Null controllability and finite-time stabilization in minimal time of one-dimensional first-order $2 \times 2$ linear hyperbolic systems

The goal of this article is to present the minimal time needed for the null controllability and finite-time stabilization of one-dimensional first-order $2 \times 2$ linear hyperbolic systems. The main technical point is to show that we cannot obtain a better time. The proof combines the backstepping method with the Titchmarsh convolution theorem.

Finally, a, b, c, d ∈ L ∞ (0, 1) couple the equations of the system inside the domain (the matrix a b c d will also be referred in the sequel to as the internal coupling matrix).
Obviously, finite-time stability implies finite-time stabilization, which in turn implies null controllability.
Remark 1.2. As we are trying to bring the solution of the system (1) to the state zero, let us first mention that, in general, u = 0 does not work. Not only this, but in fact any static boundary output feedback laws, that is of the form u(t) = ky 2 (t, 1) with k ∈ R, does not work either in general. A simple example is provided by the following 2 × 2 system with constant coefficients (see also [BC16,Section 5.6] when y 2 (t, 0) = y 1 (t, 0)): y 1 (t, 1) = ky 2 (t, 1), y 2 (t, 0) = 0, Indeed, for this system we can always construct a smooth initial data (y 0 1 , y 0 2 ) which is an eigenfunction of the operator associated with (4) and whose corresponding eigenvalue σ is a positive real number, which makes the system (4) exponentially unstable. This can be done as follows. We take (so that the second equation in (4) will always be satisfied) and • If k < 1 + 1/π, then we take σ = π √ 1 − θ 2 and y 0 2 (x) = sin(θπx), where θ ∈ (0, 1) is any solution to the equation √ 1 − θ 2 + θ cot(θπ) = k.
• If k > 1 + 1/π, then we take σ = π √ 1 + θ 2 and y 0 2 (x) = 2 sinh(θπx), where θ > 0 is any solution to the equation The goal of this work is to establish a necessary and sufficient condition on the time T for the system (1) to be null controllable in time T (resp. finite-time stabilizable with settling time T ).
Let us now introduce some notations that will be used all along the rest of this article. Let φ 1 , φ 2 ∈ C 1,1 ([0, 1]) be the increasing functions defined for every x ∈ [0, 1] by We then denote by dξ.
Finally, we set The naming of the notations in (6) will be explained in Remark 1.8 below.

Literature
Boundary null controllability and stabilization of hyperbolic systems of balance laws have attracted numerous attention of both mathematicians and engineers during the last decades. In the pioneering work [Rus78b], the author established the null controllability of general n × n coupled linear hyperbolic systems of the form (1) in a control time that is given by the sum of the two largest times from the states convecting in opposite directions ([Rus78b, Theorem 3.2]). It was also observed that this time can be shorten in some cases ([Rus78b, Proposition 3.4]), and the problem to find the minimal control time for hyperbolic partial differential equations (PDEs) was then raised ([Rus78b, Remark p. 656]). For systems of linear conservation laws (i.e. when no internal coupling matrix is present in the system), this problem was completely solved few years later in [Wec82], where the minimal control time has been characterized in terms of the boundary coupling matrix, that is the matrix coupling the equations at the boundary on the uncontrolled side. For systems of balance laws, the story is far from over. A first improvement of the control time of [Rus78b] was recently obtained in [CN19b] thanks to the introduction of some rank condition on the boundary coupling matrix. However, this was first done for some generic internal coupling matrices or under rather stringent conditions ([CN19b, Theorem 1.1 and 1.5]). The same authors were then able to remove some of these restrictions in [CN19a]. For the present paper it is especially important to emphasize that the new time introduced in [CN19b,CN19a] is only shown to be sufficient for the null controllability in these works. On the other hand, the minimal control time needed to achieve the exact controllability property (that is when we want to reach any final data and not only zero), was completely characterized in [HO19, Theorem 1.9] by a simple and calculable formula. It is also pointed out that null and exact controllability are equivalent properties if the boundary coupling matrix has a full row rank. For quasilinear systems, it has been shown in [Li10,Theorem 3.2] that the time of [Rus78b] yields the (local) exact controllability of such systems if the boundary coupling matrix has a full row rank in a neighborhood of the state zero. For homogeneous quasilinear systems, a smaller control time was then obtained in [Hu15, Theorem 1.1].
Concerning now the stabilization property, the first works seem [GL84,Qin85] for the exponential stabilization of homogeneous quasilinear hyperbolic systems in a C 1 framework by using the method of characteristics. To the best of our knowledge, the weakest sufficient condition using this technique can be found in [Li94, Theorem 1.3, p. 173]. This condition was then improved in [CBdN08, Theorem 2.3] in a H 2 framework thanks to the construction of an explicit strict Lyapunov function. In all the previous references, the feedback laws were static boundary output feedback laws (that is, depending only on the state values at the boundaries). However, due to the locality of such kind of feedback laws, these two strategies may not be effective to deal with general systems of balance laws ([BC16, Section 5.6] and Remark 1.2). Another method was then used to address this problem, the backstepping method. For PDEs, this method now consists in transforming our initial system into another system -called target system -for which the stabilization properties are simpler to study. The transformation used is usually a Volterra transformation of the second kind. One can refer to the tutorial book [KS08] to design boundary feedback laws stabilizing systems modeled by various PDEs and to the introduction of [CHOS21] for a complementary state of the art on this method. This technique turned out to be a powerful tool to stabilize general coupled hyperbolic systems, moreover in finite time. In [CVKB13] the authors adapted this technique to obtain the first finite-time stabilization result for 2 × 2 linear hyperbolic system. This method was then developed, notably with a more careful choice of the target system, to treat 3 × 3 systems in [HDM15] and then to treat general n × n systems in [HDMVK16,HVDMK19]. However, the control time obtained in these works was larger than the one in [Rus78b] and it was only shown in [ADM16,CHO17] that we can stabilize with the same time as the one of [Rus78b]. These works have recently been generalized to time-dependent systems in [CHOS21]. Finally, let us also mention the two recent works [CN20a, CN20b] concerning the finite-time stabilization of homogeneous quasilinear systems, with the same control time as in [CN19b,CN19a].
In spite of quite a number of contributions dealing with these two problems (controllability and stabilization), we see that there are no references concerning the optimality of the control time for systems of linear balance laws with spatial-varying internal coupling matrix, especially when null and exact controllability are not equivalent, so that the results in [CVKB13,HO19] cannot be considered. This is of course a nontrivial task and it requires the addition of new techniques as we shall see below. The goal of this article is to fill this gap, at least for 2 × 2 systems. We will provide an explicit formula of the minimal control time for any 2 × 2 system of linear balance laws with spacial-varying internal coupling matrix. We will see that one of the main differences between null and exact controllability is that such a critical time is sensitive to the behavior of the internal coupling matrix for the null controllability, whereas it is known to never be the case for the exact controllability ([CVKB13, HO19]).

Main result and comments
The important quantity in the present work is the following: The quantity ℓ ε (f ) is the length of the largest interval of the form (0, ℓ) where the function f vanishes.
In particular, if f has an analytic extension in a neighborhood of x = 0, then ℓ ε (f ) = 0.
(E3) An example of smooth function f with ℓ ε (f ) = 0 but that does not satisfy the previous conditions is The main result of this article is the following complete characterization of the controllability properties of the system (1): (ii) If the time T satisfies (8), then the system (1) is finite-time stabilizable with settling time T .
Note in particular that the system (1) is then null controllable in time T if, and only if, it is finite-time stabilizable with settling time T .
Example 1.6. For c satisfying the properties in (E2) or given by the function in (E3), this result shows that the time T unif (Λ) cannot be improved. This is not trivial, especially when c is given by the function in (E3).
In particular, we see that we can possibly obtain any intermediate time between T opt (Λ) and T unif (Λ). Moreover, note that the value T opt (Λ) is reachable even when c is not identically equal to zero.
Remark 1.8. As we shall see in the proof below, the most difficult part of this result is the necessary condition, that is the item (i). It is also thanks to this part that we can call the time on the right-hand side of (8) the minimal control time. It is sometimes find in the literature that the time T unif (Λ) is "the theoretical lower bound for control time" or "the optimal time". However, the failure of the controllability before this time is never proved in these works (and, in fact, it cannot be in general), which brings some confusion to our point of view. This is why we carefully introduced a different naming and use the notations • T unif (Λ) as "uniform time", for the smallest time after which all the systems of the form (1) are null controllable, • T opt (Λ) as "optimal time", for the smallest control time that can be obtained among all the possible control times for the systems of the form (1).
Remark 1.9. Let us comment other possibilities for the boundary conditions at x = 0: (i) When the boundary condition y 2 (t, 0) = 0 is replaced by y 2 (t, 0) = qy 1 (t, 0) with boundary coupling "matrix" q = 0, the result [CVKB13, Theorem 3.2] shows that the time T unif (Λ) is the minimal control time (more precisely, it is shown that the system (1) with such a boundary condition is equivalent to the same system with no internal coupling matrix, for which T unif (Λ) is clearly minimal). However, when q = 0, we see that our time is smaller than the one obtained in this reference.
(ii) When a second control is applied at the boundary x = 0, i.e. the boundary condition y 2 (t, 0) = 0 is replaced by y 2 (t, 0) = v(t) with v ∈ L 2 loc (0, +∞) a second control at our disposal, then the time T opt (Λ) is the minimal control time. The null controllability for T ≥ T opt (Λ) can be shown using for instance the well-known constructive method developed in [Li10, Theorem 3.1]. On the other hand, the failure of the null controllability for T < T opt (Λ) follows from the backstepping method (by means of Volterra transformation of the second kind) and a simple adaptation of Lemma 3.3 below.
Therefore, combining the previous results of the literature with the new results of the present paper, we see that all the following possibilities for the boundary conditions have been handled: y 1 (t, 1) = py 2 (t, 1) + ru(t), y 2 (t, 0) = qy 1 (t, 0) + sv(t), p, q, r, s ∈ R with (r, s) = (0, 0).
The rest of this article is organized as follows. In Section 2, we use the backstepping method to show that our initial system (1) is equivalent to a canonical system from a controllability point of view. In Section 3 we use the Titchmarsh convolution theorem to completely characterize the minimal control time for this canonical system. In Section 4 we characterize this time in terms of the parameters of the initial system. Finally, in Section 5 we discuss possible extensions to systems with more than two equations.

Reduction to a canonical form
In this section, we perform some changes of unknown to transform our initial system (1) into a new system whose controllability properties will be simpler to study, this is the so-called backstepping method for PDEs. The content of section is quite standard by now, we refer for instance to [CVKB13, Section 3.2] for more details on the computations below.
First of all, we remove the diagonal terms in the system (1). Using the invertible spatial transformation (seen as an operator from L 2 (0, 1) 2 onto itself) with we easily see that the system (1) is null controllable in time T (resp. finite-time stabilizable with settling time T ) if, and only if, so is the system whereb Let us now remove the coupling term on the first equation of (11) thanks to a second transformation. Set Let k 11 , k 12 , k 21 , k 22 ∈ L ∞ (T ). Using the spatial transformation which is invertible since it is a Volterra transformation of the second kind (see e.g. [Hoc73, Chapter 2, Theorem 5]), we see that the system (11) is null controllable in time T (resp. finitetime stabilizable with settling time T ) if, and only if, so is the system with g given by provided that the kernels k 11 , k 12 , k 21 , k 22 satisfy the so-called kernel equations: , and Note that (16) and (17)  In the aforementioned reference this result is stated in a C 0 framework (assuming that a, b, c, d ∈ C 0 ([0, 1])) but its proof readily shows that it is valid in L ∞ as well. As before, the notion of solution is to be understood in the sense of solution along the characteristics. The boundary terms such as k 21 (x, 0), which defines g (see (15)), or k 11 (1, ξ), k 12 (1, ξ), that will appear shortly below in our feedback law (see (19)), etc. are also understood in this sense. We refer for instance to the formula (42) below for the precise meaning of k 21 (x, 0).

Study of the canonical system
We call the system (14) the "control canonical form of the system (1)" or "canonical system" in short, by analogy with [Bru70,Rus78a] and since we will see in this section that we are able to directly read its controllability properties (a task that seems impossible on the initial system (1)).
The goal of this section is to establish the following result: Theorem 3.1. Let T > 0 and g ∈ L ∞ (0, 1).
(i) If the system (14) is null controllable in time T , then necessarily (ii) If the time T satisfies (18), then the system (14) is finite-time stable with settling time T .
Let us emphasize once again that the difficult point is the first item.

The characteristics
Before proving Theorem 3.1 we need to introduce the characteristic curves associated with the system (14) and recall some useful properties. First of all, it is convenient to extend λ 1 , λ 2 to functions of R (still denoted by the same) such that λ 1 , λ 2 ∈ C 0,1 (R) and for some ε > 0 small enough. Since all the results of the present paper depend only on the values of λ 1 , λ 2 in [0, 1], they do not depend on such an extension.
In what follows, i ∈ {1, 2}. Let χ i be the flow associated with λ i , i.e. for every (t, Let us now introduce the entry and exit times s in i (t, x), s out i (t, x) ∈ R of the flow χ i (·; t, x) inside the domain [0, 1], i.e. the respective unique solutions to Their existence and uniqueness are guaranteed by the condition (20). It readily follows from (22) and the uniqueness of s in i that By the implicit function theorem we have s in i ∈ C 1 (R 2 ) with (using (23)) Combined with the group property (24), this yields the following inverse formula for every s, t ∈ R: s < s out 1 (t, 1) ⇐⇒ s in 1 (s, 0) < t, s < s out 2 (t, 0) ⇐⇒ s in 2 (s, 1) < t. (26) Finally, since λ i does not depend on time, we have an explicit formula for the inverse function , ∀θ ∈ R, which gives This also yields an explicit formula for s in 1 , s in 2 and s out 1 , s out 2 and, in particular,

Proof of Theorem 3.1
First of all, the solution of the canonical system (14) is explicitly given by: Next, we show a uniform lower bound for the control time: Lemma 3.3. Let T > 0. If the system (14) is null controllable in time T , then necessarily This result states that the control time cannot be better than the one of the case g = 0.
Since x ∈ ω 2 −→ χ 2 (0; T, x) is bijective (it is increasing by (23) and ω 2 is an interval by (25)), this implies that the bounded linear operator K : is surjective. This is impossible since its range is clearly a subset of L ∞ (ω 2 ), which is a proper subset of L 2 (ω 2 ) (alternatively, one could note that K is compact and therefore it cannot be surjective over an infinite dimensional space, see e.g. [Rud91, Theorem 4.18 (b)]).
Remark 3.5. The difficulty in the proof of this result is the necessary condition, i.e. the implication "=⇒", just like it is the case for our main result. Let us however mention that its proof is easy in case α satisfies the condition in (E2) of Example 1.4 (by taking derivatives of (31) and using the injectivity of Volterra transformations of the second kind). It does not seem trivial for functions of the form (7) though.
We are now ready to prove the main result of Section 3: Proof of Theorem 3.1.
Now we use the fact that g (χ 2 (s; t, 0)) is actually a function of s − t. Indeed, by uniqueness to the solution to the ODE (21), we see that the characteristics take the form Using the change of variables σ = s − t and introducing whereτ = T − s in 2 (T, 1).
Remark 3.6. Let us point out that the space dependence of the speeds brings up more technical difficulties than the case of constant speeds (especially the step 2)).

Proof of the main result
In this section we show how to deduce our main result from Theorem 3.1.
Proof of Theorem 1.5.
1) First of all, let us recall that the initial system (1) is null controllable in time T (resp. finite-time stabilizable with settling time T ) if, and only if, so is the canonical system (14) (with g given by (15)). Therefore, thanks to Theorem 3.1 it suffices to show that which amounts to characterize ℓ 1 (g) in terms of ℓ xΛ (c) (we recall that x Λ is defined in the statement of Theorem 1.5). To this end, we are going to prove the identity where we recall that φ 1 , φ 2 ∈ C 1,1 ([0, 1]) are defined in (5).
We can check thath ∈ L ∞ (T ) (recall (40)). It follows from the injectivity of Volterra transformations of the second kind that which is equivalent to (41) since s in is increasing with s in (0) = 0.
Remark 4.1. In the proof of Theorem 1.5, we have not used the apparent freedom for the boundary data of k 22 provided by Theorem 2.1.