Nonlinear stabilization of abstract systems via a linear observability inequality and application to vibrating PDE's

This Note is concerned with the links between nonlinear stabilization of hyperbolic systems and linear observability for the unforced corresponding linear system, for locally distributed and boundary feedbacks as well. We show that if the linear system is observable through a locally distributed (resp. boundary) observation, then any dissipative nonlinear feedback locally distributed (resp. active only on a part of the boundary) stabilize the system and we give a general energy decay formula. Our results generalize previous results by Haraux (1989) and Ammari and Tucsnak (2001) for linear feedbacks. We show by this way that for the locally distributed case, one can combine the optimal geometric conditions of Bardos et al. (1992) and the method of Alabau-Boussouira (2005) to deduce energy decay rates for nonlinear damped systems.     

Stabilization of the nonlinear damped wave equation via linear weak observability

We consider the problem of energy decay rates for nonlinearly damped abstract infinite dimensional systems. We prove sharp, simple and quasi-optimal energy decay rates through an indirect method, namely a weak observability estimate for the corresponding undamped system. One of the main advantage of these results is that they allow to combine the optimal-weight convexity method of Alabau-Boussouira (Appl Math Optim 51:61–105, 2005) and a methodology of Ammari and Tucsnak (ESAIM COCV 6:361–386, 2001) for weak stabilization by observability. Our results extend to nonlinearly damped systems, those of Ammari and Tucsnak (ESAIM COCV 6:361–386, 2001). At the end, we give an appendix on the weak stabilization of linear evolution systems.     

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

The Liapunov method is celebrated for its strength to establish strong decay of solutions of damped equations. Extensions to infinite dimensional settings have been studied by several authors (see e.g. Haraux, 1991 [11], and Komornik and Zuazua, 1990 [17] and references therein). Results on optimal energy decay rates under general conditions of the feedback is far from being complete. The purpose of this paper is to show that general dissipative vibrating systems have structural properties due to dissipation. We present a general approach based on convexity arguments to establish sharp optimal or quasi-optimal upper energy decay rates for these systems, and on comparison principles based on the dissipation property, and interpolation inequalities (in the infinite dimensional case) for lower bounds of the energy. We stress the fact that this method works for finite as well as infinite dimensional vibrating systems and as well as for applications to semi-discretized nonlinear damped vibrating PDE's. A part of this approach has been introduced in Alabau-Boussouira (2004, 2005) [1] and [2]. In the present paper, we identify a new, simple and explicit criteria to select a class of nonlinear feedbacks, for which we prove a simplified explicit energy decay formula comparatively to the more general but also more complex formula we give in Alabau-Boussouira (2004, 2005) [1] and [2]. Moreover, we prove optimality of the decay rates for this class, in the finite dimensional case. This class includes a wide range of feedbacks, ranging from very weak nonlinear dissipation (exponentially decaying in a neighborhood of zero), to polynomial, or polynomial-logarithmic decaying feedbacks at the origin. In the infinite dimensional case, we establish a comparison principle on the energy of sufficiently smooth solutions through the dissipation relation. This principle relies on suitable interpolation inequalities. It allows us to give lower bounds for the energy of smooth initial data for the one-dimensional wave equation with a distributed polynomial damping, which improves Haraux (1995) [12] lower estimate of the energy for this case. We also establish lower bounds in the multi-dimensional case for sufficiently smooth solutions when such solutions exist. We further mention applications of these various results to several classes of PDE's, namely: the locally and boundary damped multi-dimensional wave equation, the locally damped plate equation and the globally damped coupled Timoshenko beams system but it applies to several other examples. Furthermore, we show that these optimal energy decay results apply to finite dimensional systems obtained from spatial discretization of infinite dimensional damped systems. We illustrate these results on the one-dimensional locally damped wave and plate equations discretized by finite differences and give the optimal energy decay rates for these two examples. These optimal rates are not uniform with respect to the discretization parameter. We also discuss and explain why optimality results have to be stated differently for feedbacks close to linear behavior at the origin.     

Strong lower energy estimates for nonlinearly damped Timoshenko beams and Petrowsky equations

The purpose of this paper is to establish strong lower energy estimates for strong solutions of nonlinearly damped Timoshenko beams, Petrowsky equations in two and three dimensions and wave-like equations for bounded one-dimensional domains or annulus domains in two or three dimensions. We also establish weak lower velocity estimates for strong solutions of the nonlinearly damped Petrowsky equation in two and three dimensions. The feedbacks in consideration have arbitrary growth close to the origin. These results improve the strong lower energy decay rates obtained in our previous papers (Alabau-Boussouira in J Differ Equ 249:1145–1178, 2010; J Differ Equ 248:1473–1517, 2010) for strong solutions of the nonlinearly locally damped wave equation and extend to systems and to Petrowsky equation the method of Alabau-Boussouira (J Differ Equ 249:1145–1178, 2010; J Differ Equ 248:1473–1517, 2010). These results are the first ones for Timoshenko beams and Petrowsky equations.     

Global existence and asymptotic behavior for a coupled hyperbolic system with localized damping

In this paper we investigate the global existence and energy decay rate for the solution of a coupled hyperbolic system. The semi-explicit energy decay rate is established by using piecewise multiplier techniques and weighted integral inequality. We extend the energy decay result in Alabau-Boussouira [F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim. 51 (2005) 61-105] for a single equation to the coupled hyperbolic system.     

A general method for proving sharp energy decay rates for memory-dissipative evolution equations

This Note is concerned with stabilization of hyperbolic systems by a distributed memory feedback. We present here a general method which gives energy decay rates in terms of the asymptotic behavior of the kernel at infinity. This method, which allows us to recover in a natural way the known cases (exponential, polynomial, …), applies to a large quasi-optimal class of kernels. It also provides sharp energy decay rates compared to the ones that are available in the literature. We give a general condition under which the energy of solutions is shown to decay at least as fast as the kernel at infinity. To cite this article: F. Alabau-Boussouira, P. Cannarsa, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Stabilisation faible interne locale de système élastique de Bresse

In [1], Alabau-Boussouira et al. (2011) studied the exponential and polynomial stability of the Bresse system with one globally distributed dissipation law. In this Note, our goal is to extend the results from Alabau-Boussouira et al. (2011) [1], by taking into consideration the important case when the dissipation law is locally distributed and to improve the polynomial energy decay rate. We then study the energy decay rate of the Bresse system with one locally internal distributed dissipation law acting on the equation about the shear angle displacement. Under the equal speed wave propagation condition, we show that the system is exponentially stable. On the contrary, we establish a new polynomial energy decay rate.     

General decay of solutions to a viscoelastic wave equation with linear damping,nonlinear damping and source term

ABSTRACT

This paper is concerned with the decay property of a nonlinear viscoelastic wave equation with linear damping, nonlinear damping and source term. Under weaker assumption on the relaxation function, we establish a general decay result, which extends the result obtained in Messaoudi [Exponential decay of solutions of a nonlinearly damped wave equation. Nodea-Nonlinear Differ Equat Appl. 2005;12:391–399].     

Affine dual frames and Extension Principles

In this work we provide three new characterizations of affine dual frames constructed from refinable functions. The first one is similar to Daubechies et al. (2003) [10, Proposition 5.2] but without any decay assumptions on the generators of a pair of affine systems. The second one reveals the geometric significance of the Mixed Fundamental function and the third one shows that the Mixed Oblique Extension Principle actually characterizes dual framelets. We also extend recent results on the characterization of affine Parseval frames obtained in Stavropoulos (2012) [27, Theorem 2.3].     

Actions modérées de schémas en groupes affines et champs modérés

We compare two notions of tameness: one introduced in Chinburg et al. (1996) [4] for actions of affine group schemes and one introduced in Abramovich et al. (2008) [1] for stacks. From this comparison, we deduce results on the structure of inertia groups in these tame situations.     

On stabilization and control for the critical Klein-Gordon equation on a 3-D compact manifold

In this article, we study the internal stabilization and control of the critical nonlinear Klein-Gordon equation on 3-D compact manifolds. Under a geometric assumption slightly stronger than the classical geometric control condition, we prove exponential decay for some solutions bounded in the energy space but small in a lower norm. The proof combines profile decomposition and microlocal arguments. This profile decomposition, analogous to the one of Bahouri and Gérard (1999) [2] on R³, is performed by taking care of possible geometric effects. It uses some results of S. Ibrahim (2004) [21] on the behavior of concentrating waves on manifolds.     

Exponential and polynomial stability of a wave equation for boundary memory damping with singular kernels

This work is concerned with stabilization of a wave equation stabilized by a boundary feedback. When the feedback is both frictional and with memory, we prove exponential stability of the solutions. In case of a boundary feedback which is only of memory type, uniform stability is not expected. We prove in this latter case, that the solutions decay polynomially. The method is new and uses the method of higher order energies (see [F. Alabau-Boussouira, J. Prüss, R. Zacher, Exponential and polynomial stabilization of wave equations subjected to boundary-memory dissipation with singular kernels, in preparation; F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 1015–1020; F. Alabau, P. Cannarsa, V. Komornik, Indirect internal damping of coupled systems, J. Evolution Equations 2 (2002) 127–150; F. Alabau, Indirect boundary stabilization of weakly coupled systems, SIAM J. Control Optim. 41 (2002) 511–541]), the multiplier method and the properties of a large class of singular kernels. Moreover, our method can be extended to include cases of nonsingular kernels (see [V. Vergara, R. Zacher, Lyapunov functions and convergence to steady state for differential equations of fractional order, Math. Z. 259 (2008) 287–309; R. Zacher, Convergence to equilibrium for second order differential equations with weak damping of memory type, preprint.]). To cite this article: F. Alabau-Boussouira et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Precise decay rate estimates for time-dependent dissipative systems

We consider the wave equation damped with a nonlinear time-dependent distributed dissipation. By generalizing a method recently introduced to study autonomous systems, we show that the energy of the system decays to zero with an explicit and precise decay rate estimate under sharp assumptions on the feedback. Then we prove that our estimates are optimal for the problem of the one dimensional wave equation damped by a nonlinear time-dependent boundary feedback. This extends and improves several earlier results of E. Zuazua and M. Nakao, and completes strong stability results of P. Pucci and J. Serrin.     

A general formula for decay rates of nonlinear dissipative systems

This work is concerned with stabilization of hyperbolic systems by a nonlinear feedback which can be localized on part of the boundary or locally distributed. We present here a general formula which gives the energy decay rates in terms of the behavior of the nonlinear feedback close to the origin. This formula allows us to unify for instance the cases where the feedback has a polynomial growth at the origin, with the cases where it goes exponentially fast to zero at the origin. We give also two other significant examples of nonpolynomial growth at the origin. We also show that we either obtain or improve significantly the decay rates of Lasiecka and Tataru (Differential Integral Equations 8 (1993) 507–533) and Martinez (Rev. Mat. Comput. 12 (1999) 251–283). To cite this article: F. Alabau-Boussouira, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Strong stability and uniform decay of solutions to a wave equation with semilinear porous acoustic boundary conditions

We consider a wave equation with semilinear porous acoustic boundary conditions. This is a coupled system of second and first order in time partial differential equations, with possibly semilinear boundary conditions on the interface. The results obtained are (i) strong stability for the linear model, (ii) exponential decay rates for the energy of the linear model, and (iii) local exponential decay rates for the energy of the semilinear model. This work builds on a previous result showing generation of a well-posed dynamical system. The main tools used in the proofs are (i) the Stability Theorem of Arendt-Batty, (ii) energy methods used in the study of a wave equation with boundary damping, and (iii) an abstract result of I. Lasiecka applicable to hyperbolic-like systems with nonlinearly perturbed boundary conditions.     

Decay rates at low and high frequencies for a plate equation with feedback concentrated in interior curves

In this paper we study the stabilization of plate vibrations by means of piezoelectric actuators. In this situation the geometric control condition of Bardos, Lebeau and Rauch [6] is not satisfied. We prove that we have exponential stability for the low frequencies but not for the high frequencies. We give an explicit decay rate for regular initial data at high frequencies while clarifying the behavior of the constant which intervenes in this estimation there function of the frequency of cut n. The method used is based on some trace regularity which reduces stability to some observability inequalities for the corresponding undamped problem. Moreover, we show numerically at low frequencies, that the optimal location of the actuator is the center of the domain Ω.     

On the Hughes' model for pedestrian flow: The one-dimensional case

In this paper we investigate the mathematical theory of Hughes' model for the flow of pedestrians (cf. Hughes (2002) [17]), consisting of a non-linear conservation law for the density of pedestrians coupled with an eikonal equation for a potential modelling the common sense of the task. For such an approximated system we prove existence and uniqueness of entropy solutions (in one space dimension) in the sense of Kru?kov (1970) [22], in which the boundary conditions are posed following the approach of Bardos et al. (1979) [7]. We use BV estimates on the density ρ and stability estimates on the potential ? in order to prove uniqueness. Furthermore, we analyze the evolution of characteristics for the original Hughes' model in one space dimension and study the behavior of simple solutions, in order to reproduce interesting phenomena related to the formation of shocks and rarefaction waves. The characteristic calculus is supported by numerical simulations.     

外区域上半线性波动方程解的整体存在性

该文研究带耗散项的线性和半线性波动方程外问题. 首先利用一个Sobolev型不等式得到了线性耗散波动方程在外区域上的整体能量衰减估计, 此结果用来证明非线性项为|u|^p (2+) 的半线性波动方程解的整体存在性. 为此, 该文主要研究N维(3≤ N≤7)外区域上球对称解的情形.     

Decay rates at low and high frequencies for a plate equation with feedback concentrated in interior curves

In this paper we study the stabilization of plate vibrations by means of piezoelectric actuators. In this situation the geometric control condition of Bardos, Lebeau and Rauch [6] is not satisfied. We prove that we have exponential stability for the low frequencies but not for the high frequencies. We give an explicit decay rate for regular initial data at high frequencies while clarifying the behavior of the constant which intervenes in this estimation there function of the frequency of cut n. The method used is based on some trace regularity which reduces stability to some observability inequalities for the corresponding undamped problem. Moreover, we show numerically at low frequencies, that the optimal location of the actuator is the center of the domain Ω. Research supported by the RIP program of Oberwolfach Institut and by the Tunisian Ministry for Scientific Research and Technology (MRST) under Grant 02/UR/15-01. Research supported by the RIP program of Oberwolfach Institut. (Received: September 17, 2003; revised: February 26, 2004)     

Boundary stabilization of a 3-dimensional structural acoustic model

The main result of this paper provides uniform decay rates obtained for the energy function associated with a three-dimensional structural acoustic model described by coupled system consisting of the wave equation and plate equation with the coupling on the interface between the acoustic chamber and the wall. The uniform stabilization is achieved by introducing a nonlinear dissipation acting via boundary forces applied at the edge of the plate and viscous or boundary damping applied to the wave equation. The results obtained in this paper extend, to the non-analytic, hyperbolic-like setting, the results obtained previously in the literature for acoustic problems modeled by structurally damped plates (governed by analytic semigroups). As a bypass product, we also obtain optimal uniform decay rates for the Euler Bernoulli plate equations with nonlinear boundary dissipation acting via shear forces only and without (i) any geometric conditions imposed on the domain ,(ii) any growth conditions at the origin imposed on the nonlinear function. This is in contrast with the results obtained previously in the literature ([22] and references therein).