Dirichlet Boundary Control of Semilinear Parabolic Equations Part 1: Problems with No State Constraints

This paper is concerned with distributed and Dirichlet boundary controls of semilinear parabolic equations, in the presence of pointwise state constraints. The paper is divided into two parts. In the first part we define solutions of the state equation as the limit of a sequence of solutions for equations with Robin boundary conditions. We establish Taylor expansions for solutions of the state equation with respect to perturbations of boundary control (Theorem 5.2). For problems with no state constraints, we prove three decoupled Pontryagin's principles, one for the distributed control, one for the boundary control, and the last one for the control in the initial condition (Theorem 2.1). Tools and results of Part 1 are used in the second part to derive Pontryagin's principles for problems with pointwise state constraints. Accepted 12 July 2001. Online publication 21 December 2001.     

Pontryagin's Principle for Time-Optimal Problems

We consider time-optimal control problems for semilinear parabolic equations with pointwise state constraints and unbounded controls. A Pontryagin's principle is obtained in nonqualified form without any qualification condition. The terminal time, which is a control variable, satisfies an optimality condition, which seems to be new in the context of control problems for partial differential equations.     

半线性抛物方程支配系统的最优性条件

本文讨论了控制变量含于高阶导数项系数中的抛物型偏微分方程支配系统的最优控制问题,给出了最优控制的必要条件．     

Dirichlet Boundary Control of Semilinear Parabolic Equations Part 2: Problems with Pointwise State Constraints

This paper is the continuation of the paper ``Dirichlet boundary control of semilinear parabolic equations. Part 1: Problems with no state constraints.' It is concerned with an optimal control problem with distributed and Dirichlet boundary controls for semilinear parabolic equations, in the presence of pointwise state constraints. We first obtain approximate optimality conditions for problems in which state constraints are penalized on subdomains. Next by using a decomposition theorem for some additive measures (based on the Stone—Cech compactification), we pass to the limit and recover Pontryagin's principles for the original problem. Accepted 21 July 2001. Online publication 21 December 2001.     

Maximum Principle for State-Constrained Control of Some Semilinear Parabolic Differential Equations

This paper is concerned with the Pontryagin maximum principle for optimal control problems of some semilinear equations, which may be governed by nonmonotone operators and in particular might be singular. Some state constraints are considered.     

The Carleman Inequality and Its Application to Periodic Optimal Control Governed by Semilinear Parabolic Differential Equations

This paper deals with optimal control problems for semilinear parabolic differential equations, which may be governed by nonmonotone operators and have no global solution, with periodic inputs. The Pontryagin maximum principle is obtained and the Carleman inequality for the backward linearized adjoint system associated with the state system is established.     

Existence of Optimal Controls for Semilinear Parabolic Equations without Cesari-Type Conditions

   Abstract. Optimal control problems governed by semilinear parabolic partial differential equations are considered. No Cesari-type conditions are assumed. By proving the existence theorem and the Pontryagin maximum principle of optimal ``state-control" pairs for the corresponding relaxed problems, an existence theorem of optimal pairs for the original problem is established.     

分布控制中一类半线性抛物方程的差分格式

本文讨论了在分布控制中出现的一类半线性抛物方程的有限差分方法.构造了一个线性化隐式差分格式.证明了差分格式解的存在唯一性、收敛性和无条件稳定性.并给出了L2和L∞范数意义下的收敛阶数为O(h2 τ2).数值例子验证了理论分析结果.     

Existence of Optimal Controls for Semilinear Parabolic Equations without Cesari-Type Conditions

Abstract. Optimal control problems governed by semilinear parabolic partial differential equations are considered. No Cesari-type conditions are assumed. By proving the existence theorem and the Pontryagin maximum principle of optimal ``state-control" pairs for the corresponding relaxed problems, an existence theorem of optimal pairs for the original problem is established.     

On the Approximate Controllability of Some Semilinear Parabolic Boundary-Value Problems

We prove the approximate controllability of several nonlinear parabolic boundary-value problems by means of two different methods: the first one can be called a Cancellation method and the second one uses the Kakutani fixed-point theorem. Accepted 10 June 1996     

一类半线性抛物边值问题的最大值原理

文中构造了一类具有Dirichlet或Neumann边界条件的半线性抛物方程u_t=Δu+f(x,u,q,t) (q=|u|^2)的解的一个辅助函数，对其使用Hopf最大值原理和黎曼几何理论，从而获得了该函数的最大值原理，据此原理获得了梯度q和解u的估计.     

Residual Based a Posteriori Error Estimates for Convex Optimal Control Problems Governed by Stokes-Darcy Equations

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.     

Newton Linearized Methods for Semilinear Parabolic Equations

In this study, Newton linearized finite element methods are presented for solving semi-linear parabolic equations in two- and three-dimensions. The proposed scheme is a one-step, linearized and second-order method in temporal direction, while the usual linearized second-order schemes require at least two starting values. By using a temporal-spatial error splitting argument, the fully discrete scheme is proved to be convergent without time-step restrictions dependent on the spatial mesh size. Numerical examples are given to demonstrate the efficiency of the methods and to confirm the theoretical results.     

A Free Boundary Problem Governed by Nonlinear Degenerate Equations of Parabolic Type

We consider a free boundary problem connected with non-Newtonian fluid motion, i.e. the flow of power law fluids with the yield stress. We obtain the solution of the relevant approximation problem by means of a parabolic quasi-variational inequality, and then obtain the weak solution of the original problem after a passage to the limit. Finally, we study the regularity of the weak solution.     

一类半线性抛物型方程的紧差分格式

本文构造了一类半线性抛物方程初边值问题的紧差分格式．利用离散能量估计证明了差分格式解的存在唯一性、收敛性和无条件稳定性，并给出了在离散L^∞模意义下收敛阶数为O（h^4＋τ^2）．数值例子验证了理论分析结果。     

Superconvergence for Optimal Control Problem Governed by Nonlinear Elliptic Equations

In this article, we investigate the superconvergence of the finite element approximation for optimal control problem governed by nonlinear elliptic equations. The state and co-state are discretized by piecewise linear functions and control is approximated by piecewise constant functions. We give the superconvergence analysis for both the control variable and the state variables. Finally, the numerical experiments show the theoretical results.     

Optimal Control Problems with State Constraint Governed by Magnetohydrodynamic Equations

This work deals with the existence of optimal solution and the maximum principle for optimal control problem governed by MHD equations with state constraint of pointwise type in three-dimension. Strong results in two-dimension also are given.     

Optimal Control Problems for a Class of Semilinear Multisolution Elliptic Equations

We study the optimal control problem for a class of elliptic problems that may possess multiple solutions. We obtain necessary conditions for optimal control by constructing a related parabolic problem and using known results for the parabolic problem.