Multigrid Methods for Elliptic Optimal Control Problems with Pointwise State Constraints

An elliptic optimal control problem with constraints on the state variable is considered. The Lavrentiev-type regularization is used to treat the constraints on the state variable. To solve the problem numerically, the multigrid for optimization (MGOPT) technique and the collective smoothing multigrid (CSMG) are implemented. Numerical results are reported to illustrate and compare the efficiency of both multigrid strategies.     

Optimal Control for Fractional Diffusion Equations with Incomplete Data

We are concerned with the optimal control of time-fractional diffusion equations with missing boundary condition. Using the notion of no-regret control and least (or low) regret control developed by Lions, we first prove that the least regret control problem associated with the boundary fractional diffusion equation has a unique solution. Then we show that this solution converges to the no-regret control which we characterize by a singular optimality system.     

Correction to: Optimal Control Involving Sweeping Processes

Set-Valued and Variational Analysis - We would like to make corrections to a result, Lemma 2, in the above paper.     

Optimal Control of PDEs with Regularized Pointwise State Constraints

This paper addresses the regularization of pointwise state constraints in optimal control problems. By analyzing the associated dual problem, it is shown that the regularized problems admit Lagrange multipliers in L²-spaces. Under a certain boundedness assumption, the solution of the regularized problem converges to the one of the original state constrained problem. The results of our analysis are confirmed by numerical tests. Supported by the DFG Research Center “Mathematics for key technologies” (FZT 86) in Berlin.     

Mixed Optimal Stopping and Stochastic Control Problems with Semicontinuous Final Reward for Diffusion Processes

We consider mixed control problems for diffusion processes, i.e. problems which involve both optimal control and stopping. The running reward is assumed to be smooth, but the stopping reward need only be semicontinuous. We show that, under suitable conditions, the value function w has the same regularity as the final reward g, i.e. w is lower or upper semicontinuous if g is. Furthermore, when g is l.s.c., we prove that the value function is a viscosity solution of the associated variational inequality.     

Probabilistic Solutions to Optimal Control Problems

Let ( x 1 ( t ), x 2 ( t )) be a controlled two-dimensional diffusion process. The problem of minimizing, or maximizing, the time spent by ( x 1 ( t ), x 2 ( t )) in a given subset of 2 is solved, in two particular instances, by transforming the optimal control problems into purely probabilistic problems. In Section 2 , ( x 1 ( t ), x 2 ( t )) is a two-dimensional Wiener process and the optimal control is obtained by transforming a nonlinear dynamic programming equation into the Kolmogorov backward equation for a two-dimensional geometric Brownian motion. In Section 3 , the converse problem is solved. The problem of finding the maximal instantaneous reward that we can give for survival in the continuation region is also treated.     

年龄相关的种群扩散系统的最优分布控制

本文讨论年龄相关的种群扩散系统的最优分布控制 ,证明了最优分布控制的存在性 ,得到了分布控制为最优的充分必要条件及确定最优控制的最优性组 .     

年龄相关的种群系统的最优扩散控制

运用扰动方法研究了由积分-偏微分方程所描述的年龄相关的种群系统的最优扩散控制问题.首先,在证明了原系统及其扰动系统正则广义解存在唯一等预备结果的基础上,用先验估计和紧性定理,证明了系统最优扩散控制的存在性.其次,利用Gteax微分和变分不等式等理论,得到了控制为最优的必要条件和由偏微分方程与变分不等式所构成的最优性组,由最优性组确定最优控制.     

Mixed Methods for Optimal Control Problems

In this paper, we investigate a posteriori error estimates of amixed finite elementmethod for elliptic optimal control problems with an integral constraint. The gradient for ourmethod belongs to the square integrable space instead of the classical H(div; Ω) space. The state and co-state are approximated by the P ₀ ² -P₁ (velocity–pressure) pair and the control variable is approximated by piecewise constant functions. Using duality argument method and energy method, we derive the residual a posteriori error estimates for all variables.     

Optimal Control Problems with Integral Functional and Phase Constraints: Reduction to Optimal Consistency Parameter Problems

An iteration method to solve a class of optimal control problems with integral functional and phase constraints is developed in the paper.     

Exact Solutions in Optimal Design Problems for Stationary Diffusion Equation

We consider two-phase multiple state optimal design problems for stationary diffusion equation. Both phases are taken to be isotropic, and the goal is to find the optimal distribution of materials within domain, with prescribed amounts, that minimizes a weighted sum of energies. In the case of one state equation, it is known that the proper relaxation of the problem via the homogenization theory is equivalent to a simpler relaxed problem, stated only in terms of the local proportion of given materials.

We prove an analogous result for multiple state problems if the number of states is less than the space dimension. In spherically symmetric case, the result holds for arbitrary number of states, and the optimality conditions of a simpler relaxation problem, which are necessary and sufficient, enable us to explicitly calculate the unique solution of proper relaxation for some examples. In contrary to maximization problems, these solutions are not classical.

     

一类具有年龄结构的非线性种群扩散系统的最优收获控制

讨论了一类与年龄相关的非线性种群扩散系统的最优收获控制问题,证明了最优收获控制的存在性,并且给出了控制为最优的必要条件及其由偏微分方程组和变分不等式组成的最优性组.这些结果可为种群扩散系统最优控制问题的实际研究提供理论基础.     

Second-Order Optimality Conditions for a Semilinear Elliptic Optimal Control Problem with Mixed Pointwise Constraints

This paper studies second-order optimality conditions for a semilinear elliptic optimal control problem with mixed pointwise constraints. We show that in some cases, there is a common critical cone under which the second-order necessary and sufficient optimality conditions for the problem are valid. Our results approach to a theory of no-gap second-order conditions. In order to obtain such results, we reduce the problem to a special mathematical programming problem with polyhedricity constraint set. We then use some tools of variational analysis and techniques of semilinear elliptic equations to analyze second-order conditions.     

Sufficiency for Optimal Control Problems Involving Parameters

The optimal control problem is extended to the case where the performance index, the differential constraints, and the prescribed final conditions contain parameters. The sufficient condition for a minimum is derived for nonsingular problems using the sweep method. As expected, it involves the finiteness of a matrix or the location of the conjugate point. The minimum-time navigation problem is solved as a fixed final time problem to illustrate the application of the theory.     

Meshfree Approximation for Stochastic Optimal Control Problems

In this work,we study the gradient projection method for solving a class of stochastic control problems by using a mesh free approximation ap-proach to implement spatial dimension approximation.Our main contribu-tion is to extend the existing gradient projection method to moderate high-dimensional space.The moving least square method and the general radial basis function interpolation method are introduced as showcase methods to demonstrate our computational framework,and rigorous numerical analysis is provided to prove the convergence of our meshfree approximation approach.We also present several numerical experiments to validate the theoretical re-sults of our approach and demonstrate the performance meshfree approxima-tion in solving stochastic optimal control problems.     

Wavelet Collocation Method for Optimal Control Problems

A Haar wavelet technique is discussed as a method for discretizing the nonlinear system equations for optimal control problems. The technique is used to transform the state and control variables into nonlinear programming (NLP) parameters at collocation points. A nonlinear programming solver can then be used to solve optimal control problems that are rather general in form. Here, general Bolza optimal control problems with state and control constraints are considered. Examples of two kinds of optimal control problems, continuous and discrete, are solved. The results are compared to those obtained by using other collocation methods.     

Optimal Potentials for Problems with Changing Sign Data

The main purpose of this paper is to study the duality and penalty method for a constrained nonconvex vector optimization problem. Following along with the image space analysis, a Lagrange-type duality for a constrained nonconvex vector optimization problem is proposed by virtue of the class of vector-valued regular weak separation functions in the image space. Simultaneously, some equivalent characterizations to the zero duality gap property are established including the Lagrange multiplier, the Lagrange saddle point and the regular separation. Moreover, an exact penalization is also obtained by means of a local image regularity condition and a class of particular regular weak separation functions in the image space.