Global stabilization of a Korteweg-de Vries equation with saturating distributed control††thanks: This work has been partially supported by Fondecyt 1140741, MathAmsud COSIP, and Basal Project FB0008 AC3E.
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
In recent decades, a great effort has been made to take into account input saturations in control designs (see e.g ,  or more recently ). 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 steps design. First the design is carried out without taking into account the saturation. In a 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 .
In the literature, there are few papers studying this topic in the infinite dimensional case. Among them, we can cite , , where a wave equation equipped with a saturated distributed actuator is studied, and , where a coupled PDE/ODE system 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 latter paper. There exist also some papers using the nonlinear semigroup theory and focusing on abstract systems (,,).
Let us note that in ,  and , the study of a priori bounded controller is tackled using abstract nonlinear theory. To be more specific, for bounded (,) and unbounded () 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.  or ).
The Korteweg-de Vries equation (KdV for short)
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
where stands for the state and for the control. As studied in , if and
then, there exist solutions of the linearized version of (LABEL:nlkdv), written as follows,
for which the -energy does not decay to zero. For instance, if and for all , then is a stationary solution of (LABEL:lkdv) conserving the energy for any time . Note however that, if and , the origin of (LABEL:nlkdv) is locally asymptotically stable as stated in . It is worth to mention that there is no hope to obtain global stability, as established in  where an equilibrium with arbitrary large amplitude is built.
In the literature there are some methods stabilizing the KdV equation (LABEL:nlkdv) 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 , where is a positive function whose support is a nonempty open subset of , makes the origin an exponentially stable solution.
In , in which it is considered a linear Korteweg-de Vries equation with a saturated distributed control, we use a nonlinear semigroup theory. In the case of the present paper, since the term 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 . In finite dimension, a way to describe this constraint is to use the classical saturation function (see  for a good introduction on saturated control problems) defined by
where, for all sufficiently smooth function and for all , is defined as follows
Such a saturation is called localized since its image depends only on the value of at .
In this work, we also use a saturation operator in , denoted by , and defined by
Note that this definition is borrowed from  (see also  or ) where the saturation 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 for applications. Figure LABEL:diff-sat illustrates how different these saturations are.
Our first main result states that using either the localized saturation (LABEL:sat-linf) or using the saturation map (LABEL:function-saturation) the KdV equation (LABEL:nlkdv) in closed loop with a saturated control is well-posed (see Theorem LABEL:nl-theorem-wp below for a precise statement). Our second main result states that the origin of the KdV equation (LABEL:nlkdv) 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 (LABEL:function-saturation), if the initial conditions are bounded in norm, then the solution converges exponentially with a decay rate that can be estimated (see Theorem LABEL:glob_as_stab below for a precise statement).
This article is organized as follows. In Section LABEL:sec_mainresults, we present our main results about the well posedness and the stability of (LABEL:nlkdv) in presence of saturating control. Sections LABEL:sec_wp and LABEL:sec_stab are devoted to prove these results by using the Banach fixed-point theorem, Lyapunov techniques and a contradiction argument. In Section LABEL:sec_simu, we provide a numerical scheme for the nonlinear equation and give some simulations of the equation looped by a saturated feedback. Section LABEL:sec_conc collects some concluding remarks and possible further research lines.
Notation: A function is said to be a class function if is nonnegative, increasing, vanishing at and such that .
2 Main results
We first give an analysis of our system (LABEL:nlkdv) when there is no constraint on the control . To do that, letting in (LABEL:nlkdv), where is a nonnegative function satisfying
then, following , we get that the origin of (LABEL:nlkdv) is globally asymptotically stabilized. If , then any solution to (LABEL:nlkdv) satisfies
which ensures an exponential stability with respect to the -norm. Note that the decay rate can be selected as large as we want by tuning the parameter . Such a result is refered to as a rapid stabilization result.
Let us consider the KdV equation controlled by a saturated distributed control as follows
where or . Since these two operators have properties in common, we will use the notation all along 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.
(Well posedness) For any initial condition , there exists a unique mild solution to (LABEL:nlkdv_sat).
(Global asymptotic stability) Given a nonempty open subset and the positive values and , there exist a positive value and a class function such that for any , the mild solution of (LABEL:nlkdv_sat) satisfies
Moreover, in the case where and we can estimate locally the decay rate of the solution. In other words, for all , for any initial condition such that , the mild solution to (LABEL:nlkdv_sat) satisfies
where is defined as follows
The remaining part of this paper is devoted to the proof of these results (see Sections LABEL:sec_wp and LABEL:sec_stab, respectively) and to numerical simulations to illustrate Theorem LABEL:glob_as_stab (see Section LABEL:sec_simu).
3.1 Linear system
Before proving the well-posedness of (LABEL:nlkdv_sat), let us recall some useful results on the linear system (LABEL:lkdv). To do that, consider the operator defined by
It can be proved that this operator and its adjoint operator defined by
are both dissipative, which means that, for all , and for all , .
For any initial condition , there exists a unique strong solution to (LABEL:lkdv).
For any initial condition , there exists a unique mild solution to (LABEL:lkdv). Moreover, there exists such that the solution to (LABEL:lkdv) satisfies
and the extra trace regularity
To ease the reading, let us denote the following Banach space, for all ,
endowed with the norm
Before studying the well-posedness of (LABEL:nlkdv_sat), we need a well-posedness result with a right-hand side. Given , let us consider the unique solution 111With , the existence and the unicity of are insured since generates a -semigroup of contractions. It follows from the semigroup theory the existence and the unicity of when (see ). to the following nonhomogeneous problem:
Note that we need the following property on the saturation function, which will allow us to state that this type of nonlinearity belongs to the space .
For all , we have
Thus, we get
which concludes the proof of Lemma LABEL:lipschitz-satl2.
We have the following proposition borrowed from [30, Proposition 4.1].
Proposition 3.3 ()
If , then and the map is continuous
We have also the following proposition.
Assume satisfies (LABEL:gain-control). If , then and the map is continuous;
Proof: Let . We have, using Lemma LABEL:lipschitz-satl2 and Hölder inequality
Plugging in (LABEL:regularity-sat-l1) yields and (LABEL:regularity-sat-l1) implies the continuity of the map . It concludes the proof of Proposition LABEL:proposition-reg.
Let us study the non-homogenenous linear KdV equation with . For any , it is described with the following equation
It can be rewritten as follows
By standard semigroup theory (see ), for any positive value and any function , the solution to (LABEL:kdv-zero) can be expressed as follows
Finally, we have the following result borrowed from [31, Lemma 2.2]
Proposition 3.5 ()
There exists a positive value such that for any positive value and any function the solution to (LABEL:kdv-zero) satisfies the following inequality,
3.2 Proof of Theorem LABEL:nl-theorem-wp
Let us begin this section with a technical lemma.
() For any and ,
The following is a local well-posedness result.
(Local well-posedness) Let be given. For any , there exists depending on such that (LABEL:nlkdv_sat) admits a unique mild solution .
Solution to (LABEL:kdv-fixed-point) can be written in its integral form
For given , let and be positive constants to be chosen later. We define
which is a closed, convex and bounded subset of . Consequently, is a complete metric space in the topology induced from . We define a map on by, for all
We aim at proving that there exists a unique fixed point to this operator. It follows from Proposition LABEL:prop-w(t-tau), Lemma LABEL:zhang-regularity and the linear estimates given in Theorem LABEL:lkdv-wp that for every , there exists a positive value such that it holds
where the first line has been obtained with the linear estimates given in Theorem LABEL:lkdv-wp and the estimate given in Proposition LABEL:prop-w(t-tau) and the second line with Lemma LABEL:zhang-regularity and Proposition LABEL:proposition-reg. We choose and such that
in order to obtain
Thus, with such and , maps to . Moreover, one can prove with Proposition LABEL:prop-w(t-tau), Lemma LABEL:zhang-regularity and the linear estimates given in Theorem LABEL:lkdv-wpthat
The existence of mild solutions to the Cauchy problem (LABEL:nlkdv_sat) follows by using the Banach fixed-point theorem [1, Theorem 5.7].
Before proving the global well-posedness, we need the following lemma inspired by  and  which implies that if there exists a solution for some then the solution is unique.
Let and satisfying (LABEL:gain-control). There exists such that for every for which there exist mild solutions and of
these solutions satisfy
We follow the strategy of  and . Let us assume that for given , there exist and two different solutions and to (LABEL:kdv-y) and (LABEL:kdv-z), respectively, defined on . Then defined on is a mild solution of
Integrating by parts in
and using the boundary conditions of (LABEL:kdv-delta), we readily get
By the boundary conditions and the continuous Sobolev embedding , there exists such that
Moreover, since is globally Lipschitz with constant 3 (as stated in Lemma LABEL:lipschitz-satl2) and for all , , we use a Hölder inequality to get
Note that, from [10, Lemma 16], for every with , and every ,
Thus, from (LABEL:lemma16-cc) there exists such that
Moreover, with the boundary conditions of and the Sobolev embedding , there exists such that
Hence, using the boundary conditions of and (LABEL:lemma16-cc) with , there exists such that
Finally, there exists such that
Using the Grönwal Lemma, the last inequality and the initial conditions of , we get, for every ,
and thus, we obtain the existence of such that
Similarly, integrating by parts in
we get, using the boundary conditions of ,
Thanks to the continuous Sobolev embedding , (LABEL:lipschitz-sat) and (LABEL:unicity-linf), there exists such that
Thus applying the Grönwall Lemma, we get the existence of such that