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Continuous dependence for solution classes of Euler-Lagrange equations generated by linear growth energies

Ken Shirakawa (2009)

Banach Center Publications

In this paper, a one-dimensional Euler-Lagrange equation associated with the total variation energy, and Euler-Lagrange equations generated by approximating total variations with linear growth, are considered. Each of the problems presented can be regarded as a governing equation for steady-states in solid-liquid phase transitions. On the basis of precise structural analysis for the solutions, the continuous dependence between the solution classes of approximating problems and that of the limiting...

Continuous transformation groups on spaces

K. Spallek (1991)

Annales Polonici Mathematici

A differentiable group is a group in the category of (reduced and nonreduced) differentiable spaces. Special cases are the rationals ℚ, Lie groups, formal groups over ℝ or ℂ; in general there is some mixture of those types, the general structure, however, is not yet completely determined. The following gives as a corollary a first essential answer. It is shown, more generally,that a locally compact topological transformation group, operating effectively on a differentiable space X (which satisfies...

Control for Schrödinger operators on 2-tori: rough potentials

Jean Bourgain, Nicolas Burq, Maciej Zworski (2013)

Journal of the European Mathematical Society

For the Schrödinger equation, ( i ∂ t + ∇ ) u = 0 on a torus, an arbitrary non-empty open set Ω provides control and observability of the solution: u t = 0 L 2 ( 𝕋 2 ) ≤ K T u L 2 ( [ 0 , T ] × Ω ) . We show that the same result remains true for ( i ∂ t + ∇ - V ) u = 0 where V ∈ L 2 ( 𝕋 2 ) , and 𝕋 2 is a (rational or irrational) torus. That extends the results of [1], and [8] where the observability was proved for V ∈ C ( 𝕋 2 ) and conjectured for V ∈ L ∞ ( 𝕋 2 ) . The higher dimensional generalization remains open for V ∈ L ∞ ( 𝕋 n ) .

Control systems on semi-simple Lie groups and their homogeneous spaces

Velimir Jurdjevic, Ivan Kupka (1981)

Annales de l'institut Fourier

In the present paper, we consider the class of control systems which are induced by the action of a semi-simple Lie group on a manifold, and we give a sufficient condition which insures that such a system can be steered from any initial state to any final state by an admissible control. The class of systems considered contains, in particular, essentially all the bilinear systems. Our condition is semi-algebraic but unlike the celebrated Kalman criterion for linear systems, it is not necessary. In...

Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria

Felipe Alvarez, Hedy Attouch (2001)

ESAIM: Control, Optimisation and Calculus of Variations

It is established convergence to a particular equilibrium for weak solutions of abstract linear equations of the second order in time associated with monotone operators with nontrivial kernel. Concerning nonlinear hyperbolic equations with monotone and conservative potentials, it is proved a general asymptotic convergence result in terms of weak and strong topologies of appropriate Hilbert spaces. It is also considered the stabilization of a particular equilibrium via the introduction of an asymptotically...

Convergence and asymptotic stabilization for some damped hyperbolic equations with non-isolated equilibria

Felipe Alvarez, Hedy Attouch (2010)

ESAIM: Control, Optimisation and Calculus of Variations

It is established convergence to a particular equilibrium for weak solutions of abstract linear equations of the second order in time associated with monotone operators with nontrivial kernel. Concerning nonlinear hyperbolic equations with monotone and conservative potentials, it is proved a general asymptotic convergence result in terms of weak and strong topologies of appropriate Hilbert spaces. It is also considered the stabilization of a particular equilibrium via the introduction of an asymptotically...

Currently displaying 281 – 300 of 386