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Wave Equation With Cone-Bounded Control Laws
Christophe Prieur, Sophie Tarbouriech, João M Gomes da Silva
To cite this version:
Christophe Prieur, Sophie Tarbouriech, João M Gomes da Silva. Wave Equation With Cone-Bounded
Control Laws. IEEE Transactions on Automatic Control, Institute of Electrical and Electronics En-
gineers, 2016, 61 (11), pp.3452-3463. �10.1109/TAC.2016.2519759�. �hal-01448483�
1
Wave equation with cone-bounded control laws
Christophe Prieur, Sophie Tarbouriech, Jo
˜
ao M. Gomes da Silva Jr
Abstract—This paper deals with a wave equation with
a one-dimensional space variable, which describes the
dynamics of string deflection. Two kinds of control are
considered: a distributed action and a boundary control.
It is supposed that the control signal is subject to a
cone-bounded nonlinearity. This kind of feedback laws
includes (but is not restricted to) saturating inputs. By
closing the loop with such a nonlinear control, it is thus
obtained a nonlinear partial differential equation, which is
the generalization of the classical 1D wave equation. The
well-posedness is proven by using nonlinear semigroups
techniques. Considering a sector condition to tackle the
control nonlinearity and assuming that a tuning parameter
has a suitable sign, the asymptotic stability of the closed-
loop system is proven by Lyapunov techniques. Some
numerical simulations illustrate the asymptotic stability of
the closed-loop nonlinear partial differential equations.
I. INTRODUCTION
The general problem in this paper is the study of the
wave in a one-dimensional media, as considered e.g.
when modeling the dynamics of an elastic slope vibrat-
ing around its rest position. To be more specific, it is
considered the wave equation describing the dynamics of
the deformation denoted by z(x, t). The control is either
defined by an external force f(x, t), or by a boundary
action g(t), where the force and the deformation may
depend on the space and the time variables. A scheme
of the considered problem is depicted in Figure 1.
Depending on the control action, two classes of partial
differential equations (PDEs) are obtained. In the pres-
ence of an external distributed force f when the slope is
attached at both extremities, the dynamic of the vibration
is described by the following (see e.g. [14, Chap. 5.3])
for all t ≥ 0, x ∈ (0, 1),
z
tt
(x, t) = z
xx
(x, t) + f(x, t)
(1)
Christophe Prieur is with Gipsa-lab, Grenoble Campus, 11 rue des
Math
´
ematiques, BP 46, 38402 Saint Martin d’H
`
eres Cedex, France,
christophe.prieur@gipsa-lab.fr.
S. Tarbouriech is with CNRS, LAAS, 7 avenue du colonel Roche,
F-31400 Toulouse, France and Univ de Toulouse, LAAS, F-31400
Toulouse, France. tarbour@laas.fr.
J.M. Gomes da Silva Jr is with the Department of Automa-
tion and Energy, UFRGS, 90035-190 Porto Alegre-RS, Brazil,
jmgomes@ece.ufrgs.br
This work is supported by CAPES-COFECUB ”Ma 590/08”,
CAPES-PVE grant 129/12 and STIC-AmSud ”ADNEC” projects. C.
Prieur and S. Tarbouriech are also supported by ANR project LimICoS
12-BS03-005-01. J.M. Gomes da Silva is also supported by CNPQ-
Brazil.
f(x, t)
z(x, t)
x = 0 x = 1
g(t)
Fig. 1. Vibrating slope subject to a external distributed action f (x, t)
and to a boundary action g(t).
where z stands for the state (the length of the string and
other physical parameters are normalized), and f(x, t) ∈
R is the control. The control f is distributed (in contrast
to boundary control), and is given by a bounded control
operator. Let us equip this system with the following
boundary conditions, for all t ≥ 0,
z(0, t) = 0 , (2a)
z(1, t) = 0 , (2b)
and with the following initial condition, for all x in
(0, 1),
z(x, 0) = z
0
(x) ,
z
t
(x, 0) = z
1
(x) ,
(3)
where z
0
and z
1
stand respectively for the initial deflec-
tion of the slope and the initial deflection speed.
When the control action is only at the boundary, it is
necessary to consider the following string equation, for
all t ≥ 0, x ∈ (0, 1),
z
tt
(x, t) = z
xx
(x, t)
(4)
with the boundary conditions, for all t ≥ 0,
z(0, t) = 0 (5a)
z
x
(1, t) = g(t) (5b)
where g(t) is the boundary action at time t.
When closing the loop with a linear state feedback
law, the control problem of such a 1D wave equation is
considered in many works, see e.g. [8] where, in par-
ticular, stabilizing linear controllers and optimal linear
feedback laws are computed respectively by an appli-
cation of linear semigroup theory and LQR techniques.
The aim of this paper is to investigate the well-posedness
and the asymptotic stability of these classes of PDEs by
means of nonlinear control laws, and more precisely of
cone-bounded nonlinear control laws.
2
Neglecting the presence of nonlinearity in the input
can be source of undesirable and even catastrophic
behaviors for the closed-loop system. See e.g., [3],
where it is shown that in presence of magnitude actuator
limitation, even for finite-dimensional systems, the fact
of disregarding the nonlinearity in the control loop may
yield an unstable system. For an introduction of such
nonlinear finite-dimensional systems and techniques on
how to estimate the basin of attraction for locally asymp-
totically stable equilibrium, see [25], [28] among other
references. Similarly, in the context of systems with
cone-bounded nonlinearities, a quite natural approach
to tackle the stability problem consists in combining
Lyapunov theory with cone-bounded sector conditions
(see e.g., [13], [4]). This allows to provide an estimate
of the basin of attraction of the nonlinear systems in
appropriate Sobolev spaces. This estimate can be either
a neighborhood of the equilibrium in the local case or
the whole space in the global case.
To our best knowledge, the well-posedness and the
asymptotic stability of PDE by means of a cone-bounded
input signal is less investigated than for corresponding
finite-dimensional systems. This class of feedback laws
includes classical saturation and deadzone nonlinearities
as well as more general nonlinear maps. The aim of this
work is to study the wave equation in presence of such
nonlinear control laws. To prove first the existence and
uniqueness of solutions to the PDE (1) (respectively (4))
with the boundary conditions (2) (resp. (5)) and initial
conditions (3), when the loop is closed with an odd,
Lipschitz and cone-bounded control law (see Theorems 1
and 2 below for precise statements), the nonlinear semi-
group theory in appropriate Sobolev spaces is rigorously
applied. The second contribution is to prove the global
(resp. local) asymptotic stability of the corresponding
closed-loop system by exploiting a cone-boundedness
assumption on the control as introduced in [13], [25]
for finite-dimensional systems, see Theorem 1 (resp.
Theorem 2) below for a precise statement. In other
words, this paper combines techniques that are usual
for finite dimensional systems in closed loop with cone-
bounded nonlinear control laws (see e.g. [12], [26]) with
Lyapunov theory for PDEs (see e.g. [10], [5], [21]).
It is worth noticing that for both PDEs (1)-(2) and (4)-
(5), Lyapunov techniques are applied, but the stability
proofs are quite different. To be more specific, the proof
of the stability of the PDE (1) with the boundary condi-
tions (2) when closing the loop with a nonlinear feedback
law is done by using the LaSalle invariance principle,
which needs to state a precompactness property of the
solutions. On the other hand, since a strict decreasing
Lyapunov function is computed, we do not need to use
the LaSalle invariance principle for the PDE (4) with a
nonlinear boundary control.
To our best knowledge, the first work considering
infinite dimensional systems with bounded control is
[24, Thm 5.1] where only compact and bounded control
operators with an a priori constraint are considered. In
[23] only the case of a distributed saturating control
has been considered. On the other hand, the paper [17]
suggests to use an observability assumption for the study
of PDE in closed loop with saturating controllers. In
particular the contraction semigroup obtained from the
saturating closed-loop system is compared to the corre-
sponding semigroup without saturation. In the present
paper, more general cone-bounded input nonlinearities
are considered and different techniques are used, in
particular the LaSalle invariance principle. Then, the
paper can be considered as complementary to [17] by
extending not only the class of nonlinear control laws but
also the nature of the used tools. Note finally that both
papers [24], [17] do not consider the case of saturation
of the value of the solution, but rather saturation of
the norm of the solution (compare [24, Eq. (2.8)] and
[17, Eq. (1.6)] with the definition of nonlinear controller
(8) below). Dealing with saturation of the value of
the solution is more complex and is more relevant for
applications, since it yields to a locally defined PDE.
However it needs further developments (in particular
more regularity is required to ensure that the nonlinear
map is well-defined).
The rest of the paper is organized as follows. In
Section II, the nonlinear PDE (1) is introduced, and the
first main result is stated, namely the well-posedness
of the Cauchy problem, along with the global asymp-
totic stability, when closing the loop with a nonlinear
distributed control law. In Section III, the main result
is stated for the PDE (4), where a nonlinear boundary
action is considered. The proof of the two main results
are given respectively in Sections IV and V. Section VI
presents numerical simulations to illustrate both main
results. Some concluding remarks and possible further
research lines are presented in Section VII.
Notation: z
t
(resp. z
x
) stands for the partial derivative
of the function z with respect to t (resp. x) (this is
a shortcut for
∂z
∂t
, resp.
∂z
∂x
). When there is only one
independent variable, ˙z and z
0
stand respectively for
the time and the space derivative. For a matrix A, A
>
denotes the transpose, and for a partitioned symmetric
matrix, the symbol F stands for symmetric blocks. <(s)
and =(s) stand respectively for the real and imaginary
part of a complex value s in C, s is the conjugate
of s, and |s| its modulus. k k
L
2
denotes the norm
in L
2
(0, 1) space, defined by kuk
2
L
2
(0,1)
=
R
1
0
|u|
2
dx
for all functions u ∈ L
2
(0, 1). Similarly, H
2
(0, 1)
is the set of all functions u ∈ H
2
(0, 1) such that
R
1
0
(|u|
2
+ |u
x
|
2
+ |u
xx
|
2
)dx is finite. Finally H
1
0
(0, 1)
is the closure in L
2
(0, 1) of the set of smooth functions
3
that are vanishing at x = 0 and at x = 1. It is equipped
with the norm kuk
2
H
1
0
(0,1)
=
R
1
0
|u
x
|
2
dx. The associate
inner products are denoted h·, ·i
L
2
(0,1)
and h·, ·i
H
1
0
(0,1)
.
II. WAVE EQUATION WITH A NONLINEAR
DISTRIBUTED CONTROL
Consider the PDE (1), with the boundary conditions
(2) and the initial condition (3).
Letting for the control, for all t ≥ 0 and all x ∈ (0, 1),
f(x, t) = −az
t
(x, t), (6)
where a is a constant value, and exploiting properties of
the following energy function:
V
1
=
1
2
Z
(z
2
x
+ z
2
t
)dx, (7)
for any solution z to (1) and (2), when closing the loop
with the linear controller (6), allow to show that the
closed-loop system is (globally) exponentially stable in
H
1
0
(0, 1) × L
2
(0, 1).
This can be formally shown by considering the time
derivative of V along the solutions to (1)-(2), which
yields
˙
V
1
=
R
1
0
(z
x
z
xt
− az
2
t
+ z
t
z
xx
)dx
= −
R
1
0
az
2
t
dx + [z
t
z
x
]
x=1
x=0
= −
R
1
0
az
2
t
dx
where an integration by parts is performed to get the
second line from the first one, and the boundary condi-
tions (2) are applied for the last line. Therefore, for any
positive value a, it is formally obtained that the energy
is decreasing at long as z
t
is non vanishing in [0, 1]. In
other words, V
1
is a (non strict) Lyapunov function.
Due to actuator limitations or imperfections, the actual
control law applied to the system, instead of (6), can be
modeled as follows
f(x, t) = −σ
1
(az
t
(x, t)) (8)
with σ
1
: R → R being a bounded and continuous
nonlinear function satisfying, for a constant value L > 0
and for all (s, es) ∈ R
2
,
(σ
1
(s) − σ
1
(es))(s − es) ≥ 0 , (9a)
|σ
1
(s)| ≤ L|s| . (9b)
Note that (9a) generalizes the odd property. Examples of
such functions σ
1
include the saturation maps, and are
considered in Section VI-A below.
Equation (1) in closed loop with the control (8)
becomes
z
tt
= z
xx
− σ
1
(az
t
) .
(10)
A formal computation gives, along the solutions to
(10) and (2),
˙
V
1
= −
Z
1
0
z
t
σ
1
(az
t
)dx (11)
which asks to handle the nonlinearity z
t
σ
1
(az
t
). A con-
vergence result is stated below, where the well-posedness
is separate from the asymptotic stability property.
Theorem 1. For all positive values a, and for all
bounded and continuous functions σ
1
satisfying (9), the
model (10) with the boundary conditions (2) is glob-
ally asymptotically stable. More precisely the following
properties hold:
• [Well-posedness] For all (z
0
, z
1
) in
H
2
(0, 1) ∩ H
1
0
(0, 1)
× H
1
0
(0, 1), there exists a
unique solution z: [0, ∞) → H
2
(0, 1) ∩ H
1
0
(0, 1) to
(10), with the boundary conditions (2) and the initial
condition (3), that is differentiable from [0, ∞) on
H
1
0
(0, 1).
• [Global asymptotic stability] Moreover, for all initial
conditions (z
0
, z
1
) in
H
2
(0, 1) ∩ H
1
0
(0, 1)
×H
1
0
(0, 1),
the solution to (10), with the boundary conditions (2) and
the initial condition (3), satisfies the following stability
property
kz(., t)k
H
1
0
(0,1)
+ kz
t
(., t)k
L
2
(0,1)
≤ kz
0
k
H
1
0
(0,1)
+ kz
1
k
L
2
(0,1)
, ∀t ≥ 0 ,
(12)
together with the attractivity property
kz(., t)k
H
1
0
(0,1)
+ kz
t
(., t)k
L
2
(0,1)
→
t→∞
0 . (13)
The proof of Theorem 1 is provided in Section IV.
III. WAVE EQUATION WITH A NONLINEAR
BOUNDARY ACTION
Consider the PDE (4), with the boundary conditions
(5) and the initial condition (3).
Letting for the control
g(t) = −bz
t
(1, t), (14)
where b is a positive tuning parameter, inspired by [16],
the following Lyapunov function candidate:
V
2
=
1
2
R
1
0
e
µx
(z
t
+ z
x
)
2
dx
+
R
1
0
e
−µx
(z
t
− z
x
)
2
dx
,
(15)
where µ > 0 will be prescribed below, is considered.
It may be proven that we have asymptotic stability of
(4) and (5). With this aim, let us formally compute the
time-derivative of V
2
along the solutions of (4) and (5)
4
as follows
˙
V
2
=
R
1
0
e
µx
(z
t
+ z
x
)(z
tt
+ z
xt
)dx
+
R
1
0
e
−µx
(z
t
− z
x
)(z
tt
− z
xt
)dx
=
R
1
0
e
µx
(z
t
+ z
x
)(z
xx
+ z
xt
)dx
−
R
1
0
e
−µx
(z
t
− z
x
)(z
xt
− z
xx
)dx
= −
µ
2
R
1
0
e
µx
(z
t
+ z
x
)
2
dx +
1
2
[e
µx
(z
t
+ z
x
)
2
]
x=1
x=0
−
µ
2
R
1
0
e
−µx
(z
t
− z
x
)
2
dx −
1
2
[e
−µx
(z
t
− z
x
)
2
]
x=1
x=0
where the partial differential equation (4) has been used
in the first equality and two integrations by parts have
been performed in the second equality.
Now, note that the boundary condition (5a) implies
that z
t
(0, t) = 0 and thus, for all t ≥ 0,
[e
µx
(z
t
+ z
x
)
2
](0, t) − [e
−µx
(z
t
− z
x
)
2
](0, t)
= z
2
x
(0, t) − z
2
x
(0, t) = 0
Therefore, with (5b), it is deduced
˙
V
2
= −µV
2
+
e
µ
2
(z
t
(1, t) + z
x
(1, t))
2
−
e
−µ
2
(z
t
(1, t) − z
x
(1, t))
2
(16)
and thus
˙
V
2
= −µV
2
+
e
µ
2
(z
t
(1, t) − bz
t
(1, t))
2
−
e
−µ
2
(z
t
(1, t) + bz
t
(1, t))
2
= −µV
2
+
1
2
e
µ
(1 − b)
2
− e
−µ
(1 + b)
2
z
2
t
(1, t)
For any positive value b, it holds |1 − b| < |1 + b|.
Now, pick any µ > 0 such that
e
µ
(1 − b)
2
≤ e
−µ
(1 + b)
2
(17)
holds.
With such a value b, we get
˙
V
2
≤ −µV
2
and thus
the partial differential equation (4), with the boundary
condition (5), is exponentially stable.
Now instead of the boundary condition (5) in closed
loop with the linear controller (14), consider the bound-
ary conditions, for all t ≥ 0,
z(0, t) = 0
z
x
(1, t) = −σ
2
(bz
t
(1, t))
(18)
resulting from the boundary condition (5) in closed loop
with a bounded and continuous map σ
2
: R → R
satisfying, for all (s, es) ∈ R,
(σ
2
(s) − σ
2
(es))(s − es) ≥ 0 , (19a)
|σ
2
(s)| ≤ u
2
, (19b)
with u
2
> 0. Assume moreover that, for all c in R, and
for all s in R, such that |(b − c)s| ≤ u
2
, it holds
ϕ
2
(bs)(ϕ
2
(bs) + cs) ≤ 0 , (19c)
where ϕ
2
(s) = σ
2
(s) − s. Such a function σ
2
includes
the nonlinear functions satisfying some sector bounded
condition, as the saturation maps of level u
2
(see [25,
Chap. 1] or [13, Chap. 7]). Since σ
2
is a function
of z
t
(1, t), it is needed in the next result a stronger
regularity on the initial condition than the one imposed
in e.g. [17], so that (18) makes sense.
The stability analysis of the corresponding nonlinear
partial differential equation (4) and (18) asks for special
care. It is done in our second main result, given below,
where, following the notation in [5, Sec. 2.4], it is
denoted H
1
(0)
(0, 1) = {u ∈ H
1
(0, 1), u(0) = 0}, and
kuk
H
1
(0)
(0,1)
=
q
R
1
0
|u
0
|
2
(x)dx, for all u ∈ H
1
(0)
.
Theorem 2. For all positive values b, and for all con-
tinuous functions σ
2
satisfying (19), the model (4) with
the boundary conditions (18) is globally asymptotically
stable. More precisely the following properties hold:
• [Well-posedness] For all (z
0
, z
1
) in
{(u, v), (u, v) ∈ H
2
(0, 1)×H
1
(0)
(0, 1), u
0
(1)+bv(1) =
0, u(0) = 0}, there exists a unique continuous solution
z: [0, ∞) → H
2
(0, 1) ∩ H
1
(0)
(0, 1) to (4), with the
boundary conditions (18) and the initial condition (3),
that is differentiable from [0, ∞) to H
1
(0)
(0, 1).
• [Global asymptotic stability] For all initial con-
ditions (z
0
, z
1
) in {(u, v), (u, v) ∈ H
2
(0, 1) ×
H
1
(0)
(0, 1), u
0
(1) + bv(1) = 0, u(0) = 0}, the solution
to (4), with the boundary conditions (18) and the initial
condition (3), satisfies the following global stability
property
kz(., t)k
H
1
(0)
(0,1)
+ kz
t
(., t)k
L
2
(0,1)
≤ kz
0
k
H
1
(0)
(0,1)
+ kz
1
k
L
2
(0,1)
, ∀t ≥ 0 ,
(20)
together with the attractivity property,
kz(., t)k
H
1
(0)
(0,1)
+ kz
t
(., t)k
L
2
(0,1)
→
t→∞
0, (21)
holds.
Remark 1. As for many nonlinear control systems, in
particular the finite-dimensional ones subject to input
saturation (see e.g., [25]), only the local exponential
stability can sometimes be obtained, requiring to prove
the exponential stability of the system only for a set of
admissible initial conditions. Regarding a similar case
for an infinite dimensional system, Theorem 2 does not
state the global exponential stability, however we are
able to prove the global asymptotic stability.
The proof of Theorem 2 is provided in Section V.
Note that perturbation arguments (as considered in e.g.,
[20, Chap. 3]) may be used to study (10) in closed
loop with a saturating controller instead of the nonlinear
function σ
2
. It yields a local asymptotic stability property
without exhibiting any estimate of the basin of attraction,