Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-Lyapunov function

For a system that is governed by the isothermal Euler equations with friction for ideal gas, the corresponding field of characteristic curves is determined by the velocity of the flow. This velocity is determined by a second-order quasilinear hyperbolic equation. For the corresponding initial-boundary value problem with Neumann-boundary feedback, we consider non-stationary solutions locally around a stationary state on a finite time interval and discuss the well-posedness of this kind of problem. We introduce a strict $H^2$-Lyapunov function and show that the boundary feedback constant can be chosen such that the $H^2$-Lyapunov function and hence also the $H^2$-norm of the difference between the non-stationary and the stationary state decays exponentially with time.


Introduction
The flow of gas through a pipeline is modelled by the isothermal Euler equations with friction. In the operation of gas pipelines, it is essential that the velocities remain below critical values where vibrations occur and noise is created, see [36]. We study a quasilinear wave equation for the gas velocity in the case of ideal gas which is derived from the isothermal Euler equations with friction. Using Neumann feedback at one end of the pipe, we stabilize the solution of the corresponding initial-boundary value problem with homogeneous Dirichlet boundary conditions at the other end of the pipe to a desired subsonic stationary state. Except for its nonlinearity, this system is of a similar form as the system with the linear wave equation which has been studied for example in [24].
The first results on the boundary feedback stabilization for a quasilinear wave equation have been obtained by M. Slemrod in [32] and J. Greenberg & T. Li in [13] by using the method of characteristics. In [8], J.-M. Coron, B. d'Andrea-Novel & G. Bastin constructed a strict H 2 -Lyapunov function for the boundary control of hyperbolic systems of conservation laws without source term. In [9], they constructed a strict H 2 -Lyapunov for quasilinear hyperbolic systems with dissipative boundary conditions without source term. More recently in [7], Coron and Bastin study the Lyapunov stability of the C 1 -norm for quasilinear hyperbolic systems of the first order. They consider W 1 p -Lyapunov functions for p < ∞ and look at the limit for p → ∞.
Based upon [8], M. Dick, M. Gugat & G. Leugering considered the isothermal Euler equations with friction with Dirichlet boundary feedback at both ends of the system and introduced a strict H 1 -Lyapunov function, which is a weighted and squared H 1 -norm of the difference between the nonstationary and the stationary state. They developed Dirichlet boundary feedback conditions which guarantee that the H 1 -norm of the difference between the non-stationary and the stationary state decays exponentially with time (see [11]). In [18], we have defined a strict H 2 -Lyapunov function for this stabilization problem. In contrast to [8], [11] and [18] in the present paper a Neumann boundary feedback law is used at one end of the interval for the stabilization of the system. This is motivated by the nice properties of the corresponding Neumann feedback for the linear wave equation that leads to finite-time stabilization for a certain feedback parameter, see [24], [1].
In our paper, by constructing a strict H 2 -Lyapunov function and choosing suitable boundary feedback conditions, we give results about the boundary feedback stabilization for a second-order quasilinear hyperbolic equation with source term. The exponential decay of the solution of a second-order quasilinear hyperbolic equation is established. This solution measures the difference between the present state and a desired stationary state, which is in general not constant for our system. This paper is organized as follows: In Section 2 we consider the isothermal Euler equations both in physical variables and in terms of Riemann invariants. Then we transform the isothermal Euler equations to a second-order quasilinear hyperbolic equation. In Section 3 we state a result about the well-posedness of general secondorder quasilinear hyperbolic systems on a finite time interval (see Lemma 3.1). Our main results about the exponential decay of the H 2 -norm and C 1 -norm are presented in Theorem 4.5 and Corollary 4.7 in Section 4.2. The proofs of Theorem 4.5 and Corollary 4.7 are given in Section 5. The infinite time horizon case is studied in Section 6. We show that due to the stabilization, the solution exists globally in time.

The isothermal Euler equations and a quasilinear wave equation
In this section, we present the isothermal Euler equations with friction for a single pipe both in terms of the physical variables and in terms of Riemann invariants. Let a finite time T > 0 be given. The system dynamics for the gas flow in a single pipe can be modeled by a hyperbolic system, which is described by the isothermal Euler equations (see [4], [5], [11]): where ρ = ρ(t, x) > 0 is the density of the gas, q = q(t, x) is the mass flux, the constant f g > 0 is a friction factor, δ > 0 is the diameter of the pipe and a > 0 is the sonic velocity in the gas. We consider the equations on the domain Ω := [0, T ] × [0, L]. Equation (1) states the conservation of mass and equation (2) is the momentum equation. We use the notation θ = f g δ .
In this paper, we consider positive gas flow in subsonic or subcritical states, that is, The isothermal Euler equations (1) and (2) give rise to the second-order equatioñ whereũ is the unknown function and satisfies that isũ is the velocity of the gas. The lower order term is From the velocityũ, the density ρ can be obtained from the initial value and the differential equation Then q can be obtained from the equation q = ρũ. To stabilize the system governed by the quasilinear wave equation (4) locally around a given stationary stateū(x), we use the boundary feedback law with a feedback parameter k ∈ (0, ∞).
In terms of the physical variables (q, ρ), the boundary feedback law is Sufficient conditions for the exponential stability of this system will be presented in Theorem 4.5 in Section 4.2.

2.1
The Riemann invariants and a differential equation for ρ in terms of the velocity For classical solutions the isothermal Euler equations (1) and (2) can be equivalently written as the following system with the matrixÂ System (8) has two eigenvaluesλ − (ρ, q),λ + (ρ, q) and in the subsonic case we havẽ In terms of the Riemann invariants R ± = R ± (ρ, q) = − q ρ ∓ a ln(ρ) the system (8) has the diagonal form whereD In terms of R ± , for the physical variables ρ and q we have A gas flow is positive and subsonic (i.e. 0 < q/ρ < a) if and only if For the velocityũ =ũ(ρ, q) defined in (5) we havẽ Due to (9), we can express the velocity in terms of the eigenvalues as Due to equation (15), (10) yields the second-order equation (4). A detailed derivation can be found in [21]. The second-order quasilinear equation (4) is hyperbolic with the eigenvaluesλ Using the isothermal Euler equations (1) and (2), we obtain the partial derivatives ofũ with respect to t and x, respectively, Multiplyingũ t andũ x byũ andũ 2 − a 2 , respectively, by adding the two equations we obtain (7), which means that ρ and q can be obtained fromũ and the initial data. Note that sinceũ = q ρ , we have the same value forũ for λq and λρ where λ ∈ (0, 1]. So we cannot expect to recover the values of (q, ρ) fromũ without additional information on (q, ρ). In a similar way as (7), we obtain the equation Thus ifũ is known, the values of ρ can be determined from the value of ρ at a boundary point (x = 0 or x = L) and (17) by integration.

Stationary states of the system
In [12] the existence, uniqueness and the properties of stationary subsonic C 1solutions (ρ(x),q(x)) of the isothermal Euler equations have been discussed. The stationary states of the system on networks are studied in [16].
Here we focus on the stationary states of (4). Letū =ū(x) denote a stationary state for the second-order equation (4). Then (4) yields the following second-order ordinary differential equations forū(x): This implies that equation (4) has constant stationary statesū ∈ (−∞, ∞) that can attain arbitrary real values. In contrast to this situation, the isothermal Euler equations with friction (that is (1), (2)) do not have constant stationary states except for the case of velocity zero. The stationary states of (1), (2) have been studied in [16]. Now we consider the question: Given a constant stateū = λ ∈ (0, ∞), is there a solution (q, ρ) of (1), (2) that corresponds to the constant velocitȳ u? For λ = 0 we obtain the constant solution of (1), (2) where q = 0. For λ > 0 there is a corresponding solution of travelling wave type (in particular the corresponding solution of (1), (2) is not stationary), namely where the function α is given by and C > 0 is a positive constant. Equation (18) can be rewritten in the form Thus for every stationary stateū of (4) there exists a constant λ ∈ (−∞, ∞) such thatū satisfies the first order ordinary differential equation We use the notationū 0 :=ū(0). Assume thatū 0 ∈ (0, a). Let [0, x 0 ) denote the maximal existence interval of the solution. For the solutions that are not constant, we have two cases: If λ + θ If λ + θ 2ū 3 0 < 0,ū is strictly decreasing on [0, x 0 ) and For stationary ρ andũ, equation (7) implies (22) with λ = 0. The stationary states that correspond to λ = 0 cannot be deduced from the stationary states of (1), (2). Thus all the stationary solutions of (4) that correspond to a stationary state of (1), (2) must satisfy the equation The following Lemma contains an explicit representation for these stationary velocities.
Proof 2.1 Separation of variables yields Now the definition of W −1 as the inverse function of z exp(z) for z ∈ (−∞, −1) yields the assertion.
Since for the stationary states (q, ρ) of (1), (2) the flow rate q is constant, by (5) we get the corresponding density as ρ(x) = q u(x) .
3 Well-posedness of the system locally around stationary states Now we consider non-stationary solutions locally around a subsonic stationary statē u(x) > 0 on Ω that satisfies (22) with λ = 0, that is that corresponds to a stationary state of (1), (2). For a solutionũ(t, x) of (4), define Then (4), (18) and (22) yield the equation where F := F (x, u, u x , u t ) satisfies Ifū ≥ 0 andū + u ≥ 0, we havẽ withF as defined in (6). For the second-order quasilinear hyperbolic equation (25), we consider the initial conditions and the boundary feedback conditions where k > 0 is a real constant. We work in the framework of classical semi-global solutions. To apply the theory presented in [35], the second order equation is written as a first order system (see the proof of Theorem 1 in [28]). In this way the following result can be obtained (see Lemma 1 in [28]): Lemma 3.1 Let a subsonic stationary stateū(x) > 0 as in Lemma 2.1 be given. Choose T > 0 arbitrarily large. There exist constants ε 0 (T ) > 0 and C T > 0, such that if the initial data (32) and the C 2 -compatibility conditions are satisfied at the points (t, x) = (0, 0) and (0, L), then the initial-boundary problem (25), (29),(30)-(31) has a unique solution . Moreover the following estimate holds:

Exponential stability
In this section, we introduce a strict H 2 -Lyapunov function for the closed-loop system consisting of the quasilinear wave equation (25) and the boundary conditions (30), (31). To motivate the choice of the Lyapunov function, let us reconsider the classical energy for systems governed by the linear wave equation In our quasilinear wave equation (25), instead of the square of the wave speed c 2 the term (a 2 − (ū + u) 2 ) appears as a factor in front of u xx , so it makes sense to replace c 2 by this expression in the definition of our Lyapunov function. In the same line of reasoning, if our quasilinear equation would be would be a candidate for a Lyapunov function. However, in our wave equation also the term 2(ū + u)u tx appears. In order to deal with this term, we introduce an additional quantity in our Lyapunov function in such a way that, via equation (25), we can find an upper bound for its time-derivative. For this purpose, it makes sense to introduce a term that contains the product u t u x in the integral defining the first part of our Lyapunov function. As a further motivation, we return to the linear wave equation u tt − u xx = 0 with the associated boundary conditions u(t, 0) = 0 and u t (t, L) = −ku x (t, L) with k > 0. For a number λ ∈ (0, 2k L(1+k 2 ) ), the quantity For many hyperbolic systems exponential weights in the Lyapunov function have been used successfully, see various examples in [10]. We define the weights In the sequel we consider since according to the previous considerations, this is a natural candidate to define a Lyapunov function for our system. To show the exponential decay with respect to the H 2 -norm, it is necessary to deal with the second order derivatives. Therefore we also introduce E 2 (t) which is defined analogously to E 1 to show the decay of the partial derivatives of second order. We define We define the Lyapunov function E(t) as In the following subsection we show that our Lyapunov function E(t) as defined in (36) is bounded above and below by the product of appropriate constants and the square of the H 2 -norm of u.

Equivalence of E(t) with E(t) as in (36) and the H 2norm of the state
In this section we show that E(t) with E(t) as in (36) is equivalent to the H 2norm of the state. This is a an essential property of a Lyapunov function since we want to use it to show the exponential decay of the H 2 -norm. Note that the constants in Lemma 4.1 are independent of the length T of the time interval.
be given. Choose a real number k > 0 such that Assume thatū is such that we haveū Then for the weights defined in (34), (35) on the interval [0, L] we have the strict inequality In addition, we assume thatū is sufficiently small in the sense that Then for the weights we have the inequality For a real number z define Define the matrixB For a real number z define Define the matrixÃ (47) Then there exists ε 1 (υ) > 0 such that for all z with |z| ≤ 2 ε 1 (υ) the matrixB 3 (z) is positive definite and the matrixÃ 3 (z) is positive definite.
Now we come to the assertion for the symmetric matrixB 3 . Due to (38) we have b 11 (0) = 1 − 1 k 2 a 2 > 0. Due to the continuity of b 11 (·) this implies that there exists a constant ε 1 > 0 such that for all |z| ≤ 2 ε 1 we have b 11 (z) > 0. We have Hence (44) implies detB 3 (0) > 0. Due to the continuity of detB 3 (·) this implies that we can choose the constant ε 1 > 0 in such a way that for all |z| ≤ 2 ε 1 we have detB 3 (z) > 0, and thusB 3 (z) is positive definite. We can choose the constant ε 1 > 0 in such a way that for all |z| ≤ 2 ε 1 for the 2 × 2 matrixÃ 3 (z) the upper left element in the matrix is greater than zero. We have Due to (48) we can assume that ε 1 > 0 is sufficiently small such that for all |z| ≤ 2 ε 1 we have detÃ 3 (z) > 0, and thusÃ 3 (z) is positive definite.
In Lemma 4.2 we show several inequalities that we need to show that E(t) as in (36) can be bounded above and below by the squared H 2 -norm. Note that also in Lemma 4.2 the constants are independent of the length T of the time interval.
If ε 2 > 0 is chosen sufficiently small, we have Assume in the sequel that ε 2 > 0 is chosen such that (50) and (51) hold.
Then χ x can be represented in the form which implies (50). This in turn implies (51).
The representations (55) and (56) follow directly from the definition of χ x .
In the sequel we assume that the assumptions from Lemma 4.1 hold. With χ x as defined in Lemma 4.2 we have With these representations of E 1 and E 2 , Lemma 4.2 yields lower and upper bounds for E 1 (t) and E 2 (t).
where ε 2 is chosen as in Lemma 4.2. For E 1 defined in (57) and k 1 defined in (49) we have the lower bounds and Moreover, we have the upper bounds and For E 2 defined in (58), we have the lower bounds and Moreover, we have the upper bounds Proof 4.3 Equation (55) and (50) imply the lower bound (60) for E 1 .
The representation (56) and (51)  Now we can show that E(t) can be bounded above and below by the squared H 2 -norm. Define the number If (59) holds, by (62) and (66) Lemma 4.3 implies the inequality Define By the definition of E and (60), (61), (64), (65) we also have the lower bound The Poincaré-Friedrichs inequality states that if (31) holds, we have Using this inequality and (25), inequality (71) implies that if E(t) is small, also the H 2 -norm of u(t, x) is small. Similarly E 1 (t) can be bounded above and below by the squared H 1 -norm.

Exponential Decay of the H 2 -Lyapunov Function
In this section we present our main result about the exponential decay of the Lyapunov function that we have introduced in (36). Consider the system withF as defined in (6). In Theorem 4.5 we present our main result about the stabilization of (73) forũ. For the analysis we use the fact that (73) is equivalent to (25), (29), (30),(31) that is stated in terms of u which is defined in (24) as the difference betweenũ and the stationary stateū. In Theorem 4.5 we state that the function E(t) defined in (36) is a strict Lyapunov function. In Theorem 4.5 it is assumed thatū > 0 is sufficiently small and k is sufficiently large. Before we state the theorem, in the following remark we comment on condition (76) that appears in the statement of the Theorem and explain why it can be satisfied for all a > 0 if u > 0 is sufficiently small and k is sufficiently large.
Remark 4.6 Theorem 4.5 states that ifū > 0 is sufficiently small and k is sufficiently large for sufficiently small initial data the Lyapunov function decays exponentially and the decay rate is at least µ 0 = 1 4 e L k which is independent ofū and T , since the conditions on k do not depend on T . For arbitrarily large T , we can always achieve this decay rate µ 0 for sufficiently small initial data. With this decay rate, it is possible to determine a time T 0 > 0 when the size of the H 2 × H 1 -norm of the solution is reduced at least by a factor 1/3. In fact let with K max from (68) and K min from (70). Then due to (71) and (72) we have If the assuption of (86) holds. Furthermore, there exists a constant η 2 > 0 that is independent of T such that for any t ∈ [0, T ] the C 1 -norm of the solution satisfies Due to (82), this implies that for T sufficiently large we have (u(T, ·), u t (T, ·)) H 2 (0,L)×H 1 (0,L) ≤ 1 2 (ϕ(x), ψ(x)) H 2 (0,L)×H 1 (0,L) and The proofs of Theorem 4.5 and Corollary 4.7 are given in Section 5.

Proofs of Theorem 4.and Corollary 4.7
In this section we prove Theorem 4.5 and Corollary 4.7 from Section 4.2. For the proof, we consider the time derivative of the Laypunov function E(t).

Time derivative of the Lyapunov function
First we consider the evaluation of the time derivative of the Lyapunov function E(t).
Lemma 5.1 Let the assumptions of Theorem 4.5 hold. Then the time-derivative of E 1 is given by the following equation: Hence differentiation yields Now integration by parts for the termd u x u xt = d u x (u t ) x yields the equation .
Hence we get the equation . By the partial differential equation (25) we have u tt −d u xx = F − 2(ū + u)u tx and obtain .

(93)
Using integration by parts we obtain the identities Using these identities we obtain the equation d dt E 1 (t) = I 1 + I 2 + I 3 . Here, I 3 contains all the terms coming from the boundary and I 1 = L 0 h 2xd u 2 x + h 2x u 2 t dx contains all the terms where h 2x appears. The remaining terms appear in I 2 .
Similarly the next lemma is proved, where the time derivative of E 2 is considered.
Lemma 5.2 Let the assumption of Theorem 4.5 hold. Then the following equation holds: Integration by parts for the termd u xx u xxt = d u xx (u tx ) x yields the equation Hence we get the equation By the partial differential equation (25) we have Using integration by parts we obtain the identities Using these identities we obtain d dt E 2 (t) =Ĩ 1 +Ĩ 2 +Ĩ 3 whereĨ 3 contains all the terms coming from the boundary andĨ 1 = L 0 h 2xd u 2 xx + h 2x u 2 tx dx contains all the terms where h 2x appears.

Proof of Theorem 4.5
Proof 5.3 In the proof, we use Lemma 4.1. Therefore we assume thatū is sufficiently small in the sense that (41) holds. Moreover, we use Lemma 4.2. Therefore we assume that ε 0 (T ) > 0 is sufficiently small such that (78) holds. We have First we consider d dt E 1 (t) = I 1 + I 2 + I 3 . Define µ 2 = 1 L . By the definition of h 2 in (35) we have (h 2 ) x = −µ 2 h 2 and thus For all x ∈ [0, L] we have µ 2 h 2 (x) ≥ 1 e L hence we have Thus, by (63) we have Now we consider the term I 2 as defined in (91). Note that due to (28), each of the terms that are added in I 2 , in particular F u t and F u x , contains a second order term of u, u t , u x as a factor, that is u u t , u u x , u x u t , u 2 x or u 2 t . More precisely, the terms that appear as factors are either third order terms u t u 2 x , u x u 2 t , u 3 x , uu 2 x , uu x u t , uu 2 t or terms of the form θū uu t , θū uu x , θū u x u t , θū u 2 x or θū u 2 t . Since we have h 1 (x) = k and max x∈[0,L] |h 2 (x)| = 1, the definition of I 2 implies that there exists a continuous function P 0 with P 0 (0) = 0 such that we have an estimate of the form In fact, the definitions of I 2 , F andF imply that we can choose Using (72), and then (60), (61) we obtain the inequality Now we focus on the boundary term I 3 . We use the notationū 0 :=ū(0) and u L :=ū(L). Since k > 0, by the boundary conditions (30), (31) we have Since (79) holds, we have |ū 0 + u(t, 0)| ≤ |ū 0 | + γ a − |ū 0 | = γa ≤ a − 1 k . Hence we have I 0 3 ≥ 0. Since I L 3 ≤ 0, due to (103) this implies Then inequalities (100), (102) and (106) yield With (80) this implies that E 1 (t) is a strict Lyapunov function and (83) holds. Similarly, forĨ 1 , we infer Hence (67) yieldsĨ Now we considerĨ 2 as defined in (96). All the terms that are added inĨ 2 are contain factors u xx u tx , u 2 xx , u 2 tx , F x u xx , F x u tx . Except for F x u xx , F x u tx , it can easily be seen that the coefficients that are multiplied with these quadratic terms become arbitrarily small if if T Li (t) as defined in (81) is sufficiently small. Now we have a closer look at F x . From (28), we have Also in F x u xx , F x u tx all the terms that are added contain quadratic factors u xx u tx , u 2 xx , u 2 tx , u x u xx , u x u tx , u t u xx , u t u tx , u u xx , u u tx and the coefficients that are multiplied with these factors become arbitrarily small if if T Li (t) as defined in (81) is sufficiently small. Thus similar as in the estimate of I 2 , we can find a continuous function P 1 (t) with P 1 (0) = 0 such that using (72), and then (60), (61) we obtain the inequalityĨ In fact if we replace in the representation of F x in each of the terms that are added except for one factor the expressions u, u x , u t , u xx , u tx ,ū by t, and treat the other terms from the definition ofĨ 2 in a similar way since h 1 = k and |h 2 | ≤ 1 we can choose P 1 (t) = 8 t + 2 t + 8 t 2 + 4 k t 2 + 4 k t For the boundary termĨ 3 , we use (25) in the form (a 2 − (ū + u) 2 ) u xx = u tt + 2(ū + u)u tx − F.

Proof of Corollary 4.7
Now we present the proof of Corollary 4.7.
Inequality (69) implies for t = 0: With the positive constants Theorem 4.5 implies that for all T > 0 the decay rate µ = 1 4 eLk can be achieved for sufficiently small initial data. With this decay rate, for and i ∈ {1, 2} we have the inequality η i exp − µ 2 T ≤ 1 2 . Hence the inequalities (86) and (87) imply the inequalities (88) and (89).

Global solutions
In this section we show that the exponential decay of the Lyapunov function defined in (36) implies that the solution exists global in time without losing regularity, that is it keeps the regularity of the initial state.
Let us first observe that (9) implies that the eigenvalues λ − = (−a +ū + u) and λ + = (a +ū + u) do not depend on the derivatives of u. Therefore for a given value of T > 0, (s, x, t) ∈ [0, T ] × [0, L] × [0, T ], the field of characteristic curves ξ u ± (s, x, t) corresponding to u ∈ C 1 ([0, T ] × [0, L]) defined by the integral equation 127) is well-defined for a C 1 -function u ifū and u have sufficiently small C 1 -norm. In order to obtain a semi-global C 1 -solution of (25), (30), (31) in the sense of integral equations along these characteristic curves, the boundary condition (30) at x = 0 is written in the form of the integral equation To be precise we define (r + , r − ) = (R + −R + , R − −R − ). Then we have u = − 1 2 (r + + r − ). Thus the boundary condition (31) at x = L is equivalent to and (128)  Due to (10), (r + , r − ) satisfies the system in diagonal form Let t u ± (x, t) ≤ t denote the time where ξ u ± (s, x, t) hits the boundary of [0, T ] × [0, L]. Then (11) implies that (r + , r − ) satisfy the integral equations Now we consider the initial boundary value problem with initial data for (r + , r − ) at t = 0, the equation in diagonal form (131)-(133) and the boundary conditions (129), (130).
Let a time T > 0 be given such that (88) holds. With initial data for (r + , r − ) in [C 1 ([0, L])] 2 at t = 0 that are sufficiently small (with respect to the C 1 -norm), compatible to u and satisfy the C 1 -compatibility conditions for (129), (130), as in [27] we obtain a semi-global classical solution (r + , r − ) ∈ C 1 ([0, T ] × [0, L]). Thus we also get a continuously differentiable function u = − 1 If our initial data for (r + , r − ) at t = 0 are more regular, namely in [H 2 (0, L)] 2 and satisfy the assumptions that we just mentioned, they generate a solution that is more regular than the classical solution in general: For all t ∈ [0, T ] the second partial derivatives (∂ xx r + , ∂ xx r − ) are in L 2 (0, L). This can be seen as follows.
For initial data with sufficiently small H 2 -norm also the C 1 -norm is small. Thus we know that a classical semi-global solution exists on [0, T ] and we can fix the corresponding characteristic curves. As a consequence, we obtain a semilinear evolution for (∂ xx r + , ∂ xx r − ) with fixed characteristic curves. The evolution of ∂ xx r ± is governed by the integral equation with a polynomial P ± (with C 1 -coefficients, similar to p ± ) that is affine linear with respect to ∂ xx r ± . We can consider t u ± (x, t), ξ u ± (s, x, t), r + , r − , ∂ x r + , ∂ x r − as given continuous functions. Then ∂ xx r ± (t, ·) ∈ L 2 (0, L) is given as the solution of a family of linear integral equations. Thus we can show that ∂ xx r ± (t, ·) ∈ L 2 (0, L) for all t ∈ [0, T ]. This implies that the lifespan of the H 2 -solution only depends on the C 1 -norm of (r + , r − ), so in particular it is well-defined on the time interval [0, T ].
Thus we can construct a global H 2 -solution as follows: Since we have chosen T > 0 such that (88) holds, for nonzero initial data (ϕ, ψ) with sufficiently small H 2 × H 1 -norm, we obtain an H 2 -solution on [0, T ] as described above and due to (88) the H 2 ×H 1 norm of (u(T, ·), u t (T, ·)) is less than the H 2 × H 1 norm of (u(0, ·), u t (0, ·)). Thus we can start our construction again with initial data (u(T, ·), u t (T, ·)) where the H 2 × H 1 norm has been decreased at least by a factor 1 2 to obtain an H 2 -solution on the time interval [T, 2 T ]. By repeating the procedure iteratively, for initial data (ϕ, ψ) with sufficiently small H 2 × H 1norm, we thus obtain a solution that is well defined for all t > 0. Moreover, Lemma 2 in [20] implies that the H 2 × H 1 -norm of the solution decays exponentially with time. In addition, Theorem 4.5 implies that also the Lyapunov function E(t) decays exponentially with time. The above considerations yield the following result. Theorem 6.1 (Global Exponential Decay of the H 2 -Lyapunov Function). Let a stationary subsonic stateū(x) ∈ C 2 (0, L) be given that satisfies (23). Let γ ∈ (0, 1/2] be given. Assume that for all x ∈ L we haveū(x) ∈ (0, γ a). Choose a real number k > 1 (1−γ) a . Assume thatū is sufficiently small and k sufficiently large such that for K ∂ (k,ū(0)) as defined in (74) condition (76) holds. Assume that ū C 2 ([0,L]) is sufficiently small such that ū C([0,L]) < ε 1 (2 k 2 ) and (41) holds. Choose ε 2 as in Lemma 4.2. Define T as in (126).

Summary and outlook
In this paper we have considered a quasilinear wave equation for the velocity of a gas flow that is governed by the isothermal Euler equations with friction. We have presented a method of boundary feedback stabilization to stabilize the velocity locally around a given stationary state. For the proof, we have introduced a strict H 2 -Lyapunov function (see (36)). We have shown that, for initial conditions with sufficiently small C 2 × C 1 -norm and for appropriate boundary feedback conditions, the H 2 ×H 1 -norm of the solution decays exponentially with time. In addition, we have shown that with our velocity feedback law, for initial data with sufficiently small H 2 × H 1 -norm the solution exists globally in time and the H 2 × H 1 -norm of the solution decays exponentially.
In this paper, the strict H 2 -Lyapunov function is used to prove the stability of the solution. It would also be interesting to consider other types of Lyapunov functions, such as weak Lyapunov functions. Moreover, when a disturbance is considered, Input-to-State Stability Lyapunov functions should be studied (see [29], [30]).
We have presented our stabilization method for a single pipe applying an active control at an end of the pipe. Some additional work is required to extend this method to more complicated gas networks. For the stabilization of networks it is often necessary to apply an active control in the interior of the networks. The well-posedness of systems of balance laws on networks is studied in [17]. For a star-shaped network of vibrating strings governed by the wave equation, a method of boundary feedback stabilization is presented in [19], where not for each string an active control is necessary. A related open problem is the feedback stabilization of more complicated pipe networks with leaks. Moreover, also feedback stabilization of second-order hyperbolic equations with time-delayed controls is worth to be studied. For wave equations, this has been done in [14] and in [31] and for the isothermal Euler equations with an L 2 -Lyapunov function in [15]. In the current paper we have considered an ideal gas with constant sound speed. It would be interesting to look at more realistic models of gas where the sound speed also depends on the pressure.