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Control of the wave equation by time-dependent coefficient

Antonin Chambolle, Fadil Santosa (2002)

ESAIM: Control, Optimisation and Calculus of Variations

We study an initial boundary-value problem for a wave equation with time-dependent sound speed. In the control problem, we wish to determine a sound-speed function which damps the vibration of the system. We consider the case where the sound speed can take on only two values, and propose a simple control law. We show that if the number of modes in the vibration is finite, and none of the eigenfrequencies are repeated, the proposed control law does lead to energy decay. We illustrate the rich behavior...

Control of the Wave Equation by Time-Dependent Coefficient

Antonin Chambolle, Fadil Santosa (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We study an initial boundary-value problem for a wave equation with time-dependent sound speed. In the control problem, we wish to determine a sound-speed function which damps the vibration of the system. We consider the case where the sound speed can take on only two values, and propose a simple control law. We show that if the number of modes in the vibration is finite, and none of the eigenfrequencies are repeated, the proposed control law does lead to energy decay. We illustrate the rich behavior of...

Control structure in optimization problems of bar systems

Leszek Mikulski (2004)

International Journal of Applied Mathematics and Computer Science

Optimal design problems in mechanics can be mathematically formulated as optimal control tasks. The minimum principle is employed in solving such problems. This principle allows us to write down optimal design problems as Multipoint Boundary Value Problems (MPBVPs). The dimension of MPBVPs is an essential restriction that decides on numerical difficulties. Optimal control theory does not give much information about the control structure, i.e., about the sequence of the forms of the right-hand sides...

Control variational method approach to bending and contact problems for Gao beam

Jitka Machalová, Horymír Netuka (2017)

Applications of Mathematics

This paper deals with a nonlinear beam model which was published by D. Y. Gao in 1996. It is considered either pure bending or a unilateral contact with elastic foundation, where the normal compliance condition is employed. Under additional assumptions on data, higher regularity of solution is proved. It enables us to transform the problem into a control variational problem. For basic types of boundary conditions, suitable transformations of the problem are derived. The control variational problem...

Convergence of dual finite element approximations for unilateral boundary value problems

Ivan Hlaváček (1980)

Aplikace matematiky

A semi-coercive problem with unilateral boundary conditions of the Signoriti type in a convex polygonal domain is solved on the basis of a dual variational approach. Whereas some strong regularity of the solution has been assumed in the previous author’s results on error estimates, no assumption of this kind is imposed here and still the L 2 -convergence is proved.

Convergence of mass redistribution method for the one-dimensional wave equation with a unilateral constraint at the boundary

Farshid Dabaghi, Adrien Petrov, Jérôme Pousin, Yves Renard (2014)

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

This paper focuses on a one-dimensional wave equation being subjected to a unilateral boundary condition. Under appropriate regularity assumptions on the initial data, a new proof of existence and uniqueness results is proposed. The mass redistribution method, which is based on a redistribution of the body mass such that there is no inertia at the contact node, is introduced and its convergence is proved. Finally, some numerical experiments are reported.

Convergenza per l'equazione degli integrali primi associata al problema del rimbalzo

Michele Carriero, Antonio Leaci, Eduardo Pascali (1982)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In this paper we present a few results on convergence for the prime integrals equations connected with the bounce problem. This approach allows both to prove uniqueness for the one-dimensional bounce problem for almost all permissible Cauchy data (see also [6]) and to deepen previous results (see [3], [5], [7]).

Crack in a solid under Coulomb friction law

Victor A. Kovtunenko (2000)

Applications of Mathematics

An equilibrium problem for a solid with a crack is considered. We assume that both the Coulomb friction law and a nonpenetration condition hold at the crack faces. The problem is formulated as a quasi-variational inequality. Existence of a solution is proved, and a complete system of boundary conditions fulfilled at the crack surface is obtained in suitable spaces.

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