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The weak solution of an antiplane contact problem for electro-viscoelastic materials with long-term memory

Ammar Derbazi, Mohamed Dalah, Amar Megrous (2016)

Applications of Mathematics

We study a mathematical model which describes the antiplane shear deformation of a cylinder in frictionless contact with a rigid foundation. The material is assumed to be electro-viscoelastic with long-term memory, and the friction is modeled with Tresca's law and the foundation is assumed to be electrically conductive. First we derive the classical variational formulation of the model which is given by a system coupling an evolutionary variational equality for the displacement field with a time-dependent...

Théorème de minimax sans topologie ni convexité

Andrzej Granas, Jin-Rong Lee, Fon-Che Liu (1992)

Colloquium Mathematicae

Dans cete note, nous présentons un théorème de minimax (Théorème A) formulé seulement en langage de la théorie des ensembles. Ce résultat permet de déduire de façon immédiate (en utilisant un lemme de topologie générale) plusieurs théorèmes de minimax bien connus.

Tilt stability in nonlinear programming under Mangasarian-Fromovitz constraint qualification

Boris S. Mordukhovich, Jiří V. Outrata (2013)

Kybernetika

The paper concerns the study of tilt stability of local minimizers in standard problems of nonlinear programming. This notion plays an important role in both theoretical and numerical aspects of optimization and has drawn a lot of attention in optimization theory and its applications, especially in recent years. Under the classical Mangasarian-Fromovitz Constraint Qualification, we establish relationships between tilt stability and some other stability notions in constrained optimization. Involving...

Time optimal control of the heat equation with pointwise control constraints

Karl Kunisch, Lijuan Wang (2013)

ESAIM: Control, Optimisation and Calculus of Variations

Time optimal control problems for an internally controlled heat equation with pointwise control constraints are studied. By Pontryagin’s maximum principle and properties of nontrivial solutions of the heat equation, we derive a bang-bang property for time optimal control. Using the bang-bang property and establishing certain connections between time and norm optimal control problems for the heat equation, necessary and sufficient conditions for the optimal time and the optimal control are obtained....

Time-optimal boundary control of a parabolic system with time lags given in integral form

Adam Kowalewski, Anna Krakowiak (2006)

International Journal of Applied Mathematics and Computer Science

In this paper, the time-optimal boundary control problem for a distributed parabolic system in which time lags appear in integral form in both the state equation and the boundary condition is presented. Some particular properties of optimal control are discussed.

Time-optimal boundary control of an infinite order parabolic system with time lags

Adam Kowalewski, Anna Krakowiak (2008)

International Journal of Applied Mathematics and Computer Science

In this paper the time-optimal boundary control problem is presented for a distributed infinite order parabolic system in which time lags appear in the integral form both in the state equation and in the boundary condition. Some specific properties of the optimal control are discussed.

Time-optimal control of infinite order hyperbolic systems with time delays

Adam Kowalewski (2009)

International Journal of Applied Mathematics and Computer Science

In this paper, the time-optimal control problem for infinite order hyperbolic systems in which time delays appear in the integral form both in state equations and in boundary conditions is considered. Optimal controls are characterized in terms of an adjoint system and shown to be unique and bang-bang. These results extend to certain cases of nonlinear control problems. The particular properties of optimal control are discussed.

Topological degree, Jacobian determinants and relaxation

Irene Fonseca, Nicola Fusco, Paolo Marcellini (2005)

Bollettino dell'Unione Matematica Italiana

A characterization of the total variation T ⁢ V ⁢ u , Ω of the Jacobian determinant det ⁡ D ⁢ u is obtained for some classes of functions u : Ω → R n outside the traditional regularity space W 1 , n ⁢ Ω ; R n . In particular, explicit formulas are deduced for functions that are locally Lipschitz continuous away from a given one point singularity x 0 ∈ Ω . Relations between T ⁢ V ⁢ u , Ω and the distributional determinant Det ⁢ D ⁢ u are established, and an integral representation is obtained for the relaxed energy of certain polyconvex functionals at maps u ∈ W 1 , p ⁢ Ω ; R n ∩ W 1 , ∞ ⁢ Ω x 0 ; R n .

Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices

Carl-Friedrich Kreiner, Johannes Zimmer (2006)

ESAIM: Control, Optimisation and Calculus of Variations

Continuing earlier work by Székelyhidi, we describe the topological and geometric structure of so-called T4-configurations which are the most prominent examples of nontrivial rank-one convex hulls. It turns out that the structure of T4-configurations in ℝ 2 × 2 is very rich; in particular, their collection is open as a subset of ( ℝ 2 × 2 ) 4 . Moreover a previously purely algebraic criterion is given a geometric interpretation. As a consequence, we sketch an improved algorithm to detect T4-configurations. ...

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