dissipative boundary conditions for one-dimensional quasi-linear ...
DISSIPATIVE BOUNDARY CONDITIONS FOR ONE-DIMENSIONAL QUASI-LINEAR HYPERBOLIC SYSTEMS : LYAPUNOV STABILITY FOR THE C 1 -NORM∗ JEAN-MICHEL CORON† AND GEORGES BASTIN‡ Abstract. This paper is concerned with boundary dissipative conditions that guarantee the exponential stability of classical solutions of one-dimensional quasi-linear hyperbolic systems. We present a comprehensive review of the results that are available in the literature. The main result of the paper is then to supplement these previous results by showing how a new Lyapunov stability approach can be used for the analysis of boundary conditions that are known to be dissipative for the C 1 -norm. Key words. hyperbolic systems, Lyapunov stability
1. Introduction. We are concerned with one-dimensional quasi-linear strictly hyperbolic systems of the form (1.1)
ut + A(u)ux = 0, x ∈ [0, 1], t ∈ [0, +∞),
where u : [0, ∞) × [0, 1] → Rn and A : Rn → Mn,n (R), Mn,n (R) denoting, as usual, the set of n × n real matrices. Since the system is strictly hyperbolic, the matrix A(0) has, by definition, n real eigenvalues denoted Λ1 , . . . , Λn satisfying (1.2)
Λi 6= Λj , ∀i, j ∈ {1, . . . , n} such that i 6= j.
We assume that none of these eigenvalues is zero. Then, possibly after a suitable linear change of variables, we may assume that there exists m ∈ {0, 1 . . . , n} such that Λi > 0, ∀i ∈ {1, . . . , m}, Λi < 0, ∀i ∈ {m + 1, . . . , n}, and that A(0) is a diagonal matrix: A(0) , diag{Λ1 , Λ2 , . . . , Λn }. Our concern is to analyse the asymptotic convergence of the classical solutions of the system (1.1) under a boundary condition of the form u+ (t, 0) u+ (t, 1) (1.3) =G , t ∈ [0, +∞), u− (t, 1) u− (t, 0) where the map G : Rn → Rn vanishes at 0, while u+ ∈ Rm , u− ∈ Rn−m are defined by requiring that u , (uT+ , uT− )T . The challenge is to give explicit conditions on the ∗ This work was supported by the ERC advanced grant 266907 (CPDENL) of the 7th Research Framework Programme (FP7) and by the Belgian Programme on Interuniversity Attraction Poles (IAP VII/19) † Sorbonne Universit´ es, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 4 place Jussieu, 75252 Paris cedex 05, France. ([email protected]). ‡ Department of Mathematical Engineering, ICTEAM, Universit´ e catholique de Louvain, 4, avenue G. Lemaitre, 1348 Louvain-la-Neuve, Belgium. ([email protected]). Questions, comments, or corrections to this document may be directed to that email address.
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map G such that the boundary condition (1.3) is exponentially dissipative, i.e. implies that the equilibrium solution u ≡ 0 of system (1.1) with the boundary condition (1.3) is exponentially stable. In order to state our results, we first introduce the functions ρp : Mn,n (R) → R defined by n o (1.4) ρp (M ) , inf k∆M ∆−1 kp , ∆ ∈ Dn+ , 1 6 p 6 ∞, where Dn+ denotes the set of diagonal n×n real matrices with strictly positive diagonal entries and with " n # p1 X p , kξk∞ , max{|ξi |; i ∈ {1, . . . , }}, ∀ξ , (ξ1 , . . . , ξn )T ∈ Rn , kξkp , |ξi | i=1
and kM kp , max kM ξkp ∀M ∈ Mn,n (R). kξkp =1
In this paper, our main result (Section 3) is a new proof, using a Lyapunov function approach, that a sufficient condition for the exponential stability of the steadystate u ≡ 0 for the C 1 -norm is to have a map G such that: ρ∞ (G0 (0)) < 1, where G0 denotes the Jacobian matrix of the map G. This new proof supplements the previous results which are already available in the literature and are reviewed in Section 2. 2. Literature review. The well-posedness of the Cauchy problem associated to nonlinear hyperbolic systems of the kind (1.1) and (1.3) has been studied in 1985 by Li and Yu[25] in the framework of C 1 -solutions. For such systems, the issue of finding sufficient dissipative boundary conditions has been addressed in the literature for more than thirty years. To our knowledge, first results were published by Slemrod[39] in 1983 and by Greenberg and Li[16] in 1984 for the special case of systems of size n = 2. A generalization to systems of size n was then progressively elaborated by the Ta-Tsien Li school, in particular by Qin[34] in 1985 and by Zhao[44] in 1986. All these contributions deal with the particular case of ‘local’ boundary conditions having the specific form (2.1)
u+ (t, 0) = G0 (u− (t, 0)),
u− (t, 1) = G1 (u+ (t, 1)).
With these boundary conditions, the analysis can be based on the method of characteristics which can exploit an explicit computation of the ‘reflection’ of the solutions at the boundaries along the characteristic curves. This has given rise to the sufficient condition !! 0 G00 (0) (2.2) ρ∞ < 1, G01 (0) 0 for the dissipativity of the boundary conditions (2.1) for the C 1 -norm. This result is given for instance by Li[23, Theorem 1.3, page 173] in his seminal book of 1994 on the stability of the classical solutions of quasi-linear hyperbolic systems. Finally, by using an appropriate dummy doubling of the system size, de Halleux et al.[12, Theorem 4] have shown in 2003 how the general dissipative boundary condition ρ∞ (G0 (0)) < 1 can be established for systems with the general ‘non local’ boundary condition (1.3). This
Dissipative boundary conditions for hyperbolic systems: Lyapunov stability
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dummy doubling has also been used by Li et al.[24] to prove the well-posedness of the Cauchy problem associated to (1.1) and (1.3) still in the framework of C 1 -solutions. Another approach of the analysis of dissipative boundary conditions is based on the use of Lyapunov functions. The first attempts were in using entropies as Lyapunov functions, as done for instance by Coron et al.[8] in 1999 or by Leugering and Schmidt[22] in 2002. The drawback of this approach was however that the time derivatives of such entropy-based Lyapunov functions are necessarily only semidefinite negative. Hence one would like to conclude with the LaSalle invariant set principle. However, this principle requires the precompactness of the trajectories, a property which is difficult to get in the case of nonlinear partial differential equations. In order to overcome this difficulty, Coron et al.[9] in 2007 have proposed, for systems of size n = 2, a strict Lyapunov function whose time derivative is strictly negative definite when (2.3)
ρ∞ (G0 (0)) < 1.
The advantage is that the proof is less elaborated than the one using the method of characteristics because it uses more direct computations. Furthermore, another advantage of a Lyapunov analysis is to directly induce robustness properties with respect to small uncertainties and disturbances. Then, in the paper[7], Coron et al. have generalized in 2008 this Lyapunov approach to general nonlinear hyperbolic systems of the kind (1.1) and (1.3). In particular, they emphasize a new weaker dissipative boundary condition which is formulated as follows: A sufficient condition for the exponential stability of the steady-state u ≡ 0 for the H 2 -norm is to have a map G such that: (2.4)
ρ2 (G0 (0)) < 1.
Moreover, it is also shown in the same paper that ρ2 (G0 (0)) 6 ρ∞ (G0 (0)) ∀n > 1 and that this inequality can be strict if n > 2. However, it has been proved in [10] that ρ2 (G0 (0)) < 1 is not a sufficient condition for the exponential stability of the steady-state u ≡ 0 for the C 1 -norm. Various recent contributions and extensions of the previous results are also worth to be mentioned. • Dos Santos and Maschke[37] in 2009 have given an Hamiltonian perspective to the stabilization of systems of two conservation laws. • Using the Lyapunov approach, Castillo et al.[6] in 2013 give sufficient conditions for the existence of exponentially stable observers in the case where all eigenvalues Λi are positive. • Perrollaz and Rosier[29] have shown in 2013 that there exist boundary dissipative conditions for systems of size n = 2 that achieve finite-time stabilization. • Using an approach via time delay equations, Coron and Nguyen[10] have shown in 2014 that ρp (G0 (0)) < 1 implies the exponential stability in the Sobolev norm W 2,p . For so-called inhomogeneous quasi-linear hyperbolic systems (i.e. with additional zero-order terms): ut + A(u)ux + B(u) = 0,
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the analysis of dissipative boundary conditions is much more intricate and only very partial results are known. • Using the method of characteristics, Prieur et al.[33] in 2008 have shown that the stability condition (2.2) holds for inhomogeneous systems when kB(u)k is small enough. Prieur[30] has shown in 2009 how this result can be extended to deal with differential or integral boundary errors. • Using the Lyapunov approach, Gugat and Herty[18] in 2009 and Gugat et al.[17] in 2011 analyze the boundary feedback stabilization of gas flow in fan-shaped networks described by isentropic Euler equations. • Using the Lyapunov approach, Pavel and Chang[18] in 2012 analyze the boundary control of hyperbolic Lotka-Volterra systems, with application to pumped Raman amplifiers on optical fibres (see also Dower and Farrell[15] in 2006). • For inhomogeneous systems of size n = 2 with m = 1, Coron et al.[11] have shown in 2012 the existence of full-state feedback law which achieve exponential stability for the H 2 -norm. The proof uses a backstepping transformation (see [21]) to find new variables for which a strict Lyapunov function can be constructed. Linear hyperbolic systems. A linear hyperbolic system, (2.5)
ut + Aux = 0, x ∈ [0, 1], t ∈ [0, +∞),
with linear boundary conditions u+ (t, 0) u+ (t, 1) (2.6) =K , t ∈ [0, +∞), u− (t, 1) u− (t, 0) is a special case of (1.1) and (1.3) where the matrix A ∈ Dn (R) with non-zero distinct diagonal entries, Dn (R) denoting the set of n × n diagonal real matrices, and the matrix K ∈ Mn,n (R). As it can be expected, the analysis of dissipative boundary conditions is both simpler and more comprehensive for linear than for quasi-linear systems. The linear hyperbolic system (2.5)-(2.6) is equivalent to a system of interconnected scalar time-delays τi , Λ−1 i . For such systems, it follows from a theorem by Silkowski[38] (quoted in[19, Chapter 9, Theorem 6.1], see also [28]), which relies on the Kronecker density theorem (e.g.[5]), that the boundary conditions (2.6) are, for any Lp -norm, robustly dissipative with respect to arbitrary small perturbations on the Λi ’s if and only if (2.7)
ρ¯(K) , max{ρ(diag(eιθ1 , . . . , eιθn )K; (θ1 , . . . , θn )T ∈ Rn } < 1,
where ι ∈ C is such that ι2 = −1 and ρ(M ) is the spectral radius of M ∈ Mn,n (C). It is shown in [3, 7] that ρ¯(K) = ρ2 (K) if there exists a permutation matrix P such ˜ = P KP −1 is a block diagonal matrix K ˜ = diag{K ˜ 1, K ˜ 2, . . . , K ˜ p} that the matrix K ˜ where each block Ki is a real ni × ni matrix with ni ∈ {1, 2, 3, 4, 5}. Moreover, when n > 5, there always exist K such that ρ¯(K) < ρ2 (K). The exponential stability of non-uniform inhomogeneous linear hyperbolic systems has also received a lot of attention in the literature for a long time and can be
Dissipative boundary conditions for hyperbolic systems: Lyapunov stability
5
traced back up to Rauch and Taylor[35, Section 4] in 1974 or Russell[36, Section 3] in 1978. These systems are written as (2.8)
ut + A(x)ux + B(x)u = 0,
where A(x) ∈ Dn (R) and B(x) ∈ Mn,n (R), ∀x ∈ [0, 1]. Using the Lyapunov approach (as for instance Xu and Sallet[43] in 2002 or Diagne et al.[13] in 2011 for the uniform case), it can be shown that the equilibrium u ≡ 0 is exponentially stable for the L2 -norm if there exist a positive diagonal matrix P such that: |A+ (1)| 0 |A+ (0)| 0 (a) P − KT P K is positive definite; 0 |A− (0)| 0 |A− (1)| (b) P B(x) + BT (x)P − P ∂x |A(x)| is semi positive definite for all x ∈ [0, 1], where, for every x ∈ [0, 1], A+ (x) ∈ Dm (R) and A− (x) ∈ Dn−m (R) are defined by requiring A+ (x) 0 A(x) = . 0 A− (x) Obviously conditions (a) and (b) are rather conservative. Bastin and Coron[?] have given in 2011 a less restrictive sufficient condition for inhomogeneous linear systems of size n = 2 with m = 1 under the form of the existence of the solution of an associated ordinary differential equation. Other recent significant contributions are worth to be mentioned. • Tchousso et al.[41] have shown in 2009 how the Lyapunov approach can be extended to inhomogeneous linear hyperbolic systems of higher spatial dimension. • Litrico and Fromion[26] in 2009 use a frequency domain method for the boundary control of homogeneous systems of size n = 2. They show that the transfer functions belong to the Callier-Desoer algebra, which opens the way to necessary and sufficient conditions for closed loop stability and the use of a Nyquist type test. • Amin et al.[2] and Prieur et al.[31] have shown in 2012 how the Lyapunov approach can be extended to switched linear hyperbolic systems. • Prieur and Mazenc[32] in 2012 have shown how time varying strict Lyapunov functions can be defined to get input-to-state stability for time varying inhomogeneous linear hyperbolic systems. • Dick et al.[14] in 2012 address the feedback stabilization of quasi-linear hyperbolic systems with varying delays. • The use of a backstepping method for boundary stabilization by full state feedback and observer design for non uniform inhomogeneous linear systems has also been extensively studied recently: Krstic et al.[20] in 2008 and Smyshlyaev et al.[40] in 2010 for unstable wave equations, Vazquez et al.[42] in 2011 for general systems of size n = 2 with m = 1, Di Meglio et al.[27] in 2012 for systems with m = n−1 and a single controller at the right boundary, and Aamo[1] in 2013 for the rejection of differential boundary disturbances. • Using a frequency domain approach, Bastin et al.[4] in 2014 give necessary and sufficient conditions for the stability of linear density-flow systems under proportional-integral control.
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Our contribution in this paper fills a small gap in this panorama of the analysis of boundary dissipative conditions for one-dimensional linear and quasi-linear hyperbolic systems. We show how the Lyapunov approach can be used to establish the dissipativity of the boundary condition ρ∞ (G0 (0)) < 1. 3. Main result. We consider hyperbolic systems of the form ut + A(u)ux = 0, x ∈ [0, 1], t ∈ [0, +∞),
(3.1)
under a boundary condition of the form u+ (t, 0) u+ (t, 1) (3.2) =G , t ∈ [0, +∞). u− (t, 1) u− (t, 0) For σ, let Bσ be the open ball with radius σ in Rn for the norm |u|0 , max{|uj |; j ∈ {1, . . . , n}}, u = (u1 , . . . , un )T ∈ Rn .
(3.3)
We assume that, for some σ ∈ (0, +∞), A : Bσ → Mn,n (R) and G : Bσ → Rn are of class C 1 . We define A+ (u) ∈ Mm,n (R), A− (u) ∈ M(n−m),n (R), G+ (u) ∈ Rm and G− (u) ∈ Rn−m by requiring A+ (u) G+ (u) A(u) = , G(u) = . A− (u) G− (u) Our concern is to analyze the exponential stability of this system for the C 1 -norm under an initial condition u(0, x) = uo (x), x ∈ (0, 1)
(3.4)
which satisfies the compatibility conditions o o u+ (0) u+ (1) (3.5) =G , uo− (1) uo− (0)
(3.6)
(3.7)
o u+ (1) A+ (uo (1))uox (1) A+ (uo (0))uox (0) = G0+u+ uo− (0) o u+ (1) + G0+u− A− (uo (0))uox (0), uo− (0)
o
A− (u
(1))uox (1)
o u+ (1) 0 = G−u+ A+ (uo (1))uox (1) uo− (0) o u+ (1) 0 + G−u− A− (uo (0))uox (0). uo− (0)
Let us now define some norms which will be useful to state and prove our results. For u ∈ C 0 ([0, 1]; Rn ) (resp. in C 0 ([0, 1] × [T1 , T2 ]; Rn ), one defines (3.8)
|u|0 , max{|u(x)|0 ; x ∈ [0, 1]},
(3.9)
(resp. |u|0 , max{|u(t, x)|0 ; t ∈ [T1 , T2 ], x ∈ [0, 1]}).
Dissipative boundary conditions for hyperbolic systems: Lyapunov stability
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For u ∈ C 1 ([0, 1]; Rn ) (resp. in C 1 ([T1 , T2 ] × [0, 1]; Rn )), we define (3.10)
|u|1 , |u|0 + |u0 |0 ,
(3.11)
(resp. |u|1 , |u|0 + |∂t u|0 + |∂x u|0 ).
Let us first recall the following theorem, due to Li and Yu[25], on the wellposedness of the Cauchy problem (3.1)–(3.2)–(3.4). Theorem 3.1. Let T > 0. There exists C1 > 0 and ε1 > 0 such that, for every uo ∈ C 1 ([0, 1]; Rn ) satisfying the compatibility conditions (3.5) to (3.7) and such that (3.12)
|uo |1 6 ε1 ,
there exists one and only one solution on [0, T ] × [0, 1] to the Cauchy problem (3.1)– (3.2)–(3.4). Moreover, this solutions satisfies (3.13)
|u|1 6 C1 |u0 |1 .
(In fact, [25] deals with the case of local boundary conditions (2.1); however the general case follows from this particular case by using the dummy doubling of the system size introduced in de Halleux et al.[12], already mentioned in Section 2: see [24].) Our definition of exponential stability for the C 1 -norm is as follows. Definition 3.2. The steady state x ∈ [0, 1] 7→ 0 ∈ Rn of the system (3.1)–(3.2) is exponentially stable for the C 1 -norm if there exist ε > 0, ν > 0 and C > 0 such that, for every uo such that |uo |1 < ε and satisfying the compatibility conditions (3.5) to (3.7), the Cauchy problem (3.1)–(3.2)–(3.4) has a unique C 1 -solution which satisfies |u(t, .)|1 6 Ce−νt |uo |1 , ∀t ∈ [0, +∞).
The main goal of this article is to give a new proof of the following theorem due to Qin[34] and Zhao[44] (see also Li[23, Theorem 1.3, page 173] and de Halleux et al.[12, Theorem 4]) . Theorem 3.3. If (3.14)
ρ∞ (G0 (0)) < 1,
the steady state x ∈ [0, 1] 7→ 0 ∈ Rn of the system (3.1)–(3.2) is exponentially stable for the C 1 -norm. In the remaining part of this section we give our proof of Theorem 3.3. This proof is divided in two parts: • In Section 3.1, we study the case where all the Λi are positive, i.e. the case where m = n, • In Section 3.2, we explain how to modify the Lyapunov function introduced in Section 3.1 in order to treat the general case.
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3.1. Proof of Theorem 3.3 in the special case where m = n. For the clarity of the demonstration, we shall first prove the theorem in the special case where m = n, which means the matrix A(0) is the positive diagonal matrix diag{Λ1 , . . . , Λn } with Λi > 0 ∀i = 1, . . . , n. In that case, the boundary condition (3.2) and the compatibility conditions (3.5) to (3.7) are simply rewritten (3.15) (3.16)
u(t, 0) = G(u(t, 1)), A(uo (0))uox (0) = G0 (uo (1))A(uo (1))uox (1).
uo (0) = G(uo (1)),
As already mentioned above, our proof is based on a Lyapunov approach. In order to define an appropriate Lyapunov function for the analysis, we need the following technical classical lemma, for which our assumption (1.2) is crucial and which also holds if m 6= n. Lemma 3.4. Let D(u) be the diagonal matrix whose diagonal entries are the eigenvalues λi (u), i = 1, . . . , n, of the matrix A(u). There exist a positive real number η < σ and a map M : Bη → Mn,n (R) of class C 1 such that (3.17)
M (u)A(u) = D(u)M (u), ∀u ∈ Bη ,
(3.18)
M (0) = Idn ,
where Idn is the identity matrix of Mn,n (R). Let W1 : C 1 ([0, 1]; Rn ) → R and W2 : C 1 ([0, 1]; Rn ) → R be defined by (3.19)
Z W1 (u) ,
n 1X
0
(3.20)
Z W2 (u) ,
i=1
n 1X
0
ppi
i=1
ppi
n X
1 2p
!2p mij (u)uj
e−2pµx dx ,
j=1
n X
1 2p
!2p mij (u)∂t uj
e−2pµx dx ,
j=1
with p ∈ N+ , {1, 2, 3, . . . } and pi > 0 ∀i ∈ {1, . . . , n}. In (3.19) and (3.20), mij (u) denotes the (i, j)-th entry of the matrix M (u) and, in (3.20), ∂t uj is defined by (∂t u1 , . . . , ∂t un )T = ∂t u, with (3.21)
∂t u , −A(u)∂x u, ∀u ∈ C 1 ([0, 1]; Rn ) with |u|0 small enough.
We use this slight abuse of (useful) notation (3.21) at other places in this article. Throughout all this section, u : [0, T ] → Rn denotes a C 1 -solution of (3.1) and (3.15). We define W1 : [0, T ] → R and W2 : [0, T ] → R by (3.22)
W1 (t) , W1 (u(t)), W2 (t) , W2 (u(t)), ∀t ∈ [0, T ].
In (3.22) and in the following, for t ∈ [0, T ], u(t) : [0, 1] → Rn is defined by (3.23)
u(t)(x) , u(t, x), ∀x ∈ [0, 1].
Of course W1 (t) and W2 (t) depend also on u. So we should in fact write W1u (t) and W2u (t). But, to simplify the notations, we omit this dependence on u in the notations W1 (t) and W2 (t).
Dissipative boundary conditions for hyperbolic systems: Lyapunov stability
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The proof of the theorem will be based on two preliminary lemmas. These lemmas provide estimates on dW1 /dt and dW2 /dt. Lemma 3.5. If ρ∞ (G0 (0)) < 1, there exist pi > 0 ∀i ∈ {1, . . . , n}, positive real constants α, β1 and δ1 such that, for every µ ∈ (0, δ1 ), for every p ∈ (1/δ1 , +∞), for every C 1 -solution u : [0, T ] × [0, 1] → Rn of (3.1) and (3.15) satisfying |u|0 < δ1 , we have dW1 (3.24) (t) 6 − µα + β1 |ux (t)|0 W1 (t), ∀t ∈ [0, T ]. dt Proof. Let u : [0, T ] × [0, 1] → Rn be a C 1 -solution of (3.1) and (3.15). The time derivative of W1 is:
(3.25)
1 1−2p dW1 = W dt 2p 1
Z 1X n
2p−1 n X 2p ppi mij (u)uj
0 i=1
j=1
"
n X
# mij (u)∂t uj +
j=1
n X
∂t mij (u) uj e−2pµx dx.
j=1
Using (3.1), the term between brackets can be written as n X
(3.26)
mij (u)∂t uj = −
j=1
n X
mij (u)
j=1
=−
n X k=1
n X
! ajk (u)∂x uk
k=1
n X mij (u)ajk (u)∂x uk , j=1
where ajk (u) is the (j, k)-th entry of the matrix A(u). Now, from (3.17), we have (3.27)
n X
mij (u)ajk (u)∂x uk =
n X
dij (u)mjk (u)∂x uk = λi (u)mik (u)∂x uk ,
j=1
j=1
where dij (u) is the (i, j)-th entry of the matrix D(u). From (3.26) and (3.27), we have (3.28)
n X
mij (u)∂t uj = −λi (u)
j=1
n X
mik (u)∂x uk = −λi (u)
n X
mij (u)∂x uj .
j=1
k=1
By substituting this expression for the term between brackets in (3.25), we get 1 1−2p dW1 = W dt 2p 1
n 1X
Z 0
2p−1 n X 2p ppi mij (u)uj
i=1
− λi (u)
j=1 n X j=1
mij (u)∂x uj +
n X j=1
! −2pµx ∂t mij (u) uj e dx,
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J-M Coron & G. Bastin
which leads to (3.29)
1 1−2p dW1 = W dt 2p 1 Z
1
2p
+ 0
n X
"Z
1
− 0
i=1
n X
ppi λi (u)
!2p ! e−2pµx dx
mij (u)uj x
j=1
i=1
n X
ppi
n X
!2p−1 mij (u)uj
λi (u)
n X
j=1
(∂x mij (u))uj
j=1
+
# ! −2pµx ∂t mij (u) uj e dx .
n X j=1
Using integrations by parts, we now get dW1 = T 1 + T 2 + T3 , dt
(3.30) with (3.31)
!2p 1 n n X W11−2p X p , T1 , − pi λi (u) mij (u)uj e−µx 2p i=1 j=1 0
(3.32)
T2 ,
−µW11−2p
Z
n 1X
0
(3.33)
T3 ,
W11−2p
Z 0
+
n X j=1
ppi λi (u)
i=1
ppi
e−2pµx dx,
mij (u)uj
j=1
i=1
n 1X
n X
!2p
n X
!2p−1 mij (u)uj
λi (u)
j=1
n X
(∂x mij (u))uj
j=1
! n ∂λ 1 X i mij (u)uj (u)∂x u e−2pµx dx ∂t mij (u) uj + 2p j=1 ∂u
Analysis of the first term T1 . From (3.31), we have !2p n n X W11−2p X p pi λi (u(t, 1)) (3.34) T1 = − mij (u(t, 1))uj (t, 1)e−µ 2p i=1 j=1 −
n X
ppi λi (u(t, 0))
i=1
n X
!2p mij (u(t, 0))uj (t, 0)
.
j=1
Let K , G0 (0). By the definition of ρ∞ (see (1.4)) and (3.14), there exist ∆i > 0, i ∈ {1, . . . , n}, such that (3.35)
θ,
n X
|Kij |
j=1
∆i < 1. ∆j
The parameters pi are selected such that (3.36)
ppi Λi = ∆2p i ,
i = 1, . . . , n.
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Dissipative boundary conditions for hyperbolic systems: Lyapunov stability
We define ξi : [0, T ] → R, i = 1, . . . , n, by ξi (t) , ∆i ui (t, 1), ∀t ∈ [0, T ].
(3.37)
From (3.34), (3.36) and (3.37), we have n W11−2p X λi (u(t, 1)) (3.38) T1 = − 2p Λi i=1 −
n X
∆i mij (u(t, 1)) ξj (t)e−µ ∆ j j=1
n X λi (u(t, 0)) i=1
Λi
n X
!2p
!2p mij (u(t, 0))∆i uj (t, 0)
.
j=1
Let t ∈ [0, T ]. Without loss of generality, we may assume that ξ12 (t) = max{ξi2 (t), i = 1, . . . , n}.
(3.39)
Let us denote by δ and C various positive constants which may vary from place to place but are independent of t ∈ [0, T ], u and p ∈ N+ . From (3.18) and (3.39), we have, for |u(t, 1)|0 6 δ, !2p ∆i −µ mij (u(t, 1)) ξj (t)e Λi ∆j j=1 i=1 !2p n λ1 (u(t, 1)) X ∆1 −µ > m1j (u(t, 1)) ξj (t)e Λ1 ∆j j=1 2p > e−2pµ (1 − C|ξ1 (t)|) |ξ1 (t)| − C|ξ1 (t)|2
n X λi (u(t, 1))
(3.40)
n X
= e−2pµ (1 − C|ξ1 (t)|)
2p+1
(ξ1 (t))
2p
.
From (3.15), (3.35), (3.37) and (3.39), we have, for |u(t, 0)|0 6 δ,
(3.41)
n X λi (u(t, 0)) i=1
Λi
n X
!2p mij (u(t, 0))∆i uj (t, 0)
j=1
2p n n X X ∆i C|ξ1 (t)|2 + 6 (1 + C|ξ1 (t)|) |Kij | |ξj (t)| ∆ j i=1 j=1
6 n (1 + C|ξ1 (t)|) θ|ξ1 (t)| + C|ξ1 (t)|2
2p
.
From (3.35), (3.38), (3.39) and (3.41), there exists δ11 ∈ (0, 1), independent of u, such that, for every µ ∈ (0, δ11 ), for every p ∈ (1/δ11 , +∞) ∩ N+ and for every u, we have (3.42)
T1 (t) 6 0 if |u(t)|0 < δ11 .
Analysis of the second term T2 . Let (3.43)
α , min(Λ1 , . . . , Λn )/2.
12
J-M Coron & G. Bastin
From (3.19) and (3.32) there is a δ12 ∈ (0, η] such that, for every µ ∈ (0, +∞), for every p ∈ N+ and for every u, T2 6 −µαW1 if |u|0 < δ12 .
(3.44)
Analysis of the third term T3 . Using (3.1) and (3.33), we have !2p−1 Z 1X n n X (3.45) T3 = W11−2p ppi mij (u)uj 0
1 ∂λi ux 2p ∂u
n X
i=1
j=1
! mij (u)uj
+
j=1
n X ∂mij j=1
∂u
!
− A(u) + λi (u) ux uj e−2pµx dx.
From (3.18), (3.19) and (3.45) one gets the existence of β1 > 0 and δ13 > 0 such that, for every µ ∈ (0, +∞), for every p ∈ N+ and for every u, (3.46)
T3 6 β1 |ux |0 W1 if |u|0 < δ13 .
Let δ1 , min{δ11 , δ12 , δ13 }. From (3.30), (3.42), (3.44) and (3.46), we conclude that dW1 6 −µα1 W1 + β1 |ux |0 W1 dt provided that u is such that |u|0 < δ1 , that p ∈ (1/δ1 , +∞) ∩ N+ and that µ ∈ (0, δ1 ). This completes the proof of Lemma 3.5. Lemma 3.6. Let pi (i = 1, . . . , n) be given by (3.36). If ρ∞ (G0 (0)) < 1, there exist β2 and δ2 such that, for every µ ∈ (0, δ2 ), for every p ∈ (1/δ2 , +∞) ∩ N+ and for every C 1 -solution u : [0, T ] × [0, 1] → Rn of (3.1) and (3.15) such that |u|0 < δ2 , we have, in the sense of distributions in (0, T ), dW2 6 − µα + β2 |ux (t)|0 W2 (t), ∀t ∈ [0, T ], dt with α defined by (3.43) Proof. We first deal with the case where A, G and u are of class C 2 . By time differentiation of (3.1) and (3.15), we see that ut satisfy the following hyperbolic dynamics for t ∈ [0, T ] and x ∈ [0, 1]: (3.47) (3.48)
(ut )t + A(u)(ut )x + A0 (u, ut )ux = 0, ∂t u(t, 0) =
∂G(u(t, 1)) ∂t u(t, 1), ∂u(t, 1)
where A0 (u, ut ) is a compact notation for the matrix whose entries are ∂aij (u) ut , i ∈ {1, . . . , n}, j ∈ {1, . . . , n}. ∂u Using (3.47)–(3.48), we see that the time derivative of W2 is: 2p−1 Z 1X n n X dW2 1 1−2p (3.49) = W 2p ppi mij (u)∂t uj dt 2p 2 0 i=1 j=1 " # n n X X mij (u)∂tt uj + ∂t mij (u) ∂t uj e−2pµx dx. A0 (u, ut )i,j ,
j=1
j=1
13
Dissipative boundary conditions for hyperbolic systems: Lyapunov stability
From (3.17), similarly as for (3.28), it can be shown that n X
mij (u)∂tt uj = −λi (u)
j=1
n X
mij (u)∂x ∂t uj +
j=1
n X
n X
mij (u)
j=1
! a ˜jk (u, ut )∂t uk ,
k=1
˜ ut ) , A0 (u, ut )A−1 (u). Then, where a ˜ij (u, ut ) is the (i, j)-th entry of the matrix A(u, by substituting this expression for the term between brackets in (3.49), we get 1 1−2p dW2 = W dt 2p 2
n 1X
Z 0
2p ppi
i=1
n X
!2p−1 mij (u)∂t uj
j=1
" − λi (u)
n X
mij (u)∂x ∂t uj
j=1 n n X X
+
j=1
! # mik (u)˜ akj (u, ut ) + ∂t mij (u) ∂t uj e−2pµx dx.
k=1
Using integration by parts as in the proof of Lemma 3.5, we get dW2 = U1 + U2 + U3 , dt with !2p 1 n n X −µx W21−2p X p , − pi λi (u) mij (u) ∂t uj e U1 , 2p i=1 j=1 0
U2 ,
−µW21−2p
Z
n 1X
0
U3 ,
W21−2p
0
+
i=1
n 1X
Z
i=1
n X
ppi λi (u)
ppi
n X
!2p mij (u) ∂t uj
e−2pµx dx,
j=1 n X
!2p−1 " mij (u) ∂t uj
j=1
n X
λi (u)(∂x mij (u))
j=1
mik (u)˜ akj (u, ut ) + ∂t mij (u)
! ∂t uj
k=1
# n ∂λ 1 X i mij (u)∂t uj (u)ux e−2pµx dx. + 2p j=1 ∂u Analysis of the first term U1 . Using the boundary conditions (3.15) and (3.48), we have !2p Pn W21−2p Pn p U1 = − e−2pµ i=1 pi λi (u(t, 1)) j=1 mij (u(t, 1)) ∂t uj (t, 1) 2p !2p n X Pn ∂Gj (u(t, 1)) ∂t u(t, 1) − i=1 ppi λi (G(u(t, 1))) mij (G(u(t, 1))) ∂u(t, 1) j=1
14
J-M Coron & G. Bastin
Then, in a way similar to the analysis of T1 in the proof of Lemma 3.5, we can show that, since ρ∞ (K) < 1, there exists δ21 ∈ (0, 1), such that, for every µ ∈ (0, δ21 ), for every p ∈ (1/δ21 , +∞) ∩ N+ and for every u, we have U1 6 0 provided that |u|0 < δ21 . Analysis of the second term U2 . Proceeding as in the proof of (3.44), we get the existence of δ22 ∈ (0, η] such that, for every µ ∈ (0, +∞), for every p ∈ N+ and for every u, U2 6 −µαW2 if |u|0 < δ22 ,
(3.50) with α defined as in (3.43).
Analysis of the third term U3 . Proceeding as in the proof of (3.46), we get the existence of β2 > 0 and δ23 > 0 such that, for every µ ∈ (0, +∞), for every p ∈ N+ and for every u, U3 6 β2 |ux |0 W2 if |u|0 < δ23 . From the analysis of U1 , U2 and U3 , we conclude that, with δ2 , min{δ21 , δ22 , δ23 }, dW2 6 −µαW2 + β2 |ux |0 W2 dt for all u such that |u|0 < δ2 provided that µ ∈ (0, δ2 ) and that p ∈ (1/δ2 , +∞) ∩ N+ . The above estimates were obtained for A, G and u of class C 2 . But their proofs show that they do not depend on the C 2 -norm of A, G and u. Hence, by density arguments, they remain valid with A, G and u only of class C 1 . This completes the proof of Lemma 3.6. Proof. We now prove Theorem 3.3. Let us choose µ ∈ R such that (3.51)
0 < µ < min {δ1 , δ2 } ,
where δ1 and δ2 are as in Lemma 3.5 and Lemma 3.6 respectively. Let us define two functionals V1 : C 1 ([0, 1]; Rn ) → R and V2 : C 1 ([0, 1]; Rn ) → R by T n n X X V1 (u) , ∆21 m1j (u)uj e−µx , . . . , ∆2n mnj (u)uj e−µx , (3.52) j=1 j=1 0 T n n X 2 X mnj (u)(∂t uj )e−µx , (3.53) V2 (u) , ∆1 m1j (u)(∂t uj )e−µx , . . . , ∆2n j=1 j=1 0 for every u ∈ C 1 ([0, 1]; Rn ) with |u|0 < ε0 , where ε0 > 0 is chosen small enough. In (3.53), we again use (3.21) for every u ∈ C 1 ([0, 1]; Rn ) with |u|0 < ε0 . This is a valid definition if, again, ε0 > 0 is chosen small enough. We consider the Lyapunov function candidate V : C 1 ([0, 1]; Rn ) → R defined by (3.54)
V(u) , V1 (u) + V2 (u), ∀u ∈ C 1 ([0, 1]; Rn ) such that |u|0 < ε0 .
Still if ε0 > 0 is small enough, which will be always assumed, we can select γ ∈ (1, +∞) such that 1 V(u) 6 |u|1 6 γV(u) for every u ∈ C 1 ([0, 1]; Rn ) such that |u|0 < ε0 . (3.55) γ
Dissipative boundary conditions for hyperbolic systems: Lyapunov stability
15
Let us now choose T > 0 large enough so that γ 2 e−µαT /2 6
(3.56)
1 2
with α defined by (3.43). Let ε2 ∈ (0, +∞) be such that δ 1 δ 2 ε0 , , , ε1 , (3.57) ε2 < min C1 C1 C1 where, again, δ1 and δ2 are as in Lemma 3.5 and Lemma 3.6 respectively. Let uo be in C 1 ([0, 1]; Rn ), satisfy the compatibility conditions (3.16) and be such that |uo |1 6 ε2 .
(3.58)
Let u : [0, T ] × [0, 1] → Rn be the solution of class C 1 of (3.1) and (3.15) satisfying the initial condition u(0, ·) = uo (·). By Theorem 3.1, (3.57) and (3.58), u is well defined and it satisfies |u|0 < min{δ1 , δ2 , ε0 }.
(3.59)
In order to emphasize the dependence of W1 and W2 on p ∈ N+ , we now write them W1,p and W2,p respectively. For t ∈ [0, T ], let us define (3.60)
V (t) , V(u(t)), V1 (t) , V1 (u(t)), V2 (t) , V2 (u(t)),
(3.61)
W1,p (t) , W1,p (u(t)), W2,p (t) , W2,p (u(t)).
Let us point out that, for every continuous function f = (f1 , . . . , fn )T : [0, 1] → Rn , we have !1/(2p) Z 1X n 2p (3.62) |f |0 = lim |fi (x)| dx , p→∞
Z (3.63) 0
n 1X
0
i=1
!1/(2p) |fi (x)|2p dx
6 n1/(2p) |f |0 , ∀p ∈ [1, +∞).
i=1
By the definition of W1 and W2 , (3.60), (3.61), (3.62) and (3.63), we have (3.64) (3.65)
V1 (t) = lim W1,p (t) p→∞
V2 (t) = lim W2,p (t), ∀t ∈ [0, T ],
and
p→∞
∃M > 0 such that W1,p (t) + W2,p (t) 6 M |u|1 , ∀p ∈ [1, +∞), ∀t ∈ [0, T ].
In particular ∗
(3.66)
W1,p * V1 in the weak∗ topology σ(L∞ (0, T ), L1 (0, T )) as p → +∞,
(3.67)
W2,p * V2 in the weak∗ topology σ(L∞ (0, T ), L1 (0, T )) as p → +∞.
∗
(Let us recall that, for a bounded sequence (ϕp )p∈N+ of elements of L∞ (0, T ) and ∗ ϕ ∈ L∞ (0, T ) one says that ϕp * ϕ in the weak∗ topology σ(L∞ (0, T ), L1 (0, T )) if, 1 for every ψ ∈ L (0, T ), ZT ZT ϕp (t)ψ(t)dt → ϕ(t)ψ(t)dt as p → +∞.) 0
0
16
J-M Coron & G. Bastin
From Lemmas 3.5 and 3.6, (3.59), (3.66) and (3.67), we have, in the distribution sense in (0, T ), dV1 6 −µαV1 + β1 |ux |0 V1 , dt dV2 6 −µαV2 + β2 |ux |0 V2 . dt
(3.68) (3.69)
Summing (3.68) and (3.69), we get, in the distribution sense in (0, T ), dV 6 −µαV + β|u|1 V, dt
(3.70)
with β , max {β1 , β2 }. Let us impose on ε2 , besides (3.57), that (3.71)
ε2 6
µα . 2βC1
From (3.13), (3.58) and (3.71), we get that (3.72)
β|u|1 6
µα . 2
From (3.70) and (3.72), we have, in the distribution sense in (0, T ), dV µ 6 − αV, dt 2
(3.73) which implies that
V (T ) 6 e−αµT /2 V (0).
(3.74)
From (3.55) and (3.74), we obtain that |u(T )|1 6 γ 2 e−αµT /2 |u(0)|1 ,
(3.75)
which, together with (3.56), implies that |u(T )|1 6
(3.76)
1 |u(0)|1 . 2
This completes the proof of Theorem 3.3. Remark 3.7. Instead of using time derivatives in the distribution sense as, for instance in (3.70), we could have directly derived inequality (3.74) on V (t) by establishing the same inequality for W (t) , W1 (t) + W2 (t) and let p → +∞. 3.2. Proof of Theorem 3.3 in the general case where 0 < m < n. In this section, we explain the modifications of the proof that must be used to deal with the case 0 < m < n (of course, the case m = 0 is equivalent to the case m = n by considering u(t, 1 − x) instead of u(t, x). The functionals W1 and W2 are now defined as follows: !2p Z 1"X m n X p W1 (u) , pi mij (u)uj e−2pµx 0
i=1
j=1
+
n X i=m+1
ppi
n X j=1
!2p mij (u)uj
#
1 2p
e2pµx dx ,
17
Dissipative boundary conditions for hyperbolic systems: Lyapunov stability
Z W2 (u) ,
1
"
m X
0
n X
ppi
i=1
mij (u) ∂t uj
!2p e−2pµx
j=1 n X
+
ppi
i=m+1
n X
mij (u) ∂t uj
!2p
#
1 2p
e2pµx dx .
j=1
Let u : [0, T ]×[0, 1] → Rn be a solution of (3.1) - (3.2) of class C 1 . With computations similar to those of Lemma 3.5, it is readily seen that the time derivative of W1 (t) , W1 (u(t)) is given by: dW1 = T1 + T2 + T3 dt with !2p m n X W11−2p X p T1 , pi λi (u) mij (u)uj e−µx − 2p i=1 j=1 n X
−
ppi λi (u)
n X
i=m+1
T2 ,
−µW11−2p
Z
1
"
0
m X
ppi λi (u)
n X
i=1
,
j=1
0
!2p mij (u)uj
e−2pµx
m X
ppi |λi (u)|
i=m+1
n X
!2p mij (u)uj
# e
2pµx
dx,
j=1
!2p−1
" T3 ,
mij (u)uj eµx
j=1
+
R1 W11−2p 0
!2p 1
Pm
p i=1 pi
Pn
j=1
mij (u)uj
λi (u)
Pn
j=1 (∂x mij (u))uj
! n ∂λ 1 X i mij (u)uj (u)∂x u e−2pµx + ∂t mij (u) uj + 2p ∂u j=1 j=1 !2p−1 n n n X X X + ppi mij (u)uj λi (u) (∂x mij (u))uj i=m+1 j=1 j=1 ! # n P ∂λ 1 X i n mij (u)uj (u)∂x u e2pµx dx. + j=1 ∂t mij (u) uj + 2p j=1 ∂u n X
The parameters pi > 0 are now selected such that (to be compared with (3.36)) (3.77)
ppi |Λi | = ∆2p i ,
i = 1, . . . , n.
Concerning the ξi , i = 1, . . . , n, they are now defined by (to be compared with (3.37)) ξi (t) , ∆i ui (t, 1) for i = 1, . . . , m
and
ξi (t) , ∆i ui (t, 0) for i = m + 1, . . . , n.
18
J-M Coron & G. Bastin
It is then a straightforward exercise to verify that Theorem 3.3 can be established for the case 0 < m < n in a manner completely parallel to the one we have followed in the case m = n. Remark 3.8. Looking at our proof of Theorem 3.3 we see that, for every (3.78)
ν ∈ (0, − min(|Λ1 |, . . . , |Λn |) ln (ρ (|G0 (0)|))) ,
there exist ε > 0 and C > 0 such that, for every uo such that |uo |1 < ε and satisfying the compatibility conditions (3.5) to (3.7), the Cauchy problem (3.1)–(3.2)–(3.4) has a unique C 1 -solution which satisfies |u(t, .)|1 6 Ce−νt |uo |1 , ∀t ∈ [0, +∞). See in particular (3.40) and (3.41), replace the definition of α given in (3.43) by the condition (3.79) α ∈ 0, min(|Λ1 |, . . . , |Λn |) and modify a little bit some arguments in order to deal with (3.79) instead of (3.43). 4. Conclusion and final remark. In this article, we have addressed the issue of exponential stability for the C 1 -norm of quasi-linear hyperbolic systems of the form (3.1), (3.2). Our main result (Section 3) has been to give a new proof, using a strict Lyapunov function, that a sufficient condition for the exponential stability of the steady-state is to have a map G such that ρ∞ (G0 (0)) < 1 for the boundary conditions (3.2). The approach followed in the paper can be easily adapted to the exponential stability for the C ` -norm with ` ∈ N+ provided the compatibility conditions (3.5), (3.6), (3.7) are adapted accordingly (see [10] for details) and that A and G are of class C ` . The analysis requires the following extensions of formulas (3.19) or (3.20) and (3.52) or (3.53):
(4.1)
Z Wk (u) ,
n 1X
0
ppi
n X
!2p mij (u)∂tk−1 uj
1 2p
e−2pµx dx ,
j=1
i=1
T n n X X (4.2) Vk (u) , ∆21 m1j (u)(∂tk−1 uj )e−µx , . . . , ∆2n mnj (u)(∂tk−1 uj )e−µx j=1 j=1 0 where ∂tk−1 u is defined by differentiating formally (1.1) with respect to time. For example ∂t2 u , (A0 (u) (A(u)∂x u)) ∂x u + A(u) ((A0 (u)∂x u) ∂x u) 2 + A(u) A(u)∂xx u , ∀u ∈ C 2 ([0, 1]; Rn ) with |u|0 small enough. Then, using the Lyapunov function (4.3)
V(u) =
`+1 X k=1
the following theorem holds.
Vk (u),
Dissipative boundary conditions for hyperbolic systems: Lyapunov stability
19
Theorem 4.1. Let assume that A and G are of class C ` . Then, if (4.4)
ρ∞ (G0 (0)) < 1,
the steady state x ∈ [0, 1] 7→ 0 ∈ Rn of the system (3.1)–(3.2) is exponentially stable for the C ` -norm. REFERENCES [1] O-M. Aamo. Disturbance rejection in 2 x 2 linear hyperbolic systems. IEEE Transactions on Automatic Control, 58(5):1095–1106, May 2013. [2] S. Amin, F. Hante, and A. Bayen. Exponential stability of switched linear hyperbolic initialboundary value problems. IEEE Transactions on Automatic Control, 56(2):291–301, February 2012. [3] G. Bastin and J-M. Coron. Stability and Boundary Stabilisation of One-Dimensional Hyperbolic Systems. Book in preparation, 2014. [4] G. Bastin, J-M. Coron, and S.O. Tamasoiu. Stability of linear density-flow hyperbolic systems under PI boundary control. Preprint, 2014. [5] D. Bridges and P. Schuster. A simple constructive proof of Kronecker’s density theorem. Elemente der Mathematik, 61:152–154, 2006. [6] F. Castillo, E. Witrant, C. Prieur, and L. Dugard. Boundary observers for linear and quasilinear hyperbolic systems with application to flow control. Automatica, 49(11):3180–3188, 2013. [7] J-M. Coron, G. Bastin, and B. d’Andr´ ea-Novel. Dissipative boundary conditions for one dimensional nonlinear hyperbolic systems. SIAM Journal of Control and Optimization, 47(3):1460–1498, 2008. [8] J-M. Coron, B. d’Andr´ ea-Novel, and G. Bastin. A Lyapunov approach to control irrigation canals modeled by Saint-Venant equations. In Proceedings European Control Conference, Karlsruhe, Germany, September 1999. [9] J-M. Coron, B. d’Andr´ ea-Novel, and G. Bastin. A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws. IEEE Transactions on Automatic Control, 52(1):2–11, January 2007. [10] J.-M. Coron and H.-M. Nguyen. Dissipative boundary conditions for nonlinear hyperbolic equations: sharp conditions through an approach via time delay equations. Preprint, 2014. [11] J-M. Coron, R. Vazquez, M. Krstic, and G. Bastin. Local exponential H 2 stabilization of a 2 x 2 quasilinear hyperbolic system using backstepping. SIAM Journal of Control and Optimization, 51(3):2005–2035, 2012. [12] J. de Halleux, C. Prieur, J-M. Coron, B. d’Andr´ ea-Novel, and G. Bastin. Boundary feedback control in networks of open-channels. Automatica, 39:1365–1376, 2003. [13] A. Diagne, G. Bastin, and J-M. Coron. Lyapunov exponential stability of linear hyperbolic systems of balance laws. Automatica, 48:109–114, 2012. [14] M. Dick, M. Gugat, and G. Leugering. Feedback stabilization of quasilinear hyperbolic systems with varying delays. In 17th International Conference on Methods and Models in Automation and Robotics, pages 125–130, Miedzyzdroje, Poland, August 27-30 2012. [15] P.M. Dower and P.M. Farrel. On linear control of backward pumped Raman amplifiers. In Proceedings IFAC Symposium on System Identification, pages 547–552, Newcastle, Australia, March 2006. [16] J.M. Greenberg and T. Li. The effect of boundary damping for the quasilinear wave equations. Journal of Differential Equations, 52:66–75, 1984. [17] M. Gugat, M. Dick, and G. Leugering. Gas flow in fan-shaped networks: classical solutions and feedback stabilization. SIAM Journal of Control and Optimization, 49(5):2101–20117, 2011. [18] M. Gugat and M. Herty. Existence of classical solutions and feedback stabilisation for the flow in gas networks. ESAIM Control Optimisation and Calculus of Variations, 17(1):28–51, 2009. [19] J.K. Hale and S.M. Verduyn-Lunel. Introduction to Functional-Differential Equations. Number 99 in Appl. Math. Sci. Springer-Verlag, 1993. [20] M. Krstic, B-Z Guo, A. Balogh, and A. Smyshlyaev. Output-feedback stabilization of an unstable wave equation. Automatica, 44:63–74, 2008. [21] M. Krstic and A. Smyshlyaev. Boundary control of PDEs, volume 16 of Advances in Design
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J-M Coron & G. Bastin and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. A course on backstepping designs. G. Leugering and J-P.G. Schmidt. On the modelling and stabilisation of flows in networks of open canals. SIAM Journal of Control and Optimization, 41(1):164–180, 2002. T-T. Li. Global Classical Solutions for Quasi-Linear Hyperbolic Systems. Research in Applied Mathematics. Masson and Wiley, 1994. T-T. Li, B. Rao, and Z. Wang. Exact boundary controllability and observability for first order quasilinear hyperbolic systems with a kind of nonlocal boundary conditions. Discrete Contin. Dyn. Syst., 28(1):243–257, 2010. T-T. Li and W-C. Yu. Boundary value problems for quasilinear hyperbolic systems. Duke University Mathematics Series, V. Duke University Mathematics Department, Durham, NC, 1985. X. Litrico and V. Fromion. Boundary control of hyperbolic conservation laws using a frequency domain approach. Automatica, 45:647–656, 2009. F. Di Meglio, R. Vazquez, and M. Krstic. Stabilization of a linear hyperbolic system with one boundary controlled transport PDE coupled with n counterconvecting PDEs. In Proceedings 51st IEEE Conference on Decision and Control, pages 2914–2919, December 10-13 2012. W. Michiels and T. Vyhlidal. An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type. Automatica, 41:991–998, 2005. V. Perollaz and L. Rosier. Finite-time stabilization of 2 x 2 hyperbolic systems on tree-shaped networks. SIAM Journal of Control and Optimization, 52(1):143–163, 2013. C. Prieur. Control of systems of conservation laws with boundary errors. Networks and Heterogeneous Media, 4(2):393–407, 2009. C. Prieur, A. Girard, and E. Witrant. Lyapunov functions for switched linear hyperbolic systems. In M. Lazar M. Heemels, B. De Schutter, editor, Proceedings 4th IFAC Conference on Analysis and Design of Hybrid Systems, pages 382–387, Eindhoven, The Netherlands, 2012. C. Prieur and F. Mazenc. ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws. Mathematics of Control, Signal and Systems (MCSS), 24(1-2):111–134, April 2012. C. Prieur, J. Winkin, and G. Bastin. Robust boundary control of systems of conservation laws. Mathematics of Control, Signal and Systems (MCSS), 20:173–197, 2008. T-H. Qin. Global smooth solutions of dissipative boundary value problems for first order quasilinear hyperbolic systems. Chinese Ann. Math. Ser. B, 6(3):289–298, 1985. J. Rauch and M. Taylor. Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana University Mathematics Journal, 24(1):79–86, 1974. D.L. Russell. Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Review, 20(4):639–739, October 1978. V. Dos Santos and B. Maschke. A Hamiltonian perspective to the stabilization of systems of two conservations laws. Networks and Heterogeneous Media, 4(2), June 2009. R.A. Silkowski. Star Shaped Regions of Stability in Hereditary Systems. PhD thesis, Brown University, Providence, R.I., June 1976. M. Slemrod. Boundary feedback stabilization for a quasilinear wave equation. In Control Theory for Distributed Parameter Systems, volume 54 of Lecture Notes in Control and Information Sciences, pages 221–237. Springer Verlag, 1983. A. Smyshlyaev, E. Cerpa, and M. Krstic. Boundary stabilisation of a 1-d wave equation with in-domain antidamping. SIAM Journal of Control and Optimization, 48(6):4014–4031, 2010. A. Tchousso, T. Besson, and C-Z. Xu. Exponential stability of distributed parameter systems governed by symmetric hyperbolic partial differential equations using Lyapunov’s second method. ESAIM Control Optimisation and Calculus of Variations, 15(2):403–425, 2009. R. Vazquez, M. Krstic, and J-M. Coron. Backstepping boundary stabilization and state estimation of a 2 x 2 linear hyperbolic system. In Proceedings 50th IEEE Conference on Decision and Control and European Control Conference, pages 4937–4942, Orlando, FL, USA, December 12-15 2011. C-Z. Xu and G. Sallet. Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems. ESAIM Control Optimisation and Calculus of Variations, 7:421–442, 2002. Y-C. Zhao. Classical solutions for quasilinear hyperbolic systems. Thesis, Fudan University (in Chinese), 1986.
1. Introduction. We are concerned with one-dimensional quasi-linear strictly hyperbolic systems of the form (1.1)
ut + A(u)ux = 0, x ∈ [0, 1], t ∈ [0, +∞),
where u : [0, ∞) × [0, 1] → Rn and A : Rn → Mn,n (R), Mn,n (R) denoting, as usual, the set of n × n real matrices. Since the system is strictly hyperbolic, the matrix A(0) has, by definition, n real eigenvalues denoted Λ1 , . . . , Λn satisfying (1.2)
Λi 6= Λj , ∀i, j ∈ {1, . . . , n} such that i 6= j.
We assume that none of these eigenvalues is zero. Then, possibly after a suitable linear change of variables, we may assume that there exists m ∈ {0, 1 . . . , n} such that Λi > 0, ∀i ∈ {1, . . . , m}, Λi < 0, ∀i ∈ {m + 1, . . . , n}, and that A(0) is a diagonal matrix: A(0) , diag{Λ1 , Λ2 , . . . , Λn }. Our concern is to analyse the asymptotic convergence of the classical solutions of the system (1.1) under a boundary condition of the form u+ (t, 0) u+ (t, 1) (1.3) =G , t ∈ [0, +∞), u− (t, 1) u− (t, 0) where the map G : Rn → Rn vanishes at 0, while u+ ∈ Rm , u− ∈ Rn−m are defined by requiring that u , (uT+ , uT− )T . The challenge is to give explicit conditions on the ∗ This work was supported by the ERC advanced grant 266907 (CPDENL) of the 7th Research Framework Programme (FP7) and by the Belgian Programme on Interuniversity Attraction Poles (IAP VII/19) † Sorbonne Universit´ es, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, 4 place Jussieu, 75252 Paris cedex 05, France. ([email protected]). ‡ Department of Mathematical Engineering, ICTEAM, Universit´ e catholique de Louvain, 4, avenue G. Lemaitre, 1348 Louvain-la-Neuve, Belgium. ([email protected]). Questions, comments, or corrections to this document may be directed to that email address.
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map G such that the boundary condition (1.3) is exponentially dissipative, i.e. implies that the equilibrium solution u ≡ 0 of system (1.1) with the boundary condition (1.3) is exponentially stable. In order to state our results, we first introduce the functions ρp : Mn,n (R) → R defined by n o (1.4) ρp (M ) , inf k∆M ∆−1 kp , ∆ ∈ Dn+ , 1 6 p 6 ∞, where Dn+ denotes the set of diagonal n×n real matrices with strictly positive diagonal entries and with " n # p1 X p , kξk∞ , max{|ξi |; i ∈ {1, . . . , }}, ∀ξ , (ξ1 , . . . , ξn )T ∈ Rn , kξkp , |ξi | i=1
and kM kp , max kM ξkp ∀M ∈ Mn,n (R). kξkp =1
In this paper, our main result (Section 3) is a new proof, using a Lyapunov function approach, that a sufficient condition for the exponential stability of the steadystate u ≡ 0 for the C 1 -norm is to have a map G such that: ρ∞ (G0 (0)) < 1, where G0 denotes the Jacobian matrix of the map G. This new proof supplements the previous results which are already available in the literature and are reviewed in Section 2. 2. Literature review. The well-posedness of the Cauchy problem associated to nonlinear hyperbolic systems of the kind (1.1) and (1.3) has been studied in 1985 by Li and Yu[25] in the framework of C 1 -solutions. For such systems, the issue of finding sufficient dissipative boundary conditions has been addressed in the literature for more than thirty years. To our knowledge, first results were published by Slemrod[39] in 1983 and by Greenberg and Li[16] in 1984 for the special case of systems of size n = 2. A generalization to systems of size n was then progressively elaborated by the Ta-Tsien Li school, in particular by Qin[34] in 1985 and by Zhao[44] in 1986. All these contributions deal with the particular case of ‘local’ boundary conditions having the specific form (2.1)
u+ (t, 0) = G0 (u− (t, 0)),
u− (t, 1) = G1 (u+ (t, 1)).
With these boundary conditions, the analysis can be based on the method of characteristics which can exploit an explicit computation of the ‘reflection’ of the solutions at the boundaries along the characteristic curves. This has given rise to the sufficient condition !! 0 G00 (0) (2.2) ρ∞ < 1, G01 (0) 0 for the dissipativity of the boundary conditions (2.1) for the C 1 -norm. This result is given for instance by Li[23, Theorem 1.3, page 173] in his seminal book of 1994 on the stability of the classical solutions of quasi-linear hyperbolic systems. Finally, by using an appropriate dummy doubling of the system size, de Halleux et al.[12, Theorem 4] have shown in 2003 how the general dissipative boundary condition ρ∞ (G0 (0)) < 1 can be established for systems with the general ‘non local’ boundary condition (1.3). This
Dissipative boundary conditions for hyperbolic systems: Lyapunov stability
3
dummy doubling has also been used by Li et al.[24] to prove the well-posedness of the Cauchy problem associated to (1.1) and (1.3) still in the framework of C 1 -solutions. Another approach of the analysis of dissipative boundary conditions is based on the use of Lyapunov functions. The first attempts were in using entropies as Lyapunov functions, as done for instance by Coron et al.[8] in 1999 or by Leugering and Schmidt[22] in 2002. The drawback of this approach was however that the time derivatives of such entropy-based Lyapunov functions are necessarily only semidefinite negative. Hence one would like to conclude with the LaSalle invariant set principle. However, this principle requires the precompactness of the trajectories, a property which is difficult to get in the case of nonlinear partial differential equations. In order to overcome this difficulty, Coron et al.[9] in 2007 have proposed, for systems of size n = 2, a strict Lyapunov function whose time derivative is strictly negative definite when (2.3)
ρ∞ (G0 (0)) < 1.
The advantage is that the proof is less elaborated than the one using the method of characteristics because it uses more direct computations. Furthermore, another advantage of a Lyapunov analysis is to directly induce robustness properties with respect to small uncertainties and disturbances. Then, in the paper[7], Coron et al. have generalized in 2008 this Lyapunov approach to general nonlinear hyperbolic systems of the kind (1.1) and (1.3). In particular, they emphasize a new weaker dissipative boundary condition which is formulated as follows: A sufficient condition for the exponential stability of the steady-state u ≡ 0 for the H 2 -norm is to have a map G such that: (2.4)
ρ2 (G0 (0)) < 1.
Moreover, it is also shown in the same paper that ρ2 (G0 (0)) 6 ρ∞ (G0 (0)) ∀n > 1 and that this inequality can be strict if n > 2. However, it has been proved in [10] that ρ2 (G0 (0)) < 1 is not a sufficient condition for the exponential stability of the steady-state u ≡ 0 for the C 1 -norm. Various recent contributions and extensions of the previous results are also worth to be mentioned. • Dos Santos and Maschke[37] in 2009 have given an Hamiltonian perspective to the stabilization of systems of two conservation laws. • Using the Lyapunov approach, Castillo et al.[6] in 2013 give sufficient conditions for the existence of exponentially stable observers in the case where all eigenvalues Λi are positive. • Perrollaz and Rosier[29] have shown in 2013 that there exist boundary dissipative conditions for systems of size n = 2 that achieve finite-time stabilization. • Using an approach via time delay equations, Coron and Nguyen[10] have shown in 2014 that ρp (G0 (0)) < 1 implies the exponential stability in the Sobolev norm W 2,p . For so-called inhomogeneous quasi-linear hyperbolic systems (i.e. with additional zero-order terms): ut + A(u)ux + B(u) = 0,
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J-M Coron & G. Bastin
the analysis of dissipative boundary conditions is much more intricate and only very partial results are known. • Using the method of characteristics, Prieur et al.[33] in 2008 have shown that the stability condition (2.2) holds for inhomogeneous systems when kB(u)k is small enough. Prieur[30] has shown in 2009 how this result can be extended to deal with differential or integral boundary errors. • Using the Lyapunov approach, Gugat and Herty[18] in 2009 and Gugat et al.[17] in 2011 analyze the boundary feedback stabilization of gas flow in fan-shaped networks described by isentropic Euler equations. • Using the Lyapunov approach, Pavel and Chang[18] in 2012 analyze the boundary control of hyperbolic Lotka-Volterra systems, with application to pumped Raman amplifiers on optical fibres (see also Dower and Farrell[15] in 2006). • For inhomogeneous systems of size n = 2 with m = 1, Coron et al.[11] have shown in 2012 the existence of full-state feedback law which achieve exponential stability for the H 2 -norm. The proof uses a backstepping transformation (see [21]) to find new variables for which a strict Lyapunov function can be constructed. Linear hyperbolic systems. A linear hyperbolic system, (2.5)
ut + Aux = 0, x ∈ [0, 1], t ∈ [0, +∞),
with linear boundary conditions u+ (t, 0) u+ (t, 1) (2.6) =K , t ∈ [0, +∞), u− (t, 1) u− (t, 0) is a special case of (1.1) and (1.3) where the matrix A ∈ Dn (R) with non-zero distinct diagonal entries, Dn (R) denoting the set of n × n diagonal real matrices, and the matrix K ∈ Mn,n (R). As it can be expected, the analysis of dissipative boundary conditions is both simpler and more comprehensive for linear than for quasi-linear systems. The linear hyperbolic system (2.5)-(2.6) is equivalent to a system of interconnected scalar time-delays τi , Λ−1 i . For such systems, it follows from a theorem by Silkowski[38] (quoted in[19, Chapter 9, Theorem 6.1], see also [28]), which relies on the Kronecker density theorem (e.g.[5]), that the boundary conditions (2.6) are, for any Lp -norm, robustly dissipative with respect to arbitrary small perturbations on the Λi ’s if and only if (2.7)
ρ¯(K) , max{ρ(diag(eιθ1 , . . . , eιθn )K; (θ1 , . . . , θn )T ∈ Rn } < 1,
where ι ∈ C is such that ι2 = −1 and ρ(M ) is the spectral radius of M ∈ Mn,n (C). It is shown in [3, 7] that ρ¯(K) = ρ2 (K) if there exists a permutation matrix P such ˜ = P KP −1 is a block diagonal matrix K ˜ = diag{K ˜ 1, K ˜ 2, . . . , K ˜ p} that the matrix K ˜ where each block Ki is a real ni × ni matrix with ni ∈ {1, 2, 3, 4, 5}. Moreover, when n > 5, there always exist K such that ρ¯(K) < ρ2 (K). The exponential stability of non-uniform inhomogeneous linear hyperbolic systems has also received a lot of attention in the literature for a long time and can be
Dissipative boundary conditions for hyperbolic systems: Lyapunov stability
5
traced back up to Rauch and Taylor[35, Section 4] in 1974 or Russell[36, Section 3] in 1978. These systems are written as (2.8)
ut + A(x)ux + B(x)u = 0,
where A(x) ∈ Dn (R) and B(x) ∈ Mn,n (R), ∀x ∈ [0, 1]. Using the Lyapunov approach (as for instance Xu and Sallet[43] in 2002 or Diagne et al.[13] in 2011 for the uniform case), it can be shown that the equilibrium u ≡ 0 is exponentially stable for the L2 -norm if there exist a positive diagonal matrix P such that: |A+ (1)| 0 |A+ (0)| 0 (a) P − KT P K is positive definite; 0 |A− (0)| 0 |A− (1)| (b) P B(x) + BT (x)P − P ∂x |A(x)| is semi positive definite for all x ∈ [0, 1], where, for every x ∈ [0, 1], A+ (x) ∈ Dm (R) and A− (x) ∈ Dn−m (R) are defined by requiring A+ (x) 0 A(x) = . 0 A− (x) Obviously conditions (a) and (b) are rather conservative. Bastin and Coron[?] have given in 2011 a less restrictive sufficient condition for inhomogeneous linear systems of size n = 2 with m = 1 under the form of the existence of the solution of an associated ordinary differential equation. Other recent significant contributions are worth to be mentioned. • Tchousso et al.[41] have shown in 2009 how the Lyapunov approach can be extended to inhomogeneous linear hyperbolic systems of higher spatial dimension. • Litrico and Fromion[26] in 2009 use a frequency domain method for the boundary control of homogeneous systems of size n = 2. They show that the transfer functions belong to the Callier-Desoer algebra, which opens the way to necessary and sufficient conditions for closed loop stability and the use of a Nyquist type test. • Amin et al.[2] and Prieur et al.[31] have shown in 2012 how the Lyapunov approach can be extended to switched linear hyperbolic systems. • Prieur and Mazenc[32] in 2012 have shown how time varying strict Lyapunov functions can be defined to get input-to-state stability for time varying inhomogeneous linear hyperbolic systems. • Dick et al.[14] in 2012 address the feedback stabilization of quasi-linear hyperbolic systems with varying delays. • The use of a backstepping method for boundary stabilization by full state feedback and observer design for non uniform inhomogeneous linear systems has also been extensively studied recently: Krstic et al.[20] in 2008 and Smyshlyaev et al.[40] in 2010 for unstable wave equations, Vazquez et al.[42] in 2011 for general systems of size n = 2 with m = 1, Di Meglio et al.[27] in 2012 for systems with m = n−1 and a single controller at the right boundary, and Aamo[1] in 2013 for the rejection of differential boundary disturbances. • Using a frequency domain approach, Bastin et al.[4] in 2014 give necessary and sufficient conditions for the stability of linear density-flow systems under proportional-integral control.
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Our contribution in this paper fills a small gap in this panorama of the analysis of boundary dissipative conditions for one-dimensional linear and quasi-linear hyperbolic systems. We show how the Lyapunov approach can be used to establish the dissipativity of the boundary condition ρ∞ (G0 (0)) < 1. 3. Main result. We consider hyperbolic systems of the form ut + A(u)ux = 0, x ∈ [0, 1], t ∈ [0, +∞),
(3.1)
under a boundary condition of the form u+ (t, 0) u+ (t, 1) (3.2) =G , t ∈ [0, +∞). u− (t, 1) u− (t, 0) For σ, let Bσ be the open ball with radius σ in Rn for the norm |u|0 , max{|uj |; j ∈ {1, . . . , n}}, u = (u1 , . . . , un )T ∈ Rn .
(3.3)
We assume that, for some σ ∈ (0, +∞), A : Bσ → Mn,n (R) and G : Bσ → Rn are of class C 1 . We define A+ (u) ∈ Mm,n (R), A− (u) ∈ M(n−m),n (R), G+ (u) ∈ Rm and G− (u) ∈ Rn−m by requiring A+ (u) G+ (u) A(u) = , G(u) = . A− (u) G− (u) Our concern is to analyze the exponential stability of this system for the C 1 -norm under an initial condition u(0, x) = uo (x), x ∈ (0, 1)
(3.4)
which satisfies the compatibility conditions o o u+ (0) u+ (1) (3.5) =G , uo− (1) uo− (0)
(3.6)
(3.7)
o u+ (1) A+ (uo (1))uox (1) A+ (uo (0))uox (0) = G0+u+ uo− (0) o u+ (1) + G0+u− A− (uo (0))uox (0), uo− (0)
o
A− (u
(1))uox (1)
o u+ (1) 0 = G−u+ A+ (uo (1))uox (1) uo− (0) o u+ (1) 0 + G−u− A− (uo (0))uox (0). uo− (0)
Let us now define some norms which will be useful to state and prove our results. For u ∈ C 0 ([0, 1]; Rn ) (resp. in C 0 ([0, 1] × [T1 , T2 ]; Rn ), one defines (3.8)
|u|0 , max{|u(x)|0 ; x ∈ [0, 1]},
(3.9)
(resp. |u|0 , max{|u(t, x)|0 ; t ∈ [T1 , T2 ], x ∈ [0, 1]}).
Dissipative boundary conditions for hyperbolic systems: Lyapunov stability
7
For u ∈ C 1 ([0, 1]; Rn ) (resp. in C 1 ([T1 , T2 ] × [0, 1]; Rn )), we define (3.10)
|u|1 , |u|0 + |u0 |0 ,
(3.11)
(resp. |u|1 , |u|0 + |∂t u|0 + |∂x u|0 ).
Let us first recall the following theorem, due to Li and Yu[25], on the wellposedness of the Cauchy problem (3.1)–(3.2)–(3.4). Theorem 3.1. Let T > 0. There exists C1 > 0 and ε1 > 0 such that, for every uo ∈ C 1 ([0, 1]; Rn ) satisfying the compatibility conditions (3.5) to (3.7) and such that (3.12)
|uo |1 6 ε1 ,
there exists one and only one solution on [0, T ] × [0, 1] to the Cauchy problem (3.1)– (3.2)–(3.4). Moreover, this solutions satisfies (3.13)
|u|1 6 C1 |u0 |1 .
(In fact, [25] deals with the case of local boundary conditions (2.1); however the general case follows from this particular case by using the dummy doubling of the system size introduced in de Halleux et al.[12], already mentioned in Section 2: see [24].) Our definition of exponential stability for the C 1 -norm is as follows. Definition 3.2. The steady state x ∈ [0, 1] 7→ 0 ∈ Rn of the system (3.1)–(3.2) is exponentially stable for the C 1 -norm if there exist ε > 0, ν > 0 and C > 0 such that, for every uo such that |uo |1 < ε and satisfying the compatibility conditions (3.5) to (3.7), the Cauchy problem (3.1)–(3.2)–(3.4) has a unique C 1 -solution which satisfies |u(t, .)|1 6 Ce−νt |uo |1 , ∀t ∈ [0, +∞).
The main goal of this article is to give a new proof of the following theorem due to Qin[34] and Zhao[44] (see also Li[23, Theorem 1.3, page 173] and de Halleux et al.[12, Theorem 4]) . Theorem 3.3. If (3.14)
ρ∞ (G0 (0)) < 1,
the steady state x ∈ [0, 1] 7→ 0 ∈ Rn of the system (3.1)–(3.2) is exponentially stable for the C 1 -norm. In the remaining part of this section we give our proof of Theorem 3.3. This proof is divided in two parts: • In Section 3.1, we study the case where all the Λi are positive, i.e. the case where m = n, • In Section 3.2, we explain how to modify the Lyapunov function introduced in Section 3.1 in order to treat the general case.
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3.1. Proof of Theorem 3.3 in the special case where m = n. For the clarity of the demonstration, we shall first prove the theorem in the special case where m = n, which means the matrix A(0) is the positive diagonal matrix diag{Λ1 , . . . , Λn } with Λi > 0 ∀i = 1, . . . , n. In that case, the boundary condition (3.2) and the compatibility conditions (3.5) to (3.7) are simply rewritten (3.15) (3.16)
u(t, 0) = G(u(t, 1)), A(uo (0))uox (0) = G0 (uo (1))A(uo (1))uox (1).
uo (0) = G(uo (1)),
As already mentioned above, our proof is based on a Lyapunov approach. In order to define an appropriate Lyapunov function for the analysis, we need the following technical classical lemma, for which our assumption (1.2) is crucial and which also holds if m 6= n. Lemma 3.4. Let D(u) be the diagonal matrix whose diagonal entries are the eigenvalues λi (u), i = 1, . . . , n, of the matrix A(u). There exist a positive real number η < σ and a map M : Bη → Mn,n (R) of class C 1 such that (3.17)
M (u)A(u) = D(u)M (u), ∀u ∈ Bη ,
(3.18)
M (0) = Idn ,
where Idn is the identity matrix of Mn,n (R). Let W1 : C 1 ([0, 1]; Rn ) → R and W2 : C 1 ([0, 1]; Rn ) → R be defined by (3.19)
Z W1 (u) ,
n 1X
0
(3.20)
Z W2 (u) ,
i=1
n 1X
0
ppi
i=1
ppi
n X
1 2p
!2p mij (u)uj
e−2pµx dx ,
j=1
n X
1 2p
!2p mij (u)∂t uj
e−2pµx dx ,
j=1
with p ∈ N+ , {1, 2, 3, . . . } and pi > 0 ∀i ∈ {1, . . . , n}. In (3.19) and (3.20), mij (u) denotes the (i, j)-th entry of the matrix M (u) and, in (3.20), ∂t uj is defined by (∂t u1 , . . . , ∂t un )T = ∂t u, with (3.21)
∂t u , −A(u)∂x u, ∀u ∈ C 1 ([0, 1]; Rn ) with |u|0 small enough.
We use this slight abuse of (useful) notation (3.21) at other places in this article. Throughout all this section, u : [0, T ] → Rn denotes a C 1 -solution of (3.1) and (3.15). We define W1 : [0, T ] → R and W2 : [0, T ] → R by (3.22)
W1 (t) , W1 (u(t)), W2 (t) , W2 (u(t)), ∀t ∈ [0, T ].
In (3.22) and in the following, for t ∈ [0, T ], u(t) : [0, 1] → Rn is defined by (3.23)
u(t)(x) , u(t, x), ∀x ∈ [0, 1].
Of course W1 (t) and W2 (t) depend also on u. So we should in fact write W1u (t) and W2u (t). But, to simplify the notations, we omit this dependence on u in the notations W1 (t) and W2 (t).
Dissipative boundary conditions for hyperbolic systems: Lyapunov stability
9
The proof of the theorem will be based on two preliminary lemmas. These lemmas provide estimates on dW1 /dt and dW2 /dt. Lemma 3.5. If ρ∞ (G0 (0)) < 1, there exist pi > 0 ∀i ∈ {1, . . . , n}, positive real constants α, β1 and δ1 such that, for every µ ∈ (0, δ1 ), for every p ∈ (1/δ1 , +∞), for every C 1 -solution u : [0, T ] × [0, 1] → Rn of (3.1) and (3.15) satisfying |u|0 < δ1 , we have dW1 (3.24) (t) 6 − µα + β1 |ux (t)|0 W1 (t), ∀t ∈ [0, T ]. dt Proof. Let u : [0, T ] × [0, 1] → Rn be a C 1 -solution of (3.1) and (3.15). The time derivative of W1 is:
(3.25)
1 1−2p dW1 = W dt 2p 1
Z 1X n
2p−1 n X 2p ppi mij (u)uj
0 i=1
j=1
"
n X
# mij (u)∂t uj +
j=1
n X
∂t mij (u) uj e−2pµx dx.
j=1
Using (3.1), the term between brackets can be written as n X
(3.26)
mij (u)∂t uj = −
j=1
n X
mij (u)
j=1
=−
n X k=1
n X
! ajk (u)∂x uk
k=1
n X mij (u)ajk (u)∂x uk , j=1
where ajk (u) is the (j, k)-th entry of the matrix A(u). Now, from (3.17), we have (3.27)
n X
mij (u)ajk (u)∂x uk =
n X
dij (u)mjk (u)∂x uk = λi (u)mik (u)∂x uk ,
j=1
j=1
where dij (u) is the (i, j)-th entry of the matrix D(u). From (3.26) and (3.27), we have (3.28)
n X
mij (u)∂t uj = −λi (u)
j=1
n X
mik (u)∂x uk = −λi (u)
n X
mij (u)∂x uj .
j=1
k=1
By substituting this expression for the term between brackets in (3.25), we get 1 1−2p dW1 = W dt 2p 1
n 1X
Z 0
2p−1 n X 2p ppi mij (u)uj
i=1
− λi (u)
j=1 n X j=1
mij (u)∂x uj +
n X j=1
! −2pµx ∂t mij (u) uj e dx,
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J-M Coron & G. Bastin
which leads to (3.29)
1 1−2p dW1 = W dt 2p 1 Z
1
2p
+ 0
n X
"Z
1
− 0
i=1
n X
ppi λi (u)
!2p ! e−2pµx dx
mij (u)uj x
j=1
i=1
n X
ppi
n X
!2p−1 mij (u)uj
λi (u)
n X
j=1
(∂x mij (u))uj
j=1
+
# ! −2pµx ∂t mij (u) uj e dx .
n X j=1
Using integrations by parts, we now get dW1 = T 1 + T 2 + T3 , dt
(3.30) with (3.31)
!2p 1 n n X W11−2p X p , T1 , − pi λi (u) mij (u)uj e−µx 2p i=1 j=1 0
(3.32)
T2 ,
−µW11−2p
Z
n 1X
0
(3.33)
T3 ,
W11−2p
Z 0
+
n X j=1
ppi λi (u)
i=1
ppi
e−2pµx dx,
mij (u)uj
j=1
i=1
n 1X
n X
!2p
n X
!2p−1 mij (u)uj
λi (u)
j=1
n X
(∂x mij (u))uj
j=1
! n ∂λ 1 X i mij (u)uj (u)∂x u e−2pµx dx ∂t mij (u) uj + 2p j=1 ∂u
Analysis of the first term T1 . From (3.31), we have !2p n n X W11−2p X p pi λi (u(t, 1)) (3.34) T1 = − mij (u(t, 1))uj (t, 1)e−µ 2p i=1 j=1 −
n X
ppi λi (u(t, 0))
i=1
n X
!2p mij (u(t, 0))uj (t, 0)
.
j=1
Let K , G0 (0). By the definition of ρ∞ (see (1.4)) and (3.14), there exist ∆i > 0, i ∈ {1, . . . , n}, such that (3.35)
θ,
n X
|Kij |
j=1
∆i < 1. ∆j
The parameters pi are selected such that (3.36)
ppi Λi = ∆2p i ,
i = 1, . . . , n.
11
Dissipative boundary conditions for hyperbolic systems: Lyapunov stability
We define ξi : [0, T ] → R, i = 1, . . . , n, by ξi (t) , ∆i ui (t, 1), ∀t ∈ [0, T ].
(3.37)
From (3.34), (3.36) and (3.37), we have n W11−2p X λi (u(t, 1)) (3.38) T1 = − 2p Λi i=1 −
n X
∆i mij (u(t, 1)) ξj (t)e−µ ∆ j j=1
n X λi (u(t, 0)) i=1
Λi
n X
!2p
!2p mij (u(t, 0))∆i uj (t, 0)
.
j=1
Let t ∈ [0, T ]. Without loss of generality, we may assume that ξ12 (t) = max{ξi2 (t), i = 1, . . . , n}.
(3.39)
Let us denote by δ and C various positive constants which may vary from place to place but are independent of t ∈ [0, T ], u and p ∈ N+ . From (3.18) and (3.39), we have, for |u(t, 1)|0 6 δ, !2p ∆i −µ mij (u(t, 1)) ξj (t)e Λi ∆j j=1 i=1 !2p n λ1 (u(t, 1)) X ∆1 −µ > m1j (u(t, 1)) ξj (t)e Λ1 ∆j j=1 2p > e−2pµ (1 − C|ξ1 (t)|) |ξ1 (t)| − C|ξ1 (t)|2
n X λi (u(t, 1))
(3.40)
n X
= e−2pµ (1 − C|ξ1 (t)|)
2p+1
(ξ1 (t))
2p
.
From (3.15), (3.35), (3.37) and (3.39), we have, for |u(t, 0)|0 6 δ,
(3.41)
n X λi (u(t, 0)) i=1
Λi
n X
!2p mij (u(t, 0))∆i uj (t, 0)
j=1
2p n n X X ∆i C|ξ1 (t)|2 + 6 (1 + C|ξ1 (t)|) |Kij | |ξj (t)| ∆ j i=1 j=1
6 n (1 + C|ξ1 (t)|) θ|ξ1 (t)| + C|ξ1 (t)|2
2p
.
From (3.35), (3.38), (3.39) and (3.41), there exists δ11 ∈ (0, 1), independent of u, such that, for every µ ∈ (0, δ11 ), for every p ∈ (1/δ11 , +∞) ∩ N+ and for every u, we have (3.42)
T1 (t) 6 0 if |u(t)|0 < δ11 .
Analysis of the second term T2 . Let (3.43)
α , min(Λ1 , . . . , Λn )/2.
12
J-M Coron & G. Bastin
From (3.19) and (3.32) there is a δ12 ∈ (0, η] such that, for every µ ∈ (0, +∞), for every p ∈ N+ and for every u, T2 6 −µαW1 if |u|0 < δ12 .
(3.44)
Analysis of the third term T3 . Using (3.1) and (3.33), we have !2p−1 Z 1X n n X (3.45) T3 = W11−2p ppi mij (u)uj 0
1 ∂λi ux 2p ∂u
n X
i=1
j=1
! mij (u)uj
+
j=1
n X ∂mij j=1
∂u
!
− A(u) + λi (u) ux uj e−2pµx dx.
From (3.18), (3.19) and (3.45) one gets the existence of β1 > 0 and δ13 > 0 such that, for every µ ∈ (0, +∞), for every p ∈ N+ and for every u, (3.46)
T3 6 β1 |ux |0 W1 if |u|0 < δ13 .
Let δ1 , min{δ11 , δ12 , δ13 }. From (3.30), (3.42), (3.44) and (3.46), we conclude that dW1 6 −µα1 W1 + β1 |ux |0 W1 dt provided that u is such that |u|0 < δ1 , that p ∈ (1/δ1 , +∞) ∩ N+ and that µ ∈ (0, δ1 ). This completes the proof of Lemma 3.5. Lemma 3.6. Let pi (i = 1, . . . , n) be given by (3.36). If ρ∞ (G0 (0)) < 1, there exist β2 and δ2 such that, for every µ ∈ (0, δ2 ), for every p ∈ (1/δ2 , +∞) ∩ N+ and for every C 1 -solution u : [0, T ] × [0, 1] → Rn of (3.1) and (3.15) such that |u|0 < δ2 , we have, in the sense of distributions in (0, T ), dW2 6 − µα + β2 |ux (t)|0 W2 (t), ∀t ∈ [0, T ], dt with α defined by (3.43) Proof. We first deal with the case where A, G and u are of class C 2 . By time differentiation of (3.1) and (3.15), we see that ut satisfy the following hyperbolic dynamics for t ∈ [0, T ] and x ∈ [0, 1]: (3.47) (3.48)
(ut )t + A(u)(ut )x + A0 (u, ut )ux = 0, ∂t u(t, 0) =
∂G(u(t, 1)) ∂t u(t, 1), ∂u(t, 1)
where A0 (u, ut ) is a compact notation for the matrix whose entries are ∂aij (u) ut , i ∈ {1, . . . , n}, j ∈ {1, . . . , n}. ∂u Using (3.47)–(3.48), we see that the time derivative of W2 is: 2p−1 Z 1X n n X dW2 1 1−2p (3.49) = W 2p ppi mij (u)∂t uj dt 2p 2 0 i=1 j=1 " # n n X X mij (u)∂tt uj + ∂t mij (u) ∂t uj e−2pµx dx. A0 (u, ut )i,j ,
j=1
j=1
13
Dissipative boundary conditions for hyperbolic systems: Lyapunov stability
From (3.17), similarly as for (3.28), it can be shown that n X
mij (u)∂tt uj = −λi (u)
j=1
n X
mij (u)∂x ∂t uj +
j=1
n X
n X
mij (u)
j=1
! a ˜jk (u, ut )∂t uk ,
k=1
˜ ut ) , A0 (u, ut )A−1 (u). Then, where a ˜ij (u, ut ) is the (i, j)-th entry of the matrix A(u, by substituting this expression for the term between brackets in (3.49), we get 1 1−2p dW2 = W dt 2p 2
n 1X
Z 0
2p ppi
i=1
n X
!2p−1 mij (u)∂t uj
j=1
" − λi (u)
n X
mij (u)∂x ∂t uj
j=1 n n X X
+
j=1
! # mik (u)˜ akj (u, ut ) + ∂t mij (u) ∂t uj e−2pµx dx.
k=1
Using integration by parts as in the proof of Lemma 3.5, we get dW2 = U1 + U2 + U3 , dt with !2p 1 n n X −µx W21−2p X p , − pi λi (u) mij (u) ∂t uj e U1 , 2p i=1 j=1 0
U2 ,
−µW21−2p
Z
n 1X
0
U3 ,
W21−2p
0
+
i=1
n 1X
Z
i=1
n X
ppi λi (u)
ppi
n X
!2p mij (u) ∂t uj
e−2pµx dx,
j=1 n X
!2p−1 " mij (u) ∂t uj
j=1
n X
λi (u)(∂x mij (u))
j=1
mik (u)˜ akj (u, ut ) + ∂t mij (u)
! ∂t uj
k=1
# n ∂λ 1 X i mij (u)∂t uj (u)ux e−2pµx dx. + 2p j=1 ∂u Analysis of the first term U1 . Using the boundary conditions (3.15) and (3.48), we have !2p Pn W21−2p Pn p U1 = − e−2pµ i=1 pi λi (u(t, 1)) j=1 mij (u(t, 1)) ∂t uj (t, 1) 2p !2p n X Pn ∂Gj (u(t, 1)) ∂t u(t, 1) − i=1 ppi λi (G(u(t, 1))) mij (G(u(t, 1))) ∂u(t, 1) j=1
14
J-M Coron & G. Bastin
Then, in a way similar to the analysis of T1 in the proof of Lemma 3.5, we can show that, since ρ∞ (K) < 1, there exists δ21 ∈ (0, 1), such that, for every µ ∈ (0, δ21 ), for every p ∈ (1/δ21 , +∞) ∩ N+ and for every u, we have U1 6 0 provided that |u|0 < δ21 . Analysis of the second term U2 . Proceeding as in the proof of (3.44), we get the existence of δ22 ∈ (0, η] such that, for every µ ∈ (0, +∞), for every p ∈ N+ and for every u, U2 6 −µαW2 if |u|0 < δ22 ,
(3.50) with α defined as in (3.43).
Analysis of the third term U3 . Proceeding as in the proof of (3.46), we get the existence of β2 > 0 and δ23 > 0 such that, for every µ ∈ (0, +∞), for every p ∈ N+ and for every u, U3 6 β2 |ux |0 W2 if |u|0 < δ23 . From the analysis of U1 , U2 and U3 , we conclude that, with δ2 , min{δ21 , δ22 , δ23 }, dW2 6 −µαW2 + β2 |ux |0 W2 dt for all u such that |u|0 < δ2 provided that µ ∈ (0, δ2 ) and that p ∈ (1/δ2 , +∞) ∩ N+ . The above estimates were obtained for A, G and u of class C 2 . But their proofs show that they do not depend on the C 2 -norm of A, G and u. Hence, by density arguments, they remain valid with A, G and u only of class C 1 . This completes the proof of Lemma 3.6. Proof. We now prove Theorem 3.3. Let us choose µ ∈ R such that (3.51)
0 < µ < min {δ1 , δ2 } ,
where δ1 and δ2 are as in Lemma 3.5 and Lemma 3.6 respectively. Let us define two functionals V1 : C 1 ([0, 1]; Rn ) → R and V2 : C 1 ([0, 1]; Rn ) → R by T n n X X V1 (u) , ∆21 m1j (u)uj e−µx , . . . , ∆2n mnj (u)uj e−µx , (3.52) j=1 j=1 0 T n n X 2 X mnj (u)(∂t uj )e−µx , (3.53) V2 (u) , ∆1 m1j (u)(∂t uj )e−µx , . . . , ∆2n j=1 j=1 0 for every u ∈ C 1 ([0, 1]; Rn ) with |u|0 < ε0 , where ε0 > 0 is chosen small enough. In (3.53), we again use (3.21) for every u ∈ C 1 ([0, 1]; Rn ) with |u|0 < ε0 . This is a valid definition if, again, ε0 > 0 is chosen small enough. We consider the Lyapunov function candidate V : C 1 ([0, 1]; Rn ) → R defined by (3.54)
V(u) , V1 (u) + V2 (u), ∀u ∈ C 1 ([0, 1]; Rn ) such that |u|0 < ε0 .
Still if ε0 > 0 is small enough, which will be always assumed, we can select γ ∈ (1, +∞) such that 1 V(u) 6 |u|1 6 γV(u) for every u ∈ C 1 ([0, 1]; Rn ) such that |u|0 < ε0 . (3.55) γ
Dissipative boundary conditions for hyperbolic systems: Lyapunov stability
15
Let us now choose T > 0 large enough so that γ 2 e−µαT /2 6
(3.56)
1 2
with α defined by (3.43). Let ε2 ∈ (0, +∞) be such that δ 1 δ 2 ε0 , , , ε1 , (3.57) ε2 < min C1 C1 C1 where, again, δ1 and δ2 are as in Lemma 3.5 and Lemma 3.6 respectively. Let uo be in C 1 ([0, 1]; Rn ), satisfy the compatibility conditions (3.16) and be such that |uo |1 6 ε2 .
(3.58)
Let u : [0, T ] × [0, 1] → Rn be the solution of class C 1 of (3.1) and (3.15) satisfying the initial condition u(0, ·) = uo (·). By Theorem 3.1, (3.57) and (3.58), u is well defined and it satisfies |u|0 < min{δ1 , δ2 , ε0 }.
(3.59)
In order to emphasize the dependence of W1 and W2 on p ∈ N+ , we now write them W1,p and W2,p respectively. For t ∈ [0, T ], let us define (3.60)
V (t) , V(u(t)), V1 (t) , V1 (u(t)), V2 (t) , V2 (u(t)),
(3.61)
W1,p (t) , W1,p (u(t)), W2,p (t) , W2,p (u(t)).
Let us point out that, for every continuous function f = (f1 , . . . , fn )T : [0, 1] → Rn , we have !1/(2p) Z 1X n 2p (3.62) |f |0 = lim |fi (x)| dx , p→∞
Z (3.63) 0
n 1X
0
i=1
!1/(2p) |fi (x)|2p dx
6 n1/(2p) |f |0 , ∀p ∈ [1, +∞).
i=1
By the definition of W1 and W2 , (3.60), (3.61), (3.62) and (3.63), we have (3.64) (3.65)
V1 (t) = lim W1,p (t) p→∞
V2 (t) = lim W2,p (t), ∀t ∈ [0, T ],
and
p→∞
∃M > 0 such that W1,p (t) + W2,p (t) 6 M |u|1 , ∀p ∈ [1, +∞), ∀t ∈ [0, T ].
In particular ∗
(3.66)
W1,p * V1 in the weak∗ topology σ(L∞ (0, T ), L1 (0, T )) as p → +∞,
(3.67)
W2,p * V2 in the weak∗ topology σ(L∞ (0, T ), L1 (0, T )) as p → +∞.
∗
(Let us recall that, for a bounded sequence (ϕp )p∈N+ of elements of L∞ (0, T ) and ∗ ϕ ∈ L∞ (0, T ) one says that ϕp * ϕ in the weak∗ topology σ(L∞ (0, T ), L1 (0, T )) if, 1 for every ψ ∈ L (0, T ), ZT ZT ϕp (t)ψ(t)dt → ϕ(t)ψ(t)dt as p → +∞.) 0
0
16
J-M Coron & G. Bastin
From Lemmas 3.5 and 3.6, (3.59), (3.66) and (3.67), we have, in the distribution sense in (0, T ), dV1 6 −µαV1 + β1 |ux |0 V1 , dt dV2 6 −µαV2 + β2 |ux |0 V2 . dt
(3.68) (3.69)
Summing (3.68) and (3.69), we get, in the distribution sense in (0, T ), dV 6 −µαV + β|u|1 V, dt
(3.70)
with β , max {β1 , β2 }. Let us impose on ε2 , besides (3.57), that (3.71)
ε2 6
µα . 2βC1
From (3.13), (3.58) and (3.71), we get that (3.72)
β|u|1 6
µα . 2
From (3.70) and (3.72), we have, in the distribution sense in (0, T ), dV µ 6 − αV, dt 2
(3.73) which implies that
V (T ) 6 e−αµT /2 V (0).
(3.74)
From (3.55) and (3.74), we obtain that |u(T )|1 6 γ 2 e−αµT /2 |u(0)|1 ,
(3.75)
which, together with (3.56), implies that |u(T )|1 6
(3.76)
1 |u(0)|1 . 2
This completes the proof of Theorem 3.3. Remark 3.7. Instead of using time derivatives in the distribution sense as, for instance in (3.70), we could have directly derived inequality (3.74) on V (t) by establishing the same inequality for W (t) , W1 (t) + W2 (t) and let p → +∞. 3.2. Proof of Theorem 3.3 in the general case where 0 < m < n. In this section, we explain the modifications of the proof that must be used to deal with the case 0 < m < n (of course, the case m = 0 is equivalent to the case m = n by considering u(t, 1 − x) instead of u(t, x). The functionals W1 and W2 are now defined as follows: !2p Z 1"X m n X p W1 (u) , pi mij (u)uj e−2pµx 0
i=1
j=1
+
n X i=m+1
ppi
n X j=1
!2p mij (u)uj
#
1 2p
e2pµx dx ,
17
Dissipative boundary conditions for hyperbolic systems: Lyapunov stability
Z W2 (u) ,
1
"
m X
0
n X
ppi
i=1
mij (u) ∂t uj
!2p e−2pµx
j=1 n X
+
ppi
i=m+1
n X
mij (u) ∂t uj
!2p
#
1 2p
e2pµx dx .
j=1
Let u : [0, T ]×[0, 1] → Rn be a solution of (3.1) - (3.2) of class C 1 . With computations similar to those of Lemma 3.5, it is readily seen that the time derivative of W1 (t) , W1 (u(t)) is given by: dW1 = T1 + T2 + T3 dt with !2p m n X W11−2p X p T1 , pi λi (u) mij (u)uj e−µx − 2p i=1 j=1 n X
−
ppi λi (u)
n X
i=m+1
T2 ,
−µW11−2p
Z
1
"
0
m X
ppi λi (u)
n X
i=1
,
j=1
0
!2p mij (u)uj
e−2pµx
m X
ppi |λi (u)|
i=m+1
n X
!2p mij (u)uj
# e
2pµx
dx,
j=1
!2p−1
" T3 ,
mij (u)uj eµx
j=1
+
R1 W11−2p 0
!2p 1
Pm
p i=1 pi
Pn
j=1
mij (u)uj
λi (u)
Pn
j=1 (∂x mij (u))uj
! n ∂λ 1 X i mij (u)uj (u)∂x u e−2pµx + ∂t mij (u) uj + 2p ∂u j=1 j=1 !2p−1 n n n X X X + ppi mij (u)uj λi (u) (∂x mij (u))uj i=m+1 j=1 j=1 ! # n P ∂λ 1 X i n mij (u)uj (u)∂x u e2pµx dx. + j=1 ∂t mij (u) uj + 2p j=1 ∂u n X
The parameters pi > 0 are now selected such that (to be compared with (3.36)) (3.77)
ppi |Λi | = ∆2p i ,
i = 1, . . . , n.
Concerning the ξi , i = 1, . . . , n, they are now defined by (to be compared with (3.37)) ξi (t) , ∆i ui (t, 1) for i = 1, . . . , m
and
ξi (t) , ∆i ui (t, 0) for i = m + 1, . . . , n.
18
J-M Coron & G. Bastin
It is then a straightforward exercise to verify that Theorem 3.3 can be established for the case 0 < m < n in a manner completely parallel to the one we have followed in the case m = n. Remark 3.8. Looking at our proof of Theorem 3.3 we see that, for every (3.78)
ν ∈ (0, − min(|Λ1 |, . . . , |Λn |) ln (ρ (|G0 (0)|))) ,
there exist ε > 0 and C > 0 such that, for every uo such that |uo |1 < ε and satisfying the compatibility conditions (3.5) to (3.7), the Cauchy problem (3.1)–(3.2)–(3.4) has a unique C 1 -solution which satisfies |u(t, .)|1 6 Ce−νt |uo |1 , ∀t ∈ [0, +∞). See in particular (3.40) and (3.41), replace the definition of α given in (3.43) by the condition (3.79) α ∈ 0, min(|Λ1 |, . . . , |Λn |) and modify a little bit some arguments in order to deal with (3.79) instead of (3.43). 4. Conclusion and final remark. In this article, we have addressed the issue of exponential stability for the C 1 -norm of quasi-linear hyperbolic systems of the form (3.1), (3.2). Our main result (Section 3) has been to give a new proof, using a strict Lyapunov function, that a sufficient condition for the exponential stability of the steady-state is to have a map G such that ρ∞ (G0 (0)) < 1 for the boundary conditions (3.2). The approach followed in the paper can be easily adapted to the exponential stability for the C ` -norm with ` ∈ N+ provided the compatibility conditions (3.5), (3.6), (3.7) are adapted accordingly (see [10] for details) and that A and G are of class C ` . The analysis requires the following extensions of formulas (3.19) or (3.20) and (3.52) or (3.53):
(4.1)
Z Wk (u) ,
n 1X
0
ppi
n X
!2p mij (u)∂tk−1 uj
1 2p
e−2pµx dx ,
j=1
i=1
T n n X X (4.2) Vk (u) , ∆21 m1j (u)(∂tk−1 uj )e−µx , . . . , ∆2n mnj (u)(∂tk−1 uj )e−µx j=1 j=1 0 where ∂tk−1 u is defined by differentiating formally (1.1) with respect to time. For example ∂t2 u , (A0 (u) (A(u)∂x u)) ∂x u + A(u) ((A0 (u)∂x u) ∂x u) 2 + A(u) A(u)∂xx u , ∀u ∈ C 2 ([0, 1]; Rn ) with |u|0 small enough. Then, using the Lyapunov function (4.3)
V(u) =
`+1 X k=1
the following theorem holds.
Vk (u),
Dissipative boundary conditions for hyperbolic systems: Lyapunov stability
19
Theorem 4.1. Let assume that A and G are of class C ` . Then, if (4.4)
ρ∞ (G0 (0)) < 1,
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