Nonlinear damped partial differential equations and their uniform discretizations

We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time discretization parameters, by adding appropriate numerical viscosity terms. Our main arguments use the optimal-weight convexity method and uniform observability inequalities with respect to the discretization parameters. We establish our results, first in the continuous setting, then for space semi-discrete models, and then for time semi-discrete models. The full discretization is inferred from the previous results. Our results cover, for instance, the Schr\"odinger equation with nonlinear damping, the nonlinear wave equation, the nonlinear plate equation, as well as certain classes of equations with nonlocal terms.

[1]  E. Trélat,et al.  Controllability of partial differential equations , 2006 .

[2]  M. Tucsnak,et al.  Uniformly exponentially stable approximations for a class of second order evolution equations , 2007 .

[3]  Zhuangyi Liu,et al.  Semigroups Associated with Dissipative Systems , 1999 .

[4]  Louis Roder Tcheugoué Tébou,et al.  Uniform boundary stabilization of the finite difference space discretization of the 1−d wave equation , 2007, Adv. Comput. Math..

[5]  Localized solutions for the finite difference semi-discretization of the wave equation , 2010, 1008.0197.

[6]  Marius Tucsnak,et al.  Internal stabilization of the plate equation in a square: the continuous and the semi-discretized problems , 2006 .

[7]  Fatiha Alabau-Boussouira.  Convexity and Weighted Integral Inequalities for Energy Decay Rates of Nonlinear Dissipative Hyperbolic Systems , 2005 .

[8]  Sylvain Ervedoza,et al.  Spectral conditions for admissibility and observability of wave systems: applications to finite element schemes , 2009, Numerische Mathematik.

[9]  Fatiha Alabau-Boussouira.  Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation , 2006 .

[10]  Enrique Zuazua,et al.  Propagation, Observation, and Control of Waves Approximated by Finite Difference Methods , 2005, SIAM Rev..

[11]  H. Banks,et al.  Exponentially stable approximations of weakly damped wave equations , 1991 .

[12]  V. Komornik.  Exact Controllability and Stabilization: The Multiplier Method , 1995 .

[13]  R. Glowinski,et al.  On the numerical implementation of eht Hilbert uniqueness method for the exact boundary controllability of the wave equation , 1990 .

[14]  A. Haraux.  Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps , 1989 .

[15]  Serge Nicaise,et al.  UNIFORMLY EXPONENTIALLY OR POLYNOMIALLY STABLE APPROXIMATIONS FOR SECOND ORDER EVOLUTION EQUATIONS AND SOME APPLICATIONS , 2013 .

[16]  Arnaud Münch,et al.  UNIFORM STABILIZATION OF A VISCOUS NUMERICAL APPROXIMATION FOR A LOCALLY DAMPED WAVE EQUATION , 2007 .

[17]  E. Zuazua,et al.  Decay Rates for Dissipative Wave equations , 1999 .

[18]  Elena Panteley,et al.  Advanced Topics in Control Systems Theory , 2005 .

[19]  Patrick Martinez,et al.  A new method to obtain decay rate estimates for dissipative systems , 1999 .

[20]  C. Laurent,et al.  Stabilization for the semilinear wave equation with geometric control condition , 2012, 1205.2459.

[21]  E. Zuazua,et al.  Stability and decay for a class of nonlinear hyperbolic problems , 1988 .

[22]  Louis Roder Tcheugoué Tébou,et al.  Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity , 2003, Numerische Mathematik.

[23]  Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation , 2012, 1211.7202.

[24]  Enrique Zuazua,et al.  Exponential Decay for The Semilinear Wave Equation with Locally Distributed Damping , 1990 .

[25]  Emmanuel Trélat,et al.  Uniform controllability of semidiscrete approximations of parabolic control systems , 2006, Syst. Control. Lett..

[26]  R. Glowinski,et al.  A numerical approach to the exact boundary controllability of the wave equation (I) Dirichlet controls: Description of the numerical methods , 1990 .

[27]  E. Zuazua,et al.  Decay estimates for some semilinear damped hyperbolic problems , 1988 .

[28]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[29]  C. Bardos,et al.  Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary , 1992 .

[30]  Kaïs Ammari,et al.  Sharp energy estimates for nonlinearly locally damped PDEs via observability for the associated undamped system , 2011 .

[31]  R. Triggiani,et al.  Control Theory for Partial Differential Equations: Continuous and Approximation Theories , 2000 .

[32]  C. Dafermos.  Asymptotic stability in viscoelasticity , 1970 .

[33]  Luc Miller,et al.  Resolvent conditions for the control of unitary groups and their approximations , 2012 .

[34]  E. Zuazua,et al.  Stabilization and control for the subcritical semilinear wave equation , 2003 .

[35]  R. Nagel,et al.  One-parameter semigroups for linear evolution equations , 1999 .

[36]  E. Zuazua,et al.  Uniformly exponentially stable approximations for a class of damped systems , 2009 .

[37]  C. Giorgi,et al.  A New Approach to Equations with Memory , 2009, 0901.4063.

[38]  Patrick Martinez,et al.  Optimality of Energy Estimates for the Wave Equation with Nonlinear Boundary Velocity Feedbacks , 2000, SIAM J. Control. Optim..

[39]  Vilmos Komornik,et al.  Indirect internal stabilization of weakly coupled evolution equations , 2002 .

[40]  Enrique Zuazua,et al.  TIME DISCRETE WAVE EQUATIONS: BOUNDARY OBSERVABILITY AND CONTROL , 2008 .

[41]  G. Lebeau.  Contrôle de l'équation de Schrödinger , 1992 .

[42]  Fatiha Alabau-Boussouira,et al.  Indirect Boundary Stabilization of Weakly Coupled Hyperbolic Systems , 2002, SIAM J. Control. Optim..

[43]  F. Alabau-Boussouira,et al.  A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems , 2010 .

[44]  Irena Lasiecka,et al.  Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping , 1993, Differential and Integral Equations.

[45]  Fatiha Alabau-Boussouira,et al.  Partial Differential Equations/Optimal Control A general method for proving sharp energy decay rates for memory-dissipative evolution equations , 2009 .

[46]  Fatiha Alabau,et al.  Stabilisation frontière indirecte de systèmes faiblement couplés , 1999 .

[47]  Mitsuhiro Nakao,et al.  Decay of solutions of the wave equation with a local nonlinear dissipation , 1996 .

[48]  김정기,et al.  Propagation , 1994, Encyclopedia of Evolutionary Psychological Science.

[49]  P. Martinez.  A new method to obtain decay rate estimates for dissipative systems with localized damping , 1999 .

[50]  A. Haraux,et al.  An Introduction to Semilinear Evolution Equations , 1999 .