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Stability analysis of a 1D wave equation with a nonmonotone distributed damping
11th IFAC Symposium on Nonlinear Control Systems 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Available online at www.sciencedirect.com 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 4-6, 2019 Vienna, Austria, Sept. 4-6, 2019
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IFAC PapersOnLine 52-16 (2019) 36–41
Stability analysis of a 1D wave equation Stability analysis of a 1D wave equation Stability analysis of a 1D wave equation with a nonmonotone damping Stability analysis of distributed a 1D wave equation with a nonmonotone distributed with a nonmonotone distributed damping damping with a nonmonotone distributed damping ∗ ∗∗ Swann Marx ∗ , Yacine Chitour ∗∗ , Christophe Prieur ∗∗∗ ∗∗∗
Swann Marx ∗ , Yacine Chitour ∗∗ , Christophe Prieur ∗∗∗ Swann Marx ∗ , Yacine Chitour ∗∗ , Christophe Prieur ∗∗∗ ∗ Swann Marx , Yacine Chitour , Christophe Prieur ee de ∗ LAAS-CNRS, Universit´ de Toulouse, Toulouse, CNRS, CNRS, 7 7 avenue avenue du du colonel colonel ∗ LAAS-CNRS, Universit´ LAAS-CNRS, Universit´ e de Toulouse, CNRS, 7 avenue du.. colonel Roche, 31400, Toulouse, France [email protected] Roche, 31400, Toulouse, France [email protected] ∗∗∗ LAAS-CNRS, Universit´ eetde Toulouse, CNRS, 7 avenue du. colonel Roche,des 31400, Toulouse, [email protected] Signaux mes CNRS ∗∗ Laboratoire des Signaux et Syst` Syst`eeFrance mes (L2S), (L2S), CNRS -- CentraleSupelec CentraleSupelec ∗∗ Laboratoire Roche, 31400, Toulouse, France [email protected] . Laboratoire des Signaux et Syst` e mes (L2S), CNRS -Gif-sur-Yvette, CentraleSupelec Universit´ e Paris-Sud, 3, rue Joliot Curie, 91192, ∗∗ - Universit´ e Paris-Sud, 3, rue Joliot Curie, 91192, Gif-sur-Yvette, Laboratoire des Signaux et Syst` e mes (L2S), CNRS CentraleSupelec - Universit´e Paris-Sud, 3, rue Joliot Curie, 91192, .Gif-sur-Yvette, France, France, [email protected] [email protected] ∗∗∗ - Universit´ e Paris-Sud, 3,CNRS, rue Joliot Curie,INP, 91192, ..Gif-sur-Yvette, France, [email protected] Univ. Grenoble Alpes, F-38000 ∗∗∗ Univ. Grenoble Alpes, CNRS, Grenoble Grenoble INP, Gipsa-lab, Gipsa-lab, F-38000 ∗∗∗ France, [email protected] . Univ. Grenoble Alpes, CNRS, Grenoble INP, Gipsa-lab, F-38000 Grenoble, France, [email protected] ∗∗∗ Grenoble, France, [email protected] Univ. Grenoble CNRS, Grenoble INP, Gipsa-lab, F-38000 Grenoble,Alpes, France, [email protected] Grenoble, France, [email protected] Abstract: Abstract: This This paper paper is is concerned concerned with with the the asymptotic asymptotic stability stability analysis analysis of of aa one one dimensional dimensional Abstract: This paper is concerned with the asymptotic stability analysis of a one dimensional wave equation subject to a nonmonotone distributed damping. A well-posedness result wave equation subject to a nonmonotone distributed damping. A well-posedness result is is Abstract: Thissubject paper isto concerned with the distributed asymptotic stability analysis ofofathe onetrajectories dimensional wave equation a nonmonotone damping. A well-posedness result of is provided together with a precise characterization of the asymptotic behavior provided together with a to precise characterization of the asymptotic behavior of the trajectories of p wave equation subject a nonmonotone distributed damping. A well-posedness result is provided together with a precise characterization of the asymptotic behavior of the trajectories of the system under consideration. The well-posedness is proved in the nonstandard L p functional the system under consideration. The well-posedness is proved in the nonstandard Lp functional provided together with a precise characterization of the asymptotic behavior of the trajectories of the system The mostly well-posedness proved collected in the nonstandard functional spaces, with p [2, on in (2009). The spaces, withunder p ∈ ∈ consideration. [2, ∞], ∞], and and relies relies on some someis results results collected in Haraux HarauxL The p the system The mostly well-posedness proved in the nonstandard L(2009). functional spaces, withunder p ∈ consideration. [2, analysis ∞], andisrelies mostly on someis results collected in Haraux (2009). The asymptotic behavior based on an attractivity result on a specific infinite-dimensional asymptotic behavior analysis is based on an attractivity result on a specific infinite-dimensional spaces, withbehavior p ∈ [2, ∞], andisrelies on some results in Haraux (2009). The asymptotic analysis basedmostly on an attractivity resultcollected on a specific infinite-dimensional linear time-variant system. linear time-variant system. asymptotic behaviorsystem. analysis is based on an attractivity result on a specific infinite-dimensional linear time-variant © 2019,time-variant IFAC (International linear system.Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: 1D 1D wave wave equation, equation, nonlinear nonlinear control, control, Lyapunov Lyapunov functionals. functionals. Keywords: 1D wave equation, nonlinear control, Lyapunov functionals. Keywords: 1D wave equation, nonlinear control, Lyapunov functionals. 1. INTRODUCTION INTRODUCTION Marx al. (2018), (2018), 1. Marx et et al. al. (2017a). (2017a). More More recently, recently, in in Marx Marx et et al. 1. INTRODUCTION Marx et al. (2017a). More recently, in Marx et al. (2018), trajectory of such systems have been characterized via trajectory of such systems have been characterized via 1. INTRODUCTION Marx et al.techniques. (2017a). More recently, in Marx et al. (2018), trajectory of such systems have been characterized via Lyapunov Let us also mention Prieur et al. This paper is concerned with the asymptotic behavior of a techniques. Let us also mention Prieur et al. This paper is concerned with the asymptotic behavior of a Lyapunov trajectory ofMarx suchetsystems have been characterized via Lyapunov techniques. Let us also mention Prieur et al. (2016) and al. (2017b), where a wave equation This paper is concerned with the asymptotic behavior of a one-dimensional wave equation subject to a nonmonotone (2016) and Marx et al. (2017b), where a wave equation one-dimensional wave equation subject to a nonmonotone Lyapunov techniques. Let us also mention Prieur et al. This paper is concerned with the asymptotic behavior of a (2016) and Marx et al. (2017b), where a wave equation and a nonlinear Korteweg-de Vries, respectively, subject to one-dimensional wave equation subject to a always nonmonotone nonlinear damping. For control systems, it is crucial a nonlinear Korteweg-de Vries,where respectively, subject to nonlinear damping. Forequation control systems, it isa always crucial and (2016) and Marx et al. (2017b), a wave equation one-dimensional wave subject to nonmonotone and a nonlinear Korteweg-de Vries, respectively, subject to aa nonlinear monotone damping are considered and where nonlinear damping. For feedback control systems, it isparticular, always crucial to consider nonlinear laws. In the nonlinear monotone damping are considered and where to consider nonlinear feedback laws. In particular, the a nonlinear Korteweg-de Vries, respectively, subject to nonlinear damping. For feedback controlunder systems, it isparticular, always crucial athe nonlinear monotone damping are considered and where global asymptotic stability is proved. to consider nonlinear laws. In the and definition of the nonlinearity consideration in this global asymptotic proved. definition of nonlinear the nonlinearity under consideration in this athe nonlinear monotone stability dampingis considered and where to consider feedback laws. In particular, the the global asymptotic stability isare proved. definition of the nonlinearity under consideration in this paper includes the saturation, which models amplitude paper includes the saturation,under which models amplitude The case nonmonotone nonlinear damping asymptotic stability is proved. definition ofonthe nonlinearity consideration in this the Theglobal case of of nonmonotone nonlinear damping have have been been paper includes the saturation, models amplitude limitations the actuator. Suchwhich a phenomenon phenomenon appears limitations on the actuator. Such a appears The case of nonmonotone nonlinear damping haveglobal been considered in few papers. In Feireisl (1993), the paper includes the saturation, which models amplitude considered innonmonotone few papers. nonlinear In Feireisldamping (1993), the global limitations on the systems, actuator.and Such acan phenomenon appears The in most of control it lead to undesirable case of have been in most of control systems, and it can lead to undesirable considered in few papers. In Feireisl (1993), the global asymptotic stability of a one-dimensional wave equation limitations on the of actuator. Such phenomenon appears asymptotic stability of a one-dimensional wavethe equation in most ofincontrol systems, and it acan lead to undesirable behavior terms stability. Moreover, considering that considered in few papers. In Feireisl (1993), global behavior terms of stability. considering that asymptotic of a one-dimensional waveis equation subject to nonmonotone nonlinear proved, in most ofin systems, andMoreover, it canislead to undesirable to aastability nonmonotone nonlinear damping damping proved, behavior incontrol terms of Moreover, considering that subject such aa nonlinearity is nonmonotone for asymptotic stability of a one-dimensional waveis equation such nonlinearity isstability. nonmonotone is crucial crucial for control control subject to a nonmonotone nonlinear damping is proved, thanks to some compensated compactness technique. The behavior in terms of stability. Moreover, considering that thanks to some compensated compactness technique. The such a nonlinearity ismodel nonmonotone is crucial might for control systems, because the of the nonlinearity have tosome a nonmonotone nonlinear damping isnot proved, systems, because theismodel of the nonlinearity might have subject thanks to compensated compactness technique. The characterization of the trajectories is, however, prosuch a nonlinearity nonmonotone is crucial for control characterization of the trajectories is, however, not prosystems, because the model of the nonlinearity might have thanks to some compensated compactness technique. The some error. some error. characterization of the trajectories is, however, not provided. In Martinez and Vancostenoble (2000), the systems, because the model of the nonlinearity might have vided. In Martinez and Vancostenoble (2000), the tratrasome error. of equation the is, however, not vided. In Martinez andtrajectories Vancostenoble (2000), suject the protrajectory of aa wave in two dimensions to However, as illustrated illustrated in in many many papers papers (Alabau-Boussouira (Alabau-Boussouira characterization some error. jectory of wave equation in two dimensions suject to However, as vided. In Martinez and Vancostenoble (2000), the trajectory of a wave equation in two dimensions suject toa aa nonmonotone damping is characterized, but for only However, as illustrated in many papers (Alabau-Boussouira (2012), Zuazua (1990), Haraux (2009), Marx et al. (2018), nonmonotone damping is characterized, but for only (2012), Zuazua (1990), Haraux (2009), Marx et al. (2018), jectory of a wave equation in two dimensions suject toa However, as illustrated in many papers (Alabau-Boussouira aspecific nonmonotone damping is characterized, but for only a nonmonotone damping. In this paper, we rather (2012), Zuazua (1990), Haraux (2009),property Marx etofal.the (2018), Slemrod (1989), etc.), the monotone nonspecific nonmonotone damping. In this paper, we rather Slemrod (1989), etc.), the monotone property of the nona nonmonotone damping is characterized, but for only (2012), Zuazua (1990), Haraux (2009), Marx etofal.the (2018), nonmonotone damping. Innonmonotone this paper, we rathera focus on a more general nonlinear damping, Slemrod (1989), etc.), the monotone property non- specific linearity is crucial to first prove the asymptotic stability onnonmonotone a more generaldamping. nonlinearInnonmonotone damping, linearity is crucial to first prove the property asymptotic stability specific this paper,We we rather Slemrod (1989), etc.), the monotone of the non- focus focus on afor more general nonlinear nonmonotone damping, but only aa one dimensional wave equation. are linearity is crucial toconsideration first prove the stability of the system system under andasymptotic second characterize characterize but only for one dimensional wave equation. We are able able of the under consideration and second focus on a more general nonlinear nonmonotone damping, linearity is crucial to first prove the asymptotic stability but only for a one dimensional wave equation. Westudying are able to characterize the trajectories of the system by of system under second thethe trajectory of the the consideration latter system. system. and In order order to characterize to characterize the trajectories of the system by studying the trajectory of latter In to characterize p only for a one wave equation. We are able of the system under and second to characterize thedimensional trajectories the functional L the trajectory of behavior the consideration latterofsystem. In order to characterize asymptotic the trajectory trajectory of characterize the system system but the functional spaces spaces Lpp (0, (0, 1). 1). of the system by studying the asymptotic ofsystem. the of the to the trajectories the trajectory of behavior the latter In order to characterize thecharacterize functional spaces Lp (0, 1). of the system by studying the asymptotic behavior of the trajectory ofconditions the system under consideration, we consider the initial in under consideration, we consider the initialofconditions in There exists few papers the functional spaces (0, 1).dealing the another asymptotic behavior of the trajectory the system exists also also few L papers dealing with with the the one one dimendimenunder consideration, we consider the conditions in There an functional setting than theinitial classical one, that an another functional setting than the classical one, that There exists also few papers dealing with setting. the one dimensional wave equation in this functional Let p under consideration, we consider the initial conditions in sional wave equation in this functional setting. Let us us an another functional setting than the classical one, that There exists also few papers dealing with the one dimenp ∈ [2, ∞]. is L p (0, 1), with is L (0, 1), with p ∈ [2, ∞]. sional wave equation in this functional setting. Let us mention Haraux (2009), which derives a well-posedness p anLanother functional setting than the classical one, that mention Haraux (2009), which derives a well-posedness is (0, 1), with p ∈ [2, ∞]. sional wave in which this wave functional Let us mention Haraux (2009), derives a setting. well-posedness of aa equation one equation in There a about 1), with p ∈litterature [2, ∞]. is Lp (0,exists analysis of one dimensional dimensional wave equation in this this funcfuncThere exists a vast vast litterature about linear linear PDEs PDEs subject subject analysis mention Haraux (2009), which derives a well-posedness analysis of a one dimensional wave equation in this funcsetting. Moreover, an optimal decay rate is obtained There exists nonlinear a vast litterature about linear PDEs subject tional to monotone dampings. For instance, in Slemrod tional setting. Moreover, an optimal decay rate is obtained to monotone nonlinear dampings. For instance, in Slemrod analysis of a one dimensional wave equation inisthis funcThere exists a vast stability litterature about linear PDEs subject tional setting. Moreover, an optimal decay rate obtained for this equation. Recently, in Amadori et al. (2019), a to monotone nonlinear dampings. For instance, in Slemrod (1989), asymptotic of the origin of abstract conthis equation. Recently, in Amadori et al.is(2019), a (1989), asymptotic stability of theFor origin of abstract con- for tional setting. Moreover, an optimal decay rate obtained to monotone nonlinear dampings. instance, in Slemrod for this equation. Recently, in Amadori et al. (2019), a similar result have been obtained, using techniques coming (1989), asymptotic stability of the origin of abstract control systems subject to monotone nonlinear dampings is trol systems subject to monotone nonlinear dampings is similar result have been obtained, using techniques coming this equation. Recently, in Amadori etboth al. (2019), a (1989), asymptotic stability of the origin of abstract consimilar result have been obtained, using techniques coming from conservation laws theory. Note that of these trol systems subject to monotone nonlinear dampings is for proved, using an infinite-dimensional version of LaSalle’s from conservation laws obtained, theory. Note that both ofcoming these proved, usingsubject an infinite-dimensional version dampings of LaSalle’s similar result have been using techniques trol systems to monotone nonlinear is from conservation laws theory. Note that both of these results hold true only for monotone nonlinear damping, proved, using an infinite-dimensional version of LaSalle’s invariance principle. These been then hold true only for monotone invariance principle. These results results have have been then exex- results from laws theory. Note nonlinear that bothdamping, of these proved, using an general infinite-dimensional version of LaSalle’s resultsconservation hold truecase only for monotone nonlinear damping, is not of our paper. invariance principle. These results have been then extended to more infinite-dimensional systems in tended to more general infinite-dimensional systems in which which ishold not the the case offor ourmonotone paper. nonlinear damping, results true only invariance principle. These results have been then exwhich is not the case of our paper. tended to more general infinite-dimensional systems in This works has been partially supported by Advanced Grant after aa general This works which not the caseintroducing of our paper. tended to more general infinite-dimensional systems in In In our our ispaper, paper, after introducing general nonlinear nonlinear nonnonhas been partially supported by Advanced Grant DYCON (Dynamic Control) of the European Research Council In our paper, after introducing general nonlinear nonThis works has been partially supported by Advanced Grant monotone damping, we propose a well-posedness analysis DYCON (Dynamic Control) of the European Research Council monotone damping, we propose aa well-posedness analysis In our paper, after introducing general nonlinear nonThis works has been partially supported by Advanced Grant Executive Agency. DYCON (Dynamic monotone damping, we propose a well-posedness analysis Executive Agency. Control) of the European Research Council DYCON monotone damping, we propose a well-posedness analysis Executive (Dynamic Agency. Control) of the European Research Council Executive 2405-8963 Agency. © 2019 2019, IFAC IFAC (International Federation of Automatic Control) Copyright © 38 Hosting by Elsevier Ltd. All rights reserved. Copyright 2019 IFAC 38 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2019 IFAC 38 10.1016/j.ifacol.2019.11.752 Copyright © 2019 IFAC 38
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of a one-dimensional wave equation subject to such a damping. Futhermore, using Lyapunov techniques, we are able to characterize the trajectory of this system in the Lp , p ∈ [2, ∞), provided that the initial conditions are in L∞ . This proof is mainly based on a result about linear timevarying infinite-dimensional system, which is introduced and proved in Appendix A.
We refer the reader to Marx et al. (2018) for a complete discussion on this topic. Example 1. (Example of nonlinear dampings). Below are listed some examples of nonlinear dampings:
This paper is organized as follows. In Section 2, the main results of the paper are collected. To be more specific, we propose a well-posedness and an asymptotic stability theorems. The second result proposes futhermore a precise characterization of the trajectories of the system. Section 3 is devoted to the proof of the main results. Finally, Section 4 collects some concluding remarks, together with further research lines to be followed. Appendix A introduces a result of independant interest about exponential convergence of a specific time-varying infinite-dimensional linear systems, but which is instrumental for the proof of our asymptotic stability theorems.
(2)
1. The classical saturation, defined as follows: if |s| ≤ 1, s s σ(s) = sat(s) := if |s| ≥ 1, |s| satisfies all the properties of Definition 1. 2. The following nonlinearity 1 1 s− sin(10s) σ(s) = sat 4 30
(3)
is also a nonlinear damping. Note moreover that it is not monotone, as illustrated by Figure 1.
1
Notation: For any p ∈ [2, ∞), the space Lp (0, 1) denotes p1 1 < +∞. the space of functions f satisfying 0 |f (x)|p dx ∞ The space L (0, 1) denotes the space of functions satisfying ess supx∈[0,1] |f (x)| ≤ +∞. For any p ∈ [2, ∞], the Sobolev space W 1,p (0, 1) (resp. W 2,p (0, 1)) is defined as follows W01,p (0, 1) := {f ∈ Lp (0, 1) | f ∈ Lp (0, 1) and f (0) = f (1) = 0} (resp. W 2,p (0, 1) := {f ∈ Lp (0, 1) | f , f ∈ Lp (0, 1)}).
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
Acknowledgment: The two first authors of the paper would like to thank Prof. Enrique Zuazua for having kindly invited them in DeustoTech and for having pointed out the reference Haraux (2009). We would like also to thank Nicolas Burq for interesting discussions.
-0.8 -1 -5
0
5
Fig. 1. For any s ∈ [−5, 5], the figure illustrates the function σ given by (3)
2. MAIN RESULTS As illustrated in Marx et al. (2018), some regularity is needed to obtain a characterization of the asymptotic stability of (1). To be more precise, we need the state zt to be bounded in L∞ (0, 1). With a monotone nonlinearity σ, one would have this regularity result thanks to some nonlinear semigroup theorems. In the case of the system under consideration in this paper, we need to follow another strategy.
The aim of this paper is to provide an asymptotic stability analysis of the following system: ztt (t, x) = zxx (t, x) − a(x)σ( a(x)zt (t, x)), (t, x) ∈ R+ × [0, 1] (1) z(t, 0) = z(t, 1) = 0, t ∈ R+ z(0, x) = z0 (x), zt (0, x) = z1 (x), x ∈ [0, 1],
where z denotes the state, a : [0, 1] → R+ is measurable and bounded by some positive constant a∞ and σ is a scalar nonlinear damping which satisfies the following properties: Definition 1. [Scalar nonlinear damping] A function σ : R → R is said to be a scalar damping function if 1. 2. 3. 4.
Our strategy relies on the introduction of these functional spaces: Hp (0, 1) := W01,p (0, 1) × Lp (0, 1)
Dp (0, 1) := W 2,p (0, 1) ∩ W01,p (0, 1) × W01,p (0, 1),
(4)
with p ∈ [2, ∞]. The first functional space is equipped with the following norm: 1 p1 p (z, zt )Hp (0,1) := [zx (x)| dx
It is locally Lipschitz and odd; One has σ(0) = 0; For any s ∈ R, σ(s)s > 0; The function σ is differentiable at s = 0 with σ (0) = C1 for some C1 > 0.
0
+
Due to this definition (especially, item 2), the origin is an equilibrium point for (1). Note that this nonlinearity is not assumed to be monotone. The monotone property is in many cases really useful for either the well-posedness of the equation or the asymptotic stability of the origin.
1
p
0
|zt (x)| dx
p1
, ∀p ∈ [2, ∞),
(z, zt )H∞ (0,1) :=zx L∞ (0,1) + zt L∞ (0,1) , for p = ∞. (5) The second one is equipped with the following norm 39
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(z, zt ) Dp (0,1) :=
+
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1 p
0
|zxx (x)| dx 1 p
0
p1
|ztx (x)| dx
p1
, ∀p ∈ [2, ∞)
(z, zt ) D∞ (0,1) := zxx L∞ (0,1) + zt,x L∞ (0,1) , for p = ∞. (6) Considering that (z0 , z1 ) ∈ H∞ (0, 1), our aim is to prove an L∞ regularity on the state zt so that we can obtain a characterization of the asymptotic behavior of the trajectory in Hp (0, 1), with p ∈ [2, ∞). But before stating such a result, we need a suitable notion of solution for systems modeled with (1). Such a notion is provided by the following well-posedness theorem: Theorem 1. [Well-posedness] For any initial conditions (z0 , z1 ) ∈ H∞ (0, 1), there exists a unique solution (z, zt ) ∈ L∞ (R+ ; W 1,∞ (0, 1)) × W 1,∞ (R+ ; L∞ (0, 1)) to (1). Moreover, the following inequality is satisfied, for all t ≥ 0 (z, zt ) H∞ (0,1) ≤ 2 max( z0 L∞ (0,1) , z1 L∞ (0,1) ). 1 (7) Now that the functional setting is introduced, we are in position to state our asymptotic stability result Theorem 2. [Semi-global exponential stability] Consider initial conditions (z0 , z1 ) ∈ H∞ (0, 1) satisfying: (8) (z0 , z1 ) H∞ (0,1) ≤ R,
where R is a positive constant. Then, for any p ∈ [2, ∞), there exist two positive constants K := K(R) and β := β(R) such that (z, zt ) Hp (0,1) ≤ Ke−βt (z0 , z1 ) Hp (0,1) ,
∀t ≥ 0.
(9)
where a is a 2-periodic extension of a, y ∈ L2 (0, T ; L2 (0, 1)) and σ is a scalar nonlinear damping (which is odd, due to Item 1 of Definition 1). In particular, it means that h ∈ L2 (R+ ; L2 (0, 1)). The proof of Theorem 1 consists first in applying a fixed-point theorem, which will allow us to prove the well-posedness of (1) for a small time T > 0 and second in using a stability result in Haraux (2009), stated as follows Proposition 1. Let us consider initial condition z0 , z1 ∈ H∞ (0, 1). If there exists a solution to (1), then the time derivative of the following functional along the trajectories of (1) 1 (F (zt + zx ) + F (zt − zx )) dx, (16) φ(z, zt ) :=
3. PROOF OF THE MAIN RESULTS 3.1 Proof of Theorem 1 The proof of Theorem 1 relies on the following wave equation with a source term ztt (t, x) = zxx (t, x) + h(t, x), (t, x) ∈ R+ × [0, 1] z(t, 0) = z(t, 1) = 0, t ∈ R+ (10) z(0, x) = z0 (x), zt (0, x) = z1 (x), x ∈ [0, 1],
where h denotes the source term. From (Haraux, 2018, Theorem 1.3.8), we know that, provided that z0 , z1 ∈ H2 and that h ∈ L2 (R+ ; L2 (0, 1)), there exists a unique solution z ∈ C(R+ ; H01 (0, 1)) ∩ C 1 (R+ ; L2 (0, 1)) to (10). In particular, since H∞ (0, 1) ⊂ H2 , this result holds true also for initial conditions (z0 , z1 ) ∈ H∞ . The first step of our analysis in this section is to prove that picking initial conditions in H∞ (0, 1) and the source term h ∈ L2 (R+ ; L∞ (0, 1)) improves also the regularity of the solution z itself.
0
with F any even and convex function, satisfies d φ(z, zt ) ≤ 0. dt
The latter proposition implies that (17) φ(z, zt ) ≤ φ(z0 , z1 ), ∀t ≥ 0. In particular, following the discussion in the proof of Corollary 2.3. in Haraux (2009), if one picks 2 F (s) = [Pos(|s|− 2 max( z0 L∞ (0,1) , z1 L∞ (0,1) ))]2 , then one obtains that
To do so, our aim is to give an explicit formula for the latter equation, using the reflection method surveyed in Strauss (1992). Roughly speaking, this method consists in extending the explicit formulation of trajectory of the wave equation in an unbounded domain to a bounded domain. To do so, we extend the initial datas to the whole line to be odd with respect to both x = 0 and x = 1, that is z0 (x) and z0 (2 − x) = − z0 (x), (11) z0 (−x) = − 1
where z0 denotes the 2-periodic odd extension of z0 . A way to do this is to define z0 as follows: z0 (x), 0 < x < 1 − z0 (−x), −l < x < 0 (12) z0 (x) = extended to be of period 2 We can define similarly a 2-periodic odd extension of z1 (resp. h), denoted by z1 (resp. h). Thanks to (Strauss, 1992, Theorem 1, Page 69), we can therefore define the explicit trajectory z of (10) (known as the D’Alembert formula) as follows: 1 x+ct 1 z0 (x + t) + z0 (x − t)] + z1 (s)ds z(t, x) = [ 2 2 x−ct (13) 1 t x+(t−s) + h(w, s)dwds. 2 0 x−(t−s) We can further define zt as follows 1 zt (t, x) = ( z (x + t) − z0 (x − t)) 2 0 1 z1 (x + t) − z1 (x − t)) + ( 2 1 t + h(s, x + (t − s)) − h(s, x − (t − s)) ds 2 0 (14) It is clear from these two latter equations that, when picking (z0 , z1 ) ∈ H∞ (0, 1) and h ∈ L2 (R+ ; L∞ (0, 1)), then z ∈ C(R+ ; W 1,∞ (0, 1)) ∩ C 1 (R+ ; L∞ (0, 1)). We assume now that h is written as follows h(t, x) := −σ( a(x)y(t, x)), (15)
φ(z, zt ) = 0,
2
The function Pos : s ∈ R → Pos(s) ∈ R+ is defined as follows: Pos(s) =
The term z0 denotes the derivative of z0 with respect to x.
40
(18)
s if s > 0 0 if s ≤ 0.
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which implies that, for all t ≥ 0 max( zx (t, ·) L∞ (0,1) , zt (t, ·) L∞ (0,1) ) ≤ 2 max( z0 L∞ (0,1) , z1 L∞ (0,1) ). (19)
39
since BK (y0 ) is complete, any Cauchy sequence converges in this set. By induction, one can prove that (27) yn+1 − yn T ≤ (C(y0 , K)T )n y0 − y1 T and (28) yn ∈ BK (y0 ). Indeed, thanks to the choice of T in (26), these two properties are easily proved for n = 0 and, moreover, we have with (24), for all n ≥ 1 yn+1 − yn T = φ(yn ) − φ(yn−1 ) T ≤C(y0 , K)T φ(yn−1 ) − φ(yn−2 ) T (29) ≤(C(y0 , K)T )n y0 − y1 T .
Noticing that (z, zt ) H∞ (0,1) ≤ max( zx (t, ·) L∞ (0,1) , zt (t, ·) L∞ (0,1) ), it is clear then that, for all t ≥ 0 (z, zt ) H∞ (0,1) ≤ 2 max( z0 L∞ (0,1) , z1 L∞ (0,1) ) (20) This estimate implies that, once one is able to prove that there exists a solution (z, zt ) of (1) in L∞ ([0, T ]; W 1,∞ (0, 1)) ∩ W 1,∞ ([0, T ]; L∞ (0, 1)), for a small time T > 0, then the well-posedness of (1) is ensured in L∞ (R+ ; W 1,∞ (0, 1)) ∩ W 1,∞ (R+ ; L∞ (0, 1)).
The inequality (27) can be deduced from the above inequality. The property (28) can be proved as follows: yn+1 ≤ yn T + y1 T + y0 T ≤ y0 T + φT (y0 ) T + φT (y0 ) T + y0 T (30) ≤K
Now, we are in position to prove Theorem 1.
Proof. Let us define FT the space of measurable functions defined on [0, T ]×R which are bounded, odd and 2-periodic in space. We endow FT with the L∞ -norm so that it becomes a Banach space. Hence, denoting by · T the norm of the latter functional space, we have, for every y ∈ FT y T := sup |y(t, x)|. (21)
where in the first line we have used (27) and, in the second line, we have used the fact that yn ∈ BK (y0 ).
The two properties (27) and (28) show that the sequence yn is a Cauchy sequence. Since BK (y0 ) is a complete set, this sequence is therefore convergent. This means in particular that there exists a fixed-point to the mapping φT (y0 ), which implies that, for sufficiently small time T , there exists a unique solution z ∈ L∞ (0, T ; W 1,∞ (0, 1)) and zt ∈ L∞ (0, T ; L∞ (0, 1)). Thanks to (20), we can deduce that there exists unique solution z ∈ L∞ (R+ ; W 1,∞ (0, 1)) and zt ∈ L∞ (R+ ; L∞ (0, 1)). This concludes the proof of Theorem 1. •
(t,x)∈[0,T ]×R
Let us consider BK (y) the closed ball in FT centered at y ∈ FT of radius K ≥ 0, where K remains to be defined.
We can now define the mapping with which we will apply a fixed-point φT : FT → FT (22) y → φT (y),
3.2 Proof of Theorem 2
where
1 1 φT (y) = ( z0 (x + t) − z 0 (x − t)) + ( z1 (x + t) − z 1 (x − t)) 2 t 2 1 − a(x + t − s)σ( a(x + t − s)y(s, x − (t − s)) 2 0 + a(−x + t − s)σ( a(−x + t − s)y(s, −x − (t − s)) ds (23)
Using the fact that σ is locally Lipschitz, note that, for every y0 ∈ FT and K > 0, there exists a positive constant C(y0 , K) such that, for every y, y ∈ BK (y0 ) φT (y) − φT ( y ) T ≤ C(y0 , K)T y − y T . (24) Pick K such that K := 2 ( φT (y0 ) T + y0 T ) (25) and T sufficiently small such that C(y0 , K)T ≤ 1. (26) Consider a sequence (yn )n∈N defined as yn+1 = φT (yn ).
The proof of Theorem 2 is divided into two steps: first, we transform the system (1) into a system in the form (A.1), which is in particular a linear time-variant system, and apply Theorem 3 for the space H2 , which is indeed the only Hilbert space among all the spaces Hp (0, 1). Second, using the fact that the solutions are bounded in H∞ (0, 1) thanks to Theorem 1, and invoking an interpolation theorem, which is the Riesz-Thorin theorem, we conclude. Proof. First step: semi-global exponential stability in H2 . We first fix p = 2, but still consider the initial conditions (z0 , z1 ) ∈ H∞ (0, 1). Moreover, we suppose that there exists a positive constant R such that the initial conditions satisfy (31) (z0 , z1 ) H∞ (0,1) ≤ R. In particular, due to Theorem 1, one has (32) (z, zt ) H∞ (0,1) ≤ 2R, ∀t ≥ 0.
If we prove that this sequence converges to y , then we can deduce that y = φT (y ). This means that there exists a fixed-point for the mapping φT , which implies in particular that (1) is well-posed in the desired functional spaces. To prove the convergence of this sequence, we prove that it is a Cauchy sequence. Indeed, 41
The system (1) may be rewritten as follows ztt = zxx − d(t, x)a(x)zt , z(t, 0) = z(t, 1) = 0, z(0, x) = z0 (x), zt (0, x) = z1 (x), with σ( a(x)zt ) , if a(x)zt = 0, d(t, x) = a(x)zt if a(x)zt = 0. C1 ,
(33)
(34)
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This function comes from the function t →
√ √ a(x)zt ,
for every (z0 , z1 ) ∈ H∞ (0, 1), one therefore has, for every t≥0 W (t, 0)(z0 , z1 )Hp (0,1) ≤W (t, 0)Hp (0,1) (z0 , z1 )Hp (0,1) p2 2β K ≤2 e− p t (z0 , z1 )Hp (0,1) . 2 This concludes the proof of Theorem 2. •
σ(
a(x)zt
extended at 0, which is possible due to the differentiability of the function σ at 0.
This system is in the form (A.1), with H = H2 , U = H2 , D(A) = D2 , and the operators A and B defined as follows A : D(A) ⊂ H → H (35) [v1 v2 ] → v2 v1 and B = 0 a(x) . It is well known that A generates a strongly continuous semigroup of contractions (see Haraux (2018)). Now, let us check whether d(t, ·) satisfies (A.2), for all t ≥ 0.
4. CONCLUSION In this paper, we have provided a well-posedness analysis of a one-dimensional wave equation subject to a nonlinear nonmonotone damping. Futhermore, a characterization of the asymptotic behavior of the latter system is given. It is proved with Lyapunov techniques. This work paves the way to many others. For instance, a first open question would be the case of multidimensional wave equations, for which there does not exist any proof of well-posedness in the functional setting introduced in this paper.
Since the initial conditions (z0 , z1 ) ∈ H∞ (0, 1) and are bounded by R in the H∞ (0, 1) norm, then invoking Theorem 1 we have, for all t ≥ 0 (36) sup |zt (t, x)| ≤ 2R. x∈[0,1]
√ Moreover, since a(x) ≤ a∞ , for all x ∈ [0, 1], and because d is a continuous function, there exist two positive constants d0 and d1 , depending on R and a∞ such that σ(ξ) ≤ d(t, x) d0 := √ min √ ξ∈[−2 a∞ R,2 a∞ R] ξ σ(ξ) ≤ := d1 . √ max √ ξ∈[−2 a∞ R,2 a∞ R] ξ (37)
Appendix A. EXPONENTIAL CONVERGENCE RESULT FOR A LINEAR TIME-VARIANT SYSTEM This appendix is devoted to the statement and the proof of a theorem dealing with a time-variant linear infinitedimensional system. Indeed, as it is illustrated in Section 3.2, we can transform (1) as a time-variant linear infinitedimensional system. To define it, let us introduce H (resp. U ), a Hilbert space which is equipped with the norm · H and the scalar product ·, ·H . The system under study in this section is the following: d v = (A − d(t)BB )v := Ad (t)v (A.1) dt v(τ ) = v , τ
Note moreover that the origin of the following system zt = zxx − d0 zt z(t, 0) = z(t, 1) = 0 (38) z(0, x) = z0 (x), zt (0, x) = z1 (x).
where A : D(A) ⊂ H → H, with D(A) the domain of the operator A that we suppose densely defined in H, B ∈ L(U, H). We assume moreover that A and its adjoint A are dissipative, which implies in particular that A generates a strongly continuous semigroup of contractions. Let us also assume that d is a strongly continuous mapping from any interval of time [0, T ], with T > 0, into L(H) (resp. L(D(A)), meaning that d(·)v belongs to C([0, T ]; H) for any v ∈ H (resp. C([0, T ]; D(A)) for any v ∈ D(A)). This latter implies in particular, from (Bensoussan et al., 2007, Proposition 3.6., page 138), that there exists a unique mild (resp. strong) solution to (A.1) if vτ ∈ H (resp. vτ ∈ D(A)), and that Ad (t) generates an evolution family (W (θ, τ ))θ≥τ 3 . This means that every trajectories of (A.1) can be expressed as follows v = W (θ, τ )vτ .
is exponentially stable in H2 for any initial conditions (z0 , z1 ) ∈ H2 (see e.g. Prieur et al. (2016)). The related operator of this system is A − d0 BB , with domain D(Ad0 ) = D2 . Therefore, all the properties required in Theorem 3 are satisfied. Hence, there exist two positive constants K := K(R) and β := β(R) such that (z, zt )H2 ≤ Ke−βt (z0 , z1 )H2 ,
∀t ≥ 0.
(39)
Second step: Semi-global exponential stability in Hp (0, 1). From Theorem 1, we know that, for every initial conditions (z0 , z1 ) ∈ H∞ (0, 1) satisfying (z0 , z1 )H∞ (0,1) ≤ R, and noticing that the trajectory of (1) can be expressed with the evolution family W (t, 0), one has (40) W (t, 0)(z0 , z1 )H∞ (0,1) ≤ 2R, ∀t ≥ 0.
Assume moreover that there exists two positive constants d0 and d1 such that, for all t ≥ 0 d0 ≤ d(t) ≤ d1 . (A.2)
Now, fix t > 0. Note that W (t, 0) is an operator from (L2 (0, 1))2 (resp. (L∞ (0, 1))2 ) to (L2 (0, 1))2 (resp. (L∞ (0, 1))2 ), if it associates (z0 , z1 ) ∈ L2 (0, 1)2 (resp. (z0 , z1 ) ∈ L∞ (0, 1)2 ) to (zx , zt ) ∈ L2 (0, 1) (resp. (zx , zt ) ∈ L∞ (0, 1)2 ). Hence, we can apply the so-called Riesz-Thorin theorem (Bergh and L¨ ofstr¨ om, 2012, Theorem 1.1.1, Page 2) and conclude that p2 2β K e− p t , ∀t ≥ 0, (41) W (t, 0)Hp (0,1) ≤ 2 2 where W (t, 0) corresponds exactly to the trajectory of (1) with the initial condition z0 , z1 ∈ H∞ (0, 1). In particular,
We are now in position to state the following result. Theorem 3. Consider the system given by (A.1). Supposing that the origin of the following system d v = (A − d0 BB )v := Ad0 v, (A.3) dt v(0) = v 0
3 See Chicone and Latushkin (1999) for more details on the concept of evolution families.
42
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is globally exponentially stable, then, for any initial conditions v0 ∈ H, the origin of (A.1) converges exponentially to 0 with τ = 0. Remark 1. The global asymptotic stability is not ensured for the linear time-variant system (A.1). Indeed, for such systems, the global asymptotic stability has to be proven for every initial condition vτ and for every τ ≥ 0. This is not an issue for us, since the stability of (1) already holds thanks to Theorem 1. ◦
one can conclude that P L(H) + M C −ε 2 vH ≤ exp − t v0 2H , M P L(H) + M (A.13) which ends the proof of Theorem 3. • REFERENCES
Proof. Since the origin of (A.3) is globally exponentially stable, then, due to Datko (1970), there exist a self-adjoint operator P ∈ L(H) and a positive constant C such that the following inequality holds true P Ad0 v, vH + P v, Ad0 vH ≤ −Cv2H , ∀v ∈ D(A) (A.4) Moreover, note that Ad0 is also a dissipative operator, which means in particular that (A.5) Ad0 v, vH + v, Ad0 vH ≤ 0, ∀v ∈ D(A). Now, consider the following candidate Lyapunov functional for (A.1): (A.6) V (v) := P v, vH + M v2H , where M is a positive constant which has to be defined. The time derivative of V along the trajectories of (A.1) yields d V (v) =(P + M IH )Ad(t) v, vH + (P + M IH )v, Ad(t) vH dt =P Ad0 v, vH + P v, Ad0 vH + M (Ad0 v, vH + v, Ad0 vH ) − (d(t) − d0 IH )BB v, (P + M IH )vH − (P + M IH )v, (d(t) − d0 IH )BB vH . ≤ − Cv2H − d0 M B v2U + 2(d1 − d0 )P L(H) BB L(H) B vU vH − 2M (d(t) − d0 )B v, B vU ,
(A.7) Since d(t) ≥ d0 and M > 0, one has 2M (d(t) − d0 )B v, B vU . Moreover, applying the Young inequality, we known that there exists a positive number ε such that d V (v) ≤ − Cv2H − d0 M B v2U dt (d1 − d0 ) BB L(H) P L(H) v2H + ε + ε(d1 − d0 )
BB L(H) P L(H) B v2U (A.8)
Setting (d1 − d0 )P L(H) ε> C and
BB L(H)
,
ε(d1 − d0 )P L(H) BB L(H) M> , d0 one obtains d V (v) ≤ −(C − ε)v2H , ∀v ∈ D(A). dt Since V satisfies the following inequalities M v2H ≤ V (z) ≤ (P L(H) + M )v2H
41
(A.9)
(A.10)
(A.11) (A.12) 43
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Stability analysis of a 1D wave equation Stability analysis of a 1D wave equation Stability analysis of a 1D wave equation with a nonmonotone damping Stability analysis of distributed a 1D wave equation with a nonmonotone distributed with a nonmonotone distributed damping damping with a nonmonotone distributed damping ∗ ∗∗ Swann Marx ∗ , Yacine Chitour ∗∗ , Christophe Prieur ∗∗∗ ∗∗∗
Swann Marx ∗ , Yacine Chitour ∗∗ , Christophe Prieur ∗∗∗ Swann Marx ∗ , Yacine Chitour ∗∗ , Christophe Prieur ∗∗∗ ∗ Swann Marx , Yacine Chitour , Christophe Prieur ee de ∗ LAAS-CNRS, Universit´ de Toulouse, Toulouse, CNRS, CNRS, 7 7 avenue avenue du du colonel colonel ∗ LAAS-CNRS, Universit´ LAAS-CNRS, Universit´ e de Toulouse, CNRS, 7 avenue du.. colonel Roche, 31400, Toulouse, France [email protected] Roche, 31400, Toulouse, France [email protected] ∗∗∗ LAAS-CNRS, Universit´ eetde Toulouse, CNRS, 7 avenue du. colonel Roche,des 31400, Toulouse, [email protected] Signaux mes CNRS ∗∗ Laboratoire des Signaux et Syst` Syst`eeFrance mes (L2S), (L2S), CNRS -- CentraleSupelec CentraleSupelec ∗∗ Laboratoire Roche, 31400, Toulouse, France [email protected] . Laboratoire des Signaux et Syst` e mes (L2S), CNRS -Gif-sur-Yvette, CentraleSupelec Universit´ e Paris-Sud, 3, rue Joliot Curie, 91192, ∗∗ - Universit´ e Paris-Sud, 3, rue Joliot Curie, 91192, Gif-sur-Yvette, Laboratoire des Signaux et Syst` e mes (L2S), CNRS CentraleSupelec - Universit´e Paris-Sud, 3, rue Joliot Curie, 91192, .Gif-sur-Yvette, France, France, [email protected] [email protected] ∗∗∗ - Universit´ e Paris-Sud, 3,CNRS, rue Joliot Curie,INP, 91192, ..Gif-sur-Yvette, France, [email protected] Univ. Grenoble Alpes, F-38000 ∗∗∗ Univ. Grenoble Alpes, CNRS, Grenoble Grenoble INP, Gipsa-lab, Gipsa-lab, F-38000 ∗∗∗ France, [email protected] . Univ. Grenoble Alpes, CNRS, Grenoble INP, Gipsa-lab, F-38000 Grenoble, France, [email protected] ∗∗∗ Grenoble, France, [email protected] Univ. Grenoble CNRS, Grenoble INP, Gipsa-lab, F-38000 Grenoble,Alpes, France, [email protected] Grenoble, France, [email protected] Abstract: Abstract: This This paper paper is is concerned concerned with with the the asymptotic asymptotic stability stability analysis analysis of of aa one one dimensional dimensional Abstract: This paper is concerned with the asymptotic stability analysis of a one dimensional wave equation subject to a nonmonotone distributed damping. A well-posedness result wave equation subject to a nonmonotone distributed damping. A well-posedness result is is Abstract: Thissubject paper isto concerned with the distributed asymptotic stability analysis ofofathe onetrajectories dimensional wave equation a nonmonotone damping. A well-posedness result of is provided together with a precise characterization of the asymptotic behavior provided together with a to precise characterization of the asymptotic behavior of the trajectories of p wave equation subject a nonmonotone distributed damping. A well-posedness result is provided together with a precise characterization of the asymptotic behavior of the trajectories of the system under consideration. The well-posedness is proved in the nonstandard L p functional the system under consideration. The well-posedness is proved in the nonstandard Lp functional provided together with a precise characterization of the asymptotic behavior of the trajectories of the system The mostly well-posedness proved collected in the nonstandard functional spaces, with p [2, on in (2009). The spaces, withunder p ∈ ∈ consideration. [2, ∞], ∞], and and relies relies on some someis results results collected in Haraux HarauxL The p the system The mostly well-posedness proved in the nonstandard L(2009). functional spaces, withunder p ∈ consideration. [2, analysis ∞], andisrelies mostly on someis results collected in Haraux (2009). The asymptotic behavior based on an attractivity result on a specific infinite-dimensional asymptotic behavior analysis is based on an attractivity result on a specific infinite-dimensional spaces, withbehavior p ∈ [2, ∞], andisrelies on some results in Haraux (2009). The asymptotic analysis basedmostly on an attractivity resultcollected on a specific infinite-dimensional linear time-variant system. linear time-variant system. asymptotic behaviorsystem. analysis is based on an attractivity result on a specific infinite-dimensional linear time-variant © 2019,time-variant IFAC (International linear system.Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: 1D 1D wave wave equation, equation, nonlinear nonlinear control, control, Lyapunov Lyapunov functionals. functionals. Keywords: 1D wave equation, nonlinear control, Lyapunov functionals. Keywords: 1D wave equation, nonlinear control, Lyapunov functionals. 1. INTRODUCTION INTRODUCTION Marx al. (2018), (2018), 1. Marx et et al. al. (2017a). (2017a). More More recently, recently, in in Marx Marx et et al. 1. INTRODUCTION Marx et al. (2017a). More recently, in Marx et al. (2018), trajectory of such systems have been characterized via trajectory of such systems have been characterized via 1. INTRODUCTION Marx et al.techniques. (2017a). More recently, in Marx et al. (2018), trajectory of such systems have been characterized via Lyapunov Let us also mention Prieur et al. This paper is concerned with the asymptotic behavior of a techniques. Let us also mention Prieur et al. This paper is concerned with the asymptotic behavior of a Lyapunov trajectory ofMarx suchetsystems have been characterized via Lyapunov techniques. Let us also mention Prieur et al. (2016) and al. (2017b), where a wave equation This paper is concerned with the asymptotic behavior of a one-dimensional wave equation subject to a nonmonotone (2016) and Marx et al. (2017b), where a wave equation one-dimensional wave equation subject to a nonmonotone Lyapunov techniques. Let us also mention Prieur et al. This paper is concerned with the asymptotic behavior of a (2016) and Marx et al. (2017b), where a wave equation and a nonlinear Korteweg-de Vries, respectively, subject to one-dimensional wave equation subject to a always nonmonotone nonlinear damping. For control systems, it is crucial a nonlinear Korteweg-de Vries,where respectively, subject to nonlinear damping. Forequation control systems, it isa always crucial and (2016) and Marx et al. (2017b), a wave equation one-dimensional wave subject to nonmonotone and a nonlinear Korteweg-de Vries, respectively, subject to aa nonlinear monotone damping are considered and where nonlinear damping. For feedback control systems, it isparticular, always crucial to consider nonlinear laws. In the nonlinear monotone damping are considered and where to consider nonlinear feedback laws. In particular, the a nonlinear Korteweg-de Vries, respectively, subject to nonlinear damping. For feedback controlunder systems, it isparticular, always crucial athe nonlinear monotone damping are considered and where global asymptotic stability is proved. to consider nonlinear laws. In the and definition of the nonlinearity consideration in this global asymptotic proved. definition of nonlinear the nonlinearity under consideration in this athe nonlinear monotone stability dampingis considered and where to consider feedback laws. In particular, the the global asymptotic stability isare proved. definition of the nonlinearity under consideration in this paper includes the saturation, which models amplitude paper includes the saturation,under which models amplitude The case nonmonotone nonlinear damping asymptotic stability is proved. definition ofonthe nonlinearity consideration in this the Theglobal case of of nonmonotone nonlinear damping have have been been paper includes the saturation, models amplitude limitations the actuator. Suchwhich a phenomenon phenomenon appears limitations on the actuator. Such a appears The case of nonmonotone nonlinear damping haveglobal been considered in few papers. In Feireisl (1993), the paper includes the saturation, which models amplitude considered innonmonotone few papers. nonlinear In Feireisldamping (1993), the global limitations on the systems, actuator.and Such acan phenomenon appears The in most of control it lead to undesirable case of have been in most of control systems, and it can lead to undesirable considered in few papers. In Feireisl (1993), the global asymptotic stability of a one-dimensional wave equation limitations on the of actuator. Such phenomenon appears asymptotic stability of a one-dimensional wavethe equation in most ofincontrol systems, and it acan lead to undesirable behavior terms stability. Moreover, considering that considered in few papers. In Feireisl (1993), global behavior terms of stability. considering that asymptotic of a one-dimensional waveis equation subject to nonmonotone nonlinear proved, in most ofin systems, andMoreover, it canislead to undesirable to aastability nonmonotone nonlinear damping damping proved, behavior incontrol terms of Moreover, considering that subject such aa nonlinearity is nonmonotone for asymptotic stability of a one-dimensional waveis equation such nonlinearity isstability. nonmonotone is crucial crucial for control control subject to a nonmonotone nonlinear damping is proved, thanks to some compensated compactness technique. The behavior in terms of stability. Moreover, considering that thanks to some compensated compactness technique. The such a nonlinearity ismodel nonmonotone is crucial might for control systems, because the of the nonlinearity have tosome a nonmonotone nonlinear damping isnot proved, systems, because theismodel of the nonlinearity might have subject thanks to compensated compactness technique. The characterization of the trajectories is, however, prosuch a nonlinearity nonmonotone is crucial for control characterization of the trajectories is, however, not prosystems, because the model of the nonlinearity might have thanks to some compensated compactness technique. The some error. some error. characterization of the trajectories is, however, not provided. In Martinez and Vancostenoble (2000), the systems, because the model of the nonlinearity might have vided. In Martinez and Vancostenoble (2000), the tratrasome error. of equation the is, however, not vided. In Martinez andtrajectories Vancostenoble (2000), suject the protrajectory of aa wave in two dimensions to However, as illustrated illustrated in in many many papers papers (Alabau-Boussouira (Alabau-Boussouira characterization some error. jectory of wave equation in two dimensions suject to However, as vided. In Martinez and Vancostenoble (2000), the trajectory of a wave equation in two dimensions suject toa aa nonmonotone damping is characterized, but for only However, as illustrated in many papers (Alabau-Boussouira (2012), Zuazua (1990), Haraux (2009), Marx et al. (2018), nonmonotone damping is characterized, but for only (2012), Zuazua (1990), Haraux (2009), Marx et al. (2018), jectory of a wave equation in two dimensions suject toa However, as illustrated in many papers (Alabau-Boussouira aspecific nonmonotone damping is characterized, but for only a nonmonotone damping. In this paper, we rather (2012), Zuazua (1990), Haraux (2009),property Marx etofal.the (2018), Slemrod (1989), etc.), the monotone nonspecific nonmonotone damping. In this paper, we rather Slemrod (1989), etc.), the monotone property of the nona nonmonotone damping is characterized, but for only (2012), Zuazua (1990), Haraux (2009), Marx etofal.the (2018), nonmonotone damping. Innonmonotone this paper, we rathera focus on a more general nonlinear damping, Slemrod (1989), etc.), the monotone property non- specific linearity is crucial to first prove the asymptotic stability onnonmonotone a more generaldamping. nonlinearInnonmonotone damping, linearity is crucial to first prove the property asymptotic stability specific this paper,We we rather Slemrod (1989), etc.), the monotone of the non- focus focus on afor more general nonlinear nonmonotone damping, but only aa one dimensional wave equation. are linearity is crucial toconsideration first prove the stability of the system system under andasymptotic second characterize characterize but only for one dimensional wave equation. We are able able of the under consideration and second focus on a more general nonlinear nonmonotone damping, linearity is crucial to first prove the asymptotic stability but only for a one dimensional wave equation. Westudying are able to characterize the trajectories of the system by of system under second thethe trajectory of the the consideration latter system. system. and In order order to characterize to characterize the trajectories of the system by studying the trajectory of latter In to characterize p only for a one wave equation. We are able of the system under and second to characterize thedimensional trajectories the functional L the trajectory of behavior the consideration latterofsystem. In order to characterize asymptotic the trajectory trajectory of characterize the system system but the functional spaces spaces Lpp (0, (0, 1). 1). of the system by studying the asymptotic ofsystem. the of the to the trajectories the trajectory of behavior the latter In order to characterize thecharacterize functional spaces Lp (0, 1). of the system by studying the asymptotic behavior of the trajectory ofconditions the system under consideration, we consider the initial in under consideration, we consider the initialofconditions in There exists few papers the functional spaces (0, 1).dealing the another asymptotic behavior of the trajectory the system exists also also few L papers dealing with with the the one one dimendimenunder consideration, we consider the conditions in There an functional setting than theinitial classical one, that an another functional setting than the classical one, that There exists also few papers dealing with setting. the one dimensional wave equation in this functional Let p under consideration, we consider the initial conditions in sional wave equation in this functional setting. Let us us an another functional setting than the classical one, that There exists also few papers dealing with the one dimenp ∈ [2, ∞]. is L p (0, 1), with is L (0, 1), with p ∈ [2, ∞]. sional wave equation in this functional setting. Let us mention Haraux (2009), which derives a well-posedness p anLanother functional setting than the classical one, that mention Haraux (2009), which derives a well-posedness is (0, 1), with p ∈ [2, ∞]. sional wave in which this wave functional Let us mention Haraux (2009), derives a setting. well-posedness of aa equation one equation in There a about 1), with p ∈litterature [2, ∞]. is Lp (0,exists analysis of one dimensional dimensional wave equation in this this funcfuncThere exists a vast vast litterature about linear linear PDEs PDEs subject subject analysis mention Haraux (2009), which derives a well-posedness analysis of a one dimensional wave equation in this funcsetting. Moreover, an optimal decay rate is obtained There exists nonlinear a vast litterature about linear PDEs subject tional to monotone dampings. For instance, in Slemrod tional setting. Moreover, an optimal decay rate is obtained to monotone nonlinear dampings. For instance, in Slemrod analysis of a one dimensional wave equation inisthis funcThere exists a vast stability litterature about linear PDEs subject tional setting. Moreover, an optimal decay rate obtained for this equation. Recently, in Amadori et al. (2019), a to monotone nonlinear dampings. For instance, in Slemrod (1989), asymptotic of the origin of abstract conthis equation. Recently, in Amadori et al.is(2019), a (1989), asymptotic stability of theFor origin of abstract con- for tional setting. Moreover, an optimal decay rate obtained to monotone nonlinear dampings. instance, in Slemrod for this equation. Recently, in Amadori et al. (2019), a similar result have been obtained, using techniques coming (1989), asymptotic stability of the origin of abstract control systems subject to monotone nonlinear dampings is trol systems subject to monotone nonlinear dampings is similar result have been obtained, using techniques coming this equation. Recently, in Amadori etboth al. (2019), a (1989), asymptotic stability of the origin of abstract consimilar result have been obtained, using techniques coming from conservation laws theory. Note that of these trol systems subject to monotone nonlinear dampings is for proved, using an infinite-dimensional version of LaSalle’s from conservation laws obtained, theory. Note that both ofcoming these proved, usingsubject an infinite-dimensional version dampings of LaSalle’s similar result have been using techniques trol systems to monotone nonlinear is from conservation laws theory. Note that both of these results hold true only for monotone nonlinear damping, proved, using an infinite-dimensional version of LaSalle’s invariance principle. These been then hold true only for monotone invariance principle. These results results have have been then exex- results from laws theory. Note nonlinear that bothdamping, of these proved, using an general infinite-dimensional version of LaSalle’s resultsconservation hold truecase only for monotone nonlinear damping, is not of our paper. invariance principle. These results have been then extended to more infinite-dimensional systems in tended to more general infinite-dimensional systems in which which ishold not the the case offor ourmonotone paper. nonlinear damping, results true only invariance principle. These results have been then exwhich is not the case of our paper. tended to more general infinite-dimensional systems in This works has been partially supported by Advanced Grant after aa general This works which not the caseintroducing of our paper. tended to more general infinite-dimensional systems in In In our our ispaper, paper, after introducing general nonlinear nonlinear nonnonhas been partially supported by Advanced Grant DYCON (Dynamic Control) of the European Research Council In our paper, after introducing general nonlinear nonThis works has been partially supported by Advanced Grant monotone damping, we propose a well-posedness analysis DYCON (Dynamic Control) of the European Research Council monotone damping, we propose aa well-posedness analysis In our paper, after introducing general nonlinear nonThis works has been partially supported by Advanced Grant Executive Agency. DYCON (Dynamic monotone damping, we propose a well-posedness analysis Executive Agency. Control) of the European Research Council DYCON monotone damping, we propose a well-posedness analysis Executive (Dynamic Agency. Control) of the European Research Council Executive 2405-8963 Agency. © 2019 2019, IFAC IFAC (International Federation of Automatic Control) Copyright © 38 Hosting by Elsevier Ltd. All rights reserved. Copyright 2019 IFAC 38 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2019 IFAC 38 10.1016/j.ifacol.2019.11.752 Copyright © 2019 IFAC 38
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37
of a one-dimensional wave equation subject to such a damping. Futhermore, using Lyapunov techniques, we are able to characterize the trajectory of this system in the Lp , p ∈ [2, ∞), provided that the initial conditions are in L∞ . This proof is mainly based on a result about linear timevarying infinite-dimensional system, which is introduced and proved in Appendix A.
We refer the reader to Marx et al. (2018) for a complete discussion on this topic. Example 1. (Example of nonlinear dampings). Below are listed some examples of nonlinear dampings:
This paper is organized as follows. In Section 2, the main results of the paper are collected. To be more specific, we propose a well-posedness and an asymptotic stability theorems. The second result proposes futhermore a precise characterization of the trajectories of the system. Section 3 is devoted to the proof of the main results. Finally, Section 4 collects some concluding remarks, together with further research lines to be followed. Appendix A introduces a result of independant interest about exponential convergence of a specific time-varying infinite-dimensional linear systems, but which is instrumental for the proof of our asymptotic stability theorems.
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1. The classical saturation, defined as follows: if |s| ≤ 1, s s σ(s) = sat(s) := if |s| ≥ 1, |s| satisfies all the properties of Definition 1. 2. The following nonlinearity 1 1 s− sin(10s) σ(s) = sat 4 30
(3)
is also a nonlinear damping. Note moreover that it is not monotone, as illustrated by Figure 1.
1
Notation: For any p ∈ [2, ∞), the space Lp (0, 1) denotes p1 1 < +∞. the space of functions f satisfying 0 |f (x)|p dx ∞ The space L (0, 1) denotes the space of functions satisfying ess supx∈[0,1] |f (x)| ≤ +∞. For any p ∈ [2, ∞], the Sobolev space W 1,p (0, 1) (resp. W 2,p (0, 1)) is defined as follows W01,p (0, 1) := {f ∈ Lp (0, 1) | f ∈ Lp (0, 1) and f (0) = f (1) = 0} (resp. W 2,p (0, 1) := {f ∈ Lp (0, 1) | f , f ∈ Lp (0, 1)}).
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6
Acknowledgment: The two first authors of the paper would like to thank Prof. Enrique Zuazua for having kindly invited them in DeustoTech and for having pointed out the reference Haraux (2009). We would like also to thank Nicolas Burq for interesting discussions.
-0.8 -1 -5
0
5
Fig. 1. For any s ∈ [−5, 5], the figure illustrates the function σ given by (3)
2. MAIN RESULTS As illustrated in Marx et al. (2018), some regularity is needed to obtain a characterization of the asymptotic stability of (1). To be more precise, we need the state zt to be bounded in L∞ (0, 1). With a monotone nonlinearity σ, one would have this regularity result thanks to some nonlinear semigroup theorems. In the case of the system under consideration in this paper, we need to follow another strategy.
The aim of this paper is to provide an asymptotic stability analysis of the following system: ztt (t, x) = zxx (t, x) − a(x)σ( a(x)zt (t, x)), (t, x) ∈ R+ × [0, 1] (1) z(t, 0) = z(t, 1) = 0, t ∈ R+ z(0, x) = z0 (x), zt (0, x) = z1 (x), x ∈ [0, 1],
where z denotes the state, a : [0, 1] → R+ is measurable and bounded by some positive constant a∞ and σ is a scalar nonlinear damping which satisfies the following properties: Definition 1. [Scalar nonlinear damping] A function σ : R → R is said to be a scalar damping function if 1. 2. 3. 4.
Our strategy relies on the introduction of these functional spaces: Hp (0, 1) := W01,p (0, 1) × Lp (0, 1)
Dp (0, 1) := W 2,p (0, 1) ∩ W01,p (0, 1) × W01,p (0, 1),
(4)
with p ∈ [2, ∞]. The first functional space is equipped with the following norm: 1 p1 p (z, zt )Hp (0,1) := [zx (x)| dx
It is locally Lipschitz and odd; One has σ(0) = 0; For any s ∈ R, σ(s)s > 0; The function σ is differentiable at s = 0 with σ (0) = C1 for some C1 > 0.
0
+
Due to this definition (especially, item 2), the origin is an equilibrium point for (1). Note that this nonlinearity is not assumed to be monotone. The monotone property is in many cases really useful for either the well-posedness of the equation or the asymptotic stability of the origin.
1
p
0
|zt (x)| dx
p1
, ∀p ∈ [2, ∞),
(z, zt )H∞ (0,1) :=zx L∞ (0,1) + zt L∞ (0,1) , for p = ∞. (5) The second one is equipped with the following norm 39
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(z, zt ) Dp (0,1) :=
+
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1 p
0
|zxx (x)| dx 1 p
0
p1
|ztx (x)| dx
p1
, ∀p ∈ [2, ∞)
(z, zt ) D∞ (0,1) := zxx L∞ (0,1) + zt,x L∞ (0,1) , for p = ∞. (6) Considering that (z0 , z1 ) ∈ H∞ (0, 1), our aim is to prove an L∞ regularity on the state zt so that we can obtain a characterization of the asymptotic behavior of the trajectory in Hp (0, 1), with p ∈ [2, ∞). But before stating such a result, we need a suitable notion of solution for systems modeled with (1). Such a notion is provided by the following well-posedness theorem: Theorem 1. [Well-posedness] For any initial conditions (z0 , z1 ) ∈ H∞ (0, 1), there exists a unique solution (z, zt ) ∈ L∞ (R+ ; W 1,∞ (0, 1)) × W 1,∞ (R+ ; L∞ (0, 1)) to (1). Moreover, the following inequality is satisfied, for all t ≥ 0 (z, zt ) H∞ (0,1) ≤ 2 max( z0 L∞ (0,1) , z1 L∞ (0,1) ). 1 (7) Now that the functional setting is introduced, we are in position to state our asymptotic stability result Theorem 2. [Semi-global exponential stability] Consider initial conditions (z0 , z1 ) ∈ H∞ (0, 1) satisfying: (8) (z0 , z1 ) H∞ (0,1) ≤ R,
where R is a positive constant. Then, for any p ∈ [2, ∞), there exist two positive constants K := K(R) and β := β(R) such that (z, zt ) Hp (0,1) ≤ Ke−βt (z0 , z1 ) Hp (0,1) ,
∀t ≥ 0.
(9)
where a is a 2-periodic extension of a, y ∈ L2 (0, T ; L2 (0, 1)) and σ is a scalar nonlinear damping (which is odd, due to Item 1 of Definition 1). In particular, it means that h ∈ L2 (R+ ; L2 (0, 1)). The proof of Theorem 1 consists first in applying a fixed-point theorem, which will allow us to prove the well-posedness of (1) for a small time T > 0 and second in using a stability result in Haraux (2009), stated as follows Proposition 1. Let us consider initial condition z0 , z1 ∈ H∞ (0, 1). If there exists a solution to (1), then the time derivative of the following functional along the trajectories of (1) 1 (F (zt + zx ) + F (zt − zx )) dx, (16) φ(z, zt ) :=
3. PROOF OF THE MAIN RESULTS 3.1 Proof of Theorem 1 The proof of Theorem 1 relies on the following wave equation with a source term ztt (t, x) = zxx (t, x) + h(t, x), (t, x) ∈ R+ × [0, 1] z(t, 0) = z(t, 1) = 0, t ∈ R+ (10) z(0, x) = z0 (x), zt (0, x) = z1 (x), x ∈ [0, 1],
where h denotes the source term. From (Haraux, 2018, Theorem 1.3.8), we know that, provided that z0 , z1 ∈ H2 and that h ∈ L2 (R+ ; L2 (0, 1)), there exists a unique solution z ∈ C(R+ ; H01 (0, 1)) ∩ C 1 (R+ ; L2 (0, 1)) to (10). In particular, since H∞ (0, 1) ⊂ H2 , this result holds true also for initial conditions (z0 , z1 ) ∈ H∞ . The first step of our analysis in this section is to prove that picking initial conditions in H∞ (0, 1) and the source term h ∈ L2 (R+ ; L∞ (0, 1)) improves also the regularity of the solution z itself.
0
with F any even and convex function, satisfies d φ(z, zt ) ≤ 0. dt
The latter proposition implies that (17) φ(z, zt ) ≤ φ(z0 , z1 ), ∀t ≥ 0. In particular, following the discussion in the proof of Corollary 2.3. in Haraux (2009), if one picks 2 F (s) = [Pos(|s|− 2 max( z0 L∞ (0,1) , z1 L∞ (0,1) ))]2 , then one obtains that
To do so, our aim is to give an explicit formula for the latter equation, using the reflection method surveyed in Strauss (1992). Roughly speaking, this method consists in extending the explicit formulation of trajectory of the wave equation in an unbounded domain to a bounded domain. To do so, we extend the initial datas to the whole line to be odd with respect to both x = 0 and x = 1, that is z0 (x) and z0 (2 − x) = − z0 (x), (11) z0 (−x) = − 1
where z0 denotes the 2-periodic odd extension of z0 . A way to do this is to define z0 as follows: z0 (x), 0 < x < 1 − z0 (−x), −l < x < 0 (12) z0 (x) = extended to be of period 2 We can define similarly a 2-periodic odd extension of z1 (resp. h), denoted by z1 (resp. h). Thanks to (Strauss, 1992, Theorem 1, Page 69), we can therefore define the explicit trajectory z of (10) (known as the D’Alembert formula) as follows: 1 x+ct 1 z0 (x + t) + z0 (x − t)] + z1 (s)ds z(t, x) = [ 2 2 x−ct (13) 1 t x+(t−s) + h(w, s)dwds. 2 0 x−(t−s) We can further define zt as follows 1 zt (t, x) = ( z (x + t) − z0 (x − t)) 2 0 1 z1 (x + t) − z1 (x − t)) + ( 2 1 t + h(s, x + (t − s)) − h(s, x − (t − s)) ds 2 0 (14) It is clear from these two latter equations that, when picking (z0 , z1 ) ∈ H∞ (0, 1) and h ∈ L2 (R+ ; L∞ (0, 1)), then z ∈ C(R+ ; W 1,∞ (0, 1)) ∩ C 1 (R+ ; L∞ (0, 1)). We assume now that h is written as follows h(t, x) := −σ( a(x)y(t, x)), (15)
φ(z, zt ) = 0,
2
The function Pos : s ∈ R → Pos(s) ∈ R+ is defined as follows: Pos(s) =
The term z0 denotes the derivative of z0 with respect to x.
40
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s if s > 0 0 if s ≤ 0.
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which implies that, for all t ≥ 0 max( zx (t, ·) L∞ (0,1) , zt (t, ·) L∞ (0,1) ) ≤ 2 max( z0 L∞ (0,1) , z1 L∞ (0,1) ). (19)
39
since BK (y0 ) is complete, any Cauchy sequence converges in this set. By induction, one can prove that (27) yn+1 − yn T ≤ (C(y0 , K)T )n y0 − y1 T and (28) yn ∈ BK (y0 ). Indeed, thanks to the choice of T in (26), these two properties are easily proved for n = 0 and, moreover, we have with (24), for all n ≥ 1 yn+1 − yn T = φ(yn ) − φ(yn−1 ) T ≤C(y0 , K)T φ(yn−1 ) − φ(yn−2 ) T (29) ≤(C(y0 , K)T )n y0 − y1 T .
Noticing that (z, zt ) H∞ (0,1) ≤ max( zx (t, ·) L∞ (0,1) , zt (t, ·) L∞ (0,1) ), it is clear then that, for all t ≥ 0 (z, zt ) H∞ (0,1) ≤ 2 max( z0 L∞ (0,1) , z1 L∞ (0,1) ) (20) This estimate implies that, once one is able to prove that there exists a solution (z, zt ) of (1) in L∞ ([0, T ]; W 1,∞ (0, 1)) ∩ W 1,∞ ([0, T ]; L∞ (0, 1)), for a small time T > 0, then the well-posedness of (1) is ensured in L∞ (R+ ; W 1,∞ (0, 1)) ∩ W 1,∞ (R+ ; L∞ (0, 1)).
The inequality (27) can be deduced from the above inequality. The property (28) can be proved as follows: yn+1 ≤ yn T + y1 T + y0 T ≤ y0 T + φT (y0 ) T + φT (y0 ) T + y0 T (30) ≤K
Now, we are in position to prove Theorem 1.
Proof. Let us define FT the space of measurable functions defined on [0, T ]×R which are bounded, odd and 2-periodic in space. We endow FT with the L∞ -norm so that it becomes a Banach space. Hence, denoting by · T the norm of the latter functional space, we have, for every y ∈ FT y T := sup |y(t, x)|. (21)
where in the first line we have used (27) and, in the second line, we have used the fact that yn ∈ BK (y0 ).
The two properties (27) and (28) show that the sequence yn is a Cauchy sequence. Since BK (y0 ) is a complete set, this sequence is therefore convergent. This means in particular that there exists a fixed-point to the mapping φT (y0 ), which implies that, for sufficiently small time T , there exists a unique solution z ∈ L∞ (0, T ; W 1,∞ (0, 1)) and zt ∈ L∞ (0, T ; L∞ (0, 1)). Thanks to (20), we can deduce that there exists unique solution z ∈ L∞ (R+ ; W 1,∞ (0, 1)) and zt ∈ L∞ (R+ ; L∞ (0, 1)). This concludes the proof of Theorem 1. •
(t,x)∈[0,T ]×R
Let us consider BK (y) the closed ball in FT centered at y ∈ FT of radius K ≥ 0, where K remains to be defined.
We can now define the mapping with which we will apply a fixed-point φT : FT → FT (22) y → φT (y),
3.2 Proof of Theorem 2
where
1 1 φT (y) = ( z0 (x + t) − z 0 (x − t)) + ( z1 (x + t) − z 1 (x − t)) 2 t 2 1 − a(x + t − s)σ( a(x + t − s)y(s, x − (t − s)) 2 0 + a(−x + t − s)σ( a(−x + t − s)y(s, −x − (t − s)) ds (23)
Using the fact that σ is locally Lipschitz, note that, for every y0 ∈ FT and K > 0, there exists a positive constant C(y0 , K) such that, for every y, y ∈ BK (y0 ) φT (y) − φT ( y ) T ≤ C(y0 , K)T y − y T . (24) Pick K such that K := 2 ( φT (y0 ) T + y0 T ) (25) and T sufficiently small such that C(y0 , K)T ≤ 1. (26) Consider a sequence (yn )n∈N defined as yn+1 = φT (yn ).
The proof of Theorem 2 is divided into two steps: first, we transform the system (1) into a system in the form (A.1), which is in particular a linear time-variant system, and apply Theorem 3 for the space H2 , which is indeed the only Hilbert space among all the spaces Hp (0, 1). Second, using the fact that the solutions are bounded in H∞ (0, 1) thanks to Theorem 1, and invoking an interpolation theorem, which is the Riesz-Thorin theorem, we conclude. Proof. First step: semi-global exponential stability in H2 . We first fix p = 2, but still consider the initial conditions (z0 , z1 ) ∈ H∞ (0, 1). Moreover, we suppose that there exists a positive constant R such that the initial conditions satisfy (31) (z0 , z1 ) H∞ (0,1) ≤ R. In particular, due to Theorem 1, one has (32) (z, zt ) H∞ (0,1) ≤ 2R, ∀t ≥ 0.
If we prove that this sequence converges to y , then we can deduce that y = φT (y ). This means that there exists a fixed-point for the mapping φT , which implies in particular that (1) is well-posed in the desired functional spaces. To prove the convergence of this sequence, we prove that it is a Cauchy sequence. Indeed, 41
The system (1) may be rewritten as follows ztt = zxx − d(t, x)a(x)zt , z(t, 0) = z(t, 1) = 0, z(0, x) = z0 (x), zt (0, x) = z1 (x), with σ( a(x)zt ) , if a(x)zt = 0, d(t, x) = a(x)zt if a(x)zt = 0. C1 ,
(33)
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This function comes from the function t →
√ √ a(x)zt ,
for every (z0 , z1 ) ∈ H∞ (0, 1), one therefore has, for every t≥0 W (t, 0)(z0 , z1 )Hp (0,1) ≤W (t, 0)Hp (0,1) (z0 , z1 )Hp (0,1) p2 2β K ≤2 e− p t (z0 , z1 )Hp (0,1) . 2 This concludes the proof of Theorem 2. •
σ(
a(x)zt
extended at 0, which is possible due to the differentiability of the function σ at 0.
This system is in the form (A.1), with H = H2 , U = H2 , D(A) = D2 , and the operators A and B defined as follows A : D(A) ⊂ H → H (35) [v1 v2 ] → v2 v1 and B = 0 a(x) . It is well known that A generates a strongly continuous semigroup of contractions (see Haraux (2018)). Now, let us check whether d(t, ·) satisfies (A.2), for all t ≥ 0.
4. CONCLUSION In this paper, we have provided a well-posedness analysis of a one-dimensional wave equation subject to a nonlinear nonmonotone damping. Futhermore, a characterization of the asymptotic behavior of the latter system is given. It is proved with Lyapunov techniques. This work paves the way to many others. For instance, a first open question would be the case of multidimensional wave equations, for which there does not exist any proof of well-posedness in the functional setting introduced in this paper.
Since the initial conditions (z0 , z1 ) ∈ H∞ (0, 1) and are bounded by R in the H∞ (0, 1) norm, then invoking Theorem 1 we have, for all t ≥ 0 (36) sup |zt (t, x)| ≤ 2R. x∈[0,1]
√ Moreover, since a(x) ≤ a∞ , for all x ∈ [0, 1], and because d is a continuous function, there exist two positive constants d0 and d1 , depending on R and a∞ such that σ(ξ) ≤ d(t, x) d0 := √ min √ ξ∈[−2 a∞ R,2 a∞ R] ξ σ(ξ) ≤ := d1 . √ max √ ξ∈[−2 a∞ R,2 a∞ R] ξ (37)
Appendix A. EXPONENTIAL CONVERGENCE RESULT FOR A LINEAR TIME-VARIANT SYSTEM This appendix is devoted to the statement and the proof of a theorem dealing with a time-variant linear infinitedimensional system. Indeed, as it is illustrated in Section 3.2, we can transform (1) as a time-variant linear infinitedimensional system. To define it, let us introduce H (resp. U ), a Hilbert space which is equipped with the norm · H and the scalar product ·, ·H . The system under study in this section is the following: d v = (A − d(t)BB )v := Ad (t)v (A.1) dt v(τ ) = v , τ
Note moreover that the origin of the following system zt = zxx − d0 zt z(t, 0) = z(t, 1) = 0 (38) z(0, x) = z0 (x), zt (0, x) = z1 (x).
where A : D(A) ⊂ H → H, with D(A) the domain of the operator A that we suppose densely defined in H, B ∈ L(U, H). We assume moreover that A and its adjoint A are dissipative, which implies in particular that A generates a strongly continuous semigroup of contractions. Let us also assume that d is a strongly continuous mapping from any interval of time [0, T ], with T > 0, into L(H) (resp. L(D(A)), meaning that d(·)v belongs to C([0, T ]; H) for any v ∈ H (resp. C([0, T ]; D(A)) for any v ∈ D(A)). This latter implies in particular, from (Bensoussan et al., 2007, Proposition 3.6., page 138), that there exists a unique mild (resp. strong) solution to (A.1) if vτ ∈ H (resp. vτ ∈ D(A)), and that Ad (t) generates an evolution family (W (θ, τ ))θ≥τ 3 . This means that every trajectories of (A.1) can be expressed as follows v = W (θ, τ )vτ .
is exponentially stable in H2 for any initial conditions (z0 , z1 ) ∈ H2 (see e.g. Prieur et al. (2016)). The related operator of this system is A − d0 BB , with domain D(Ad0 ) = D2 . Therefore, all the properties required in Theorem 3 are satisfied. Hence, there exist two positive constants K := K(R) and β := β(R) such that (z, zt )H2 ≤ Ke−βt (z0 , z1 )H2 ,
∀t ≥ 0.
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Second step: Semi-global exponential stability in Hp (0, 1). From Theorem 1, we know that, for every initial conditions (z0 , z1 ) ∈ H∞ (0, 1) satisfying (z0 , z1 )H∞ (0,1) ≤ R, and noticing that the trajectory of (1) can be expressed with the evolution family W (t, 0), one has (40) W (t, 0)(z0 , z1 )H∞ (0,1) ≤ 2R, ∀t ≥ 0.
Assume moreover that there exists two positive constants d0 and d1 such that, for all t ≥ 0 d0 ≤ d(t) ≤ d1 . (A.2)
Now, fix t > 0. Note that W (t, 0) is an operator from (L2 (0, 1))2 (resp. (L∞ (0, 1))2 ) to (L2 (0, 1))2 (resp. (L∞ (0, 1))2 ), if it associates (z0 , z1 ) ∈ L2 (0, 1)2 (resp. (z0 , z1 ) ∈ L∞ (0, 1)2 ) to (zx , zt ) ∈ L2 (0, 1) (resp. (zx , zt ) ∈ L∞ (0, 1)2 ). Hence, we can apply the so-called Riesz-Thorin theorem (Bergh and L¨ ofstr¨ om, 2012, Theorem 1.1.1, Page 2) and conclude that p2 2β K e− p t , ∀t ≥ 0, (41) W (t, 0)Hp (0,1) ≤ 2 2 where W (t, 0) corresponds exactly to the trajectory of (1) with the initial condition z0 , z1 ∈ H∞ (0, 1). In particular,
We are now in position to state the following result. Theorem 3. Consider the system given by (A.1). Supposing that the origin of the following system d v = (A − d0 BB )v := Ad0 v, (A.3) dt v(0) = v 0
3 See Chicone and Latushkin (1999) for more details on the concept of evolution families.
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2019 IFAC NOLCOS Vienna, Austria, Sept. 4-6, 2019
Swann Marx et al. / IFAC PapersOnLine 52-16 (2019) 36–41
is globally exponentially stable, then, for any initial conditions v0 ∈ H, the origin of (A.1) converges exponentially to 0 with τ = 0. Remark 1. The global asymptotic stability is not ensured for the linear time-variant system (A.1). Indeed, for such systems, the global asymptotic stability has to be proven for every initial condition vτ and for every τ ≥ 0. This is not an issue for us, since the stability of (1) already holds thanks to Theorem 1. ◦
one can conclude that P L(H) + M C −ε 2 vH ≤ exp − t v0 2H , M P L(H) + M (A.13) which ends the proof of Theorem 3. • REFERENCES
Proof. Since the origin of (A.3) is globally exponentially stable, then, due to Datko (1970), there exist a self-adjoint operator P ∈ L(H) and a positive constant C such that the following inequality holds true P Ad0 v, vH + P v, Ad0 vH ≤ −Cv2H , ∀v ∈ D(A) (A.4) Moreover, note that Ad0 is also a dissipative operator, which means in particular that (A.5) Ad0 v, vH + v, Ad0 vH ≤ 0, ∀v ∈ D(A). Now, consider the following candidate Lyapunov functional for (A.1): (A.6) V (v) := P v, vH + M v2H , where M is a positive constant which has to be defined. The time derivative of V along the trajectories of (A.1) yields d V (v) =(P + M IH )Ad(t) v, vH + (P + M IH )v, Ad(t) vH dt =P Ad0 v, vH + P v, Ad0 vH + M (Ad0 v, vH + v, Ad0 vH ) − (d(t) − d0 IH )BB v, (P + M IH )vH − (P + M IH )v, (d(t) − d0 IH )BB vH . ≤ − Cv2H − d0 M B v2U + 2(d1 − d0 )P L(H) BB L(H) B vU vH − 2M (d(t) − d0 )B v, B vU ,
(A.7) Since d(t) ≥ d0 and M > 0, one has 2M (d(t) − d0 )B v, B vU . Moreover, applying the Young inequality, we known that there exists a positive number ε such that d V (v) ≤ − Cv2H − d0 M B v2U dt (d1 − d0 ) BB L(H) P L(H) v2H + ε + ε(d1 − d0 )
BB L(H) P L(H) B v2U (A.8)
Setting (d1 − d0 )P L(H) ε> C and
BB L(H)
,
ε(d1 − d0 )P L(H) BB L(H) M> , d0 one obtains d V (v) ≤ −(C − ε)v2H , ∀v ∈ D(A). dt Since V satisfies the following inequalities M v2H ≤ V (z) ≤ (P L(H) + M )v2H
41
(A.9)
(A.10)
(A.11) (A.12) 43
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