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Global stabilization of a Korteweg-de Vries equation
with saturating distributed control
Swann Marx, Eduardo Cerpa, Christophe Prieur, Vincent Andrieu
To cite this version:
Swann Marx, Eduardo Cerpa, Christophe Prieur, Vincent Andrieu. Global stabilization of a Korteweg-
de Vries equation with saturating distributed control. SIAM Journal on Control and Optimization,
Society for Industrial and Applied Mathematics, 2017, 55 (3), pp.1452-1480. �10.1137/16M1061837�.
�hal-01367622v3�
SIAM J. CONTROL OPTIM.
c
2017 Society for Industrial and Applied Mathematics
Vol. 55, No. 3, pp. 1452–1480
GLOBAL STABILIZATION OF A KORTEWEG–DE VRIES
EQUATION WITH SATURATING DISTRIBUTED CONTROL
∗
SWANN MARX
†
, EDUARDO CERPA
‡
, CHRISTOPHE PRIEUR
†
,
AND VINCENT ANDRIEU
§
Abstract. This article deals with the design of saturated controls in the context of partial
differential equations. It focuses on a Korteweg–de Vries equation, which is a nonlinear mathematical
model of waves on shallow water surfaces. Two different types of saturated controls are considered.
The well-posedness is proven applying a Banach fixed-point theorem, using some estimates of this
equation and some properties of the saturation function. The proof of the asymptotic stability of the
closed-loop system is separated in two cases: (i) when the control acts on all the domain, a Lyapunov
function together with a sector condition describing the saturating input is used to conclude on the
stability; (ii) when the control is localized, we argue by contradiction. Some numerical simulations
illustrate the stability of the closed-loop nonlinear partial differential equation.
Key words. Korteweg–de Vries equation, stabilization, distributed control, saturating control,
nonlinear system
AMS subject classifications. 93C20, 93D15, 35Q53
DOI. 10.1137/16M1061837
1. Introduction. In recent decades, a great effort has been made to take into
account input saturations in control designs (see, e.g., [39], [15], or more recently [17]).
In most applications, actuators are limited due to some physical constraints and the
control input has to be bounded. Neglecting the amplitude actuator limitation can
be source of undesirable and catastrophic behaviors for the closed-loop system. The
standard method to analyze the stability with such nonlinear controls follows a two-
step design. First the design is carried out without taking into account the saturation.
In the second step, a nonlinear analysis of the closed-loop system is made when adding
the saturation. In this way, we often get local stabilization results. Tackling this
particular nonlinearity in the case of finite dimensional systems is already a difficult
problem. However, nowadays, numerous techniques are available (see, e.g., [39], [41],
[37]) and such systems can be analyzed with an appropriate Lyapunov function and
a sector condition of the saturation map, as introduced in [39].
In the literature, there are few papers studying this topic in the infinite dimen-
sional case. Among them, we can cite [18], [29], where a wave equation equipped with
a saturated distributed actuator is studied, and [12], where a coupled PDE/ODE sys-
tem modeling a switched power converter with a transmission line is considered. Due
to some restrictions on the system, a saturated feedback has to be designed in the
∗
Received by the editors February 16, 2016; accepted for publication (in revised form)
January 26, 2017; published electronically May 9, 2017.
http://www.siam.org/journals/sicon/55-3/M106183.html
Funding: This work has been partially supported by Fondecyt 1140741, MathAmsud COSIP,
and Basal Project FB0008 AC3E.
†
Universit´e Grenoble Alpes, CNRS, GIPSA-lab, F-38000 Grenoble, France (marx.swann@
gmail.com, christophe.prieur@gipsa-lab.fr).
‡
Departamento de Matem´atica, Universidad T´ecnica Federico Santa Mar´ıa, Avda. Espa˜na 1680,
Valpara´ıso, Chile (eduardo.cerpa@usm.cl).
§
Universit´e Lyon 1 CNRS UMR 5007 LAGEP, France, and Fachbereich C, Mathematik und
Naturwissenschaften, Bergische Universit¨at Wuppertal, Gaussstrasse 20, 42097 Wuppertal, Germany
(vincent.andrieu@gmail.com).
1452
KdV EQUATION WITH SATURATING CONTROL 1453
latter paper. There exist also some papers using the nonlinear semigroup theory and
focusing on abstract systems [20], [34], [36].
Let us note that in [36], [34], and [20], the study of the a priori bounded controller
is tackled using abstract nonlinear theory. To be more specific, for bounded [36], [34]
and unbounded [34] control operators, some conditions are derived to deduce, from
the asymptotic stability of an infinite dimensional linear system in abstract form, the
asymptotic stability when closing the loop with saturating controller. These articles
use the nonlinear semigroup theory (see, e.g., [24] or [1]).
The Korteweg–de Vries (KdV) equation
y
t
+ y
x
+ y
xxx
+ yy
x
= 0(1.1)
is a mathematical model of waves on shallow water surfaces. Its controllability and
stabilizability properties have been deeply studied with no constraints on the control,
as reviewed in [3, 9, 32]. In this article, we focus on the following controlled KdV
equation:
y
t
+ y
x
+ y
xxx
+ yy
x
+ f = 0, (t, x) ∈ [0, +∞) × [0, L],
y(t, 0) = y(t, L) = y
x
(t, L) = 0, t ∈ [0, +∞),
y(0, x) = y
0
(x), x ∈ [0, L],
(1.2)
where y stands for the state and f for the control. As studied in [30], if f = 0 and
L ∈
(
2π
r
k
2
+ kl + l
2
3
,
k, l ∈ N
∗
)
,(1.3)
then there exist solutions of the linearized version of (1.2), written as follows,
y
t
+ y
x
+ y
xxx
= 0,
y(t, 0) = y(t, L) = y
x
(t, L) = 0,
y(0, x) = y
0
(x),
(1.4)
for which the L
2
(0, L)-energy does not decay to zero. For instance, if L = 2π and
y
0
= 1 − cos(x) for all x ∈ [0, L], then y(t, x) = 1 − cos(x) is a stationary solution
of (1.4) conserving the energy for any time t. Note, however, that if L = 2π and
f = 0, the origin of (1.2) is locally asymptotically stable as stated in [8]. It is worth
mentioning that there is no hope to obtain global stability, as established in [13],
where an equilibrium with arbitrary large amplitude is built.
In the literature there are some methods stabilizing the KdV equation (1.2)
with boundary [5], [4], [21] or distributed controls [25], [26]. Here we focus on
the distributed control case. In fact, as proven in [25], [26], the feedback control
f(t, x) = a(x)y(t, x), where a is a positive function whose support is a nonempty
open subset of (0, L), makes the origin an exponentially stable solution.
In [22], in which a linear KdV equation with a saturated distributed control is
considered, we use a nonlinear semigroup theory. In the case of the present paper,
since the term yy
x
is not globally Lipschitz, such a theory is harder to use. Thus,
we aim here at studying a particular nonlinear partial differential equation without
seeing it as an abstract control system and without using the nonlinear semigroup
theory. In this paper, we introduce two different types of saturation borrowed from
[29], [22] and [36]. In finite dimension, a way to describe this constraint is to use
the classical saturation function (see [39] for a good introduction on saturated control
problems) defined by
1454 S. MARX, E. CERPA, C. PRIEUR, AND V. ANDRIEU
sat(s) =
−u
0
if s ≤ −u
0
,
s if − u
0
≤ s ≤ u
0
,
u
0
if s ≥ u
0
(1.5)
for some u
0
> 0. As in [29] and [22] we use its extension to infinite dimension for the
feedback law
f(t, x) = sat
loc
(ay)(t, x),(1.6)
where, for all sufficiently smooth function s and for all x ∈ [0, L], sat
loc
is defined as
follows:
sat
loc
(s)(x) = sat(s(x)).(1.7)
Such a saturation is called localized since its image depends only on the value of
s at x.
In this work, we also use a saturation operator in L
2
(0, L), denoted by sat
2
, and
defined by
sat
2
(s)(x) =
(
s(x) if ksk
L
2
(0,L)
≤ u
0
,
s(x)u
0
ksk
L
2
(0,L)
if ksk
L
2
(0,L)
≥ u
0
.
(1.8)
Note that this definition is borrowed from [36] (see also [34] or [18]), where the sat-
uration is obtained from the norm of the Hilbert space of the control operator. This
saturation seems more natural when studying the stability with respect to an energy,
but it is less relevant than sat
loc
for applications. Figure 1 illustrates how different
these saturations are.
Our first main result states that using either the localized saturation (1.7) or using
the L
2
saturation map (1.8) the KdV equation (1.2) in closed loop with a saturated
control is well-posed (see Theorem 2.1 below for a precise statement). Our second main
0 0.5 1 1.5 2 2.5 3
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x
cos(x)
sat
2
(cos)(x)
sat
loc
(cos)(x)
Fig. 1. x ∈ [0, π]. Red: sat
2
(cos)(x) and u
0
= 0.5. Blue: sat
loc
(cos)(x) and u
0
= 0.5. Dotted
lines: cos(x).
KdV EQUATION WITH SATURATING CONTROL 1455
result states that the origin of the KdV equation (1.2) in closed loop with a saturated
control is globally asymptotically stable. Moreover, in the case where the control
acts on all the domain and where the control is saturated with (1.8), if the initial
conditions are bounded in L
2
norm, then the solution converges exponentially with a
decay rate that can be estimated (see Theorem 2.2 below for a precise statement).
This article is organized as follows. In section 2, we present our main results
about the well-posedness and the stability of (1.2) in presence of saturating control.
Sections 3 and 4 are devoted to proving these results by using the Banach fixed-
point theorem, Lyapunov techniques, and a contradiction argument. In section 5,
we provide a numerical scheme for the nonlinear equation and give some simulations
of the equation looped by a saturated feedback. Section 6 collects some concluding
remarks and possible further research lines.
Notation. A function α is said to be a class K
∞
function if α is nonnegative,
increasing, vanishing at 0, and such that lim
s→+∞
α(s) = +∞.
2. Main results. We first give an analysis of our system (1.2) when there is no
constraint on the control f. To do that, letting f (t, x) := ay(t, x) in (1.2), where a is
a nonnegative function satisfying
0 < a
0
≤ a(x) ≤ a
1
∀x ∈ ω,
where ω is a nonempty open subset of (0, L),
(2.1)
then, following [31], we get that the origin of (1.2) is globally asymptotically stabilized.
If ω = [0, L], then any solution to (1.2) satisfies
1
2
d
dt
Z
L
0
|y(t, x)|
2
dx = −
1
2
|y
x
(t, 0)|
2
−
Z
L
0
a(x)|y(t, x)|
2
dx ≤ −a
0
Z
L
0
|y(t, x)|
2
dx,
(2.2)
which ensures an exponential stability with respect to the L
2
(0, L)-norm. Note that
the decay rate can be selected as large as we want by tuning the parameter a
0
. Such
a result is referred to as a rapid stabilization result.
Let us consider the KdV equation controlled by a saturated distributed control
as follows:
y
t
+ y
x
+ y
xxx
+ yy
x
+ sat(ay) = 0,
y(t, 0) = y(t, L) = y
x
(t, L) = 0,
y(0, x) = y
0
(x),
(2.3)
where sat = sat
2
or sat
loc
. Since these two operators have properties in common,
we will use the notation sat throughout the paper. However, in some cases, we get
different results. Therefore, the use of a particular saturation is specified when it is
necessary.
Let us state the main results of this paper.
Theorem 2.1 (well-posedness). For any initial condition y
0
∈ L
2
(0, L), there
exists a unique mild solution y ∈ C([0, T ]; L
2
(0, L)) ∩ L
2
(0, T ; H
1
(0, L)) to (2.3).
Theorem 2.2 (global asymptotic stability). Given a nonempty open subset ω and
the positive values a
0
and u
0
, there exist a positive value µ
?
and a class K
∞
function
α
0
: R
≥0
→ R
≥0
such that for any y
0
∈ L
2
(0, L), the mild solution y of (2.3) satisfies
ky(t, .)k
L
2
(0,L)
≤ α
0
(ky
0
k
L
2
(0,L)
)e
−µ
?
t
∀t ≥ 0.(2.4)
Moreover, in the case where ω = [0, L] and sat = sat
2
we can estimate locally
the decay rate of the solution. In other words, for all r > 0, for any initial condition
y
0
∈ L
2
(0, L) such that ky
0
k
L
2
(0,L)
≤ r, the mild solution y to (2.3) satisfies
![Fig. 3. Solution y(t, x) with the control f = sat2(a0y), where ω = [0, L], u0 = 0.5.](/figures/fig-3-solution-y-t-x-with-the-control-f-sat2-a0y-where-o-0-l-1exxligp.webp)
![Fig. 2. Solution yw(t, x) with the control f = a0yw, where ω = [0, L].](/figures/fig-2-solution-yw-t-x-with-the-control-f-a0yw-where-o-0-l-1dameycq.webp)
![Fig. 4. Control f = sat2(a0y)(t, x), where ω = [0, L], u0 = 0.5.](/figures/fig-4-control-f-sat2-a0y-t-x-where-o-0-l-u0-0-5-2hqdtxuy.webp)
![Fig. 8. Control f = satloc(ay)(t, x), where ω = [ 1 3 L, 2 3 L ] , u0 = 0.5.](/figures/fig-8-control-f-satloc-ay-t-x-where-o-1-3-l-2-3-l-u0-0-5-1w0v8j18.webp)
![Fig. 9. Blue: Time evolution of the energy function ‖y‖2 L2(0,L) with a saturation u0 = 0.5, a0 = 1, and ω = [ 1 3 L, 2 3 L ] . Dotted line: Time evolution of the solution without saturation yw with a0 = 1 and ω = [ 1 3 L, 2 3 L ] .](/figures/fig-9-blue-time-evolution-of-the-energy-function-y-2-l2-0-l-j3b6hv8o.webp)