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L∞-Norm minimal control of the wave equation: on the weakness of the bang-bang principle

Martin Gugat, Gunter Leugering (2008)

ESAIM: Control, Optimisation and Calculus of Variations


For optimal control problems with ordinary differential equations where the L ∞ -norm of the control is minimized, often bang-bang principles hold. For systems that are governed by a hyperbolic partial differential equation, the situation is different: even if a weak form of the bang-bang principle still holds for the wave equation, it implies no restriction on the form of the optimal control. To illustrate that for the Dirichlet boundary control of the wave equation in general not even feasible...

Local energy decay for several evolution equations on asymptotically euclidean manifolds

Jean-François Bony, Dietrich Häfner (2012)

Annales scientifiques de l'École Normale Supérieure

Let  P be a long range metric perturbation of the Euclidean Laplacian on  ℝ d , d ≥ 2 . We prove local energy decay for the solutions of the wave, Klein-Gordon and Schrödinger equations associated to  P . The problem is decomposed in a low and high frequency analysis. For the high energy part, we assume a non trapping condition. For low (resp. high) frequencies we obtain a general result about the local energy decay for the group e i t f ( P ) where f has a suitable development at zero (resp. infinity).

Local Energy Decay in Even Dimensions for the Wave Equation with a Time-Periodic Non-Trapping Metric and Applications to Strichartz Estimates

Kian, Yavar (2010)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 35B40, 35L15.We obtain local energy decay as well as global Strichartz estimates for the solutions u of the wave equation ∂t2 u-divx(a(t,x)∇xu) = 0, t ∈ R, x ∈ Rn, with time-periodic non-trapping metric a(t,x) equal to 1 outside a compact set with respect to x. We suppose that the cut-off resolvent Rχ(θ) = χ(U(T, 0)− e−iθ)−1χ, where U(T, 0) is the monodromy operator and T the period of a(t,x), admits an holomorphic continuation to {θ ∈ C : Im(θ) ≥ 0}, for...

Local existence and estimations for a semilinear wave equation in two dimension space

Amel Atallah Baraket (2004)

Bollettino dell'Unione Matematica Italiana

In this paper we prove a local existence theorem for a Cauchy problem associated to a semi linear wave equation with an exponential nonlinearity in two dimension space. In this problem, the first Cauchy data is equal to zero, the second is in L 2 ⁢ R 2 , radially symmetric and compactly supported. To prove this theorem, we first show a Moser-Trudinger type inequality for the linear problem and then we use a fixed point method to achieve the proof of the result.

Local existence of solutions of the mixed problem for the system of equations of ideal relativistic hydrodynamics

Joanna Rencławowicz, Wojciech Zajączkowski (1998)

Applicationes Mathematicae

Existence and uniqueness of local solutions for the initial-boundary value problem for the equations of an ideal relativistic fluid are proved. Both barotropic and nonbarotropic motions are considered. Existence for the linearized problem is shown by transforming the equations to a symmetric system and showing the existence of weak solutions; next, the appropriate regularity is obtained by applying Friedrich's mollifiers technique. Finally, existence for the nonlinear problem is proved by the method...

Logarithmic stabilization of the Kirchhoff plate transmission system with locally distributed Kelvin-Voigt damping

Gimyong Hong, Hakho Hong (2022)

Applications of Mathematics

We are concerned with a transmission problem for the Kirchhoff plate equation where one small part of the domain is made of a viscoelastic material with the Kelvin-Voigt constitutive relation. We obtain the logarithmic stabilization result (explicit energy decay rate), as well as the wellposedness, for the transmission system. The method is based on a new Carleman estimate to obtain information on the resolvent for high frequency. The main ingredient of the proof is some careful analysis for the...

Long time behaviour of a Cahn-Hilliard system coupled with viscoelasticity

Irena Pawłow, Wojciech M. Zajączkowski (2010)

Annales Polonici Mathematici

The long-time behaviour of a unique regular solution to the Cahn-Hilliard system coupled with viscoelasticity is studied. The system arises as a model of the phase separation process in a binary deformable alloy. It is proved that for a sufficiently regular initial data the trajectory of the solution converges to the ω-limit set of these data. Moreover, it is shown that every element of the ω-limit set is a solution of the corresponding stationary problem.

Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations

Albert J. Milani, Hans Volkmer (2011)

Applications of Mathematics

We give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation u t t + 2 u t - a i j ( u t , ∇ u ) ∂ i ∂ j u = f corresponding to initial values and source terms of sufficiently small size, as well as of small solutions to the corresponding stationary version, i.e. the quasilinear elliptic equation - a i j ( 0 , ∇ v ) ∂ i ∂ j v = h . We then give conditions for the convergence, as t → ∞ , of the solution of the evolution equation to its stationary state.

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