{'fraction_non_alphanumeric': 0.07219636524822695, 'fraction_numerical': 0.024933510638297872, 'mean_word_length': 3.679543686803215, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 16, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Boundary stabilization of quasilinear hyperbolic systems of balance laws: Exponential decay for small source terms

We investigate the long-time behavior of solutions of quasilinear hyperbolic systems with transparent boundary conditions when small source terms are incorporated in the system. Even if the finite-time stability of the system is not preserved, it is shown here that an exponential convergence towards the steady state still holds with a decay rate which is proportional to the logarithm of the amplitude of the source term. The result is stated for a system with dynamical boundary conditions in order to deal with initial data that are free of any compatibility condition.


Introduction
Solutions of certain hyperbolic systems can reach the equilibrium state in finite time. Such a property, called finite-time stability in [1,17,18] or super-stability in [19], was first noticed in [12,15] for the (linear) wave equation. The extension of such a property to the wave equation on networks was addressed in [1,19].
Fortunately, the finite-time stability still occurs for systems of 2 × 2 quasilinear hyperbolic equations of diagonal form without source terms, as it was noticed in [13] with initial data satisfying some compatibility conditions to prevent the emergence of shockwaves, and next in [17,18] for arbitrary initial data by replacing homogeneous boundary conditions by some dynamical boundary conditions. The finite-time stabilization of a quasilinear hyperbolic system with source terms seems to be very challenging. In [4], the authors proved that a 2 × 2 linear hyperbolic system with source terms can be stabilized to the origin in finite-time by using some boundary feedback laws designed with the backstepping approach.
On the other hand, the finite-time stability of a system may be lost when a small, bounded perturbation is added to the system. A famous example is provided by the telegraph equation x y + ∂ t y = 0, (t, x) ∈ (0, +∞) × (0, L), (1.1) y(t, 0) = 0, y x (t, L) = −y t (t, L).
(1.2) equations and [5] for the loss of the stability when incorporating an arbitrarily small delay in a transparent boundary condition for the wave equation. The aim of this paper is to show that the robustness property noticed in [8] is shared by most of the finite-time stable systems. The first result in this paper shows that a linear finite-time stable system with a (small) disturbance is exponentially stable with a decay rate proportional to the logarithm of the amplitude of the perturbation. We refer the reader to [21,Theorem 4.2] for a sufficient condition involving the resolvent for the finite-time stability of a linear system. The (simple) proof of Theorem 1 is given in Appendix. It rests on the observation that the solution of a Cauchy problem can be obtained as a fixed-point of a map, derived from Duhamel formula, in a weighted space. The weight is related to the decay rate. It is unclear whether such an approach could be extended to quasilinear systems.
It should be noticed that the estimate in (1.3) is essentially sharp. Indeed, for the system x ∈ (0, L), (1.4) we shall prove that the decay rate is roughly speaking bounded from below by (c/L) ln −1 . The main aim of the paper is to investigate the application of transparent boundary conditions, or more generally of dynamical boundary conditions as in [17,18], to 2×2 quasilinear hyperbolic systems in diagonal form with small source terms where λ(u, v) > c, µ(u, v) > c for some constant c > 0, and 0 < 1. Our results are stated when f and g do not depend on , but there are still valid when f and g depend on but are bounded in W 2,∞ (0, L) for 0 < 1. We shall prove that for small enough and for initial data sufficiently close to a steady state of (1.6)-(1.7), the solution of (1.6)-(1.7) with dynamical boundary conditions converge exponentially to the steady state with a decay rate proportional to ln( −1 ).
Our result can be applied to e.g. the Saint-Venant system with sources terms (see e.g. [3,10]), which is commonly used as a model for the water flow regulation in a canal with a slowly varying topography and some damping: x) is the flow velocity in the direction parallel to the bottom, g is the gravitation constant, c f is the friction coefficient, and z = b(x) is the equation of the bottom.
Using the Riemann invariants we easily see that system (1.8)-(1.9) can be rewritten as Thus, our results can be applied when ∂ x b L ∞ (0,L) + c f 1. The above model is valid when the function ∂ x b takes "small values". A more accurate model, the so-called Savage-Hutter system (see [3]), reads Here, X denotes a curvilinear coordinate along the bottom, θ = θ(X) is the angle of the bottom tangent with some fixed horizontal axis, H = H(t, X) is the width of fluid in the normal direction at a point X of the bottom, and V (t, X) is the tangential velocity. Introducing the Riemann invariants we derive again a system of the form (1.10)- (1.11), with x = X and Again, our results can be applied when |θ| 1. The paper is outlined as follows. The main result (Theorem 3) is stated in Section 2. Its proof is displayed in Section 3. It is divided in three parts. The first one is a rephrasing of the problem. The second part establishes the existence and uniqueness of global solutions for small initial data using Schauder's fixed-point theorem. The last one introduces some Lyapunov functions with exponential weights needed to prove the exponential convergence towards the steady state. The paper ends with an Appendix which contains the proofs of Theorem 1 and of Theorem 2 and which provides some background about linear transport equations.

Stationary states and Main result
We are interested in the following system of balance laws and the boundary conditions In (2.3), the boundary data y l and y r are defined as the solutions of the initial value problem where γ ∈ (0, 1) and K ∈ (0, +∞) are any given numbers.
Here and in what follows, we assume that the functions λ, µ, f and g are of class C 2 . We fix a pair (ū,v) ∈ R 2 such that and we introduce two real numbers c > 0 and R > 0 such that Let us construct for each 0 < 1 a pair (ū ,v ) of stationary states for (2.1).
We denote by X the Banach space of continuous functions from [0, L] to R 2 equipped with the uniform norm (u, v) = sup x∈[0,L] ||(u(x), v(x))||, and by Ω the subset of X consisting of the functions taking their values in the open ball B((ū,v), R); that is, We can now define a functional F : It is easy to see that (i) the set Ω is open; We infer from the implicit function theorem the local existence and uniqueness of a map ∈ Ω which is of class C 1 and which satisfies It follows then from (2.7)-(2.8) that the functionsū andv are of class C 1 in [0, L] and that they satisfy (2.10) We are now in a position to state the main result in this paper.  Remark 1. We notice that for any fixed and therefore that the r.h.s. of (2.12) tends to 0 as → 0 + . This result, combined with the boundedness of for all T > 0, yields again the finite-time stability around (ū,v) of the limit system (without source term) x ∈ (0, L).

Proof of Theorem 3
3.1. Reduction of the problem. We aim to show that if (2.11) holds with δ small enough, then the solutions of (2.1)-(2.4) tend to (ū ,v ) as t → +∞.
To this end, we introduce the functions The original system (2.1) can be written (3.1) The boundary conditions become where y l and y r still solve (2.4), and the initial condition read Thanks to the definition of Ω, we notice that for any ∈ [− 0 , 0 ], it holds For any given r > 0 and any f : In the following, for any function Using (2.10), we see that Differentiating in (2.10), we infer that We now simplify the notations by setting (3.5) The system (3.1) can be written as It is supplemented with the initial conditions and the boundary conditions where the functions Y l and Y r solve the system The functions α, β, F and G enjoy the following properties: (3.12) 3.2. The fixed-point argument. We show in this section the existence and uniqueness of the solution of (3.1)-(3.3) for small enough. Introduce some positive constants I 0 , I 1 , P , and M such that We notice that we can assume that (i) the constants I 0 , I 1 are small by picking the numbers δ and 0 in Theorem 3 small enough; (ii) the constant P is small by adjusting thanks to (3.12); To display our fixed-point argument based on Schauder fixed-point theorem, we have to define (i) a class of functions containing the desired solution; (ii) a map whose fixed-point is the desired solution of (3.1)- (3.3). This is done in the following two definitions. Definition 1. Given two positive constants A and B, we define D as the set of functions and The following result is classical (see e.g. [?]).
We first have to show that F maps D into itself for an appropriate choice of the constants. Proof. Using Corollary 1 in Appendix, we obtain the following estimates: It is thus sufficient to show that for I 0 and I 1 small enough, one can choose P sufficiently small so that there exist some numbers A > 0 and B > 0 with 2L For given A, B, M, I 0 and I 1 in (0, +∞), it is thus sufficient to impose that as P → 0 + , the limit of the leftsided terms in (3.23)-(3.25) are strictly less than those of the corresponding rightsided terms; that is, (3.29) By Definition 2, we conclude that (Ũ ,Ṽ ) = F(U, V ). It follows that F is continuous.
We are now in a position to prove that F has a fixed-point in D.

Lyapunov Functions and Decay
Rates. Let (U, V, Y r , Y l ) denote the solution of system (3.6)-(3.9).
To simplify the computations, we denote byC a positive constant such that for all x ∈ [0, L] and all (a, b) ∈ B((0, 0), R), we have and such that for all (t, We first introduce a Lyapunov function to investigate the long-time behavior of (U, V ).
Definition 3. Given any θ > 0, let the function L θ be defined by Then we have the following result.
Proposition 6. For almost every t ∈ R + , it holds Proof. Since U and V are uniformly bounded and Lipschitz continuous, and hence differentiable almost everywhere by Rademacher theorem, the derivative of L θ (U (t, .), V (t, .)) exists almost everywhere and it is obtained by differentiating the integrand with respect to t. Thus we obtain Let us begin with I 3 .
As far I 4 is concerned, we have that Using (3.10) and (3.35), we infer that Let us evaluate the remaining terms I 1 and I 2 . We have that where we used twice in the last inequality the following estimate The proof of Proposition 6 is completed by gathering together (3.36) and (3.37).
Next, we introduce another Lyapunov function for the dynamics of the boundary conditions. Definition 4. Given any θ > 0, let the functionL θ be defined by Then the following result holds.
We are in a position to define the Lyapunov function for the full state (U, V, Y l , Y r ). Let the functional L be defined by Then the following holds. On the other hand, the best decay rate obtained by taking the minimum of the parenthesis over θ reads The estimate (3.38) will give the exponential decay of the L 2 norm of (U, V ). To derive an exponential decay for the L ∞ norm of (U, V ), we need the following result.
we see that and hence

|I| ·
From the definition of I (using x ∈ [0, L]), we have that |I| ≥ inf(L, D), so that Using the definition of D, we obtain (3.40). Assume in addition that u(0) = 0. We claim that Indeed, we have by Cauchy-Schwarz inequality and an integration w.r.t. x ∈ (0, L) yields at once (3.42). Next, if the first estimate in (3.40) holds, taking the cube of each term and using (3.42), we arrive to (3.41). If the second estimate in (3.40) holds, then (3.41) is obvious.

Conclusion
In this paper, we have shown the exponential decay of the L 2 -norm and of the L ∞ -norm of solutions of a quasilinear hyperbolic system of balance laws with sufficiently small source terms and appropriately chosen boundary controls. In fact, with vanishing source terms the decay rates become arbitrarily large, and thus in the limit, the case of finite-time stability is recovered. Since we work with solutions that are at each time t ≥ 0 in the space W 1,∞ (0, L) with respect to the space variable, the question arises whether also the L ∞ -norm of the space derivative of the solution decays exponentially. This question will be the subject of future investigations.  Let Γ(u) = v. We aim to show that for ∈ (0, 0 ) (with 0 small enough) and λ > 0 conveniently chosen, Γ has a unique fixed-point in E. First, we note that Γ maps the space E into itself. Indeed, for t ∈ [0, T ] we have and for t ≥ T , we have (using the fact that e sA w = 0 for all s ≥ T and all w ∈ H) Let us show now that Γ contracts in E. Pick any u 1 , u 2 ∈ E, and let us denote v 1 = Γ(u 1 ), v 2 = Γ(u 2 ). Then we have for all t ≥ 0 so that Γ contracts for any given λ > 0 if 1. Then by the contraction mapping theorem, Γ has a unique fixed-point in E which is nothing but the mild solution of (5. This holds if C < 1/T and ∈ (0, 0 ) with 0 > 0 small enough. On the other hand, we have that The proof of Theorem 1 is complete.

Proof of Theorem 2.
To prove (1.5), it is sufficient to find an eigenvalue λ of A + B of the form λ ∼ − c L ln −1 .
Thus, we investigate the following spectral problem Differentiating with respect to x in (5.4), replacing ∂ x v by its expression in (5.5) and next v by its expression in (5.4), we arrive to the following ODE for u Similarly, we see that v solves the ODE Let α ∈ C be such that Then u and v can be written as (5.14) Now, using (5.9), we see that (5.12) and (5.13) follow respectively from (5.14) and (5.11), so that (5.11)-(5.14) is equivalent to we wee that the solutions of (5.4)-(5.5) are the functions of the form where A, D ∈ C are arbitrary. The boundary conditions (5.6) yield the following constraints for A and D: A + µD = 0, µAe αL + De −αL = 0.
The above system admits some nontrivial solutions if and only if the determinant of the associated matrix is null: Gathering together (5.9), (5.15) and (5.16), we conclude that λ is an eigenvalue of the operator A + B if there is some α ∈ C such that or equivalently if there are some numbers α ∈ C and s ∈ {−1, 1} such that To complete the proof, it is sufficient to pick s = −1 and to limit ourselves to the solutions α ∈ R + of the equation It is easily seen that the map x → sinh x/x is (strictly) increasing on (0, +∞) and onto (1, +∞), so that for any ∈ (0, c/L) there is a unique α = α( ) ∈ (0, +∞) satisfying (5.22). Moreover, the map → α is decreasing and α → +∞ as → 0 + . Finally, e Lα /(2Lα) ∼ c/(L ) yields by taking the logarithm Lα ∼ ln −1 . This gives by using again (5.22) if κ > c/L and 0 < 1.

Transport equation.
This part follows closely the Appendix of [16].
Definition 5. Let a = a(t, x) be a Lipschitz continuous function on R + × R such that For any pair (t, x) ∈ R + × R, we denote by s → φ a (s, t, x) the maximal solution to the following Cauchy problem θ (s) = a(s, θ(s)), which is defined on R + thanks to (5.23).
Proposition 8. Assume that a = a(t, x) satisfies (5.23) and is of class C 1 on R + × R. Then the function φ a is of class C 1 on (R + ) 2 × R, and we have Proof. Using the integral form of the differential equation, we obtain We infer from the implicit function theorem (proceeding as in Section 2) that φ is of class C 1 on (R + ) 2 × R. On the other hand, differentiating in (5.24) yields Noticing that the last equation can be viewed as a linear ODE and using the fact that a(t, φ(t, t, x)) = a(t, x), we obtain the desired formula for ∂ 2 φ. The other one for ∂ 3 φ is proven in the same way. We also set e(t, x) := inf F − .

Proposition 9.
Assume that a is a function of class C 1 and that there exists c > 0 such that
Remark 3. The following holds Since the propagation speed a(t, x) is greater than c, it should be clear that Proof. Using (5.25), we see that for (t, x) ∈ G, e(t, x) is the only solution (at least locally) of the equation φ(e(t, x), t, x) = 0. (5.26) An application of the implicit function theorem gives that the function e is of class C 1 . Taking partial derivatives in (5.26) yields We shall need also to introduce the concept of weak solution of (5.27), following [16].
is a weak solution of (5.27) if for any function Then we have the following result. Proof. The first assertion follows from classical arguments. The second one is proven in the Appendix of [16].
Theorem 6. Let y n , a n , b n , y 0,n and y l,n be Lipschitz continuous functions such that (i) for any n ≥ 0, the function y n is a strong solution of      ∂ t y n + a n (t, x)∂ x y n = b n (t, x), y n (t, 0) = y l,n (t), y n (0, x) = y 0,n (x); Proof. Since y n is the unique solution of (5.34), it follows from Corollary 1 that the sequence (y n ) n≥0 is bounded in Lip([0, T ] × [0, L]). From Ascoli-Arzela theorem, we can extract a subsequence (y n k ) k≥0 which converges uniformly towards a function y. On the other hand, since (∂ x a n ) n≥0 is bounded in L ∞ ((0, T ) × (0, L)) and since the only possible weak− * limit of a convergent subsequence is ∂ x a, we have by weak− * compactness that ∂ x a n → ∂ x a weakly− * in L ∞ . But this is enough to pass to the limit as k → +∞ in the weak formulation T 0 L 0 [y n k (t, x) (∂ t ψ(t, x) + a n k (t, x)∂ x ψ(t, x) + ∂ x a n k (t, x)ψ(t, x)) + b n k (t, x)ψ(t, x)]dxdt + T 0 a n k (t, 0)y l,n k (t)ψ(t, 0)dt + L 0 y 0,n k (x)ψ(0, x)dx = 0, (5.36) where ψ is any function as in Definition 7. We arrive to T 0 L 0 [y(t, x) (∂ t ψ(t, x) + a(t, x)∂ x ψ(t, x) + ∂ x a(t, x)ψ(t, x)) + b(t, x)ψ(t, x)]dxdt + T 0 a(t, 0)y l (t)ψ(t, 0)dt + L 0 y 0 (x)ψ(0, x)dx = 0, (5.37) that is, y is a weak solution of the transport equation (5.35). As the weak solution of (5.35) is unique by Proposition 10, we infer that there is only one possible limit for any convergent subsequence of (y n ) n≥0 , so that the whole sequence (y n ) n≥0 converges uniformly towards y.
Theorem 7. Theorem 5 and Corollary 1 are still valid when we assume merely that the functions a, b, y l and y 0 are Lipschitz continuous.