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Convergence analysis of smoothing methods for optimal control of stationary variational inequalities with control constraints

Anton Schiela, Daniel Wachsmuth (2013)

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

In the article an optimal control problem subject to a stationary variational inequality is investigated. The optimal control problem is complemented with pointwise control constraints. The convergence of a smoothing scheme is analyzed. There, the variational inequality is replaced by a semilinear elliptic equation. It is shown that solutions of the regularized optimal control problem converge to solutions of the original one. Passing to the limit in the optimality system of the regularized problem...

Convergence of minimax structures and continuation of critical points for singularly perturbed systems

Benedetta Noris, Hugo Tavares, Susanna Terracini, Gianmaria Verzini (2012)

Journal of the European Mathematical Society

In the recent literature, the phenomenon of phase separation for binary mixtures of Bose–Einstein condensates can be understood, from a mathematical point of view, as governed by the asymptotic limit of the stationary Gross–Pitaevskii system - Δ u + u 3 + β u v 2 = λ u , - Δ v + v 3 + β u 2 v = μ v , u , v H 0 1 ( Ω ) , u , v > 0 , as the interspecies scattering length β goes to + . For this system we consider the associated energy functionals J β , β ( 0 , + ) , with L 2 -mass constraints, which limit J (as β + ) is strongly irregular. For such functionals, we construct multiple critical points via a common...

Convergence of optimal solutions in control problems for hyperbolic equations

S. Migórski (1995)

Annales Polonici Mathematici

A sequence of optimal control problems for systems governed by linear hyperbolic equations with the nonhomogeneous Neumann boundary conditions is considered. The integral cost functionals and the differential operators in the equations depend on the parameter k. We deal with the limit behaviour, as k → ∞, of the sequence of optimal solutions using the notions of G- and Γ-convergences. The conditions under which this sequence converges to an optimal solution for the limit problem are given.

Convergence of primal-dual solutions for the nonconvex log-barrier method without LICQ

Christian Grossmann, Diethard Klatte, Bernd Kummer (2004)

Kybernetika

This paper characterizes completely the behavior of the logarithmic barrier method under a standard second order condition, strict (multivalued) complementarity and MFCQ at a local minimizer. We present direct proofs, based on certain key estimates and few well–known facts on linear and parametric programming, in order to verify existence and Lipschitzian convergence of local primal-dual solutions without applying additionally technical tools arising from Newton–techniques.

Convergence of the Lagrange-Newton method for optimal control problems

Kazimierz Malanowski (2004)

International Journal of Applied Mathematics and Computer Science

Convergence results for two Lagrange-Newton-type methods of solving optimal control problems are presented. It is shown how the methods can be applied to a class of optimal control problems for nonlinear ODEs, subject to mixed control-state constraints. The first method reduces to an SQP algorithm. It does not require any information on the structure of the optimal solution. The other one is the shooting method, where information on the structure of the optimal solution is exploited. In each case,...

Curvature functionals for curves in the equi-affine plane

Steven Verpoort (2011)

Czechoslovak Mathematical Journal

After having given the general variational formula for the functionals indicated in the title, the critical points of the integral of the equi-affine curvature under area constraint and the critical points of the full-affine arc-length are studied in greater detail. Notice. An extended version of this article is available on arXiv:0912.4075.

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