This paper is concerned with the implementation of optimization algorithms for the solution of smooth discretized optimal control problems. The problems under consideration can be written as min f(y; u)

In this paper we analyze inexact trust-region interior-point (TRIP) sequential quadratic programming (SQP) algorithms for the solution of optimization problems with nonlinear equality constraints and simple bound constraints on some of the variables. Such problems arise in many engineering applications, in particular in optimal control problems with bounds on the control. The nonlinear constraints often come from the discretization of partial differential equations. In such cases the calculation of derivative information and the solution of linearized equations is expensive. Often, the solution of linear systems and derivatives are computed inexactly yielding nonzero residuals. This paper analyzes the effect of the inexactness onto the convergence of TRIP SQP and gives practical rules to control the size of the residuals of these inexact calculations. It is shown that if the size of the residuals is of the order of both the size of the constraints and the trust-region radius, t...

An optimal control problem governed by an elliptic equation written in variational form in an abstract functional framework is considered. The control is subject to restrictions. The optimality conditions are established and the Ritz-Galerkin discretization is introduced. If the error estimate corresponding to the elliptic equation is given as a function like O(h q ), where h is the discretization parameter and q 1 is an integer, then the error estimates for the optimal control, for the optimal state and for the optimal value are obtained. These results are applied first for a Two-Point BVP and next for a 2D/3D elliptic problem as state equation. Next a spectral method is used in the discretization process. The estimates obtained in the abstract case are applied to a distributed control problem and to a boundary control problem.

In this paper we introduce a penalty function and a corresponding multipliers method for the solution of a class of nonlinear programming problems where the equality constraints have a particular structure. The class models optimal control and engineering design problems with bounds on the state and control variables and has wide applicability. The multipliers method updates multipliers corresponding to inequality constraints (maintaining their nonnegativity) instead of dealing with multipliers associated with equality constraints. The basic local convergence properties of the method are proved and a dual framework is introduced. We also analyze the properties of the penalized problem related with the penalty function. Keywords. Nonlinear programming, optimal control, state constraints, penalty function, multipliers method, augmented Lagrangian. AMS subject classications. 49M37, 90C06, 90C30 1

We present a mathematical model for the laser surface hardening of steel. It consists of a nonlinear heat equation coupled with a system of five ordinary differential equations to describe the volume fractions of the occuring phases. Existence, regularity and stability results are discussed. Since the resulting hardness can be estimated by the volume fraction of martensite, we formulate the problem of surface hardening in terms of an optimal control problem. To avoid surface melting, which would decrease the workpiece's quality, state constraints for the temperature are included. We prove differentiability of the solution operator and derive necessary conditions for optimality.

The identification problem of a nonlinear functional coefficient in elliptic and parabolic quasilinear equations is considered. A distributed observation of the solution of the corresponding equation is assumed to be known a priori. An identification method is introduced, which needs only a linear equation to be solved in each iteration step of the optimization. Estimates of the rate of convergence for the proposed approach are proved, when the equation is discretized with the finite element method with respect to space variables. Some numerical results are given. 1 Introduction In this article we consider the homogenous, quasilinear elliptic boundary value problem \Gammar \Delta (b(u) ru(x)) = f(x) in\Omega ; uj @\Omega = 0 ; (1.1) and corresponding parabolic equation @u(t; x) @t \Gamma r \Delta (b(u) ru(t; x)) = f(t; x) in [0; T ] \Theta\Omega ; uj @\Omega = 0 in [0; T ] ; u(0; x) = u 0 (x) in\Omega ; (1.2) where\Omega is a bounded domain in R d ; d 3; with smooth b...

A Method for solving optimal control problems with general elliptic operators is presented and analyzed. Especially, estimates of the rate of convergence for the control problems with the proposed approach are derived independently of the underlying approximation method. Some numerical experiments with the proposed method are included.