Finite-time stabilization in optimal time of homogeneous quasilinear hyperbolic systems in one dimensional space

We consider the finite-time stabilization of homogeneous quasilinear hyperbolic systems with one side controls and with nonlinear boundary condition at the other side. We present time-independent feedbacks leading to the finite-time stabilization in any time larger than the optimal time for the null controllability of the linearized system if the initial condition is sufficiently small. One of the key technical points is to establish the local well-posedness of quasilinear hyperbolic systems with nonlinear, non-local boundary conditions.


Introduction and statement of the main result
Linear hyperbolic systems in one dimensional space are frequently used in modeling of many systems such as traffic flow, heat exchangers, and fluids in open channels. The stability and boundary stabilization of these hyperbolic systems have been studied intensively in the literature, see e.g. [2] and the references therein. In this paper, we investigate the finite-time stabilization in optimal time of the following homogeneous, quasilinear, hyperbolic system in one dimensional space (1.1) ∂ t w(t, x) = Σ x, w(t, x) ∂ x w(t, x) for (t, x) ∈ [0, +∞) × (0, 1).
Throughout the paper, we assume (1.4) λ i is of class C 2 with respect to x and y for 1 ≤ i ≤ n = k + m.
The main result of this paper is the following result whose proof is given in the next section. In what follows, we denote, for x ∈ [0, 1] and y ∈ R n , The compatibility conditions considered in Theorem 1.1 are: and (1.14) Null-controllability of hyperbolic systems with one side controls have been studied at least from the work of David Russell [15] even for inhomogeneous systems, i.e., instead of (1.1), one considers n×n with C(x, 0) = 0. For linear systems, i.e., Σ(x, ·) and C(x, ·) are constant for x ∈ [0, 1] and B is linear (B(·) = B· with B = ∇B(0)), the null-controllability was established in [15,Section 3] for the time τ k + τ k+1 . Using backstepping approach, feedback controls leading to finite-time stabilization in the same time were then initiated by Jean-Michel Coron et al. in [7] for m = k = 1 and later developed in [1,4] for the general case. The set B was introduced in [6] and the null-controllability for the linear systems with B ∈ B was established for T > T opt in [5,6] (see also [16] for the case C diagonal) via the backstepping approach. A tutorial introduction of backstepping approach can be found in [10]. In the quasilinear case with m ≥ k and with the linear boundary condition at x = 0, the null controllability for any time greater than τ k + τ k+1 was established for m ≥ k by Tatsien Li in [13, Theorem 3.2] (see also [11]). This work is concerned about homogeneous quasilinear hyperbolic systems with controls on one side, and with nonlinear boundary conditions on the other side: (1.1), (1.5), and (1.6). When the boundary condition is linear, the null-controllability was obtained by Long Hu [8] for m ≥ k at any time greater than max{τ k+1 , τ k + τ m+1 } if initial data are sufficiently small. In the linear case [6], for B ∈ B, we obtained time-independent feedbacks for the null controllability at the optimal time T opt and showed the optimality of T opt . Related exact controllability results can be also found in [6,8,9]. In this work, for ∇B(0) ∈ B, we present time-independent feedbacks leading to finite-time stabilization of (1.1), (1.5), and (1.6) in any time T > T opt provided that the initial data are sufficiently small. It is easy to see that B is an open subset of the set of (real) k × m matrices, and the Hausdorff dimension of its complement is min{k, m − 1}.
The feedbacks for (1.1), (1.5), and (1.6) are nonlinear and inspired from the ones in [6]. The construction is more complicated due to quasilinear nature of the system. We add auxiliary dynamics to fulfill the compatibility conditions at x = 1 since C 1 -solutions are considered. One of the key technical points is to establish the local well-posedness of quasilinear hyperbolic systems with nonlinear, non-local boundary conditions, which is interesting in itself.

Proof of the main result
This section containing two subsections is devoted to the proof of Theorem 1.1. In the first subsection, we establish the local well-posedness of quasilinear hyperbolic systems with nonlinear, non-local boundary conditions. This implies in particular the well-posedness for the feedback laws given in the proof of Theorem 1.1 associated with (1.1) and (1.5). The proof of Theorem 1.1 is given in the second subsection.
2.1. Preliminaries. The main result of this section is Lemma 2.2 on the well-posedness for quasilinear hyperbolic systems related to (1.1) and (1.5). The assumptions made are guided by our feedback controls used in Theorem 1.1. We first consider the semilinear system, with T > 0, where −λ 1 (t, x) < · · · < −λ m (t, x) < 0 < λ m+1 (t, x) < · · · < λ m+k (t, x) , We have Lemma 2.1. Assume that A is of class C 1 , f , and g are of class C 2 , and the following conditions hold, for some C > 0, a ∈ [0, 1), 1 ≤ p < +∞, and ε 0 > 0, We recall the following definition of compatibility conditions for (2.1): u 0 ∈ C 1 ([0, 1]) n is said to satisfy the compatibility conditions if Here and in what follows, the partial derivatives are taken with respect to the notations f (t, x, y), g(t, y + ), and h(t, y, u 0 ).
where L 1 and L 2 are two large, positive constants determined later. Set From now, we assume implicitly that Here and in what follows, for notational ease, we ignore the dependence of h on u 0 and denote h(t, v(t, ·)) instead of h(t, v(t, ·), u 0 ). As in the proof of [6, Lemma 3.2] by (2.4) and (2.5), and the fact that f and g are of class C 1 , one can prove that F is contracting for · 1 -norm provided that L 2 is large and L 1 is much larger than L 2 . The condition 0 ≤ a < 1 and 1 ≤ p < +∞ are essential for the existence of L 1 and L 2 . 1 The existence and uniqueness of u then follow. Moreover, there exist two constants C 1 , C 2 > 0, independent of u 0 such that for u 0 C 1 ([0,1]) ≤ C 1 ε and v 1 < ε, there exists a unique solution u ∈ C 1 ([0, T ] × [0, 1]) n and moreover, We claim that, for u 0 C 1 ([0,1]) ≤ C 3 (ε) and ε sufficiently small, , v j+k (t, 0) must be understood as (F(v)) j+k (t, 0) and (F(v)) j+k (t, 0) is then determined by the RHS of [6, (3.6) or (3.7)] as mentioned there. Related to this point, Vj(t, 0) for k + 1 ≤ j ≤ k + m in [6, (3.14)] and in the inequality just below must be replaced by (F(v) − F(v))j . The rest of the proof is unchanged.
The existence and uniqueness of solutions of (2.1) in The proof is complete.
We next establish the key result of this section. To this end, we first set, for τ > 0, and, for T > 0, and the compatibility conditions at x = 0 hold for the system (2.25) below .
The set D τ also depends on T but we ignore this dependence explicitly for notational ease.
The compatibility conditions at x = 0 considered in the context of Lemma 2.2 are Here and in what follows, we only consider the flows with x Ξ j (t, s, ξ) ∈ [0, 1] so that Ξ is well-defined. Assume that m > k. Since ∇B(0) ∈ B, by the implicit theorem and the Gaussian elimination method, there exist M k : U k → R, . . . , M 1 : U 1 → R of class C 2 for some neighborhoods U k of 0 ∈ R m−1 , . . . , U 1 of 0 ∈ R m−k such that, for y + = (y k+1 , · · · , y k+m ) T ∈ R m with sufficiently small norm, the following facts hold For T > T opt , set δ = T − T opt . Consider ζ j and η j of class C 1 for k + 1 ≤ j ≤ k + m and for t ≥ 0 satisfying (2.28) ζ j (0) = w 0,j (1), ζ j (t) = 0 for t ≥ δ/2, η j (0) = 1, η j (t) = 0 for t ≥ δ/2, , t, 1) = 0 for k + 1 ≤ j ≤ k + m. We now show that H satisfies the assumptions given in Lemma 2.2 if w 0 C 1 ([0,1]) ≤ ε and ε is sufficiently small (τ is sufficiently small as well). We first note that the solutions of the system will be a consequence of our construction η j and ζ j given later. We are next concerned about (2.21). It suffices to prove that We claim that, for 1 ≤ j ≤ k + m.
for (t, s, ξ) so that both flows are well-defined. We only consider the case k + 1 ≤ j ≤ k + m, the other cases can be proved similarly. We have it follows from (1.3) and (2.35) that Combining (2.35) and (2.36) yields (2.34). One can also verify (2.24) by direct/similar computations and by using the fact We now give the
Consider the last two equations of (1.5) and impose the condition w k (t, 0) = w k−1 (t, 0) = 0. Using (1.10) with i = 2 and the Gaussian elimination approach, one can then write these two equations under the form (2.46) and (2.47) w m+k−1 (t, 0) = M k−1 w k+1 (t, 0), · · · , w m+k−2 (t, 0) , for some C 2 nonlinear map M k−1 from U k−1 into R for some neighborhood U k−1 of 0 ∈ R m−2 with M k−1 (0) = 0 provided that |w + (t, 0)| is sufficiently small, etc. Finally, consider the k equations of (1.5) and impose the condition w k (t, 0) = · · · = w 1 (t, 0) = 0. Using (1.10) with i = k and the Gaussian elimination approach, one can then write these k equations under the form (2.46), (2.47), . . . , and for some C 2 nonlinear map M 1 from U 1 into R for some neighborhood U 1 of 0 ∈ R m−k with M 1 (0) = 0 provided that |w + (t, 0)| is sufficiently small. These nonlinear maps M 1 , . . . , M k will be used in the construction of feedbacks. We next introduce the flows along the characteristic curves. Set d dt x j (t, s, ξ) = λ j x j (t, s, ξ), w t, x j (t, s, ξ) and x j (s, s, ξ) = ξ for 1 ≤ j ≤ k, We do not precise at this stage the domain of the definition of x j . Later, we only consider the flows in the regions where the solution w is well-defined.
We are ready to construct a feedback law leading to finite-time stabilization in the time T . Let t m+k be such that x m+k (t + t m+k , t, 1) = 0. It is clear that t m+k depends only on the current state w(t, ·). Let D m+k = D m+k (t) ⊂ R 2 be the open set whose boundary is {t}×[0, 1], [t, t+t m+k ]×{0}, and (s, x m+k (s, t, 1)); s ∈ [t, t+t m+k ] . Then D m+k depends only on the current state as well. This implies x k+1 (t, t + t m+k , 0), . . . , x k+m−1 (t, t + t m+k , 0) are well-defined by the current state w(t, ·).
To complete the feedback for the system, we consider, for k + 1 ≤ j ≤ m, We will establish that the feedback constructed gives the finite-time stabilization in the time T if ε is sufficiently small. To this end, we first claim that Indeed, it is clear to see that the feedback is given by where H is given by (2.30)-(2.33). The well-posedness for the feedback law is now a consequence of Lemma 2.2 through the example mentioned and examined right after it.
The conclusion now follows by the same arguments. The details are omitted.