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A minimum effort optimal control problem for elliptic PDEs

Christian Clason, Kazufumi Ito, Karl Kunisch (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This work is concerned with a class of minimum effort problems for partial differential equations, where the control cost is of L∞-type. Since this problem is non-differentiable, a regularized functional is introduced that can be minimized by a superlinearly convergent semi-smooth Newton method. Uniqueness and convergence for the solutions to the regularized problem are addressed, and a continuation strategy based on a model function is proposed. Numerical examples for a convection-diffusion equation...

A minimum effort optimal control problem for elliptic PDEs

Christian Clason, Kazufumi Ito, Karl Kunisch (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

This work is concerned with a class of minimum effort problems for partial differential equations, where the control cost is of L∞-type. Since this problem is non-differentiable, a regularized functional is introduced that can be minimized by a superlinearly convergent semi-smooth Newton method. Uniqueness and convergence for the solutions to the regularized problem are addressed, and a continuation strategy based on a model function is proposed. Numerical examples for a convection-diffusion equation...

A new series of conjectures and open questions in optimization and matrix analysis

Jean-Baptiste Hiriart-Urruty (2009)

ESAIM: Control, Optimisation and Calculus of Variations

We present below a new series of conjectures and open problems in the fields of (global) Optimization and Matrix analysis, in the same spirit as our recently published paper [J.-B. Hiriart-Urruty, Potpourri of conjectures and open questions in Nonlinear analysis and Optimization. SIAM Review 49 (2007) 255–273]. With each problem come a succinct presentation, a list of specific references, and a view on the state of the art of the subject.

A new series of conjectures and open questions in optimization and matrix analysis

Jean-Baptiste Hiriart-Urruty (2008)

ESAIM: Control, Optimisation and Calculus of Variations

We present below a new series of conjectures and open problems in the fields of (global) Optimization and Matrix analysis, in the same spirit as our recently published paper [J.-B. Hiriart-Urruty, Potpourri of conjectures and open questions in Nonlinear analysis and Optimization. SIAM Review49 (2007) 255–273]. With each problem come a succinct presentation, a list of specific references, and a view on the state of the art of the subject.

A note on a class of equilibrium problems with equilibrium constraints

Jiří V. Outrata (2004)

Kybernetika

The paper concerns a two-level hierarchical game, where the players on each level behave noncooperatively. In this way one can model eg an oligopolistic market with several large and several small firms. We derive two types of necessary conditions for a solution of this game and discuss briefly the possibilities of its computation.

A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems

Mark Kärcher, Martin A. Grepl (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider the efficient and reliable solution of linear-quadratic optimal control problems governed by parametrized parabolic partial differential equations. To this end, we employ the reduced basis method as a low-dimensional surrogate model to solve the optimal control problem and develop a posteriori error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. We show that our approach can be applied to problems involving...

A posteriori error estimation for semilinear parabolic optimal control problems with application to model reduction by POD

Eileen Kammann, Fredi Tröltzsch, Stefan Volkwein (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider the following problem of error estimation for the optimal control of nonlinear parabolic partial differential equations: let an arbitrary admissible control function be given. How far is it from the next locally optimal control? Under natural assumptions including a second-order sufficient optimality condition for the (unknown) locally optimal control, we estimate the distance between the two controls. To do this, we need some information on the lowest eigenvalue of the reduced Hessian....

A simple trolley-like model in the presence of a nonlinear friction and a bounded fuel expenditure

Andrei Dmitruk, Ivan Samylovskiy (2013)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We consider a problem of maximization of the distance traveled by a material point in the presence of a nonlinear friction under a bounded thrust and fuel expenditure. Using the maximum principle we obtain the form of optimal control and establish conditions under which it contains a singular subarc. This problem seems to be the simplest one having a mechanical sense in which singular subarcs appear in a nontrivial way.

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