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Convergence of optimal solutions in control problems for hyperbolic equations

100%
Migórski S.
Annales Polonici Mathematici
|
1995
|
tom 62
|
nr 2
111-121
EN
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.
2
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Discrete thickness

75%
Scholtes S.
Molecular Based Mathematical Biology
|
2014
|
tom 2
|
nr 1
EN
We investigate the relationship between a discrete version of thickness and its smooth counterpart. These discrete energies are deffned on equilateral polygons with n vertices. It will turn out that the smooth ropelength, which is the scale invariant quotient of length divided by thickness, is the Γ-limit of the discrete ropelength for n → ∞, regarding the topology induced by the Sobolev norm ‖ · ‖ W1,∞(S1,ℝd). This result directly implies the convergence of almost minimizers of the discrete energies in a fixed knot class to minimizers of the smooth energy.Moreover,we show that the unique absolute minimizer of inverse discrete thickness is the regular n-gon.
3
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Effective energy integral functionals for thin films with three dimensional bending moment in the Orlicz-Sobolev space setting

51%
Laskowski W., Nguyen H.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2016
|
tom 36
|
nr 1
7-31
EN
In this paper we consider an elastic thin film ω ⊂ ℝ² with the bending moment depending also on the third thickness variable. The effective energy functional defined on the Orlicz-Sobolev space over ω is described by Γ-convergence and 3D-2D dimension reduction techniques. Then we prove the existence of minimizers of the film energy functional. These results are proved in the case when the energy density function has the growth prescribed by an Orlicz convex function M. Here M is assumed to be non-power-growth-type and to satisfy the conditions Δ₂ and ∇₂.
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