Error estimates of mixed methods for optimal control problems governed by parabolic equations

In this paper, we investigate the error estimates for the solutions of parabolic optimal control problem by mixed finite element methods. The state and co‐state are approximated by the lowest‐order Raviart–Thomas mixed finite element spaces, and the control is approximated by piecewise constant functions. The convergence for the states, co‐states and the control is demonstrated. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite element methods for an optimal steady-state control problem

An optimal steady-state control problem governed by an elliptic state equation is solved by several finite element methods. Finite element discretizations are applied to different variational formulations of the problem yielding accurate numerical results as compared with the given analytical solution. It is sated that, for minimum computational effort and high accuracy, ‘mixed’ finite elements requiring only C° continuity, and approximating the control and state functions simultaneously are better suited to similar ‘fourth order’ problems.     

Finite element and conjugate gradient methods for a nonlinear optimal heat transfer control problem

The boundary control problem of optimal heating of an infinitely long slab with tempertue-dependent thermal conductivity, subjected to a convection and radiation boundary condition, is analysed by numerical methods. In order to reformulate the optimal control problem of distributed parameter systems as a mathematical programming problem of finite dimension, a space, co-ordinate is discretized by use of the finite element method, while the Runge–Kutta method is utilized for time integrations. Finally, the performance index of the optimal control problem is minimized by the conjugate gradient method of optimization.     

Pendulum's rotational motion governed by a stochastic Mathieu equation

This paper considers rotational motion of a nonlinear Mathieu equation with a narrow-band stochastic excitation. The path integration technique is utilized to obtain the joint probability density function of the response, which is used to construct domains of rotational motion in parameter space.     

Approximated analytical solution to an Ebola optimal control problem

An analytical expression for the optimal control of an Ebola problem is obtained. The analytical solution is found as a first-order approximation to the Pontryagin Maximum Principle via the Euler–Lagrange equation. An implementation of the method is given using the computer algebra system Maple. Our analytical solutions confirm the results recently reported in the literature using numerical methods.     

Convergence of the SQP method for quasilinear parabolic optimal control problems

Optimization and Engineering - Based on the theoretical framework recently proposed by Bonifacius and Neitzel (Math Control Relat Fields 8(1):1–34, 2018. https://doi.org/10.3934/mcrf.2018001...     

A finite element collocation method for the exact control of a parabolic problem

A finite element method is given for the problem of exact control of a linear parabolic equation. The basis functions consist of piecewise bicubic polynomials and the differential equation is satisfied at Gaussian collocation points within each element. The overdetermined system of equations obtained is solved by the method of least squares, and a convergence argument is given for the complete procedure. Numerical results are given for two problems of boundary control.     

Extremal geometry for an optimal control problem with a coupled system of two symmetric rigid bodies

We consider motions of a mechanical system consisting of two completely symmetric rigid bodies constrained to rotate with opposite angular velocities, and the following optimal control problem: find the motions of this system on [0, I] with given endpoints in SO(3) x SO(3) which minimize the integral of the system's kinetic energy. We show that the extremals for this problem can be obtained explicitly as products of exponentials, and give a geometric interpretation of these extremals     

An optimal control problem in estimating the optimal control force for the force vibration system

An optimal control problem for the force vibration system based on the iterative regularization method, i.e. the conjugate gradient method (CGM), is examined to estimate the optimal control force in a damped system having time‐dependent system parameters such that the desire (or design) system displacements can be satisfied. It is assumed that no prior information is available on the functional form of the unknown control function in the present study, thus, it is classified as the function estimation. Numerical simulations are performed to test the validity of the present algorithm by using different types of the desire system displacements. Results show that an excellent estimation on the optimal control force can be obtained with arbitrary initial guesses within a couple of second's CPU time at Pentium III‐500 MHz PC. Copyright © 2001 John Wiley & Sons, Ltd.     

Application of the finite element method to an optimal control problem

The finite element method is applied to a typical problem in the optimal control theory. A finite element formulation is first set up, and then applied, as an example, to a problem of minimizing the transfer time of a low-thrust rocket vehicle between the orbits of Earth and Mars. A procedure which combines the gradient projection technique with the Newton-Raphson method is introduced to obtain numerically the solution of the minimum time transfer problem from the discretized functional. Numerical results show that the application of the finite element method to optimal control problems seems promising as well as encouraging.     

Numerical solution of a parabolic equation

A splitting scheme is proposed for the numerical solution of a parabolic equation containing singularities important on the boundary of the region of the sought function.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 40, No. 6, pp. 1090–1092, June, 1981.     

Numerical solution to a sideways parabolic equation

The inverse heat conduction problem can be considered to be a sideways parabolic equation in the quarter plane. This is a model of a problem, where one wants to determine the temperature on both sides of a thick wall, but one side is inaccessible to measurements. A numerical procedure for this severely ill‐posed problem is suggested, which consists of two steps, namely a mollification of the data and a marching difference scheme. The numerical method is proved to be stable. Several computational results are presented and discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

Modified nonlocal boundary value problem method for an ill-posed problem for the biharmonic equation

In this paper, we propose a modified nonlocal boundary value problem method for an homogeneous biharmonic equation in a rectangular domain. We show that the considered problem is ill-posed in the sense of Hadamard, i.e. the solution does not depend continuously on the given data. Convergence estimates for the regularized solution are obtained under a priori bound assumptions for the exact solution. Some numerical results are given to show the effectiveness of the proposed method.     

Resonance phenomena in the nonlinear vibration of plates governed by Duffing''s equation

This paper presents a theoretical and experimental study of the resonance phenomena of a nonlinear system, the time dependent part of which is governed by Duffing's equation. The method of Krylov, Bogoliubov and Mitropolsky is applied to analyze the transient motion near resonance, and the amplitude of the system is shown to change rapidly, during a short time duration, at certain critical driving frequencies. During this transitional period, the instantaneous response frequency of the system differs from the driving frequency, and the variation of amplitude vs response frequency is bounded by the usual steady-state nonlinear response curve. Experimental results are presented for a clamped circular plate driven into large amplitude oscillations by a magnetic force, during the downward and upward “jumps”. The transient nonlinear response curves during the jumps, and the free vibration (skelton) curves of the plate are determined experimentally. Methods for measuring the damping coefficient, linear frequency and nonlinear stiffness of the plate are also discussed.     

General boundary element method for non-linear heat transfer problems governed by hyperbolic heat conduction equation

The general Boundary Element Method (BEM) for strongly non-linear problems proposed by Liao (1995) is further applied to solve a two-dimensional unsteady non-linear heat transfer problem in the time domain, governed by the hyperbolic heat conduction equation (HHCE) with the temperature-dependent thermal conductivity coefficients which are different in the x and y directions. This paper confirms that the general BEM can be used to solve even those non-linear unsteady heat transfer problems whose governing equations do not contain any linear terms in spatial domain.     

Finite element formulation of a heat transfer problem for an axisymmetric composite structure

By using weighted residual method, the finite element formulation of a heat transfer problem for axisymmetric composite structures is established from the heat transfer differential equations expressed by heat fluid density. A few examples are included to indicate that the heat transfer anisotropy has an important effect on temperature field and to prove the accuracy and effectiveness of the finite element formulation.Aknowledgement Special thanks are due to the National Natural Science Foundation of China (No: 10272037) for supporting the present work.     

Higher order difference formulas for a fourth order parabolic partial differential equation

In this paper, we have derived some new higher order difference formulas for the solution of a fourth order parabolic partial differential equation governing transverse vibrations of a uniform flexible beam in one and two space dimensions using Richtmyer's approach and a direct approach. Two examples illustrate the utility of the new difference methods.     

Robust optimal Robin boundary control for the transient heat equation with random input data

The problem of robust optimal Robin boundary control for a parabolic partial differential equation with uncertain input data is considered. As a measure of robustness, the variance of the random system response is included in two different cost functionals. Uncertainties in both the underlying state equation and the control variable are quantified through random fields. The paper is mainly concerned with the numerical resolution of the problem. To this end, a gradient‐based method is proposed considering different functional costs to achieve the robustness of the system. An adaptive anisotropic sparse grid stochastic collocation method is used for the numerical resolution of the associated state and adjoint state equations. The different functional costs are analysed in terms of computational efficiency and its capability to provide robust solutions. Two numerical experiments illustrate the performance of the algorithm. Copyright © 2016 John Wiley & Sons, Ltd.