Superconvergence for Optimal Control Problems Governed by Semi-linear Elliptic Equations

In this paper, we will investigate the superconvergence of the finite element approximation for quadratic optimal control problem governed by semi-linear elliptic equations. The state and co-state variables are approximated by the piecewise linear functions and the control variable is approximated by the piecewise constant functions. We derive the superconvergence properties for both the control variable and the state variables. Finally, some numerical examples are given to demonstrate the theoretical results. This work is supported by National Science Foundation of China, the National Basic Research Program under the Grant 2005CB321703, Scientific Research Fund of Hunan Provincial Education Department, and Hunan Provincial Innovation Foundation for Postgraduate (No. S2008yjscx04).     

A Priori Error Estimates for Optimal Control Problems Governed by Transient Advection-Diffusion Equations

In this paper, we investigate a characteristic finite element approximation of quadratic optimal control problems governed by linear advection-dominated diffusion equations, where the state and co-state variables are discretized by piecewise linear continuous functions and the control variable is approximated by piecewise constant functions. We derive some a priori error estimates for both the control and state approximations. It is proved that these approximations have convergence order , where h _U and h are the spatial mesh-sizes for the control and state discretization, respectively, and k is the time increment. Numerical experiments are presented, which verify the theoretical results. This research was supported by the National Basic Research Program of China (No. 2007CB814906) and the National Natural Science Foundation of China (No. 10771124).     

A Strongly Conservative Hybrid DG/Mixed FEM for the Coupling of Stokes and Darcy Flow

We consider the coupling of free and porous media flow governed by Stokes and Darcy equations with the Beavers–Joseph–Saffman interface condition. This model is discretized using a divergence-conforming finite element for the velocities in the whole domain. Hybrid discontinuous Galerkin techniques and mixed methods are used in the Stokes and Darcy subdomains, respectively. The discretization achieves mass conservation in the sense of \(H(\mathrm {div},\Omega )\), and we obtain optimal velocity convergence. Numerical results are presented to validate the theoretical findings.     

A Mixed Finite Element Scheme for Optimal Control Problems with Pointwise State Constraints

In this paper, we propose a mixed variational scheme for optimal control problems with point-wise state constraints, the main idea is to reformulate the optimal control problems to a constrained minimization problem involving only the state, which is characterized by a fourth order variational inequality. Then mixed form based on this fourth order variational inequality is formulated and a direct numerical algorithm is proposed without the optimality conditions of underlying optimal control problems. The a priori and a posteriori error estimates are proved for the mixed finite element scheme. Numerical experiments confirm the efficiency of the new strategy.     

A New Nonsymmetric Discontinuous Galerkin Method for Time Dependent Convection Diffusion Equations

We propose a discontinuous Galerkin finite element method for convection diffusion equations that involves a new methodology handling the diffusion term. Test function derivative numerical flux term is introduced in the scheme formulation to balance the solution derivative numerical flux term. The scheme has a nonsymmetric structure. For general nonlinear diffusion equations, nonlinear stability of the numerical solution is obtained. Optimal kth order error estimate under energy norm is proved for linear diffusion problems with piecewise P ^k polynomial approximations. Numerical examples under one-dimensional and two-dimensional settings are carried out. Optimal (k+1)th order of accuracy with P ^k polynomial approximations is obtained on uniform and nonuniform meshes. Compared to the Baumann-Oden method and the NIPG method, the optimal convergence is recovered for even order P ^k polynomial approximations.     

Regional Optimal Control Problem Governed by Distributed Bi-linear Hyperbolic Systems

This paper considers the regional bi-linear control problem of an important class of hyperbolic systems. The objective is to bring the state solutions at time T close to a desired observations w ^d only on a sub-region ω along the spatial domain Ω. We prove the existence of solution by minimizing sequence method. The adjoint system of this problem is introduced and used to characterize the optimal control. A numerical approach is developed and illustrated successfully by simulations.     

A Superconvergent HDG Method for Distributed Control of Convection Diffusion PDEs

We consider a distributed optimal control problem governed by an elliptic convection diffusion PDE, and propose a hybridizable discontinuous Galerkin method to approximate the solution. We use polynomials of degree \(k+1\) to approximate the state and dual state, and polynomials of degree \(k \ge 0\) to approximate their fluxes. Moreover, we use polynomials of degree k to approximate the numerical traces of the state and dual state on the faces, which are the only globally coupled unknowns. We prove optimal a priori error estimates for all variables when \( k \ge 0 \). Furthermore, from the point of view of the number of degrees of freedom of the globally coupled unknowns, this method achieves superconvergence for the state, dual state, and control when \(k\ge 1\). We illustrate our convergence results with numerical experiments.     

Two Mixed Finite Element Methods for Time-Fractional Diffusion Equations

Based on spatial conforming and nonconforming mixed finite element methods combined with classical L1 time stepping method, two fully-discrete approximate schemes with unconditional stability are first established for the time-fractional diffusion equation with Caputo derivative of order \(0<\alpha <1\). As to the conforming scheme, the spatial global superconvergence and temporal convergence order of \(O(h^2+\tau ^{2-\alpha })\) for both the original variable u in \(H^1\)-norm and the flux \(\vec {p}=\nabla u\) in \(L^2\)-norm are derived by virtue of properties of bilinear element and interpolation postprocessing operator, where h and \(\tau \) are the step sizes in space and time, respectively. At the same time, the optimal convergence rates in time and space for the nonconforming scheme are also investigated by some special characters of \(\textit{EQ}_1^{\textit{rot}}\) nonconforming element, which manifests that convergence orders of \(O(h+\tau ^{2-\alpha })\) and \(O(h^2+\tau ^{2-\alpha })\) for the original variable u in broken \(H^1\)-norm and \(L^2\)-norm, respectively, and approximation for the flux \(\vec {p}\) converging with order \(O(h+\tau ^{2-\alpha })\) in \(L^2\)-norm. Numerical examples are provided to demonstrate the theoretical analysis.     

A Mixed DG Method for Linearized Incompressible Magnetohydrodynamics

We introduce and analyze a discontinuous Galerkin method for the numerical discretization of a stationary incompressible magnetohydrodynamics model problem. The fluid unknowns are discretized with inf-sup stable discontinuous ? _k ³ ?? _k?1 elements whereas the magnetic part of the equations is approximated by discontinuous ? _k ³ ?? _k+1 elements. We carry out a complete a-priori error analysis of the method and prove that the energy norm error is convergent of order k in the mesh size. These results are verified in a series of numerical experiments.     

A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex Meshes

A Lagrangian numerical scheme for solving nonlinear degenerate Fokker–Planck equations in space dimensions \(d\ge 2\) is presented. It applies to a large class of nonlinear diffusion equations, whose dynamics are driven by internal energies and given external potentials, e.g. the porous medium equation and the fast diffusion equation. The key ingredient in our approach is the gradient flow structure of the dynamics. For discretization of the Lagrangian map, we use a finite subspace of linear maps in space and a variational form of the implicit Euler method in time. Thanks to that time discretisation, the fully discrete solution inherits energy estimates from the original gradient flow, and these lead to weak compactness of the trajectories in the continuous limit. Consistency is analyzed in the planar situation, \(d=2\). A variety of numerical experiments for the porous medium equation indicates that the scheme is well-adapted to track the growth of the solution’s support.     

H_2/l_1混合优化问题的凸二次规划解法

采用上逼近算法求解 H2 / l1 混合优化问题。首先将其转化为有限维的凸二次规划问题 ,并利用L emke互补转轴算法求解 ;然后逐次进行逼近。计算示例表明所得结果是正确的     

求解最优控制问题的混合变量变分方法及其航天控制应用

针对非线性最优控制导出的Hamiltonian系统两点边值问题,提出一种以离散区段右端状态和左端协态为混合独立变量的数值求解方法, 将非线性Hamiltonian系统两点边值问题的求解通过混合独立变量变分原理转化为非线性方程组求解.所提出的算法综合了求解最优控制 的"直接法"和"间接法"的特征,既满足最优控制理论的一阶必要条件,又不需要对协态初值的准确猜测,避免了求解大规模非线性规划问题. 通过两个航天控制算例讨论了本文算法的精度和效率等问题.与近年来在航空航天控制中备受关注的高斯伪谱方法相比较,本文算法无论是在 精度还是效率上都具有明显的优势.     

Fundamental Systems of Numerical Schemes for Linear Convection–Diffusion Equations and their Relationship to Accuracy

A new approach towards the assessment and derivation of numerical methods for convection dominated problems is presented, based on the comparison of the fundamental systems of the continuous and discrete operators. In two or more space dimensions, the dimension of the fundamental system is infinite, and may be identified with a ball. This set is referred to as the true fundamental locus. The fundamental system for a numerical scheme also forms a locus. As a first application, it is shown that a necessary condition for the uniform convergence of a numerical scheme is that the discrete locus should contain the true locus, and it is then shown it is impossible to satisfy this condition with a finite stencil. This shows that results of Shishkin concerning non-uniform convergence at parabolic boundaries are also generic for outflow boundaries. It is shown that the distance between the loci is related to the accuracy of the schemes provided that the loci are sufficiently close. However, if the loci depart markedly, then the situation is rather more complicated. Under suitable conditions, we develop an explicit numerical lower bound on the attainable relative error in terms of the coefficients in the stencil characterising the scheme and the loci. Received December 10, 1999; revised August 14, 2000     

Multivariable Adaptive Control of the Rewinding Process of a Roll-to-roll System Governed by Hyperbolic Partial Differential Equations

In this paper, an active control scheme for the rewinding process of a roll-to-roll (R2R) system is investigated. The control objectives are to suppress the transverse vibration of the moving web, to track the desired velocity profile, and to keep the desired radius value of a rewind roller. The bearing coefficient in the rewind shaft is unknown and the rotating elements in the drive motor are various. The moving web is modeled as an axially moving beam system governed by hyperbolic partial differential equations (PDEs). The control scheme utilizes two control inputs: a control force exerted from a hydraulic actuator equipped with a damper, and a control torque applied to the rewind roller. Two adaptation laws are derived to estimate the unknown bearing coefficient and the bound of variations of the rotating elements. The Lyapunov method is employed to prove the robust stability of the rewind section, specifically the uniform and ultimate boundedness of all of the signals. The effectiveness of the proposed control schemes was verified by numerical simulations.     

A Semi-smooth Newton Method for Regularized State-constrained Optimal Control of the Navier-Stokes Equations

In this paper, we study semi-smooth Newton methods for the numerical solution of regularized pointwise state-constrained optimal control problems governed by the Navier-Stokes equations. After deriving an appropriate optimality system for the original problem, a class of Moreau-Yosida regularized problems is introduced and the convergence of their solutions to the original optimal one is proved. For each regularized problem a semi-smooth Newton method is applied and its local superlinear convergence verified. Finally, selected numerical results illustrate the behavior of the method and a comparison between the max-min and the Fischer-Burmeister as complementarity functionals is carried out.