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ESAIM: COCV 27 (2021) 96 ESAIM: Control, Optimisation and Calculus of Variations
https://doi.org/10.1051/cocv/2021091 www.esaim-cocv.org
NULL CONTROLLABILITY AND FINITE-TIME STABILIZATION IN
MINIMAL TIME OF ONE-DIMENSIONAL FIRST-ORDER 2 ×2
LINEAR HYPERBOLIC SYSTEMS
Long Hu1,*and Guillaume Olive2
Abstract. 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 ×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.
Mathematics Sub ject Classification. 35L40, 93B05, 93D15, 45D05.
Received April 29, 2021. Accepted September 10 , 2021.
1. Introduction and main result
1.1. Problem description
In this paper we are interested in the characterization of the minimal time needed for the controllability of
the following class of one-dimensional first-order 2 ×2 linear hyperbolic systems:
∂y1
∂t (t, x) + λ1(x)∂y1
∂x (t, x) = a(x)y1(t, x) + b(x)y2(t, x),
∂y2
∂t (t, x) + λ2(x)∂y2
∂x (t, x) = c(x)y1(t, x) + d(x)y2(t, x),
y1(t, 1) = u(t), y2(t, 0) = 0,
y1(0, x) = y0
1(x), y2(0, x) = y0
2(x),
t∈(0,+∞), x ∈(0,1),(1.1)
where (y1(t, ·), y2(t, ·)) is the state at time t, (y0
1, y0
2) is the initial data and u(t) is the control at time t. We
assume that the speeds λ1, λ2∈C0,1([0,1]) are such that
λ1(x)<0< λ2(x),∀x∈[0,1].
Keywords and phrases: Hyperbolic systems, Boundary controllability, Minimal control time, Backstepping method, Titchmarsh
convolution theorem.
1School of Mathematics, Shandong University, Jinan, Shandong 250100, PR China.
2Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Lojasiewicza 6, 30-348 Krak´ow, Poland.
*Corresponding author: hul@sdu.edu.cn
c
The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
2L. HU AND G. OLIVE
Finally, a, b, c, d ∈L∞(0,1) couple the equations of the system inside the domain (the matrix a b
c dwill also
be referred in the sequel to as the internal coupling matrix).
The boundary condition at x= 0 is a particular case of the more general boundary condition
y2(t, 0) = qy1(t, 0),(q∈R),(1.2)
and the goal of this paper is to investigate what happens when q= 0.
The aforementioned systems appear in linearized versions of various physical models of balance laws, see e.g.
Chapter 1 of [2]. For instance, the telegrapher equations of Heaviside form a linear system of the form (1.1) for
some parameters (see e.g. [2], Sect. 1.2 and (1.20) with −1 + λR0C`= 0).
We recall that the system (1.1) is well-posed: for every u∈L2
loc(0,+∞) and (y0
1, y0
2)∈L2(0,1)2, there exists
a unique solution (y1, y2)∈C0([0,+∞); L2(0,1)2) to the system (1.1). By solution we mean “solution along the
characteristics” or “broad solution” (see e.g. Appendix A of [6]). The same statement remains true if, in the
boundary condition at x= 1, uis replaced by
u(t) = Z1
0
(f1(ξ)y1(t, ξ) + f2(ξ)y2(t, ξ)) dξ, (1.3)
for any f1, f2∈L∞(0,1). The relation (1.3) is called the “feedback law”.
Let us now introduce the notions of controllability that we are interested in:
Definition 1.1. Let T > 0. We say that the system (1.1) is:
–finite-time stable with settling time Tif, for every y0
1, y0
2∈L2(0,1), the corresponding solution to
the system (1.1) with u= 0 satisfies
y1(T, ·) = y2(T, ·)=0.(1.4)
–finite-time stabilizable with settling time Tif there exist f1, f2∈L∞(0,1) such that, for every
y0
1, y0
2∈L2(0,1), the corresponding solution to the system (1.1) with ugiven by (1.3) satisfies (1.4).
–null controllable in time Tif, for every y0
1, y0
2∈L2(0,1), there exists u∈L2
loc(0,+∞) such that the
corresponding solution to the system (1.1) satisfies (1.4).
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 (1.1) 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) = ky2(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 [2], Sect. 5.6 when y2(t, 0) = y1(t, 0)):
∂y1
∂t (t, x)−∂y1
∂x (t, x) = πy2(t, x),
∂y2
∂t (t, x) + ∂y2
∂x (t, x) = πy1(t, x),
y1(t, 1) = ky2(t, 1), y2(t, 0) = 0,
y1(0, x) = y0
1(x), y2(0, x) = y0
2(x),
t∈(0,+∞), x ∈(0,1).(1.5)
Indeed, for this system we can always construct a smooth initial data (y0
1, y0
2) which is an eigenfunction of the
operator associated with (1.5) and whose corresponding eigenvalue σis a positive real number, which makes
NULL CONTROLLABILITY AND FINITE-TIME STABILIZATION IN MINIMAL TIME 3
the system (1.5) exponentially unstable. This can be done as follows. We take
y0
1(x) = 1
πσy0
2(x) + ∂y0
2
∂x (x),
(so that the second equation in (1.5) will always be satisfied) and
– If k < 1+1/π, then we take σ=π√1−θ2and y0
2(x) = sin(θπx), where θ∈(0,1) is any solution to the
equation √1−θ2+θcot(θπ) = k.
– If k= 1 + 1/π, then we take σ=πand y0
2(x) = πx.
– If k > 1 + 1/π, then we take σ=π√1 + θ2and y0
2(x) = 2 sinh(θπx), where θ > 0 is any solution to the
equation √1 + θ2+θcoth(θπ) = k.
The goal of this work is to establish a necessary and sufficient condition on the time Tfor the system (1.1)
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 φ1, φ2∈C1,1([0,1])
be the increasing functions defined for every x∈[0,1] by
φ1(x) = Zx
0
1
−λ1(ξ)dξ, φ2(x) = Zx
0
1
λ2(ξ)dξ. (1.6)
We then denote by
T1(Λ) = φ1(1) = Z1
0
1
−λ1(ξ)dξ, T2(Λ) = φ2(1) = Z1
0
1
λ2(ξ)dξ.
Finally, we set
Tmin (Λ) = max {T1(Λ), T2(Λ)}, Tmax (Λ) = T1(Λ) + T2(Λ).(1.7)
The naming of the notations in (1.7) will be explained in Remark 1.9 below.
1.2. 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 [29], the author
established the null controllability of general n×ncoupled linear hyperbolic systems of the form (1.1) in a
control time that is given by the sum of the two largest times from the states convecting in opposite directions
([29], Thm. 3.2). It was also observed that this time can be shorten in some cases ([29], Prop. 3.4), and the
problem to find the minimal control time for hyperbolic partial differential equations (PDEs) was then raised
([29], Rem. 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 [31], 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 [29] was recently obtained in [7] 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 ([7], Thms. 1.1 and 1.5). The same authors were then able to remove some of these restrictions in
[10]. For the present paper it is especially important to emphasize that the new time introduced in [7,10] 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
4L. HU AND G. OLIVE
only zero), was completely characterized in Theorem 1.12 of [19] 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 Theorem 3.2 of [22] that the time of [29] yields the
(local) exact controllability of such systems if the linearization of the boundary coupling matrix has a full row
rank in a neighborhood of the state zero (see also [23] concerning local null controllability). For homogeneous
quasilinear systems, a smaller control time was then obtained in Theorem 1.1 of [16].
Concerning now the stabilization property, the first works seem [13,26] for the exponential stabilization of
homogeneous quasilinear hyperbolic systems in a C1framework by using the method of characteristics. To the
best of our knowledge, the weakest sufficient condition using this technique can be found in Theorem 1.3, p. 173
of [24]. This condition was then improved in Theorem 2.3 of [4] in a H2framework 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 ([2], Sect. 5.6 and Rem. 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 [21] to design boundary feedback
laws stabilizing systems modeled by various PDEs and to the introduction of [6] 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 [11] 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 [17] and then to treat general n×nsystems in
[18,20]. However, the control time obtained in these works was larger than the one in [29] and it was only
shown in [1,5] that we can stabilize with the same time as the one of [29]. These works have recently been
generalized to time-dependent systems in [6]. Finally, let us also mention the two recent works [8,9] concerning
the finite-time stabilization of homogeneous quasilinear systems, with the same control time as in [7,10].
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 [11,19] 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 [11,19].
1.3. Main result and comments
The important quantity in the present work is the following:
Definition 1.3. For ε > 0 and a function f: (0, ε)−→ R, we denote by
`ε(f) = (sup Iε(f) if Iε(f)6=∅,
0 otherwise,
where Iε(f) = {`∈(0, ε)|f= 0 a.e. in (0, `)}.
The quantity `ε(f) is the length of the largest interval of the form (0, `) where the function fvanishes.
NULL CONTROLLABILITY AND FINITE-TIME STABILIZATION IN MINIMAL TIME 5
Example 1.4.
(E1) The simplest example of function fwith `ε(f) = `(`∈[0, ε]) is obviously the step function
f(x) = (0 if x≤`,
1 if x > `.
(E2) If f∈Ck([0, ε)) (k∈N) and satisfies f(k)(0) 6= 0, then `ε(f) = 0. In particular, if fhas an analytic
extension in a neighborhood of x= 0, then `ε(f) = 0.
(E3) An example of smooth function fwith `ε(f) = 0 but that does not satisfy the previous conditions is
f(x) =
0 if x≤0,
exp −1
xif x > 0.(1.8)
The main result of this article is the following complete characterization of the controllability properties of
the system (1.1):
Theorem 1.5. Let T > 0.
(i) If the system (1.1)is null controllable in time T, then necessarily
T≥max (Tmin (Λ) ,Z1
`xΛ(c)1
−λ1(ξ)+1
λ2(ξ)dξ),(1.9)
where xΛ∈(0,1) is the unique solution to φ1(xΛ) + φ2(xΛ) = T2(Λ) (=φ2(1)).
(ii) If the time Tsatisfies (1.9), then the system (1.1)is finite-time stabilizable with settling time T.
Note in particular that the system (1.1) is then null controllable in time Tif, and only if, it is finite-time
stabilizable with settling time T.
Remark 1.6. As we shall see in the proof below, the most difficult part of this result is the necessary condition,
that is the item (i).
Example 1.7. For csatisfying the properties in (E2) or given by the function in (E3), this result shows that
the time Tmax (Λ) cannot be improved. This is not trivial, especially when cis given by the function in (E3).
Remark 1.8. When λ1, λ2do not depend on space, the condition (1.9), in the situation Tmin (Λ) ≤T <
Tmax (Λ), simply becomes
c= 0 in 0,1−T
Tmax (Λ).
In particular, we see that we can possibly obtain any intermediate time between Tmin (Λ) and Tmax (Λ). Moreover,
note that the value Tmin (Λ) is reachable even when cis not identically equal to zero.
Remark 1.9. Theorem 1.5 says that the time on the right-hand side of (1.9) is the so-
called minimal control time, that is it is equal to Tinf(λ1, λ2, a, b, c, d), where Tinf (λ1, λ2, a, b, c, d) =
inf {T > 0 system (1.1) is null controllable in time T},(and the infimum being a minimum here). Let us men-
tion that it is sometimes found in the literature that the time Tmax (Λ) is called “the theoretical lower bound
for control time” or “the optimal time”. However, we see from our result that we may have
Tinf (λ1, λ2, a, b, c, d)< Tmax (Λ) .
6L. HU AND G. OLIVE
Therefore, the minimal control time can be strictly less than what is sometimes called the optimal time. This
brings some confusion to our point of view and this is why we prefer to avoid using the naming “optimal” for
Tmax (Λ). Instead, we carefully introduced a different naming and use the notations Tmin (Λ) and Tmax (Λ) since
we can easily check that
Tmin (Λ) = min {Tinf (λ1, λ2, a, b, c, d)a, b, c, d ∈L∞(0,1)},
Tmax (Λ) = max {Tinf (λ1, λ2, a, b, c, d)a, b, c, d ∈L∞(0,1)}.
Remark 1.10. Let us comment other possibilities for the boundary conditions at x= 0:
(i) When the boundary condition y2(t, 0) = 0 is replaced by (1.2) with boundary coupling “matrix” q6= 0,
the result ([11], Thm. 3.2) shows that the time Tmax (Λ) is the minimal control time (more precisely, it
is shown that the system (1.1) with such a boundary condition is equivalent to the same system with no
internal coupling matrix, for which Tmax (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 y2(t, 0) = 0 is
replaced by y2(t, 0) = v(t) with v∈L2
loc(0,+∞) a second control at our disposal, then the time Tmin (Λ)
is the minimal control time. The null controllability for T≥Tmin (Λ) can be shown using for instance
the well-known constructive method developed in Theorem 3.1 of [22]. On the other hand, the failure
of the null controllability for T < Tmin (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:
y1(t, 1) = py2(t, 1) + ru(t), y2(t, 0) = qy1(t, 0) + sv(t),
p, q, r, s ∈Rwith (r, s)6= (0,0).
Remark 1.11. As we shall see below during the proof, Theorem 1.5 remains true for more regular initial data
y0
1, y0
2∈L∞(0,1) (see in particular Rem. 3.4).
The rest of this article is organized as follows. In Section 2, we use the backstepping method to show that
our initial system (1.1) is equivalent to a canonical system from a controllability point of view. In Section 3we
use the Titchmarsh convolution theorem to completely characterize the minimal control time for this canonical
system. In Section 4we characterize this time in terms of the parameters of the initial system. Finally, in
Section 5we discuss possible extensions to systems with more than two equations.
2. Reduction to a canonical form
In this section, we perform some changes of unknown to transform our initial system (1.1) 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 Section 3.2 of [11] for more details on
the computations below.
First of all, we remove the diagonal terms in the system (1.1). Using the invertible spatial transformation
(seen as an operator from L2(0,1)2onto itself)
(˜y1(t, x) = e1(x)y1(t, x),
˜y2(t, x) = e2(x)y2(t, x),(2.1)
NULL CONTROLLABILITY AND FINITE-TIME STABILIZATION IN MINIMAL TIME 7
with
e1(x) = exp −Zx
0
a(ξ)
λ1(ξ)dξ, e2(x) = exp −Zx
0
d(ξ)
λ2(ξ)dξ,(2.2)
we easily see that the system (1.1) is null controllable in time T(resp. finite-time stabilizable with settling time
T) if, and only if, so is the system
∂˜y1
∂t (t, x) + λ1(x)∂˜y1
∂x (t, x) = ˜
b(x)˜y2(t, x),
∂˜y2
∂t (t, x) + λ2(x)∂˜y2
∂x (t, x) = ˜c(x)˜y1(t, x),
˜y1(t, 1) = ˜u(t),˜y2(t, 0) = 0,
˜y1(0, x) = ˜y0
1(x),˜y2(0, x) = ˜y0
2(x),
t∈(0,+∞), x ∈(0,1),(2.3)
where
˜
b(x) = b(x)e1(x)
e2(x),˜c(x) = c(x)e2(x)
e1(x).(2.4)
Let us now remove the coupling term on the first equation of (2.3) thanks to a second transformation. Set
T={(x, ξ)∈(0,1) ×(0,1) |x>ξ}.
Let k11, k12 , k21, k22 ∈L∞(T). Using the spatial transformation
ˆy1(t, x) = ˜y1(t, x)−Zx
0
(k11(x, ξ )˜y1(t, ξ) + k12 (x, ξ)˜y2(t, ξ)) dξ,
ˆy2(t, x) = ˜y2(t, x)−Zx
0
(k21(x, ξ )˜y1(t, ξ) + k22 (x, ξ)˜y2(t, ξ)) dξ,
(2.5)
which is invertible since it is a Volterra transformation of the second kind (see e.g. [15], Chap. 2, Thm. 5), we
see that the system (2.3) is null controllable in time T(resp. finite-time stabilizable with settling time T) if,
and only if, so is the system
∂ˆy1
∂t (t, x) + λ1(x)∂ˆy1
∂x (t, x) = 0,
∂ˆy2
∂t (t, x) + λ2(x)∂ˆy2
∂x (t, x) = g(x) ˆy1(t, 0),
ˆy1(t, 1) = ˆu(t),ˆy2(t, 0) = 0,
ˆy1(0, x) = ˆy0
1(x),ˆy2(0, x) = ˆy0
2(x),
t∈(0,+∞), x ∈(0,1),(2.6)
with ggiven by
g(x) = −k21(x, 0)λ1(0),(2.7)
8L. HU AND G. OLIVE
provided that the kernels k11, k12, k21 , k22 satisfy the so-called kernel equations:
λ1(x)∂k11
∂x (x, ξ ) + ∂k11
∂ξ (x, ξ )λ1(ξ) + k11(x, ξ)∂λ1
∂ξ (ξ) + k12 (x, ξ)˜c(ξ)=0,
λ1(x)∂k12
∂x (x, ξ ) + ∂k12
∂ξ (x, ξ )λ2(ξ) + k11(x, ξ)˜
b(ξ) + k12(x, ξ )∂λ2
∂ξ (ξ)=0,
k11(x, 0) = 0,
k12(x, x) = ˜
b(x)
λ1(x)−λ2(x),
(x, ξ)∈ T ,(2.8)
and
λ2(x)∂k21
∂x (x, ξ ) + ∂k21
∂ξ (x, ξ )λ1(ξ) + k21(x, ξ)∂λ1
∂ξ (ξ) + k22 (x, ξ)˜c(ξ)=0,
λ2(x)∂k22
∂x (x, ξ ) + ∂k22
∂ξ (x, ξ )λ2(ξ) + k21(x, ξ)˜
b(ξ) + k22(x, ξ )∂λ2
∂ξ (ξ)=0,
k21(x, x) = ˜c(x)
λ2(x)−λ1(x),
(x, ξ)∈ T .(2.9)
Note that (2.8) and (2.9) are not coupled.
From Theorem A.1 of [11], we know that the kernel equations (2.8)–(2.9) have a solution. More precisely, we
have the following result:
Theorem 2.1. For every k0∈L∞(0,1), there exists a unique solution (k11 , k12, k21 , k22)∈L∞(T)4to the
kernel equations (2.8)–(2.9)with
k22(x, 0) = k0(x), x ∈(0,1).
In the aforementioned reference this result is stated in a C0framework (assuming that a, b, c, d ∈C0([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 k21(x, 0), which defines g(see (2.7)),
or k11(1, ξ ), k12(1, ξ), that will appear shortly below in our feedback law (see (3.2)), etc. are also understood in
this sense. We refer for instance to the formula (4.6) below for the precise meaning of k21(x, 0).
3. Study of the canonical system
We call the system (2.6) the “control canonical form of the system (1.1)” or “canonical system” in short,
by analogy with [3,28] 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 (1.1)).
The goal of this section is to establish the following result:
Theorem 3.1. Let T > 0and g∈L∞(0,1).
(i) If the system (2.6)is null controllable in time T, then necessarily
T≥max (T1(Λ) + Z1
`1(g)
1
λ2(ξ)dξ, T2(Λ)).(3.1)
(ii) If the time Tsatisfies (3.1), then the system (2.6)is finite-time stable with settling time T.
Let us emphasize once again that the difficult point is the first item.
NULL CONTROLLABILITY AND FINITE-TIME STABILIZATION IN MINIMAL TIME 9
Remark 3.2. Since ˆu= 0 stabilizes the canonical system (2.6)by(ii) of Theorem 3.1, we see from the formula
(2.1) and (2.5) that our feedback for the system (1.1) is then
u(t) = Z1
0
k11(1, ξ )e1(ξ)
e1(1) y1(t, ξ) dξ+Z1
0
k12(1, ξ )e2(ξ)
e1(1) y2(t, ξ) dξ. (3.2)
Note that u∈C0([0,+∞)).
3.1. The characteristics
Before proving Theorem 3.1 we need to introduce the characteristic curves associated with the system (2.6)
and recall some useful properties.
First of all, it is convenient to extend λ1, λ2to functions of R(still denoted by the same) such that λ1, λ2∈
C0,1(R) and
λ1(x)≤ −ε < 0< ε < λ2(x),∀x∈R,(3.3)
for some ε > 0 small enough. Since all the results of the present paper depend only on the values of λ1, λ2in
[0,1], they do not depend on such an extension.
In what follows, i∈ {1,2}. Let χibe the flow associated with λi,i.e. for every (t, x)∈R×R, the function
s7−→ χi(s;t, x) is the solution to the ODE
∂χi
∂s (s;t, x) = λi(χi(s;t, x)),∀s∈R,
χi(t;t, x) = x.
(3.4)
The existence and uniqueness of a (global) solution to the ODE (3.4) follows from the (global) Cauchy-Lipschitz
theorem (see e.g. [14], Thm. II.1.1). The uniqueness also yields the important group property
χi(σ;s, χi(s;t, x)) = χi(σ;t, x),∀σ, s ∈R.(3.5)
By classical regularity results on ODEs (see e.g. [14], Thm. V.3.1), we have χi∈C1(R3) and
∂χi
∂t (s;t, x) = −λi(χi(s;t, x)),∂χi
∂x (s;t, x) = λi(χi(s;t, x))
λi(x).(3.6)
Let us now introduce the entry and exit times sin
i(t, x), sout
i(t, x)∈Rof the flow χi(·;t, x) inside the domain
[0,1], i.e. the respective unique solutions to
(χ1(sin
1(t, x); t, x)=1, χ1(sout
1(t, x); t, x)=0,
χ2(sin
2(t, x); t, x)=0, χ2(sout
2(t, x); t, x)=1.
Their existence and uniqueness are guaranteed by the condition (3.3). It readily follows from (3.5) and the
uniqueness of sin
ithat
sin
i(s, χi(s;t, x)) = sin
i(t, x),∀s∈R.(3.7)
10 L. HU AND G. OLIVE
By the implicit function theorem we have sin
i∈C1(R2) with (using (3.6))
∂sin
1
∂t (t, x)>0,∂sin
1
∂x (t, x)>0,
∂sin
2
∂t (t, x)>0,∂sin
2
∂x (t, x)<0.
(3.8)
Combined with the group property (3.7), this yields the following inverse formula for every s, t ∈R:
(s < sout
1(t, 1) ⇐⇒ sin
1(s, 0) < t,
s < sout
2(t, 0) ⇐⇒ sin
2(s, 1) < t. (3.9)
Finally, since λidoes not depend on time, we have an explicit formula for the inverse function θ7−→
χ−1
i(θ;t, x). Indeed, it solves
∂(χ−1
i)
∂θ (θ;t, x) = 1
∂χi
∂s χ−1
i(θ;t, x); t, x=1
λi(θ),∀θ∈R,
χ−1
i(x;t, x) = t,
which gives
χ−1
i(θ;t, x) = t+Zθ
x
1
λi(ξ)dξ. (3.10)
This also yields an explicit formula for sin
1, sin
2and sout
1, sout
2and, in particular,
T1(Λ) = sout
1(0,1), T2(Λ) = sout
2(0,0).
3.2. Proof of Theorem 3.1
First of all, the solution of the canonical system (2.6) is explicitly given by:
ˆy1(t, x) = (ˆy0
1(χ1(0; t, x)) if sin
1(t, x)<0,
ˆusin
1(t, x)if sin
1(t, x)>0,(3.11)
and
ˆy2(t, x) =
ˆy0
2(χ2(0; t, x)) + Zt
0
g(χ2(s;t, x)) ˆy1(s, 0) dsif sin
2(t, x)<0,
Zt
sin
2(t,x)
g(χ2(s;t, x)) ˆy1(s, 0) dsif sin
2(t, x)>0.
(3.12)
Next, we show a uniform lower bound for the control time:
Lemma 3.3. Let T > 0. If the system (2.6)is null controllable in time T, then necessarily
T≥Tmin (Λ) .
This result states that the control time cannot be better than the one of the case g= 0.
NULL CONTROLLABILITY AND FINITE-TIME STABILIZATION IN MINIMAL TIME 11
Proof. For i∈ {1,2}, let ωibe the open subset defined by
ωi=x∈(0,1) sin
i(T, x)<0.
From (3.9) and (3.8), we see that
T≥Ti(Λ) ⇐⇒ ωi=∅.(3.13)
Therefore, if T < T1(Λ), then we see from (3.11) that ˆy0
1can be chosen so that ˆy1(T , x)6= 0 for x∈ω1, whatever
ˆuis. On the other hand, if T < T2(Λ) and if the system (2.6) is null controllable in time T, then for every
ˆy0
2∈L2(0,1), there exists ˆu∈L2(0, T ) such that, for a.e. x∈ω2, we have
0 = ˆy0
2(χ2(0; T, x)) + ZT
0
g(χ2(s;T, x)) ˆy1(s, 0) ds. (3.14)
Since x∈ω27−→ χ2(0; T , x) is a C1diffeomorphism (its inverse is given by ξ7−→ χ2(T; 0, ξ) thanks to (3.5)),
this implies that the bounded linear operator K:L2(0, T )−→ L2(ω2) defined by
(Kh)(x) = −ZT
0
g(χ2(s;T, x)) h(s) ds,
is surjective. This is impossible since its range is clearly a subset of L∞(ω2), which is a proper subset of L2(ω2)
(alternatively, one could note that Kis compact and therefore it cannot be surjective over an infinite dimensional
space, see e.g. [27], Thm. 4.18 (b)).
Remark 3.4. The previous proof can be adapted to show that the condition T≥Tmin (Λ) is also necessary for
the null controllability in time Twith more regular initial data y0
1, y0
2∈L∞(0,1). Indeed, doing the change of
variable x=χ2(T;t, 0) in (3.14) we obtain the surjectivity of the operator ˜
K:L∞(0, T )−→ L∞(T−T2(Λ),0)
defined by ( ˜
Kh)(t)=(Kh)(χ2(T;t, 0)), but this is impossible since its range is in fact included in C0(T−
T2(Λ),0). To see this, we use that its kernel (t, s)7−→ g(χ2(s;t, 0)) is a convolution kernel (see Step 2in the
proof of Thm. 3.1 below) and the continuity of translations in L1.
The proof of the item (i) of Theorem 3.1 crucially relies on the Titchmarsh convolution theorem ([30],
Thm. VII) (see also [25], Chap. XV):
Theorem 3.5. Let α, β ∈L1(0,¯τ)(¯τ > 0). We have
Zτ
0
α(τ−σ)β(σ) dσ= 0,a.e. 0< τ < ¯τ , (3.15)
if, and only if,
`¯τ(α) + `¯τ(β)≥¯τ .
Remark 3.6. 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 αsatisfies
the condition in (E2) of Example 1.4 (by taking derivatives of (3.15) and using the injectivity of Volterra
transformations of the second kind). It does not seem trivial for functions of the form (1.8) though.
We are now ready to prove the main result of Section 3:
12 L. HU AND G. OLIVE
Proof of Theorem 3.1.
1) Thanks to Lemma 3.3, we can assume that T≥T1(Λ) and T≥T2(Λ). This means that sin
1(T, x)>0 and
sin
2(T, x)>0 for every x∈(0,1) (see (3.13) and (3.8)). It then follows from the explicit formula (3.11)
and (3.12) that ˆy1(T , ·) = 0 if, and only if,
ˆusin
1(T, x)= 0,0<x<1,(3.16)
and ˆy2(T, ·) = 0 if, and only if,
ZT
sin
2(T,x)
g(χ2(s;T, x)) ˆy1(s, 0) ds= 0,0<x<1.(3.17)
2) Let us focus on the second condition (3.17). Writing x=χ2(T;t, 0), which belongs to (0,1) for t∈
(sin
2(T, 1), T ) (recall in particular (3.9)), and using the group properties (3.5) and (3.7) with the identity
sin
2(t, 0) = t, we obtain that ˆy2(T , ·) = 0 if, and only if,
ZT
t
g(χ2(s;t, 0)) ˆy1(s, 0) ds= 0, sin
2(T, 1) < t < T. (3.18)
Now we use the fact that g(χ2(s;t, 0)) is actually a function of s−t. Indeed, by uniqueness to the solution
to the ODE (3.4), we see that the characteristics take the form
χi(s;t, x) = ˜χi(s−t;x),
where s7−→ ˜χi(s;x) is the unique solution to
∂˜χi
∂s (s;x) = λi( ˜χi(s;x)),∀s∈R,
˜χi(0; x) = x.
Using the change of variables σ=s−tand introducing
α(θ) = ˆy1(−θ+T , 0), β(θ) = g( ˜χ2(θ; 0)) ,0< θ < T −sin
2(T, 1),
we see that (3.18) is equivalent to (setting τ=T−t)
Zτ
0
α(τ−σ)β(σ) dσ= 0,0< τ < ¯τ , (3.19)
where
¯τ=T−sin
2(T, 1).
3) Applying the Titchmarsh convolution theorem (Thm. 3.5) we deduce that (3.19) is equivalent to
`¯τ(α) + `¯τ(β)≥¯τ .
NULL CONTROLLABILITY AND FINITE-TIME STABILIZATION IN MINIMAL TIME 13
From the explicit expression (3.11) and the inverse formula (3.9), we see that
α(θ) = (ˆy0
1(χ1(0; −θ+T, 0)) if θ > T −sout
1(0,1),
ˆusin
1(−θ+T, 0)if θ < T −sout
1(0,1).
Therefore, we can choose ˆy0
1so that
α(θ)6= 0,∀θ∈T−sout
1(0,1), T −sout
1(0,1) + ε,
for some 0 < ε < sout
1(0,1). This yields the bound
`¯τ(α)≤T−sout
1(0,1).
Consequently, we necessarily have
`¯τ(β)≥sout
1(0,1) −sin
2(T, 1).(3.20)
Since s7−→ ˜χ2(s; 0) is increasing with ˜χ2(0; 0) = 0, this is equivalent to
`1(g)≥˜χ2sout
1(0,1) −sin
2(T, 1); 0=χ2sout
1(0,1); sin
2(T, 1),0
=χ2sout
1(0,1); T, 1(by (3.5) with s=sin
2(T, 1)).
Since s7−→ χ2(s;T , 1) is increasing, this is also equivalent to
χ−1
2(`1(g) ; T, 1) ≥sout
1(0,1) = T1(Λ).
Using the explicit expression (3.10), we then obtain the desired condition T≥T1(Λ) + R1
`1(g)
1
λ2(ξ)dξ.
4) Conversely, assume that Tsatisfies this condition and T≥T2(Λ). Then, (3.20) holds by the previous
equivalences. Taking ˆu= 0, we see that α= 0 in (0, T −T1(Λ)), which yields
`¯τ(α) + `¯τ(β)≥T−T1(Λ) + sout
1(0,1) −sin
2(T, 1) = T−sin
2(T, 1) = ¯τ .
This implies (3.19) (here we only use the “easy part” of the Titchmarsh convolution theorem) and thus
ˆy2(T, ·) = 0. Finally, note that ˆu= 0 also obviously satisfies (3.16) and thus ˆy1(T , ·) = 0 as well.
Remark 3.7. 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)).
4. 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 (1.1) is null controllable in time T(resp. finite-time
stabilizable with settling time T) if, and only if, so is the canonical system (2.6) (with ggiven by (2.7)).
14 L. HU AND G. OLIVE
Therefore, thanks to Theorem 3.1 it suffices to show that
T≥T1(Λ) + Z1
`1(g)
1
λ2(ξ)dξ⇐⇒ T≥Z1
`xΛ(c)1
−λ1(ξ)+1
λ2(ξ)dξ,
which amounts to characterize `1(g) in terms of `xΛ(c) (we recall that xΛis defined in the statement of
Thm. 1.5). To this end, we are going to prove the identity
φ2(`1(g)) = φ1(`xΛ(c)) + φ2(`xΛ(c)),(4.1)
where we recall that φ1, φ2∈C1,1([0,1]) are defined in (1.6).
2) We recall that g(x) = −k21 (x, 0)λ1(0), where k21 is the solution in Tto
λ2(x)∂k21
∂x (x, ξ ) + ∂k21
∂ξ (x, ξ )λ1(ξ) + k21(x, ξ)∂λ1
∂ξ (ξ) + k22 (x, ξ)˜c(ξ)=0,
k21(x, x) = ˜c(x)
λ2(x)−λ1(x),
(4.2)
and where ˜cis defined in (2.4) and (2.2) (note that `ε(˜c) = `ε(c) for any ε∈(0,1]). Let s7−→ χ(s;x) be
the associated characteristic passing through (x, ξ) = (x, 0), i.e. the solution to the ODE
∂χ
∂s (s;x) = λ1(χ(s;x))
λ2(s),∀s∈R,
χ(x;x)=0,
(4.3)
(we recall that λ1, λ2have been extended to Rin Sect. 3.1). We have χ∈C1(R2) by classical regularity
results on ODEs with
∂χ
∂x (s;x) = −λ1(χ(s;x))
λ2(x)>0.
Since f:s7−→ s−χ(s;x) is continuous and increasing with lims→∓∞ f(s) = ∓∞, there exists a unique
solution sin(x)∈Rto
χsin(x); x=sin (x).
Besides, for every x∈(0,1), we have 0 < sin(x)< x and
(s, χ(s;x)) ∈ T ,∀s∈(sin(x), x).
By the implicit function theorem we have sin ∈C1(R) with, for every x∈R,
(sin)0(x) =
∂χ
∂x (sin (x); x)
1−∂χ
∂s (sin (x); x)>0.(4.4)
In particular, the inverse function (sin)−1: [0, sin(1)] −→ [0,1] exists. We are going to show that
`sin(1) (˜c) = sin(`1(g)).(4.5)
NULL CONTROLLABILITY AND FINITE-TIME STABILIZATION IN MINIMAL TIME 15
Along the characteristics, the solution to (4.2) satisfies, for s∈sin(x), x,
d
dsk21(s, χ(s;x)) = −∂λ1
∂ξ (χ(s;x))
λ2(s)k21(s, χ(s;x)) + −k22(s, χ(s;x))
λ2(s)˜c(χ(s;x)),
k21 sin(x), sin (x)=˜c(sin (x))
λ2(sin(x)) −λ1(sin (x)) .
Consequently,
k21(x, 0) = r(x)˜c(sin(x)) + Zx
sin(x)
h(x, σ)˜c(χ(σ;x)) dσ, (4.6)
with
r(x) = exp Zx
sin(x)
−∂λ1
∂ξ (χ(s;x))
λ2(s)ds!1
λ2(sin(x)) −λ1(sin (x)) ,
and
h(x, σ) = exp Zx
σ
−∂λ1
∂ξ (χ(s;x))
λ2(s)ds!−k22(σ, χ(σ;x))
λ2(σ).
Using the change of variable θ= (sin )−1(χ(σ;x)), we obtain
1
r(x)k21(x, 0) = ˜c(sin(x)) + Zx
0
˜
h(x, θ)˜csin(θ)dθ,
with kernel
˜
h(x, θ) = 1
r(x)hx, χ−1(sin(θ); x)(sin )0(θ)
∂χ
∂s (χ−1(sin (θ); x); x).
We can check that ˜
h∈L∞(T) (recall (4.4)). It follows from the injectivity of Volterra transformations of
the second kind that
`1(g) = `1˜c◦sin ,
which is equivalent to (4.5) since sin is increasing with sin(0) = 0.
3) To conclude the proof, it remains to observe that the solution to the ODE (4.3) satisfies
φ1(χ(s;x)) = φ2(x)−φ2(s),
for every x∈[0,1] and s∈[sin(x), x]. Taking x= 1 and s=sin(1), we see that sin(1) = xΛ(by uniqueness
of the solution to the equation φ1(xΛ) + φ2(xΛ) = φ2(1)). Taking then x=`1(g) and s=sin (`1(g)) =
`xΛ(˜c) (recall (4.5)), we obtain the desired identity (4.1).
Remark 4.1. In the proof of Theorem 1.5, we have not used the apparent freedom for the boundary data of
k22 provided by Theorem 2.1.
16 L. HU AND G. OLIVE
5. Extensions and open problems
The results of this paper can be partially extended to systems of more than 2 equations. More precisely, we
can consider the following n×nsystems (n≥2):
∂y1
∂t (t, x) + λ1(x)∂y1
∂x (t, x) = a(x)y1(t, x) + B(x)y+(t, x),
∂y+
∂t (t, x)+Λ+(x)∂y+
∂x (t, x) = C(x)y1(t, x) + D(x)y+(t, x),
y1(t, 1) = u(t), y+(t, 0) = Qy1(t, 0),
y1(0, x) = y0
1(x), y+(0, x) = y0
+(x),
t∈(0,+∞), x ∈(0,1).(5.1)
In (5.1), (y1(t, ·), y+(t, ·)) ∈R×Rn−1is the state at time t, (y0
1, y0
+) is the initial data and u(t)∈Ris the
control at time t. We assume that we have one negative speed λ1∈C0,1([0,1]) and n−1 positive speeds
λ2, . . . , λn∈C0,1([0,1]) such that:
λ1(x)<0< λ2(x)<··· < λn(x),∀x∈[0,1],(5.2)
and we use the notation Λ+= diag(λ2, . . . , λn). Finally, a∈L∞(0,1), B∈L∞(0,1)1×(n−1),C∈L∞(0,1)n−1,
D∈L∞(0,1)(n−1)×(n−1) couple the equations of the system inside the domain and the constant matrix Q∈
Rn−1couples the equations of the system on the boundary x= 0.
Let us now introduce the times defined by
T1(Λ) = Z1
0
1
−λ1(ξ)dξ, Ti(Λ) = Z1
0
1
λi(ξ)dξ, ∀i∈ {2, . . . , n}.
Note that Tn(Λ) < . . . < T2(Λ) by (5.2).
It was established in [12] and Lemma 3.1 of [18] that the system (5.1) is finite-time stabilizable with setting
time Tif T≥Tmax (Λ), where Tmax (Λ) is still given by (1.7).
Using the backstepping method (see e.g. [20], Sect. 2.2), it can be shown as before that the system (5.1) is
null controllable in time T(resp. finite-time stabilizable with settling time T) if, and only if, so is the system
∂ˆy1
∂t (t, x) + λ1(x)∂ˆy1
∂x (t, x)=0,
∂ˆy+
∂t (t, x)+Λ+(x)∂ˆy+
∂x (t, x) = G(x) ˆy1(t, 0),
ˆy1(t, 1) = ˆu(t),ˆy+(t, 0) = Qˆy1(t, 0),
ˆy1(0, x) = ˆy0
1(x),ˆy+(0, x) = ˆy0
+(x),
t∈(0,+∞), x ∈(0,1),(5.3)
for some G∈L∞(0,1)n−1depending on all the parameters λ1,Λ+,a, B, C, D and Q.
By mimicking the proof of Theorem 3.1, we can obtain the following result:
Theorem 5.1. Let T > 0.
(i) If the system (5.3)is null controllable in time T, then necessarily
T≥max T1(Λ) + max
i∈{2,...,n}T(λi, gi−1, qi−1), T2(Λ),(5.4)
NULL CONTROLLABILITY AND FINITE-TIME STABILIZATION IN MINIMAL TIME 17
where
T(λi, gi−1, qi−1) =
Z1
`1(gi−1)
1
λi(ξ)dξif qi−1= 0,
Ti(Λ) if qi−16= 0.
(ii) If the time Tsatisfies (5.4), then the system (5.3)is finite-time stable with settling time T.
However, we are unable so far to deduce from this result some explicit condition for the initial system (5.1).
The main technical problem is that Gis heavily coupled on the parameters λ1,Λ+,a, B, C, D and Q(see e.g.
[20], Sect. 2.2). We leave it as an open problem that could be investigated in future works.
Acknowledgements. The first author would like to thank Institute of Mathematics in Jagiellonian University for its
hospitality. This work was initiated while he was visiting there. This project was supported by National Natural Sci-
ence Foundation of China (Nos. 12071258 and 12122110), the Young Scholars Program of Shandong University (No.
2016WLJH52) and National Science Centre, Poland UMO-2020/39/D/ST1/01136. For the purpose of Open Access, the
authors have applied a CC-BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from
this submission.
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