposed numerical method, a numerical example is presented along with a comparison between our numerical results and those obtained using the Legendre spectral-collocation method.

scholarly journals Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint

2018 ◽  
Vol 40 (1) ◽  
pp. 377-404 ◽  
Author(s):  
Bangti Jin ◽  
Buyang Li ◽  
Zhi Zhou
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Numerical Solutions ◽  
Control Variable ◽  
Mesh Size ◽  
Discrete Analogue ◽  
Distributed Optimal Control ◽  
Fully Discrete ◽  
Backward Euler

Abstract In this work we present numerical analysis for a distributed optimal control problem, with box constraint on the control, governed by a subdiffusion equation that involves a fractional derivative of order $\alpha \in (0,1)$ in time. The fully discrete scheme is obtained by applying the conforming linear Galerkin finite element method in space, L1 scheme/backward Euler convolution quadrature in time, and the control variable by a variational-type discretization. With a space mesh size $h$ and time stepsize $\tau $ we establish the following order of convergence for the numerical solutions of the optimal control problem: $O(\tau ^{\min ({1}/{2}+\alpha -\epsilon ,1)}+h^2)$ in the discrete $L^2(0,T;L^2(\varOmega ))$ norm and $O(\tau ^{\alpha -\epsilon }+\ell _h^2h^2)$ in the discrete $L^{\infty }(0,T;L^2(\varOmega ))$ norm, with any small $\epsilon>0$ and $\ell _h=\ln (2+1/h)$. The analysis relies essentially on the maximal $L^p$-regularity and its discrete analogue for the subdiffusion problem. Numerical experiments are provided to support the theoretical results.

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