共查询到20条相似文献,搜索用时 15 毫秒
1.
A Smooth Regularization of the Projection Formula for Constrained Parabolic Optimal Control Problems
Ira Neitzel Uwe Prüfert Thomas Slawig 《Numerical Functional Analysis & Optimization》2013,34(12):1283-1315
We present a smooth, that is, differentiable regularization of the projection formula that occurs in constrained parabolic optimal control problems. We summarize the optimality conditions in function spaces for unconstrained and control-constrained problems subject to a class of parabolic partial differential equations. The optimality conditions are then given by coupled systems of parabolic PDEs. For constrained problems, a non-smooth projection operator occurs in the optimality conditions. For this projection operator, we present in detail a regularization method based on smoothed sign, minimum and maximum functions. For all three cases, that is, (1) the unconstrained problem, (2) the constrained problem including the projection, and (3) the regularized projection, we verify that the optimality conditions can be equivalently expressed by an elliptic boundary value problem in the space-time domain. For this problem and all three cases we discuss existence and uniqueness issues. Motivated by this elliptic problem, we use a simultaneous space-time discretization for numerical tests. Here, we show how a standard finite element software environment allows to solve the problem and, thus, to verify the applicability of this approach without much implementation effort. We present numerical results for an example problem. 相似文献
2.
In this paper an ultraspherical integral method is proposed to solve optimal control problems governed by ordinary differential equations. Ultraspherical approximation method reduced the problem to a constrained optimization problem. Penalty leap frog method is presented to solve the resulting constrained optimization problem. Error estimates for the ultraspherical approximations are derived and a technique that gives an optimal approximation of the problems is introduced. Numerical results are included to confirm the efficiency and accuracy of the method. 相似文献
3.
Yanping Chen Lingli Liu Zuliang Lu 《Numerical Functional Analysis & Optimization》2013,34(10):1135-1157
In this article, we shall give a brief review on the fully discrete mixed finite element method for general optimal control problems governed by parabolic equations. The state and the co-state are approximated by the lowest order Raviart–Thomas mixed finite element spaces and the control is approximated by piecewise constant elements. Furthermore, we derive a posteriori error estimates for the finite element approximation solutions of optimal control problems. Some numerical examples are given to demonstrate our theoretical results. 相似文献
4.
该文讨论了一类状态变量约束下由发展方程导出的最优控制系统,通过原问题的扰动,得到了状态变量与控制变量分离的最优性条件. 相似文献
5.
A multigrid scheme for the solution of constrained optimal control problems discretized by finite differences is presented. This scheme is based on a new relaxation procedure that satisfies the given constraints pointwise on the computational grid. In applications, the cases of distributed and boundary control problems with box constraints are considered. The efficient and robust computational performance of the present multigrid scheme allows to investigate bang-bang control problems.AMS Subject Classification: 49J20, 65N06, 65N12, 65N55Supported in part by the SFB 03 “Optimization and Control” 相似文献
6.
7.
<正>The formulation of optimal control problems governed by Fredholm integral equations of second kind and an efficient computational framework for solving these control problems is presented.Existence and uniqueness of optimal solutions is proved. A collective Gauss-Seidel scheme and a multigrid scheme are discussed.Optimal computational performance of these iterative schemes is proved by local Fourier analysis and demonstrated by results of numerical experiments. 相似文献
8.
A dynamic programming method is presented for solving constrained, discrete-time, optimal control problems. The method is based on an efficient algorithm for solving the subproblems of sequential quadratic programming. By using an interior-point method to accommodate inequality constraints, a modification of an existing algorithm for equality constrained problems can be used iteratively to solve the subproblems. Two test problems and two application problems are presented. The application examples include a rest-to-rest maneuver of a flexible structure and a constrained brachistochrone problem. 相似文献
9.
In this paper, a finite difference method is used to approximate for the solution of the parabolic partial differential equation of order 2n and error of the method is determined. The resulting system is solved by efficient implicit iterations. In some numerical examples, MAPLE modules are designed for the purpose of testing and using the method. 相似文献
10.
Buttazzo G. Drakhlin M. E. Freddi L. Stepanov E. 《Journal of Optimization Theory and Applications》1997,93(1):103-119
The paper deals with the variational convergence of a sequence of optimal control problems for functional differential state equations with deviating argument. Variational limit problems are found under various conditions of convergence of the input data. It is shown that, upon sufficiently weak assumptions on convergence of the argument deviations, the limit problem can assume a form different from that of the whole sequence. In particular, it can be either an optimal control problem for an integro-differential equation or a purely variational problem. Conditions are found under which the limit problem preserves the form of the original sequence. 相似文献
11.
The asymptotic behavior of state-constrained semilinear optimal control problems for distributed-parameter systems with variable
compact control zones is investigated. We derive conditions under which the limiting problems can be made explicit.
We gratefully acknowledge the support of the DAAD. The paper was prepared during the visit of the first author at the Institute
of Applied Mathematics II, University Erlangen-Nuremberg in 2003. 相似文献
12.
O. Mali 《Numerical Functional Analysis & Optimization》2017,38(1):58-79
In this article, functional type a posteriori error estimates are presented for a certain class of optimal control problems with elliptic partial differential equation constraints. It is assumed that in the cost functional the state is measured in terms of the energy norm generated by the state equation. The functional a posteriori error estimates developed by Repin in the late 1990s are applied to estimate the cost function value from both sides without requiring the exact solution of the state equation. Moreover, a lower bound for the minimal cost functional value is derived. A meaningful error quantity coinciding with the gap between the cost functional values of an arbitrary admissible control and the optimal control is introduced. This error quantity can be estimated from both sides using the estimates for the cost functional value. The theoretical results are confirmed by numerical tests. 相似文献
13.
The accurate solution of optimal control problems is crucial in many areas of engineering and applied science. For systems which are described by a nonlinear set of differential-algebraic equations, these problems have been shown to often contain multiple local minima. Methods exist which attempt to determine the global solution of these formulations. These algorithms are stochastic in nature and can still get trapped in local minima. There is currently no deterministic method which can solve, to global optimality, the nonlinear optimal control problem. In this paper a deterministic global optimization approach based on a branch and bound framework is introduced to address the nonlinear optimal control problem to global optimality. Only mild conditions on the differentiability of the dynamic system are required. The implementa-tion of the approach is discussed and computational studies are presented for four control problems which exhibit multiple local minima. 相似文献
14.
Meshfree Finite Volume Element Method for Constrained Optimal Control Problem Governed by Random Convection Diffusion Equations
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In this paper, we investigate a stochastic meshfree finite volume element method for an optimal control problem governed by the convection diffusion equations with random coefficients. There are two contributions of this paper. Firstly, we establish a scheme to approximate the optimality system by using the finite volume element method in the physical space and the meshfree method in the probability space, which is competitive for high-dimensional random inputs. Secondly, the a priori error estimates are derived for the state,the co-state and the control variables. Some numerical tests are carried out to confirm the theoretical results and demonstrate the efficiency of the proposed method. 相似文献
15.
In this paper, we derive a posteriori error estimates for finite element approximations of the optimal control problems governed by the Stokes-Darcy system. We obtain a posteriori error estimators for both the state and the control based on the residual of the finite element approximation. It is proved that the a posteriori error estimate provided in this paper is both reliable and efficient. 相似文献
16.
In the present paper, we investigate an approximation technique for relaxed optimal control problems. We study control processes governed by ordinary differential equations in the presence of state, target, and integral constraints. A variety of approximation schemes have been recognized as powerful tools for the theoretical studying and practical solving of Infinite-dimensional optimization problems. On the other hand, theoretical approaches to the relaxed optimal control problem with constraints are not sufficiently advanced to yield numerically tractable schemes. The explicit approximation of the compact control set makes it possible to reduce the sophisticated relaxed problem to an auxiliary optimization problem. A given trajectory of the relaxed problem can be approximated by trajectories of the auxiliary problem. An optimal solution of the introduced optimization problem provides a basis for the construction of minimizing sequences for the original optimal control problem. We describe how to carry out the numerical calculations in the context of nonlinear programming and establish the convergence properties of the obtained approximations.The authors thank the referees for helpful comments and suggestions. 相似文献
17.
Second-Order Analysis for Control Constrained Optimal Control Problems of Semilinear Elliptic Systems 总被引:2,自引:0,他引:2
J. F. Bonnans 《Applied Mathematics and Optimization》1998,38(3):303-325
This paper presents a second-order analysis for a simple model optimal control problem of a partial differential equation,
namely, a well-posed semilinear elliptic system with constraints on the control variable only. The cost to be minimized is
a standard quadratic functional. Assuming the feasible set to be polyhedric, we state necessary and sufficient second-order
optimality conditions, including a characterization of the quadratic growth condition. Assuming that the second-order sufficient
condition holds, we give a formula for the second-order expansion of the value of the problem as well as the directional derivative
of the optimal control, when the cost function is perturbed. Then we extend the theory of second-order optimality conditions
to the case of vector-valued controls when the feasible set is defined by local and smooth convex constraints. When the space
dimension n is greater than 3, the results are based on a two norms approach, involving spaces L
2
and L
s
, with s>n/2 .
Accepted 27 January 1997 相似文献
18.
A theoretical sensitivity analysis for parametric optimal control problems subject to pure state constraints has recently been elaborated in [7,8]. The articles consider both first and higher order state constraints and develop conditions for solution differentiability of optimal solutions with respect to parameters. In this paper, we treat the numerical aspects of computing sensitivity differentials via appropriate boundary value problems. In particular, numerical methods are proposed that allow to verify all assumptions underlying solution differentiability. Three numerical examples with state constraints of order one, two and four are discussed in detail. 相似文献
19.
B. Vexler 《Numerical Functional Analysis & Optimization》2013,34(7-8):957-973
We develop a priori error analysis for the finite element Galerkin discretization of elliptic Dirichlet optimal control problems. The state equation is given by an elliptic partial differential equation and the finite dimensional control variable enters the Dirichlet boundary conditions. We prove the optimal order of convergence and present a numerical example confirming our results. 相似文献
20.
Hongfei Fu & Hongxing Rui 《计算数学(英文版)》2015,33(2):113-127
In this paper, a constrained distributed optimal control problem governed by a first-order elliptic system is considered. Least-squares mixed finite element methods, which are not subject to the Ladyzhenkaya-Babuska-Brezzi consistency condition, are used for solving the elliptic system with two unknown state variables. By adopting the Lagrange multiplier approach, continuous and discrete optimality systems including a primal state equation, an adjoint state equation, and a variational inequality for the optimal control are derived, respectively. Both the discrete state equation and discrete adjoint state equation yield a symmetric and positive definite linear algebraic system. Thus, the popular solvers such as preconditioned conjugate gradient (PCG) and algebraic multi-grid (AMG) can be used for rapid solution. Optimal a priori error estimates are obtained, respectively, for the control function in $L^2(Ω)$-norm, for the original state and adjoint state in $H^1(Ω)$-norm, and for the flux state and adjoint flux state in $H$(div; $Ω$)-norm. Finally, we use one numerical example to validate the theoretical findings. 相似文献