This paper is concerned with infinite-dimensional nonlinear optimization problems in which the objective function and the constraints are subjected to Lipschitzian perturbations. In particular, continuity properties of the extremal value function are studied and bounds for the distance between local minimizers of perturbed problems and the original problem are derived.

This paper investigates local convergence properties of Newton's method for discretized generalized equations. For stable and consistent discretizations it is shown, that the local behaviour of the discretized Newton iterations is asymptotically the same as that for the original iteration and, as a consequence, a mesh-independence principle is derived. The results are applied to the Lagrange-Newton method for optimal control problems.

We implement a prototype for an application of automatic optical material testing in the manufacturing of glass panels. Based on a nonlinear optimal control approach, we present a numerical method for optimal control at run time. The algorithm will be demonstrated and tested with the help of an illustrative example where it turns out that the optimal control is of bang-bang or bang-zero-bang type, depending on the state constraints.

We investigate Euler discretization for a class of optimal control problems with a nonlinear cost functional of Mayer type, a nonlinear system equation with control appearing linearly and constraints defined by lower and upper bounds for the controls. Under the assumption that the cost functional satisfies a growth condition we prove for the discrete solutions Hölder type error estimates w.r.t. the mesh size of the discretization. If a stronger second-order optimality condition is satisfied the order of convergence can be improved. Numerical experiments confirm the theoretical findings.

We analyze the implicit Euler discretization for a class of convex linear-quadratic optimal control problems with control appearing linearly. Constraints are defined by lower and upper bounds for the controls, and the cost functional may depend on a regularization parameter.. Without any structural assumption on the optimal control we prove convergence of order 1 w.r.t. the mesh size for the discrete optimal values. Under the additional assumption that the optimal control is of bang-bang type and the switching function satisfies a growth condition around their zeros we show that the solutions are calm functions of perturbation and regularization parameters. By applying this result to the implicit Euler discretization we improve existing error estimates for discretizations based on the explicit Euler method. Numerical experiments confirm the theoretical findings and demonstrate the usefulness of implicit methods and regularization in case of bang-bang controls. (C) 2016 Elsevier Inc. All rights reserved.

We consider linear-quadratic (LQ) control problems, where the control variable appears linearly and is box-constrained. It is well-known that these problems exhibit bang–bang and singular solutions. We assume that the solution is of bang–bang type, which is computationally challenging to obtain. We employ a quadratic regularization of the LQ control problem by embedding the L2-norm of the control variable into the cost functional. First, we find a dual problem guided by the methodology of Fenchel duality. Then we prove strong duality and the saddle point property, which together ensure that the primal solution can be recovered from the dual solution. We propose a discretization scheme for the dual problem, under which a diagram depicting the relations between the primal and dual problems and their discretization commutes. The commuting diagram ensures that, given convergence results for the discrete primal variables, discrete dual variables also converge to a solution of the dual problem with a similar error bound. We demonstrate via a simple but illustrative example that significant computational savings can be achieved by solving the dual, rather than the primal, problem.

We analyze a class of linear-quadratic optimal control problems with an additional L-1-control cost depending on a parameter . To deal with this nonsmooth problem, we use an augmentation approach known from linear programming in which the number of control variables is doubled. It is shown that if the optimal control for a given *0 is bang-zero-bang and the switching function has a stable structure, the solutions are Lipschitz continuous functions of the parameter . We also show that in this case the optimal controls for (*) and a 0 with|-(*)|sufficiently small coincide except on a set of measure O(). Finally, we use the augmentation approach to derive error estimates for Euler discretizations. Copyright (c) 2014 John Wiley & Sons, Ltd.

We analyse an implicit discretization scheme for a class of linear-quadratic optimal control problems. First, we show convergence of order O(h) for the optimal values of the objective function, where h is the mesh size. Under the additional assumption that the optimal control has bang–bang structure we show that the discrete and the continuous controls coincide except on a set of measure O(√h).

We analyze L^2 -regularization of a class of linear-quadratic optimal control problems with an additional L^1 -control cost depending on a parameter β . To deal with this nonsmooth problem we use an augmentation approach known from linear programming in which the number of control variables is doubled. It is shown that if the optimal control for a given β ^*≥ 0 is bang-zero-bang, the solutions are continuous functions of the parameter β and the regularization parameter α . Moreover we derive error estimates for Euler discretization.

We analyze an implicit discretization scheme for a class of linear-quadratic optimal control problems without mixed state-control terms. Under the assumption that the optimal control has bang-bang structure we show convergence of the discrete approximation and improve existing error estimates to order 𝒪(h) .

We analyse the Euler discretization to a class of linear optimal control problems. First we show convergence of order h for the discrete approximation of the adjoint solution and the switching function, where h is the mesh size. Under the additional assumption that the optimal control has bang-bang structure we show that the discrete and the exact controls coincide except on a set of measure O(h). As a consequence, the discrete optimal control approximates the optimal control with order 1 w.r.t. the L 1-norm and with order 1/2 w.r.t. the L 2-norm. An essential assumption is that the slopes of the switching function at its zeros are bounded away from zero which is in fact an inverse stability condition for these zeros. We also discuss higher order approximation methods based on the approximation of the adjoint solution and the switching function. Several numerical examples underline the results.

We analyze the Euler discretization to a class of linear-quadratic optimal control problems. First we show convergence of order h for the optimal values of the objective function, where h is the mesh size. Under the additional assumption that the optimal control has bang-bang structure we show that the discrete and the continuous controls coincide except on a set of measure O(root h). Under a slightly stronger assumption on the smoothness of the coefficients of the system equation we obtain an error estimate of order O(root h).

We consider Euler discretizations to a class of linear-quadratic optimal control problems. Assuming that the discrete controls are of bang-bang type and define switching functions with a uniform structure independent of the discretization we show, that the discrete controls converge to a bang-bang control, which is a solution of the original problem. linear-quadratic optimal control, bang-bang solutions, Euler approximation, convergence.

A family of linear-quadratic optimal control problems with pointwise mixed state-control constraints governed by linear elliptic partial differential equations is considered. All data depend on a vector parameter of perturbations. Lipschitz stability with respect to perturbations of the optimal control, the state and adjoint variables, and the Lagrange multipliers is established.

In this paper, we analyze finite difference discretizations for a class of control constrained elliptic optimal control problems. If the optimal control has a derivative of bounded variation, we show discrete quadratic convergence in terms of the mesh size h of the discrete optimal controls. Furthermore, based on the optimality conditions, we construct a new discrete control for which we derive continuous error estimates of order h (2).

We consider linear-quadratic problems of optimal control with an elliptic state equation and control constraints. For a discretization of the state equation by the method of Finite Differences and a piecewise approximation of the control we develop error estimates for the solution of the discrete problem and, based on the optimality conditions, we construct a new feasible control for which we derive error estimates of quadratic order.

We investigate discretizations for a class of quadratic optimal control problems governed by one-dimensional elliptic differential equations. In contrast to the papers [3] dealing with finite element approximations and [2, 1] dealing with finite difference approximation, the dicretizations considered here are based on a collocation method using quadratic splines for the state equation. Under the assumption that the optimal control has bounded variation we prove discrete and continuous quadratic convergence of approximating controls.