Time-periodic solutions to quasilinear hyperbolic systems with time-periodic boundary conditions

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Journal de Mathématiques Pures et Appliquées www.elsevier.com/locate/matpur

Time-periodic solutions to quasilinear hyperbolic systems with time-periodic boundary conditions ✩ Peng Qu School of Mathematical Sciences & Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433, China

a r t i c l e

i n f o

Article history: Received 1 August 2019 Available online xxxx MSC: 35L50 35L60 35B10 35B40 35A09 93D15 Keywords: Time-periodic solution Quasilinear hyperbolic system Time-periodic boundary condition Asymptotic stability

a b s t r a c t For quasilinear hyperbolic systems with time-periodic boundary conditions possessing a dissipative structure, the existence, uniqueness and stability of the timeperiodic classical solutions are proved. Moreover, the feedback boundary control with dissipative structure can stabilize the system around the time-periodic solution, provided that the time-periodic boundary conditions are W 2,∞ smooth. © 2019 Elsevier Masson SAS. All rights reserved.

r é s u m é Pour des systèmes hyperboliques quasi linéaires avec conditions aux limites périodiques en temps et dissipatives, on démontre l’existence, l’unicité et la stabilité de solutions classiques périodiques en temps. De plus, le contrôle frontière feedback avec une structure dissipative peut stabiliser le système autour de cette solution périodique en temps, si les conditions aux limites ont la régularité W 2,∞ . © 2019 Elsevier Masson SAS. All rights reserved.

1. Introduction In this paper, we study the initial-boundary value problem for general system of first order quasilinear hyperbolic equations ∂t u + A(u)∂x u = 0,

(t, x) ∈ R × [0, L],

(1.1)

where u = (u1 (t, x), . . . , un (t, x))T ∈ C 1 (R × [0, L]; U) is the unknown vector and the domain U ⊂ Rn is a small neighborhood of u = 0. By hyperbolicity, A(u) = (aij (u))ni,j=1 is a smooth matrix-valued function on U, which has n nonzero real eigenvalues λi (u) (i = 1, . . . , n) ✩

This work is supported in part by NSFC Grants 11831011 and 11501121. E-mail address: [email protected].

https://doi.org/10.1016/j.matpur.2019.10.010 0021-7824/© 2019 Elsevier Masson SAS. All rights reserved.

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2

∀ r = 1, . . . , m; s = m + 1, . . . , n,

(1.2)

∀ u ∈ U, ∀ r = 1, . . . , m; s = m + 1, . . . , n,

(1.3)

λr (0) < 0 < λs (0), or equivalently λr (u) < 0 < λs (u),

and a complete set of left and right eigenvectors li (u) = (li1 (u), . . . , lin (u)) and ri (u) = (r1i (u), . . . , rni (u))T (i = 1, . . . , n), namely, li (u)A(u) = λi (u)li (u),

∀ u ∈ U, ∀ i = 1, . . . , n,

(1.4)

A(u)ri (u) = λi (u)ri (u),

∀ u ∈ U, ∀ i = 1, . . . , n

(1.5)

with det(lij (u))ni,j=1 = 0,

∀ u ∈ U.

(1.6)

Without loss of generality, we may assume A(0) = diag{λi (0)}ni=1 .

(1.7)

Moreover, for simplicity, we require that all the eigenvalues λi (u) and eigenvectors li (u), ri (u) (i = 1, . . . , n) are smooth, and without loss of generality, we assume that li (u)rj (u) = δij , |ri (u)| = 1,

∀ u ∈ U, ∀ i, j = 1, . . . , n,

(1.8)

∀ u ∈ U, ∀ i = 1, . . . , n,

(1.9)

∀ i = 1, . . . , n,

(1.10)

∀ i, j = 1, . . . , n.

(1.11)

i = 1, . . . , n,

(1.12)

where δij is the Kronecker’s symbol, and thus li (0) = eTi ,

ri (0) = ei ,

where ei is the i-th unit vector in Rn , namely, lij (0) = rij (0) = δij , Noting (1.3), we can further set μi (u) = λ−1 i (u), and denote μmax = max sup |μi (u)|. i=1,...,n u∈U

(1.13)

By rescaling the time variable if needed, we may assume μmax ≤ 1.

(1.14)

For system (1.1), we consider its C 1 classical solutions to the initial-boundary value problem with initial data

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t = 0 : u(0, x) = ϕ(x),

3

x ∈ [0, L]

(1.15)

and boundary conditions x = 0 : us = Gs (hs (t), u1 , . . . , um ),

s = m + 1, . . . , n,

(1.16)

x = L : ur = Gr (hr (t), um+1 , . . . , un ),

r = 1, . . . , m,

(1.17)

where Gs (hs , u1 , . . . , um ) (s = m + 1, . . . , n) and Gr (hr , um+1 , . . . , un ) (r = 1, . . . , m) are smooth functions with Gs (0, 0, . . . , 0) = 0,

s = m + 1, . . . , n,

(1.18)

Gr (0, 0, . . . , 0) = 0, r = 1, . . . , m;

(1.19)

while ϕ(x) and hi (t) (i = 1, . . . , n) are small C 1 smooth functions satisfying certain C 1 compatibility conditions at (t, x) = (0, 0) and (0, L), respectively. By redefining hi (t) if needed, we can further assume  ∂G   r  max  (0, . . . , 0) ≤ 1, r=1,...,m ∂hr

 ∂G   s  (0, . . . , 0) ≤ 1.  s=m+1,...,n ∂hs max

(1.20)

In what follows, we always suppose that assumptions (1.3)–(1.9), (1.14), (1.18)–(1.20) hold. One may refer to [15] for the local existence of the corresponding C 1 classical solutions. In this paper, we discuss the case of time-periodic boundary conditions, namely, we assume that all functions hi (t) (i = 1, . . . , n) share one constant temporal period T∗ > 0, i.e., hi (t + T∗ ) = hi (t),

∀ t ∈ R, ∀ i = 1, . . . , n.

(1.21)

In this case, u ≡ 0 is not an equilibrium to the system in general, and it is natural to guess that the system may admit time-periodic classical solutions. See for instance [22]. The time-periodic solutions to hyperbolic systems are also important in the analysis of resonance phenomenon, see for instance [17,19]. In what follows, under certain conditions, we will show first the existence of this time-periodic solution, and then prove its asymptotic stability, namely, for all other small solutions to the system with the same boundary conditions but different initial data, they would converge to the time-periodic solution asymptotically. Generally speaking, due to the nonlinearity of the system, if boundary conditions do not provide dissipation, the classical solutions to the initial-boundary value problem (1.1) and (1.15)–(1.17) would blowup in finite time, even for small and smooth initial and boundary data, see examples in [14] and Section 6 of this paper for instance. In order to discuss the asymptotic behavior of C 1 classical solutions, we need certain dissipative structures for boundary conditions. In this paper, we will apply the one given in [18] and [23] as follows, see also Chapter 5 of [14]. For the matrix ⎛

 ∂G

0

def. ⎜ Θ = (θij )ni,j=1 = ⎜  ⎝  ∂Gs (0, 0, . . . , 0) s=m+1,...,n ∂ur r=1,...,m

r

∂us

⎞

 (0, 0, . . . , 0)

r=1,...,m ⎟ s=m+1,...,n ⎟

0

⎠,

(1.22)

we define its minimal characterizing number as θ = Θmin =

where

inf

Γ=diag{γi }n i=1 γi =0

ΓΘΓ−1 ,

(1.23)

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Θ = max

i=1,...,n

n 

|θij |,

(1.24)

j=1

and through out this paper, except in Section 6, we require θ < 1.

(1.25)

By the results given in [14,18,23], under the assumption (1.25), there exists a constant ε0 > 0, such that for any given constant ε ∈ (0, ε0 ) and any given functions ϕ and hi (i = 1, . . . , n) satisfying ϕC 1 ≤ ε,

(1.26)

max hi C 1 ≤ ε,

(1.27)

i=1,...,n

there exists a global C 1 classical solution u = u(t, x) on (t, x) ∈ R+ × [0, L] to the initial-boundary value problem (1.1) and (1.15)–(1.17). These results also show the asymptotic stability of the constant equilibrium u = 0, provided that hi (t) (i = 1, . . . , n) decays in certain rate. See also [1–11,13,16] for the corresponding boundary stabilization results of hyperbolic systems under this kind of dissipative structure, and [20] for its applications to controllability problems. On the other hand, for the case that hi (t) (i = 1, . . . , n) are periodic as in (1.21), the literature on the asymptotic behavior of the solutions is quite limited. [22] discusses this problem for supersonic isentropic Euler flow in frameworks of classical solutions or entropy solutions. We note that due to the supersonic assumption, it belongs to the special case θ = 0, and one has boundary conditions only on one side of the domain. In this paper, we will study time-periodic boundary conditions under a more general requirement (1.25) but in the framework of classical solutions. Our first result is the existence of time-periodic solutions. Theorem 1 (Existence of time-periodic solutions). Under the hypothesis (1.25), there exists a small constant ε1 ∈ (0, ε0 ) and a constant CP > 0, such that for any given ε ∈ (0, ε1 ), any given T∗ ∈ R+ , and any given functions hi (t) (i = 1, . . . , n) satisfying (1.21) and (1.27), there exists a C 1 smooth function ϕ = ϕ(P ) (x) satisfying ϕC 1 ≤ CP ε,

(1.28)

such that the initial-boundary value problem (1.1) and (1.15)–(1.17) admits a C 1 classical solution u = u(P ) (t, x) on (t, x) ∈ R × [0, L], satisfying u(P ) (t + T∗ , x) = u(P ) (t, x),

∀ (t, x) ∈ R × [0, L].

(1.29)

Moreover, we can further get the asymptotic stability of such time-periodic solution, which also shows the long time behavior of solutions with general small initial data. Theorem 2 (Stability of the time-periodic solution). Under the hypothesis (1.25), there exists a small constant ε2 ∈ (0, ε1 ) and a constant CS > 0, such that for any given ε ∈ (0, ε2 ), any given T∗ ∈ R+ , and any given ϕ(x) and hi (t) (i = 1, . . . , n) satisfying (1.21) and (1.26)–(1.27) with certain compatibilities, the initialboundary value problem (1.1) and (1.15)–(1.17) admits a unique global C 1 classical solution u = u(t, x) on (t, x) ∈ R+ × [0, L], satisfying u(t, ·) − u(P ) (t, ·)C 0 ≤ CS α[t/T0 ] ε,

∀ t ∈ R+ ,

where u(P ) , depending on hi (t) (i = 1, . . . , n), is the time-periodic solution given by Theorem 1 and

(1.30)

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α=

5

2+θ ∈ (0, 1), 3

T0 = max sup 1≤i≤n u∈U

(1.31)

L = Lμmax . |λi (u)|

(1.32)

By taking t → +∞, this result directly shows the uniqueness. Corollary 3 (Uniqueness of the time-periodic solution). Under the hypothesis (1.25), there exists a constant ε3 ∈ (0, ε2 ), such that for any given ε ∈ (0, ε3 ), any given T∗ ∈ R+ , and any given functions hi (t) (i = 1, . . . , n) satisfying (1.21) and (1.27), the corresponding time-periodic solution u = u(P ) (t, x) is unique. From another point of view, we can consider a problem of boundary stabilization. As discussed in [14, 18,23] and in [1–11,13,16], when hi (t) ≡ 0 for all i = 1, . . . , n, the boundary condition (1.16)–(1.17) can be treated as a feedback boundary control, and when the dissipative structure requirement (1.25) is satisfied, this feedback control can stabilize the system around the constant equilibrium u = 0. In this paper, we will study this stabilization property for the more general case that when all hi (t) (i = 1, . . . , n) are smooth and periodic, satisfying (1.21), whether or not the feedback boundary control with (1.25) can still stabilize the system around the corresponding time-periodic solution. In order to discuss the stabilization problem in the framework of C 1 classical solutions, we first present the following regularity result. Theorem 4 (Regularity of the time-periodic solution). Under the hypothesis (1.25), if all hi (t) (i = 1, . . . , n) satisfy (1.21) and (1.27) and possess further W 2,∞ regularity with max hi L∞ ≤ M2 < +∞,

(1.33)

i=1,...,n

then there exist constants CR > 0 and ε4 ∈ (0, ε1 ), such that for any given ε ∈ (0, ε4 ), the time-periodic solution u = u(P ) (t, x) provided by Theorem 1 is also a W 2,∞ function with max{∂t2 u(P ) L∞ , ∂t ∂x u(P ) L∞ , ∂x2 u(P ) L∞ } ≤ CR < +∞.

(1.34)

In fact, one can get a more general theory as in [21], that for any given k ≥ 2 and p ∈ [1, ∞], if all hi (t) k,p (i = 1, . . . , n) are C k or Wperiod smooth with small C 1 norm, then the corresponding time-periodic solution k,p will possess the same C k or Wperiod regularity. Since for our next stabilization result, we need only the 2,∞ W regularity, we omit the proof of the general cases in this paper. Now we can show that the boundary feedback with (1.25) can stabilize the system around such W 2,∞ time-periodic solution.

Theorem 5 (Stabilization around the time-periodic solution). Under the hypotheses of Theorem 2, suppose furthermore (1.33), then there exist constants CS∗ > 0 and ε5 ∈ (0, min{ε2 , ε4 }), such that for any given ε ∈ (0, ε5 ), we have not only the C 0 convergence result (1.30) in Theorem 2, but also the C 1 convergence as ˙ C 0 , ∂x u(t, ·) − ∂x u(P ) (t, ) ˙ C 0 } ≤ C ∗ α[t/T0 ] ε, max{∂t u(t, ·) − ∂t u(P ) (t, ) S

∀ t ∈ R+ .

(1.35)

In the remaining parts of this paper, we will prove the above theorems one by one in Sections 2–5. And in Section 6, we will present a counter example to show the necessity of the dissipative structure requirement (1.25).

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2. Existence of time-periodic solutions In this section, we will use an iteration method to construct the time-periodic solution of Theorem 1. First, we apply the classical linearization as in [15], see also [12]. By multiplying the left eigenvectors li (u) (i = 1, . . . , n) to system (1.1) from left, we get

∂t ui + λi (u)∂x ui =

n 



Bij (u) ∂t uj + λi (u)∂x uj ,

i = 1, . . . , n,

(2.1)

j=1

where ⎧ l (u) ⎪ ⎨ − ij , i = j, lii (u) Bij (u) = ⎪ ⎩ 0, i = j,

(2.2)

and by (1.11), we have ∀ i, j = 1, . . . , n.

Bij (0) = 0,

(2.3)

Accordingly, we can set our linearized system as (k)

∂t ui

(k)

+ λi (u(k−1) )∂x ui

=

n 

(k−1)

Bij (u(k−1) )(∂t uj

(k−1)

+ λi (u(k−1) )∂x uj

),

i = 1, . . . , n,

(2.4)

j=1 (k−1)

x = 0 : u(k) s = Gs (hs (t), u1

(k−1) , . . . , um ),

(k−1)

(k−1) x = L : u(k) ), r = Gr (hr (t), um+1 , . . . , un

s = m + 1, . . . , n,

(2.5)

r = 1, . . . , m,

(2.6)

which allows us to perform the iteration that starts from u(0) (t, x) ≡ 0.

(2.7)

For this linearized system (2.4)–(2.6), started from (2.7), we will prove the following a priori estimates Proposition 2.1. For the iteration scheme (2.4)–(2.7), under hypothesis (1.25), for small enough ε1 > 0 and (k) large enough CP > 0, the sequence of C 1 solutions ui (t, x) (i = 1, . . . , n; k ∈ Z+ ) to system (2.4)–(2.6) satisfies u(k) (t + T∗ , x) = u(k) (t, x), def.

u(k) C 1 =

(k)

∀ (t, x) ∈ R × [0, L], ∀ k ∈ Z+ , (k)

(k)

max {ui C 0 , ∂t ui C 0 , ∂x ui C 0 } ≤ CP ε,

i=1,...,n

u(k) − u(k−1) C 0 ≤ CP εαk ,

∀ k ∈ Z+ ,

∀ k ∈ Z+

(2.8) (2.9) (2.10)

and max

i=1,...,n



(k) (k)  ω(η | ∂t ui ) + ω(η | ∂x ui ) ≤ ΩP (η),

∀ k ∈ Z+ .

(2.11)

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Here α is the constant defined by (1.31), ω(η | ·) is the modulus of continuity: ω(η | f ) =

sup |t1 −t2 |≤η |x1 −x2 |≤η

|f (t1 , x1 ) − f (t2 , x2 )|,

and ΩP (η) is a continuous function of η ∈ (0, 1), independent of k, to be determined later with lim ΩP (η) = 0.

η→0+

Similarly as in [15], once we show Proposition 2.1, we can directly prove Theorem 1. In fact, first, by (2.10), 0 0 (P ) the sequence {u(k) }∞ k=1 is a Cauchy sequence in C space, and thus converges to some C function u (P ) uniformly. Moreover by (2.8), this u is time-periodic. Then by (2.9) and (2.11), applying Arzelà–Ascoli theorem, we know that there exists a subsequence of {u(k) }, which converges uniformly in C 1 space. By the (P ) uniqueness of the limit, we know that the whole original sequence {u(k) }∞ in C 1 space. k=1 converges to u (P ) 1 Thus u is C smooth and is a classical solution to problem (1.1) and (1.16)–(1.17). Then we can take (P ) ϕ (x) = u(P ) (0, x) to get the initial data, which clearly satisfies (1.28) due to (2.9). In the next part of this section, we will establish the uniform a priori estimates (2.8)–(2.11) given in Proposition 2.1 inductively, namely, for each k ∈ Z+ , we will prove (k)

(k)

ui (t + T∗ , x) = ui (t, x),

∀ (t, x) ∈ R × [0, L], ∀ i = 1, . . . , n,

(2.12)

(k)

(2.13)

max ui C 1 ≤ CP ε,

i=1,...,n (k)

max ui

i=1,...,n

(k−1)

− ui

(k)

max ω(η | ∂t ui (·, x)) ≤

i=1,...,n

C 0 ≤ CP εαk ,

(2.14)

1 ΩP (η), 8

(2.15)

∀ x ∈ [0, L]

and (k) (k)  ω(η | ∂t ui ) + ω(η | ∂x ui ) ≤ ΩP (η),



max

i=1,...,n

(2.16)

under the assumptions (k−1)

ui

(k−1)

(t + T∗ , x) = ui

(t, x),

max

i=1,...,n (k−1)

max ui

i=1,...,n

(k−2)

− ui

(k−1)

max ω(η | ∂t ui

i=1,...,n

∀ (t, x) ∈ R × [0, L], ∀ i = 1, . . . , n,

(k−1) ui C 1

≤ CP ε,

C 0 ≤ CP εαk−1 ,

(·, x)) ≤

1 ΩP (η), 8

(2.17) (2.18)

(for k ≥ 2)

(2.19)

∀ x ∈ [0, L]

(2.20)

 ) ≤ ΩP (η).

(2.21)

and max

i=1,...,n



(k−1)

ω(η | ∂t ui

(k−1)

) + ω(η | ∂x ui

Here ω(η | f (·, x)) =

max

|t1 −t2 |≤η

|f (t1 , x) − f (t2 , x)|.

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Especially, due to (2.18), we have u(k−1) ∈ U for small ε1 , which guarantees that all our assumptions (0) (1.3)–(1.9) and (1.14) hold for (2.4)–(2.6). Since we have chosen (2.7) to start our iteration, ui (i = 1, . . . , n) satisfy (2.17)–(2.18) and (2.20)–(2.21) naturally. In the following proof, we need a series of constants in (0, α), being defined in a clearer manner. We first note that due to definition (1.23), after a linear transformation u ˜ = Γu,

(2.22)

we can assume without loss of generality that Θ = Θmin = θ < 1. Then for ε1 > 0 small enough, noting the definition of α given by (1.31), we may set

α0 = max

1

n    ∂Gr  (hr , um+1 , . . . , un ), 3 r=1,...,m |hr |≤ε u∈U s=m+1 ∂us

, max

sup sup

m   ∂Gs  1  (hs , u1 , . . . , un ) ∈ [ , α) (2.23) s=m+1,...,n |h |≤ε u∈U ∂u 3 r s r=1

max

sup sup

and accordingly set α as α =

2 − 1 + (2 + 1)α0 1 − α0 = α−1 + +1 +1 2 2

(2.24)

for all  ∈ Z+ . It is easy to check that 0 < α < α+1 <

1 + α0 < α < 1, 2

∀  ∈ N.

(2.25)

(k−1) Next, to simplify our analysis, we multiply μi (u(k−1) ) = λ−1 ) to the i-th equation of (2.4) for i (u i = 1, . . . , n, and then exchange the roles of t and x to get

(k−1) ∂x u(k) )∂t u(k) r + μr (u r =

n 

(k−1)

Brj (u(k−1) )(∂x + μr (u(k−1) )∂t )uj

, r = 1, . . . , m,

(2.26)

j=1 (k−1)

(k−1) x = L : u(k) ), r = Gr (hr (t), um+1 , . . . , un (k−1) ∂x u(k) )∂t u(k) s + μs (u s =

n 

r = 1, . . . , m,

(2.27) (k−1)

Bsj (u(k−1) )(∂x + μs (u(k−1) )∂t )uj

, s = m + 1, . . . , n,

(2.28)

j=1 (k−1)

x = 0 : u(k) s = Gs (hs (t), u1

(k−1) , . . . , um ),

s = m + 1, . . . , n.

(2.29)

The advantages of system (2.26)–(2.29) are that by exchanging the roles of t and x, we can treat the linearized time-periodic boundary information (2.5)–(2.6) as periodic initial conditions (2.27) and (2.29), and the linearized system is now decoupled in each iteration. Moreover, since we have changed the roles of t and x, we can define the characteristic curves t = (k) ti (x; t∗ , x∗ ) for i = 1, . . . , n and k ∈ Z+ by the following ODEs: ⎧ (k) ⎪ ⎨ dti (x; t , x ) = μ (u(k−1) (t(k) (x; t , x ), x)), ∗ ∗ i ∗ ∗ i dx ⎪ ⎩ (k) ti (x∗ ; t∗ , x∗ ) = t∗ .

(2.30)

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(k)

Now, we will check (2.12)–(2.16) one by one. First, by (2.17), we know that if ui (t, x) (i = 1, . . . , n) (k) solves (2.26)–(2.29), so does ui (t + T∗ , x) (i = 1, . . . , n), and thus by the uniqueness to this linear system, we get (2.12). Then in order to get the C 0 estimates part in (2.13), we first show it on the boundary. In fact, by (2.27), applying Hadamard’s formula on the boundary x = L and noting (1.19), we have 1 u(k) r (t, L)

= hr (t)

∂Gr (k−1) (γhr (t), γum+1 (t, L), . . . , γun(k−1) (t, L)) dγ ∂hr

0

+

n 

1 us(k−1) (t, L)

s=m+1

∂Gr (k−1) (γhr (t), γum+1 (t, L), . . . , γun(k−1) (t, L)) dγ, ∂vs

r = 1, . . . , m. (2.31)

0

Thus, using (1.20), (1.27), (2.23) and (2.18), for large CP and small ε1 > 0, we can get sup |u(k) r (t, L)| ≤ ε(1 + Cε) + CP α0 ε ≤ α1 CP ε,

r = 1, . . . , m.

(2.32)

t

Similarly, on x = 0 we have sup |u(k) s (t, 0)| ≤ α1 CP ε,

s = m + 1, . . . , n.

(2.33)

t

(k)

Next, we integrate (2.26) along the characteristic curve t = tr (x; t∗ , L) to get (k) (k) u(k) r (tr (x; t∗ , L), x) = ur (t∗ , L) x   n  (k−1) Brj (u(k−1) ) · (∂x + μr (u(k−1) )∂t )uj + (t(k) r (y; t∗ , L), y) dy, L

r = 1, . . . , m.

j=1

Due to (2.3), (2.18) and (2.32), this leads directly to 2 u(k) r C 0 ≤ α1 CP ε + C(CP ε) ≤ α2 CP ε,

∀ r = 1, . . . , m.

(2.34)

(k)

Similarly, one can integrate (2.28) along t = ts (x; t∗ , 0) and get u(k) s C 0 ≤ α2 CP ε,

∀ s = m + 1, . . . , n.

(2.35)

Combining (2.34) and (2.35), we have the C 0 norm estimates in (2.13). (k) In order to get the estimates for the temporal derivative of ui (i = 1, . . . , n), we set (k)

zi

(k)

= ∂t ui ,

i = 1, . . . , n; k ∈ N.

(2.36)

For k ∈ Z+ , by taking the temporal derivative on the boundary conditions (2.27) and (2.29), we have that on x = L, zr(k) (t, L) = hr (t)

∂Gr (k−1) (hr (t), um+1 (t, L), . . . , un(k−1) (t, L)) ∂hr n  ∂Gr (k−1) + zs(k−1) (t, L) (hr (t), um+1 (t, L), . . . , un(k−1) (t, L)), ∂u s s=m+1

∀ r = 1, . . . , m (2.37)

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and on x = 0, zs(k) (t, 0) = hs (t)

∂Gs (k−1) (k−1) (hs (t), u1 (t, 0), . . . , um (t, 0)) ∂hs +

m 

zr(k−1) (t, 0)

r=1

∂Gs (k−1) (k−1) (hs (t), u1 (t, 0), . . . , um (t, 0)), ∂ur

∀ s = m + 1, . . . , n. (2.38)

Then (1.20), (1.27), (2.23) and (2.18) yield sup |zr(k) (t, L)| ≤ ε(1 + Cε) + α0 CP ε ≤ α1 CP ε, t

∀ r = 1, . . . , m

(2.39)

and sup |zs(k) (t, 0)| ≤ α1 CP ε, t

∀ s = m + 1, . . . , n.

(2.40)

In order to further get the estimates in the domain, we take the temporal derivative to the equations (2.26) and (2.28) and get the corresponding formulas of wave decomposition for zi (i = 1, . . . , n) as follows: ∂x zr(k) + μr (u(k−1) )∂t zr(k) n 



(k−1) = − ∇μr (u(k−1) ) · ∂t u(k−1) zr(k) + Brj (u(k−1) ) ∇μr (u(k−1) ) · ∂t u(k−1) ∂t uj j=1

+

n 



(k−1) ∇Brj (u(k−1) ) · ∂t u(k−1) (∂x + μr (u(k−1) )∂t )uj

j=1

+

n  (k−1) (∂x + μr (u(k−1) )∂t ) Brj (u(k−1) )∂t uj j=1

−

n 

(k−1) ∇Brj (u(k−1) ) · (∂x + μr (u(k−1) )∂t )u(k−1) ∂t uj , r = 1, . . . , m,

(2.41)

j=1

∂x zs(k) + μs (u(k−1) )∂t zs(k) n 



(k−1) = − ∇μs (u(k−1) ) · ∂t u(k−1) zs(k) + Bsj (u(k−1) ) ∇μs (u(k−1) ) · ∂t u(k−1) ∂t uj j=1

+

n 



(k−1) ∇Bsj (u(k−1) ) · ∂t u(k−1) (∂x + μs (u(k−1) )∂t )uj

j=1

+

n  (k−1) (∂x + μs (u(k−1) )∂t ) Bsj (u(k−1) )∂t uj j=1

−

n 

(k−1) ∇Bsj (u(k−1) ) · (∂x + μs (u(k−1) )∂t )u(k−1) ∂t uj , s = m + 1, . . . , n.

(2.42)

j=1 (k)

Integrating (2.42) along the corresponding characteristic curve t = ts (x; t∗ , 0), and noting (2.3), we have (k) |zs(k) (t(k) s (x; t∗ , 0), x) − zs (t∗ , 0)|

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x ≤ sup |∇μs (u)|∂t u

(k−1)

u∈U

C 0

|zs(k) (t(k) s (y; t∗ , 0), y)| dy 0

+L

n 



sup |∇Bsj (u)|∂t u(k−1) C 0 ∂x u(k−1) C 0 + μmax ∂t u(k−1) C 0

j=1 u∈U

+L

n 

u(k−1) C 0  sup |∇Bsj (u)| sup |∇μs (u)|∂t u(k−1) 2C 0 u∈U

j=1

+2

n 

u∈U

sup |∇Bsj (u)|u(k−1) C 0 ∂t u(k−1) C 0

j=1 u∈U

+L

n 



sup |∇Bsj (u)| ∂x u(k−1) C 0 + μmax ∂t u(k−1) C 0 ∂t u(k−1) C 0 ,

j=1 u∈U

∀ x ∈ [0, L], ∀ t∗ ∈ R, ∀ s = m + 1, . . . , n. By (2.18), (2.40) and applying Gronwall’s inequality, we can get zs(k) C 0 ≤ α2 CP ε,

∀ s = m + 1, . . . , n.

(2.43)

∀ r = 1, . . . , m.

(2.44)

Similarly, integrating (2.41) we get zr(k) C 0 ≤ α2 CP ε,

Moreover, by the equations (2.26) and (2.28), and using the basic assumption (1.14) on the propagation speeds, the property (2.3) of Bij , and our known estimates (2.18) and (2.43)–(2.44), it is easy to get (k)

∂x ui C 0 ≤ CP ε,

∀ i = 1, . . . , n.

(2.45)

Combining the above estimates (2.34)–(2.35) and (2.43)–(2.45), we get the C 1 estimates (2.13). Next, we try to get the Cauchy sequence property (2.14). For k = 1, noting (2.7), it can be derived directly from (2.34)–(2.35). Then for k ≥ 2, we first note that by (2.27), on x = L we have (k−1) u(k) (t, L) = r (t, L) − ur

n  (k−1)

us (t, L) − us(k−2) (t, L) s=m+1

1 ·

∂Gr (k−1) (k−1) (εhr (t), γum+1 (t, L) + (1 − γ)um+1 (t, L), ∂us

0

. . . , γun(k−1) (t, L) + (1 − γ)un(k−1) (t, L)) dγ,

∀ r = 1, . . . , m.

Then, using (2.19) and (2.23) we have (k−1) sup |u(k) (t, L)| ≤ CP εαk−1 α0 , r (t, L) − ur t

Meanwhile, in the domain, equation (2.26) yields (k−1) (∂x + μr (u(k−1) )∂t )(u(k) ) r − ur

= − (μr (u(k−1) ) − μr (u(k−2) ))∂t ur(k−1)

∀ r = 1, . . . , m.

(2.46)

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+

n 

(k−1)

(Brj (u(k−1) ) − Brj (u(k−2) ))(∂x + μr (u(k−1) )∂t )uj

j=1

+

n 

(k−1) (k−2) (∂x + μr (u(k−1) )∂t ) Brj (u(k−2) )(uj − uj )

j=1

−

n 

(k−1)

uj

(k−2)

− uj

(∂x + μr (u(k−1) )∂t )Brj (u(k−2) )

j=1

+

n 

(k−2)

Brj (u(k−2) )(μr (u(k−1) ) − μr (u(k−2) ))∂t uj

,

r = 1, . . . , m.

j=1 (k)

Then we can integrate (2.47) along the corresponding characteristic curve t = tr (x; t∗ , L) to get (k) (k−1) (k) |u(k) (tr (x; t∗ , L), x)| r (tr (x; t∗ , L), x) − ur (k−1) ≤|u(k) (t∗ , L)| r (t∗ , L) − ur

+ L sup |∇μr (u)|u(k−1) − u(k−2) C 0 ∂t u(k−1) C 0 u∈U

+L

n 



sup |∇Brj (u)|u(k−1) − u(k−2) C 0 ∂x u(k−1) C 0 + μmax ∂t u(k−1) C 0

j=1 u∈U

+2

n 

sup |∇Brj (u)|u(k−2) C 0 u(k−1) − u(k−2) C 0

j=1 u∈U

+L

n 



u(k−1) − u(k−2) C 0 sup |∇Brj (u)| ∂x u(k−2) C 0 + μmax ∂t u(k−2) C 0 u∈U

j=1

+L

n 

sup |∇Brj (u)|u(k−2) C 0 sup |∇μr (u)|u(k−1) − u(k−2) C 0 ∂t u(k−2) C 0 ,

j=1 u∈U

u∈U

∀ x ∈ [0, L], ∀ t∗ ∈ R, ∀ r = 1, . . . , m. Thus, using (2.18)–(2.19) and (2.46), we have (k−1) |u(k) (t, x)| ≤ CP εαk−1 α0 + C(CP ε)2 αk−1 , r (t, x) − ur

∀ (t, x) ∈ R × [0, L], ∀ r = 1, . . . , m,

where C is a large constant independent of k. We can choose ε1 > 0 so small that CCP ε ≤ α − α0 ,

∀ k ∈ Z+ ,

which yields (k−1) u(k) C 0 ≤ CP εαk , r − ur

∀ r = 1, . . . , m.

Similarly, we have (k−1) u(k) C 0 ≤ CP εαk , s − us

Thus, we get (2.14).

∀ s = m + 1, . . . , n.

(2.47)

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Now we check the modulus of continuity for zi (i = 1, . . . , n) on the temporal direction (2.15). In fact, for η ∈ (0, 1), we may choose ΩP (η) = 20(α1 − α0 )−1 max ω(η | hi ) + 20η.

(2.48)

i=1,...,n

Since hi ∈ C 1 (R) for all i = 1, . . . , n, we have limη→0+ ΩP (η) = 0. For any given two points (t1 , L) and (t2 , L) with |t1 − t2 | ≤ η on the boundary x = L, using (2.37), we have zr(k) (t1 , L) − zr(k) (t2 , L)

∂Gr (k−1) (hr (t2 ), um+1 (t2 , L), . . . , un(k−1) (t2 , L)) = hr (t2 ) − hr (t1 ) ∂hr ∂Gr (k−1) + hr (t1 ) (hr (t2 ), um+1 (t2 , L), . . . , un(k−1) (t2 , L)) ∂hr

∂Gr (k−1) − (hr (t1 ), um+1 (t1 , L), . . . , un(k−1) (t1 , L)) ∂hr n 

∂Gr (k−1) (k−1) + (t2 , L) − zs(k−1) (t1 , L) (hr (t2 ), um+1 (t2 , L), . . . , un(k−1) (t2 , L)) zs ∂u s s=m+1 +

n 

zs(k−1) (t1 , L)

s=m+1

−

∂Gr (k−1) (hr (t2 ), um+1 (t2 , L), . . . , un(k−1) (t2 , L)) ∂us

∂Gr (k−1) (hr (t1 ), um+1 (t1 , L), . . . , un(k−1) (t1 , L)) , ∂us

r = 1, . . . , m.

Therefore, we have |zr(k) (t1 , L) − zr(k) (t2 , L)| ≤ω(η | hr )(1 + sup |∇2 Gr |nCP ε) + ε sup |∇2 Gr | · nCP εη u∈U

u∈U

1 + ΩP (η)α0 + CP ε sup |∇2 Gr | · n2 CP εη, 8 u∈U

∀ r = 1, . . . , m.

Taking ε1 > 0 small enough, we get 1 |zr(k) (t1 , L) − zr(k) (t2 , L)| ≤ α1 ΩP (η), 8

∀ r = 1, . . . , m.

(2.49)

Then, for any given two points (t1 , x∗ ) and (t2 , x∗ ) in the domain with |t1 − t2 | ≤ η and x∗ ∈ [0, L), by definition (2.30), we have (k) |t(k) r (x; t1 , x∗ ) − tr (x; t2 , x∗ )| x   (k−1) (k) (tr (y; t2 , x∗ ), y)) dy  ≤|t1 − t2 | +  μr (u(k−1) (t(k) r (y; t1 , x∗ ), y)) − μr (u x∗

 ≤|t1 − t2 | + 

x (k) |t(k) r (y; t1 , x∗ ) − tr (y; t2 , x∗ )| x∗

1 · 0







(k)  ∇μr (u(k−1) ) · ∂t u(k−1) γt(k) r (y; t1 , x∗ ) + (1 − γ)tr (y; t2 , x∗ ), y dγ dy ,

∀ r = 1, . . . , m.

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Using Gronwall’s inequality and noting (2.18), we have (k) CP ε|x−x∗ | supu∈U |∇μr | |t(k) r (x; t1 , x∗ ) − tr (x; t2 , x∗ )| ≤|t1 − t2 |e

≤η(1 + CCP ε), (k)

∀ x ∈ [0, L], ∀ r = 1, . . . , m.

(2.50)

(k)

Then we can integrate (2.41) along t = tr (x; t1 , x∗ ) and t = tr (x; t2 , x∗ ) to get zr(k) (t2 , x∗ ) − zr(k) (t1 , x∗ ) (k) (k) =zr(k) (t(k) r (L; t2 , x∗ ), L) − zr (tr (L; t1 , x∗ ), L) x∗ (t(k) (x;t ,x ),x) dx + −(∇μr (u(k−1) ) · ∂t u(k−1) )zr(k) | r(k) 2 ∗ (tr (x;t1 ,x∗ ),x)

L

+

x∗  n 



(k−1) Brj (u(k−1) ) ∇μr (u(k−1) ) · ∂t u(k−1) ∂t uj

L j=1



(k−1) + ∇Brj (u(k−1) ) · ∂t u(k−1) (∂x + μr (u(k−1) )∂t )uj

(k−1) (t(k) r (x;t2 ,x∗ ),x) − ∇Brj (u(k−1) ) · (∂x + μr (u(k−1) )∂t )u(k−1) ∂t uj dx  (k) (tr (x;t1 ,x∗ ),x)

(L;t ,x ),L) (k−1) (t2 ,x∗ ) (k−1) (t(k) , |(t1 ,x∗ ) − Brj (u(k−1) )∂t uj | r(k) 2 ∗ + Brj (u(k−1) )∂t uj (tr (L;t1 ,x∗ ),L)

r = 1, . . . , m.

(2.51)

Applying Grinwall’s inequality again and noting (2.49)–(2.50), (2.18) and (2.20), we get ω(η | zr(k) (·, x)) ≤ (k)

Similar analysis for zs

1 ΩP (η), 8

∀ x ∈ [0, L], ∀ r = 1, . . . , m.

(2.52)

∀ x ∈ [0, L], ∀ s = m + 1, . . . , n.

(2.53)

leads to

ω(η | zs(k) (·, x)) ≤

1 ΩP (η), 8

The proof of (2.15) is complete. Next, in order to show (2.16), we first discuss the special case that two given points (t1 , x1 ) and (t2 , x2 ) with |t1 − t2 | ≤ η, |x1 − x2 | ≤ η (k)

(k)

locate on the same characteristic curve t = tr (x; t∗ , x∗ ), namely, t2 = tr (x2 ; t1 , x1 ). For this case, we can (k) integrate the corresponding formulas of wave decomposition (2.41) along t = tr (x; t1 , x1 ) to get zr(k) (t2 , x2 ) − zr(k) (t1 , x1 ) x2  = − (∇μr (u(k−1) ) · ∂t u(k−1) )zr(k) x1

+

n 



(k−1) Brj (u(k−1) ) ∇μr (u(k−1) ) · ∂t u(k−1) ∂t uj

j=1

+

n  j=1

(k−1) ∇Brj (u(k−1) ) · ∂t u(k−1) (∂x + μr (u(k−1) )∂t )uj

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−

n 

15

(k−1)  (k) ∇Brj (u(k−1) ) · (∂x + μr (u(k−1) )∂t )u(k−1) ∂t uj (tr (x; t1 , x1 ), x) dx

j=1

+

n 



(k−1) Brj (u(k−1) (t2 , x2 )) − Brj (u(k−1) (t1 , x1 )) ∂t uj (t2 , x2 )

j=1

+

n 

(k−1)

Brj (u(k−1) (t1 , x1 ))(∂t uj

(k−1)

(t2 , x2 ) − ∂t uj

(t1 , x1 )),

r = 1, . . . , m.

j=1

Therefore, (2.3), (2.13), (2.18) and (2.21) yield |zr(k) (t2 , x2 ) − zr(k) (t1 , x1 )| ≤Cε2 η + CεΩP (η) ≤ CεΩP (η),

∀ r = 1, . . . , m.

(2.54)

Then, for general two points (t1 , x1 ) and (t2 , x2 ) with |t1 − t2 | ≤ η, |x1 − x2 | ≤ η, we can choose a point (t3 , x2 ) locating on the r-th characteristic curve passing through (t1 , x1 ), namely, t3 = t(k) r (x2 ; t1 , x1 ). Noting (1.14), by definition (2.30), we have |t3 − t1 | ≤ |x2 − x1 | ≤ η, and thus |t3 − t2 | ≤ 2η. Now we can combine estimates (2.52)–(2.54) to get |zr(k) (t2 , x2 ) − zr(k) (t1 , x1 )|     t2 + t 3 t2 + t 3 ≤zr(k) (t2 , x2 ) − zr(k) ( , x2 ) + zr(k) ( , x2 ) − zr(k) (t3 , x2 ) 2 2 + |zr(k) (t3 , x2 ) − zr(k) (t1 , x1 )| 1 ≤ ΩP (η) + CεΩP (η) 4 1 ≤ Ω(η), ∀ r = 1, . . . , m, 3 which yields ω(η | zr(k) ) ≤

1 ΩP (η), 3

∀ r = 1, . . . , m.

Similarly we have ω(η | zs(k) ) ≤

1 ΩP (η), 3

∀ s = m + 1, . . . , n.

At last, using equations (2.26), (2.28) and noting the basic assumption (1.14) on the propagation speeds, the property (2.3) of Bij , as well as our known estimates (2.13), (2.18), (2.21), we have (k)

ω(η | ∂x ui ) ≤

1 ΩP (η), 2

∀ i = 1, . . . , n,

which completes the proof of (2.16) and also the proof of Proposition 2.1.

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3. Stability of the time-periodic solution In this section we will prove Theorem 2. Noting that by the results of [14,18,23], for ε2 ∈ (0, ε0 ), we already know the global existence of the classical solutions u = u(t, x) and u = u(P ) (t, x) to the corresponding problem (1.1), (1.15)–(1.17) with max{uC 1 , u(P ) C 1 } ≤ Cε.

(3.1)

In order to prove (1.30) inductively, suppose we have (P )

max ui (t, ·) − ui

i=1,...,n

(t, ·)C 0 ≤ CS εαN ,

∀ t ∈ [t0 , t0 + T0 ]

(3.2)

for some t0 ≥ 0 and N ∈ N, we will show that (P )

max ui (t, ·) − ui

i=1,...,n

(t, ·)C 0 ≤ CS εαN +1 ,

∀ t ∈ [t0 + T0 , t0 + 2T0 ].

(3.3)

In order to do so, we denote (P )

ζ(t) = max sup |ui (t, x) − ui i

(t, x)|.

(3.4)

x∈[0,L]

Since ζ(t) is continuous with ζ(t0 + T0 ) ≤ CS εαN , we need only to show ζ(t) ≤ CS εαN +1 ,

∀ t ∈ [t0 + T0 , τ ]

(3.5)

under the assumption ζ(t) ≤

α1 CS εαN , α0

∀ t ∈ [t0 , τ ]

(3.6)

for each given τ ∈ [t0 + T0 , t0 + 2T0 ], where α0 and α1 are the constants defined in (2.23)–(2.24). Noting that both u = u(t, x) and u = u(P ) (t, x) solve (1.1), as in (2.1), by multiplying li (u) and li (u(P ) ) (i = 1, . . . , n) from left, respectively, we can rewrite the system into (∂t + λi (u)∂x )ui =

n 

Bij (u)(∂t + λi (u)∂x )uj ,

i = 1, . . . , n,

(3.7)

i = 1, . . . , n.

(3.8)

j=1 (P )

(∂t + λi (u(P ) )∂x )ui

=

n 

(P )

Bij (u(P ) )(∂t + λi (u(P ) )∂x )uj ,

j=1

Meanwhile, from boundary conditions (1.16)–(1.17), we can deduce that on x = L, n 

) ur (t, L) − u(P r (t, L) =

) (us (t, L) − u(P s (t, L))·

s=m+1

1 0

∂Gr (P ) (hr (t), γum+1 (t, L) + (1 − γ)um+1 (t, L), ∂us

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) . . . , γun (t, L) + (1 − γ)u(P n (t, L)) dγ,

∀ r = 1, . . . , m (3.9)

and on x = 0, ) us (t, 0) − u(P s (t, 0) =

m 

) (ur (t, 0) − u(P r (t, 0))·

r=1

1

∂Gs (P ) (hs (t), γu1 (t, 0) + (1 − γ)u1 (t, 0), ∂ur

0 ) . . . , γum (t, 0) + (1 − γ)u(P m (t, 0)) dγ,

∀ s = m + 1, . . . , n. (3.10)

) N sup |ur (t, L) − u(P r (t, L)| ≤ α1 CS εα

(3.11)

Thus, by (2.23) and (3.6), we have max

r=1,...,m t∈[t ,τ ] 0

and ) N sup |us (t, 0) − u(P s (t, 0)| ≤ α1 CS εα .

max

s=m+1,...,n t∈[t ,τ ] 0

(3.12)

Then in the domain, we use (3.7)–(3.8) to get (P )

(∂t + λi (u)∂x )(ui − ui

) (P )

=(∂t + λi (u)∂x )ui − (∂t + λi (u(P ) )∂x )ui (P )

=(λi (u(P ) ) − λi (u))∂x ui

+

n 

(P )

+ (λi (u(P ) ) − λi (u))∂x ui

Bij (u) − Bij (u(P ) ) (∂t + λi (u)∂x )uj

j=1

+

n 

Bij (u(P ) )(λi (u) − λi (u(P ) ))∂x uj +

j=1

−

n 

n  (P ) (∂t + λi (u(P ) )∂x ) Bij (u(P ) )(uj − uj ) j=1



(P ) ∇Bij (u(P ) ) · (∂t + λi (u(P ) )∂x )u(P ) (uj − uj ),

i = 1, . . . , n.

j=1

We can integrate this equation along the i-th characteristic curve x = xi (t; t∗ , x∗ ) (i = 1, . . . , n) defined by the ODE system ⎧ ⎪ ⎨ dxi (t; t∗ , x∗ ) = λi (u(xi (t; t∗ , x∗ ), t)), dt (3.13) ⎪ ⎩ x (t ; t , x ) = x . i

∗

∗

∗

∗

Here we note that due to (1.32), passing through each point (t∗ , x∗ ) ∈ [t0 + T0 , τ ] × [0.L], the backward characteristic curve x = xi (t; t∗ , x∗ ) intersects the boundary x = 0 or x = L at t ∈ [t0 , τ ], thus, applying (3.11)–(3.12) and using (2.3), (3.1) and (3.6), we have ζ(t) ≤α1 CS εαN + CCS ε2 αN , which, for small ε2 > 0, shows clearly our desired result ζ(t) ≤ CS εαN +1 .

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4. Regularity of the time-periodic solution In this section, we will prove higher regularity of the time-periodic solutions, provided that all boundary functions hi (t) (i = 1, . . . , n) possess higher regularity. A basic application of this result is the stabilization presented in Theorem 5 below, which shows that the feedback boundary controls with (1.25) can stabilize the system around time-periodic solutions u(P ) in C 1 framework, provided that u(P ) is in W 2,∞ , or equivalently, in C 1,1 space. And this is why we should focus our proof in W 2,∞ case. As mentioned before and as presented in [21], we believe that, through a similar mechanism as in this section, one can prove that for each k ≥ 2 k,p and p ∈ [1, ∞], when all hi (t) possess C k or Wperiod smoothness, so do the corresponding time-periodic solutions. In order to get the regularity of u(P ) , we recall the iteration scheme (2.4)–(2.7) introduced in Section 2, and now we will present the uniform W 2,∞ a priori estimate for the approximate sequence as follows. Proposition 4.1. For the iteration scheme (2.4)–(2.7), under hypotheses (1.25) and (1.33), for large enough CR > 0 and any given k ∈ Z+ we have (k)

(k)

(k)

max {∂t2 ui L∞ , ∂t ∂x ui L∞ , ∂x2 ui L∞ } ≤ CR < +∞

(4.1)

i=1,...,n

under the assumption (k−1)

max {∂t2 ui

i=1,...,n

(k−1)

L∞ , ∂t ∂x ui

(k−1)

L∞ , ∂x2 ui

L∞ } ≤ CR < +∞.

(4.2)

Once we get Proposition 4.1, we know the uniform W 2,∞ bound for {u(k) }∞ k=1 , and thus its weak∗ (k) ∞ (P ) 1 convergence. Noting that {u }k=1 converges strongly to u in C , this shows the W 2,∞ regularity of (P ) u . We note that, since we use exactly the same sequence constructed in Section 2, by Proposition 2.1, we already have (2.12)–(2.16) for each k, and especially, u(k) C 1 ≤ CP ε,

u(k−1) C 1 ≤ CP ε.

(4.3)

Now we denote (k)

ξi

(k)

= ∂t zi

(k)

= ∂t2 ui ,

i = 1, . . . , n; k ∈ N,

(4.4)

and first take the temporal derivative on (2.37)–(2.38) to get ξr(k) (t, L) = hr (t) +

n 

ξs(k−1) (t, L)

s=m+1

+ hr (t)2 +

n 

∂Gr (k−1) (hr (t), um+1 (t, L), . . . , un(k−1) (t, L)) ∂hr ∂Gr (k−1) (hr (t), um+1 (t, L), . . . , un(k−1) (t, L)) ∂us

∂ 2 Gr (k−1) (hr (t), um+1 (t, L), . . . , un(k−1) (t, L)) ∂h2r 2hr (t)zs(k−1) (t, L)

s=m+1

+

n  s,s =m+1

∂ 2 Gr (k−1) (hr (t), um+1 (t, L), . . . , un(k−1) (t, L)) ∂us ∂hr

(k−1)

zs(k−1) (t, L)zs

(t, L)

∂ 2 Gr (k−1) (hr (t), um+1 (t, L), . . . , un(k−1) (t, L)), ∂us ∂us

r = 1, . . . , m (4.5)

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and ξs(k) (t, 0) = hs (t) +

m 

∂Gs (k−1) (k−1) (hs (t), u1 (t, 0), . . . , um (t, 0)) ∂hs

ξr(k−1) (t, 0)

r=1

+ hs (t)2 +

m 

∂Gs (k−1) (k−1) (hs (t), u1 (t, 0), . . . , um (t, 0)) ∂ur

∂ 2 Gs (k−1) (k−1) (hs (t), u1 (t, 0), . . . , um (t, 0)) ∂h2s

2hs (t)zr(k−1) (t, 0)

r=1

+

m 

∂ 2 Gs (k−1) (k−1) (hs (t), u1 (t, 0), . . . , um (t, 0)) ∂ur ∂hs

(k−1)

zr(k−1) (t, 0)zr

(t, 0)

r,r  =1

∂ 2 Gs (k−1) (k−1) (hs (t), u1 (t, 0), . . . , um (t, 0)), ∂ur ∂ur

s = m + 1, . . . , n, (4.6)

which, by the basic assumption (1.20), definition (2.23), our known estimates (4.2)–(4.3) and the boundary regularity (1.33), yield sup |ξr(k) (t, L)| ≤ M2 (1 + Cε) + CR α0 + Cε2 ≤ α1 CR , t

∀ r = 1, . . . , m

(4.7)

and sup |ξs(k) (t, 0)| ≤ α1 CR , t

∀ s = m + 1, . . . , n

for small enough ε4 > 0 and large enough CR . To get the estimates in the domain, we take the temporal derivative on (2.41) to get ∂x ξr(k) + μr (u(k−1) )∂t ξr(k) =−

n  ∂μr j=1

+

∂uj

n  j,l,p=1

+

(k−1) (k) ξr

(u(k−1) )(2zj

(k−1) (k) zr )

+ ξj

−

n  ∂ 2 μr (k−1) (k−1) (k) (u(k−1) )zj zl zr ∂ul ∂uj

j,l=1

∂ 2 Brj (k−1) (k−1) (k−1) (k−1) (u )zl zp (∂x + μr (u(k−1) )∂t )uj ∂up ∂ul

n  ∂Brj (k−1) (k−1) (k−1) (u )ξl (∂x + μr (u(k−1) )∂t )uj ∂ul

j,l=1

+

n 

2

j,l,p=1

+

∂Brj (k−1) (k−1) ∂μr (k−1) (k−1) (k−1) (u )zl (u )zp zj ∂ul ∂up

n  ∂Brj (k−1) (k−1) (k−1) (u )zl (∂x + μr (u(k−1) )∂t )zj ∂ul

j,l=1

+

n 

Brj (u(k−1) )

j,l,p=1

+

n  j,l=1

Brj (u(k−1) )

∂ 2 μr (k−1) (k−1) (k−1) (u(k−1) )zl zp zj ∂up ∂ul

∂μr (k−1) (k−1) (k−1) (k−1) (k−1) (u ) ξl zj + 2zl ξj ∂ul

(4.8)

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+

n   ∂Brj (k−1) (k−1) (k−1)  (k−1) (∂x + μr (u(k−1) )∂t ) Brj (u(k−1) )ξj + (u )zl zj ∂ul j=1

n 

l=1

−

n  j,l=1

−

n  ∂Brj j=1

−

∂Brj (k−1) (k−1) (k−1) (u )ξj (∂x + μr (u(k−1) )∂t )ul ∂ul

∂ul

n  j,l,p=1

(k−1)

(u(k−1) )zj

(k−1)

(∂x + μr (u(k−1) )∂t )zl

∂ 2 Brj (k−1) (k−1) (k−1) (k−1) (u )zj zp (∂x + μr (u(k−1) )∂t )ul , ∂up ∂ul

r = 1, . . . , m.

(k)

By integrating along the characteristic curve t = tr (x; t∗ , L), applying Gronwall’s inequality, and noting (2.3), (4.2)–(4.3), (4.7), we get ξr(k) L∞ ≤ α2 CR ,

∀ r = 1, . . . , m.

(4.9)

∀ s = m + 1, . . . , n.

(4.10)

Similarly, we have ξs(k) L∞ ≤ α2 CR ,

Next, using equations (2.41)–(2.42) and the basic assumption (1.14) on the propagation speeds, the property (2.3) of Bij , as well as our known estimates (4.2), (4.3) and (4.9)–(4.10), we have (k)

∂x ∂t ui L∞ ≤ α3 CR ,

∀ i = 1, . . . , n.

(4.11)

Moreover, taking the spatial derivative to (2.26) and (2.28), we have (k)

∂x2 ui

(k)

= − μi (u(k−1) )∂x ∂t ui

+

−

n  ∂μi (k−1) (k−1) (k) (u )∂x uj zi ∂u j j=1

n  ∂Bij (k−1) (k−1) (k−1) (k−1) (u )∂x ul (∂x uj + μi (u(k−1) )zj ) ∂ul

j,l=1

+

n 

(k−1)

Bij (u(k−1) )(∂x2 uj

(k−1)

+ μi (u(k−1) )∂x ∂t uj

)

j=1

+

n 

Bij (u(k−1) )

j,l=1

∂μi (k−1) (k−1) (k−1) (u )∂x ul ∂x uj , ∂ul

i = 1, . . . , n.

Thus, using again (1.14), (2.3), (4.2)–(4.3) and (4.11), we get (k)

∂x2 ui L∞ ≤ α4 CR ,

∀ i = 1, . . . , n.

(4.12)

Combining (4.9)–(4.12), we get the desired result (4.1), and complete the proof of Proposition 4.1. 5. Boundary stabilization around the time-periodic solution In this section, we will give the proof of Theorem 5. First, we note that we have already got the C 0 exponential convergence in Theorem 2 as follows:

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u(t, ·) − u(P ) (t, ·)C 0 ≤ CS αN ε,

21

∀ t ∈ [N T0 , (N + 1)T0 ), ∀ N ∈ N,

(5.1)

which also shows that u(t, ·) − u(P ) (t, ·)C 0 ≤ CS αN +1 ε,

∀ t ∈ [(N + 1)T0 , (N + 2)T0 ), ∀ N ∈ N.

(5.2)

Moreover, by Theorem 1 and Theorem 4, as well as the results in [14,18,23], we have uC 1 ≤ C0 ε,

u(P ) C 1 ≤ C0 ε,

u(P ) W 2,∞ ≤ CR .

(5.3)

Accordingly, we choose CS∗ >

32LCR CS max sup |∇μi (u)| + CS . 1 − α0 i=1,...,n u∈U

(5.4)

By the continuity, we will inductively get the estimates for the convergence of the first derivatives, namely, for each N ∈ N and s ∈ [0, T0 ], we will prove (P )

max {∂t ui (t, ·) − ∂t ui

i=1,...,n

(P )

(t, ·)C 0 , ∂x ui (t, ·) − ∂x ui

(t, ·)C 0 } ≤ CS∗ αN +1 ε,

∀ t ∈ [(N + 1)T0 , τ ] (5.5)

under the assumption (P )

max {∂t ui (t, ·) − ∂t ui

i=1,...,n

(P )

(t, ·)C 0 , ∂x ui (t, ·) − ∂x ui

(t, ·)C 0 } ≤

α1 ∗ N C α ε, α0 S

∀ t ∈ [N T0 , τ ]. (5.6)

Now, denoting (P )

zi = ∂t ui , wi = ∂x ui and zi

(P )

= ∂t ui

(P )

, wi

(P )

= ∂x ui

,

(5.7)

we take the temporal derivative on boundary conditions (1.16)–(1.17) to get ∂Gs (hs (t), u1 , . . . , um ) ∂hs m  ∂Gs + zr (hs (t), u1 , . . . , um ), ∂ur r=1

x = 0 : zs =hs (t)

s = m + 1, . . . , n,

∂Gr (hr (t), um+1 , . . . , un ) ∂hr n  ∂Gr + zs (hr (t), um+1 , . . . , un ), ∂us s=m+1

(5.8)

x = L : zr =hr (t)

r = 1, . . . , m

(5.9)

and ∂Gs (P ) ) (hs (t), u1 , . . . , u(P m ) ∂hs m  ∂Gs (P ) ) + zr(P ) (hs (t), u1 , . . . , u(P m ), ∂u r r=1

x = 0 : zs(P ) =hs (t)

s = m + 1, . . . , n,

∂Gr (P ) ) (hr (t), um+1 , . . . , u(P n ) ∂hr n  ∂Gr (P ) ) + zs(P ) (hr (t), um+1 , . . . , u(P n ), ∂u s s=m+1

(5.10)

x = L : zr(P ) =hr (t)

r = 1, . . . , m.

(5.11)

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Thus, on the boundary x = 0 we have zs (t, 0) − zs(P ) (t, 0) =hs (t)

m  ) (ur (t, 0) − u(P r (t, 0))· r=1

1

∂ 2 Gs (P ) ) (hs (t), γu1 (t, 0) + (1 − γ)u1 (t, 0), . . . , γum (t, 0) + (1 − γ)u(P m (t, 0)) dγ ∂ur ∂hs

0

+

m 

(zr (t, 0) − zr(P ) (t, 0))

r=1

+

m 

∂Gs (hs (t), u1 (t, 0), . . . , um (t, 0)) ∂ur (P )

zr(P ) (t, 0)(ur (t, 0) − ur (t, 0))·

r,r  =1

1

∂ 2 Gs (P ) ) (hs (t), γu1 (t, 0) + (1 − γ)u1 (t, 0), . . . , γum (t, 0) + (1 − γ)u(P m (t, 0)) dγ, ∂ur ∂ur

s = m + 1, . . . , n.

0

Using (2.23), (5.1)–(5.3) and (5.6), on x = 0 we have |zs (t, 0) − zs(P ) (t, 0)| ≤CεCS∗ αN ε + α0

sup t∈[N T0 ,τ ]

≤α2 CS∗ αN ε,

α1 ∗ N C α ε + CεCS αN +1 ε α0 S

∀ s = m + 1, . . . , n.

(5.12)

Similarly, on x = L we have |zr (t, L) − zr(P ) (t, L)| ≤ α2 CS∗ αN ε,

sup

∀ r = 1, . . . , m.

(5.13)

t∈[N T0 ,τ ] (P )

In the domain, we will derive the formulas of wave decomposition for zi and zi (2.4). We first multiply li (u) from left to system (1.1) to get ∂x ui + μi (u)∂t ui =

n 

(i = 1, . . . , n) as in

Bij (u)(∂x uj + μi (u)∂t uj ), i = 1, . . . , n,

(5.14)

j=1

and (P )

∂x ui

(P )

+ μi (u(P ) )∂t ui

=

n 

(P )

Bij (u(P ) )(∂x uj

(P )

+ μi (u(P ) )∂t uj ), i = 1, . . . , n,

(5.15)

j=1

where Bij (u) are defined by (2.2), and thus (2.3) holds. Then we can take the temporal derivative to get

∂x zi + μi (u)∂t zi =

n 

Bij (u)(∂x zj + μi (u)∂t zj ) −

j=1

+

n  j,j  =1

and similarly,

Bij (u)

n  ∂μi (u)zj zi ∂uj j=1

n  ∂μi ∂Bij (u)zj  zj + (u)zj  (wj + μi (u)zj ), ∂uj  ∂uj   j,j =1

i = 1, . . . , n (5.16)

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(P )

∂x zi

(P )

+ μi (u(P ) )∂t zi

=

n 

(P )

Bij (u(P ) )(∂x zj

(P )

+ μi (u(P ) )∂t zj

23

)

j=1

−

n n   ∂μi (P ) (P ) (P ) ∂μi (P ) (P ) (P ) (u )zj zi + Bij (u(P ) ) (u )zj  zj ∂uj ∂uj   j=1 j,j =1

+

n  j,j  =1

∂Bij (P ) (P ) (P ) (P ) (u )zj  (wj + μi (u(P ) )zj ), ∂uj 

i = 1, . . . , n. (5.17)

Thus, noting that n 



(P ) (P ) Bij (u) ∂x zj + μi (u)∂t zj − Bij (u(P ) ) ∂x zj + μi (u(P ) )∂t zj

j=1

=

n 





(P ) ∂x + μi (u)∂t Bij (u)zj − ∂x + μi (u(P ) )∂t Bij (u(P ) )zj

j=1

−

n 

zj

j,j  =1

∂Bij (P ) ∂Bij (P ) (P ) (u)(wj  + μi (u)zj  ) − zj (u(P ) )(wj  + μi (u(P ) )zj  ), ∂uj  ∂uj 

we have (P )

∂x (zi − zi

(P )

) + μi (u)∂t (zi − zi (P )

= − (μi (u) − μi (u(P ) ))∂t zi

+

)

n 

(P )

(μi (u) − μi (u(P ) ))Bij (u(P ) )∂t zj

j=1

+

n    (P ) (∂x + μi (u)∂t ) (Bij (u) − Bij (u(P ) ))zj + Bij (u(P ) )(zj − zj ) j=1

+

n 

(μi (u) − μi (u(P ) ))

j,j  =1

−

∂Bij (P ) (P ) (P ) (u )zj  zj ∂uj 

n    ∂Bij ∂Bij (P ) ∂Bij (P ) (P ) (u) − (u ) zj wj  + (u )(zj − zj )wj  ∂uj  ∂uj  ∂uj  

j,j =1

−

n  ∂Bij (P ) (P ) (P ) (u )zj (wj  − wj  )  ∂u j 

j,j =1

−

n    ∂Bij (P ) ∂Bij (u)μi (u) − (u )μi (u(P ) ) zj zj  ∂uj  ∂uj  

j,j =1

−

n  ∂Bij (P ) (P ) (P ) (P ) (u )μi (u(P ) ) (zj − zj )zj  + zj (zj  − zj  )  ∂u j 

j,j =1

−

n  ∂μi j=1

−

∂uj

(u) −

∂μi (P ) (u ) zj zi ∂uj

n  ∂μi (P ) (P ) (P ) (P ) (u ) (zj − zj )zi + zj (zi − zi ) ∂uj j=1

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+

n  ∂μi ∂μi (P ) Bij (u) (u) − Bij (u(P ) ) (u ) zj  zj ∂uj  ∂uj  

j,j =1

+

n 

∂μi (P ) (P ) (P ) (P ) (u ) (zj  − zj  )zj + zj  (zj − zj ) ∂uj 

Bij (u(P ) )

j,j  =1

+

n  ∂Bij ∂Bij (P ) (u) − (u ) zj  (wj + μi (u)zj )  ∂uj ∂uj  

j,j =1

+

n  ∂Bij (P ) (P ) (u )(zj  − zj  )(wj + μi (u)zj ) ∂uj  

j,j =1

+

n  ∂Bij (P ) (P ) (P ) (P ) (u )zj  (wj − wj ) + μi (u)(zj − zj ) ∂uj  

j,j =1

+

n  ∂Bij (P ) (P ) (P ) (u )zj  (μi (u) − μi (u(P ) ))zj .  ∂u j 

j,j =1

Then we can integrate it along the corresponding backward characteristic curve t = ti (x; t∗ , x∗ ) defined by ⎧ ⎪ ⎨ dti (x; t∗ , x∗ ) = μi (u(ti (x; t∗ , x∗ ), x)), dx ⎪ ⎩ t (x ; t , x ) = t . i

∗

∗

∗

(5.18)

∗

Note that by (1.32), for each point (t∗ , x∗ ) ∈ [(N +1)T0 , τ ], the backward characteristic curve t = ti (x; t∗ , x∗ ) will intersect the boundary in a time interval shorter than T0 , namely, tr (L; t∗ , x∗ ) ∈ [t∗ − T0 , t∗ ] ⊆ [N T0 , τ ],

∀ (t∗ , x∗ ) ∈ [(N + 1)T0 , τ ], ∀ r = 1, . . . , m,

ts (0; t∗ , x∗ ) ∈ [t∗ − T0 , t∗ ] ⊆ [N T0 , τ ],

∀ (t∗ , x∗ ) ∈ [(N + 1)T0 , τ ], ∀ s = m + 1, . . . , n,

and thus we can use estimates (5.12)–(5.13) on the boundary. By (5.1)–(5.4), (5.6) and (2.3), we have (P )

|zi (t∗ , x∗ ) − zi

(t∗ , x∗ )|

≤α2 CS∗ αN ε +

α3 − α2 ∗ N α1 ∗ N CS α ε + C α ε · Cε + CS αN ε · Cε 2 α0 S

≤α3 CS∗ αN ε,

∀ (t∗ , x∗ ) ∈ [(N + 1)T0 , τ ] × [0, L], ∀ i = 1, . . . , n,

which leads to the first part of (5.5). Then we can use (5.14)–(5.15) to get (P )

w i − wi

(P )

= − (μi (u) − μi (u(P ) ))zi − μi (u(P ) )(zi − zi +

n n   (P ) (Bij (u) − Bij (u(P ) ))wj + Bij (u(P ) )(wj − wj ) j=1

+

)

n 

j=1



Bij (u)μi (u) − Bij (u(P ) )μi (u(P ) ) zj

j=1

+

n  j=1

(P )

Bij (u(P ) )μi (u(P ) )(zj − zj

),

i = 1, . . . , n.

(5.19)

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Using the basic assumption (1.14) on the propagation speed, the property (2.3) of Bij and our known estimates (5.1)–(5.3), (5.6) and (5.19), we have (P )

wi (t, ·) − wi

(t, ·)C 0 ≤α3 CS∗ αN ε + ≤α4 CS∗ αN ε,

α1 ∗ N C α ε · Cε + CS αN ε · Cε α∗ S ∀ t ∈ [(N + 1)T0 , τ ], ∀ i = 1, . . . , n,

(5.20)

which is the second part of (5.5) and completes the proof of Theorem 5. 6. A counter example In this section we will show the dissipative structure requirement (1.25) is sharp in the sense that, for the case Θmin ≥ 1, one can construct a quasilinear hyperbolic system with linear time-periodic boundary condition, and small and smooth initial data, such that the corresponding classical solution will blow up in a finite time even if the system is strictly hyperbolic and linearly degenerate. Consider the following system of quasilinear hyperbolic equations: ⎧ ∂t u1 − 4∂x u1 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂t u2 − 2∂x u2 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂t u3 + u2 u5 ∂x u1 − ∂x u3 + u1 u2 ∂x u5 = 0, 3 3 ⎪ ∂t u4 − u24 u6 ∂x u2 + ∂x u4 − u2 u24 ∂x u6 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂t u5 + 2∂x u5 = 0, ⎪ ⎪ ⎪ ⎪ ⎩ ∂t u6 + 4∂x u6 = 0,

(t, x) ∈ R+ × (0, 1), (t, x) ∈ R+ × (0, 1), (t, x) ∈ R+ × (0, 1), (t, x) ∈ R+ × (0, 1), (t, x) ∈ R+ × (0, 1), (t, x) ∈ R+ × (0, 1).

By a direct calculation, we find that the eigenvalues of its coefficient matrix are all distinct constants as follows: λ1 = −4, λ2 = −2, λ3 = −1, λ4 = 1, λ5 = 2, λ6 = 4. Thus the system is strictly hyperbolic and linearly degenerate. Moreover, we set the linear boundary conditions as x = 0 : u4 = θu3 ,

u5 = u2 ,

u6 = u1 ,

x = 1 : u1 = −u6 ,

u2 = −u5 ,

u3 = θu4

with Θmin = max{θ, 1}. For any θ ≥ 1 and for the small, smooth and compatible initial data π x, 2 1

−1 π u3 (0, x) = ε−1 e 1−(2x−1)2 − (1 − x) ε2 sin(πx) , 2 u4 (0, x) = 0

u1 (0, x) = u2 (0, x) = u5 (0, x) = u6 (0, x) = ε cos

with ε > 0 small enough. The corresponding classical solution blows up in a finite time. In fact, we can get π u1 (t, x) =ε cos 2πt + x , 2 π u2 (t, x) =ε cos πt + x , 2

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π u5 (t, x) =ε cos πt − x , 2 π u6 (t, x) =ε cos 2πt − x , 2 which shows that both u3 and u4 satisfy Riccati type equations: π 2

ε sin(πt + πx)) u23 , 2 π 2

∂t u4 + ∂x u4 = ε sin(πt − πx)) u24 . 2 ∂t u3 − ∂x u3 =

And we can write down the exact formulas also for u3 and u4 accordingly as u3 (t, x) = 0,

t + x ∈ [2k + 1, 2k + 2], k ∈ N,

2k  1 −2k −1

−1 π 2 1−(2t+2x−4k−1) ε e −( θ−l − x) ε2 sin(πt + πx) , u3 (t, x) = θ 2

t + x ∈ [2k, 2k + 1], k ∈ N,

l=0

u4 (t, x) = 0,

t − x ∈ [2k − 1, 2k], k ∈ N,

2k+1  1

−1 π θ−l + x) ε2 sin(πt − πx) , u4 (t, x) = θ−2k−1 ε−1 e 1−(2t−2x−4k−1)2 − ( 2

t − x ∈ [2k, 2k + 1], k ∈ N,

l=1

which shows that the solution itself will tend to the infinity at a time scale of ln ε−1 / ln θ when θ > 1, and of ε−3 when θ = 1. Acknowledgement The author would like to thank Professor Ta-Tsien Li for his encouragements, instructions and discussions. References [1] Georges Bastin, Jean-Michel Coron, Stability and Boundary Stabilization of 1-D Hyperbolic Systems, Progress in Nonlinear Differential Equations and Their Applications, vol. 88, Birkhäuser/Springer, 2016. [2] Georges Bastin, Jean-Michel Coron, Brigitte d’Andréa Novel, On Lyapunov stability of linearised Saint–Venant equations for a sloping channel, Netw. Heterog. Media 4 (2) (2009) 177–187. [3] Georges Bastin, Jean-Michel Coron, Amaury Hayat, Peipei Shang, Exponential boundary feedback stabilization of a shock steady state for the inviscid Burgers equation, Math. Models Methods Appl. Sci. 29 (2) (2019) 271–316. [4] Jean-Michel Coron, Georges Bastin, Dissipative boundary conditions for one-dimensional quasi-linear hyperbolic systems: Lyapunov stability for the C 1 -norm, SIAM J. Control Optim. 53 (3) (2015) 1464–1483. [5] Jean-Michel Coron, Georges Bastin, Brigitte d’Andréa Novel, Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems, SIAM J. Control Optim. 47 (3) (2008) 1460–1498. [6] Jean-Michel Coron, Brigitte d’Andréa Novel, Georges Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, IEEE Trans. Autom. Control 52 (1) (2007) 2–11. [7] Jonathan de Halleux, Christophe Prieur, Jean-Michel Coron, Brigitte d’Andréa Novel, Georges Bastin, Boundary feedback control in network of open channels, Automatica 39 (8) (2003) 1365–1376. [8] James M. Greenberg, Ta-Tsien Li, The effect of boundary damping for the quasilinear wave equation, J. Differ. Equ. 52 (1984) 66–75. [9] Martin Gugat, Boundary feedback stabilization of the telegraph equation: decay rates for vanishing damping term, Syst. Control Lett. 66 (2014) 72–84. [10] Martin Gugat, Günter Leugering, Ke Wang, Neumann boundary feedback stabilization for a nonlinear wave equation: a strict H 2 -Lyapunov function, Math. Control Relat. Fields 7 (3) (2017) 419–448. [11] Martin Gugat, Vincent Perrollaz, Lionel Rosier, Boundary stabilization of quasilinear hyperbolic systems of balance laws: exponential decay for small source terms, J. Evol. Equ. 18 (3) (2018) 1471–1500. [12] Long Hu, Sharp time estimates for exact boundary controllability of quasilinear hyperbolic systems, SIAM J. Control Optim. 53 (6) (2015) 3383–3410. [13] Long Hu, Rafael Vazquez, Florent Di Meglio, Miroslav Krstic, Boundary exponential stabilization of 1-dimensional inhomogeneous quasi-linear hyperbolic systems, SIAM J. Control Optim. 57 (2) (2019) 963–998.

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