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Control of a Wave Equation with a Dynamic Boundary
Condition
Nicolas Vanspranghe, Francesco Ferrante, Christophe Prieur
To cite this version:
Nicolas Vanspranghe, Francesco Ferrante, Christophe Prieur. Control of a Wave Equation with a
Dynamic Boundary Condition. CDC 2020 - 59th IEEE Conference on Decision and Control, IEEE,
Dec 2020, Jeju Island (virtual), South Korea. pp.652-657, �10.1109/CDC42340.2020.9303767�. �hal-
02987252�
Control of a Wave Equation with a Dynamic Boundary Condition
Nicolas Vanspranghe, Francesco Ferrante, Christophe Prieur
Abstract— The general problem of this paper is the anal-
ysis of wave propagation in a bounded medium where the
uncontrolled boundary obeys a coupled differential equation.
More precisely, we study a one-dimensional wave equation
with a nonlinear second-order dynamic boundary condition
and a Neuman-type boundary control acting on the other
extremity. A generic class of nonlinear collocated feedback laws
is considered. Hadamard well-posedness is established for the
closed-loop system, with initial data lying in the natural energy
space of the problem. Moreover, we investigate an example of
stabilization through a proportional controller.
I. INTRODUCTION
The aim of this paper is to study a wave equation in a
bounded one-dimensional medium supplied with a dynamic
(or kinetic) boundary condition at one end. The system is ac-
tuated via a Neuman-type control at the other end. Dynamic
boundary conditions involve second-order time derivative
and are typically obtained in physical models for which the
momentum of the boundary cannot be neglected. A prime
example is an infinite-dimensional model for the propagation
of mechanical vibrations along drilling rods. In that case, the
control is the torque applied to one extremity, and the kinetic
boundary condition is given by the behavior of the rock-tip
contact, which is subject to nonlinear friction. In particular,
stick-slip phenomena can occur at the rock-tip interface and
generate unwanted vibrations that might jeopardize the plant.
A mechanical analysis of the rock-tip dynamics is given in
[7]. The problem of stabilization and regulation of the veloc-
ity at the rock-tip contact has sparked interest in the control
commmunity, see e.g. [10]. In engineering applications, the
goal is to minimize the stick-slip vibrations through a suitable
control. Various boundary control strategies are proposed to
address this problem in [8], including backstepping design
– see also [9]. In [4], an observer-based boundary control
design is proposed. In [13], stabilization and regulation using
a proportional integral boundary controller is investigated.
In [12] and [1], a similar problem is considered, but with a
boundary anti-damping only involving first-order derivatives.
However, the aforementioned papers deal with linearized
models. We should also mention [3] and [6] where different
classes of boundary nonlinearities for distributed parameter
systems are considered.
Let us introduce the specific dynamical model under study
in this paper. Let L > 0 and Ω , (0, L) ⊂ R. We deal with
Nicolas Vanspranghe, Francesco Ferrante and Christophe Prieur are with
Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-lab, 38000 Grenoble,
France. Email: name.surname@gipsa-lab.fr.
This work has been partially supported by MIAI @ Grenoble Alpes
(ANR-19-P3IA-0003).
the following control system:
∂
tt
u − ∂
xx
u = 0 on Ω × R
+
,
∂
tt
u(0, t) − ∂
x
u(0, t) = F (∂
t
u(0, t)) for all t,
∂
x
u(L, t) = U(t) for all t,
(1)
where F is a scalar function that models the nonlinear
boundary friction, and U(t) is the control input.
Our first goal is to prove the well-posedness of the control
system (1) supplied with the following collocated feedback:
U(t) = −g(∂
t
u(L, t)), (2)
where g is a continuous nondecreasing scalar function.
The output considered here is the velocity at the boundary
where the actuator lies, meaning that an observer is not
required to implement the controller. Note that this class of
feedback laws includes nonlinearities of interest in control
applications, such as saturations or deadzones – see e.g. [5].
In this paper, we prove that, under appropriate assump-
tions, closing the loop in (1) with (2) leads to a well-
posed dynamical system. A precise variational formulation
of the problem is given and the regularity of the solutions
is rigorously investigated. The underlying control problem
is the stabilization of a possibly non-dissipative boundary
by an action on the other boundary. We prove that, under
suitable conditions, exponential stability can be achieved
using a proportional controler. To the best of our knowledge,
the stabilization of such system in the presence of nonlinear
boundary anti-damping has not been investigated so far.
This paper is organized as follows. Section II introduces
the variational formulation of control system (1) and contains
the first main result, namely the well-posedness of the closed-
loop dynamics. Section III introduces the natural energy
associated with (1) and states an exponential stability result
under appropriate assumptions on the nonlinear map F . This
is the second main contribution. Section IV contains some
illustrative numerical results. Section V gives the proof of
the well-posedness result. Section VI contains concluding
remarks.
Notation: Given a Banach space E, we denote its norm
by k·k
E
and we use the duality bracket hφ, xi
E
to write φ(x)
for any x ∈ E and φ ∈ E
0
, where E
0
is the topological dual
of E. If E is also a Hilbert space, its scalar product is written
(·, ·)
E
. The space of infinitely differentiable functions on Ω
with compact support is denoted by C
∞
c
(Ω). Also, for T > 0,
we denote by W
1,p
(0, T ; E) the subspace of L
p
(0, T ; E)
composed of (classes of) E-valued functions f such that,
for some g in L
p
(0, T ; E), f(t) = f(0) +
R
t
0
g(s) ds for a.e.
t in (0, T ). Such class f is identified with its continuous
representative and we say that f
0
= g in the sense of E-
valued distributions.
II. VARIATIONAL FORMULATION AND WELL-POSEDNESS
In this section, we establish the framework in which we
analyze system (1), and state our well-posedness result. We
start by introducing the energy spaces associated with (1) as
well as some notation. First, define
H , L
2
(Ω) × R. (3)
We endow H with the usual product Hilbertian structure.
Define now the following subspace of H:
V ,
(u, u(0)) : u ∈ H
1
(Ω)
, (4)
which is equipped with the scalar product inherited from
H
1
(Ω) × R. As stated in Section V, V is also a Hilbert
space. We also introduce the state space
X , V × H (5)
endowed with the product Hilbertian structure. For the sake
of clarity, we use parenthesis to denote elements of V or H,
and brackets to denote elements of X , as in X = [u, v] ∈ X ,
u = (u, u(0)) ∈ V , etc. Now, let us define the bilinear
continuous symmetric form a on V × V by
a(u
1
, u
2
) ,
Z
Ω
∂
x
u
1
(x)∂
x
u
2
(x) dx. (6)
Finally, we denote by δ
L
the linear form mapping u ∈ V
into u(L), which belongs to V
0
since H
1
(Ω) is continuously
embedded into C(
¯
Ω) in the one-dimensional case.
Assumption 1. The scalar function F is globally Lipschitz.
We define the nonlinear operator B on H associated with
the first-order boundary pertubation:
∀v = (v, γ) ∈ H, B(v) , (0, F (γ)). (7)
This operator is globally Lipschitz by Assumption 1. Let us
precise the meaning of weak solutions to (1).
Definition 1. A weak solution to (1) on (0, T ) is any u in
L
∞
(0, T ; V ) ∩ W
1,∞
(0, T ; H) verifying
Z
T
0
− (u
0
(t), φ
0
(t))
H
+ a(u(t), φ(t)) dt =
Z
T
0
(B(u
0
(t)), φ(t))
H
+ U(t)hδ
L
, φ(t)i
V
dt
(8)
for all test-functions φ in C
c
(0, T ; V ) ∩ C
1
c
(0, T ; H).
Definition 1 is motivated by simple formal calculations
in which one multiplies the wave equation by some smooth
test-function φ(x, t) and integrates it over Ω × (0, T ) using
integration by parts and the boundary conditions. Note that
Definition 1 makes sense if, say, U belongs to L
2
(0, T ).
Closing the loop, a hidden regularity property of the solu-
tions is needed to ensure all terms are defined.
1
1
In the language of systems theory, one may say that the Neuman input
operator is unbounded with respect to the state space X .
Assumption 2. The function g : R → R is continuous and
nondecreasing.
2
We show that the closed-loop system generates a dynam-
ical system on X by defining the operators S
t
for t ≥ 0 as
follows:
∀X = [u
0
, u
1
] ∈ X , S
t
(X) , [u(t), u
0
(t)] ∈ X , (9)
where u is the unique solution associated with initial data
[u
0
, u
1
].
Theorem 1 (Hadamard well-posedness). Let [u
0
, u
1
] ∈ X .
Under Assumptions 1 and 2, there exists a unique (weak)
solution u ∈ C(R
+
, V )∩C
1
(R
+
, H) to (1) with feedback (2)
and initial data [u
0
, u
1
]. The solution u enjoys the following
hidden regularity property: for all T > 0,
u(L, ·) ∈ H
1
(0, T ) and g(∂
t
u(L, ·)) ∈ L
2
(0, T ). (10)
Moreover, we can associate with (1) and control law (2) the
semigroup S = {S
t
}
t≥0
of nonlinear continuous operators
on X as defined in (9).
Theorem 1 is proved in Section V. In the following
proposition, we give some additional regularity when the
initial datum is smooth and verifies a compatibility condition.
We write W , [H
2
(Ω) × R] ∩ V , equipped with the scalar
product inherited from H
2
(Ω) × R.
Proposition 1 (Strong solutions). Let T > 0. If [u
0
, u
1
]
belongs to W ×V and verifies ∂
x
u
0
(L) = −g(u
1
(L)), then,
u ∈ L
∞
(0, T ; W ), u
0
∈ L
∞
(0, T ; V ). (11)
Moreover, any weak solution u is the limit of a sequence of
strong solutions [u
n
, u
0
n
] in C([0, T ], X ), and ∂
t
u
n
(L, ·) →
∂
t
u(L, ·) in L
2
(0, T ).
Proposition 1 is a byproduct of the proof of Theorem 1
and is used to justify computations performed in Section III.
III. STABILITY ANALYSIS OF THE CLOSED-LOOP SYSTEM
Next, we analyze the stability of the closed-loop system
when the feedback is linear. We introduce the energy func-
tional E defined on X by
E(u, v) ,
1
2
kvk
2
H
+ a(u, u)
, (12)
which is, in the context of abstract wave equations, the
natural “mechanical” energy. Here, for all u in V and
v = (v, γ) in H,
E(u, v) =
1
2
Z
Ω
|v(x)|
2
+ |∂
x
u(x)|
2
dx +
1
2
|γ|
2
.
(13)
For any ρ and µ in H
1
(Ω), we define a weighted energy
functional (or Lyapunov function candidate):
Γ
ρ,µ
(u, v) ,
1
2
Z
Ω
v
∂
x
u
>
µ ρ
ρ µ
v
∂
x
u
+
1
2
µ(0)|γ|
2
.
(14)
2
It is a specificity of the one-dimensional case that no additional growth
assumption on g is required to obtain weak solutions for such feedback.
It follows from the boundedness of the weights ρ and µ that
there exists a constant M
ρ,µ
> 0 such that
∀[u, v] ∈ X , Γ
ρ,µ
(u, v) ≤ M
ρ,µ
E(u, v). (15)
A sufficient condition for the converse inequality to hold is
the existence of C
1
, C
2
> 0 such that
µ(x) ≥ C
1
and µ(x)
2
− ρ(x)
2
≥ C
2
(16)
holds for all x in Ω. We also remark that E and Γ are contin-
uous on X . We note that the position does not appear in the
energy functionals and without Poincar
´
e-type inequalities,
one cannot directly infer an estimate of the norm of the
position. Stabilization shall be investigated with respect to
the following invariant set, which is the line composed of
constant solutions:
A , {[θ1, 0] ∈ X : θ ∈ R}, (17)
where 1 , ((x 7→ 1), 1). Observe that A is exactly the subset
of X where the mechanical energy E vanishes. For any [u, v]
in X ,
dist([u, v], A)
2
≤ CE(u, v), (18)
where C is some positive constant. This is a consequence
of the Poincar
´
e-Wirtinger inequality – see [2] for instance.
From now on, we are interested in stabilizing system (1)
using a proportional feedback, i.e.
U(t) = −α∂
t
u(L, t), (19)
where α is a positive gain to be tuned.
Assumption 3. The function F is q-Lipschitz for some q <
1/2. Also, F (0) = 0.
The condition F (0) = 0 is quite natural when dealing with
friction. Assumption 3 is meant to be in force in the context
of nonlinear anti-damping.
3
Theorem 2 (Exponential stability). Under Assumption 3 and
with α = 1, A uniformly and exponentially attracts the
bounded sets of X . More precisely, there exist two positive
constants M and η such that for all solutions u to (1),
E(u(t), u
0
(t)) ≤ ME(u
0
, u
1
) exp(−ηt). (20)
Moreover, for each solution u, u(t) converges in V to some
constant function u
∞
when t → +∞.
Proof. We pick a solution u and, for the sake of concision,
we denote by Γ
u
ρ,µ
the function t ∈ R
+
7→ Γ
ρ,µ
(u(t), u
0
(t)),
which is continuous. Take ρ, µ ∈ H
1
(Ω). Assume temporar-
ily that u is a strong solution. For all τ ≥ 0, denoting by
3
Alternatively, if F has a stabilizing effect, e.g. F (0) = 0, F nonincreas-
ing, it can be relaxed to F locally Lipschitz – with appropriate modifications
in the proof of well-posedness.
Q
τ
the rectangle Ω × (0, τ), we have the following identity:
Γ
u
ρ,µ
(t)
τ
0
= −
1
2
ZZ
Q
τ
∂
t
u
∂
x
u
>
ρ
0
µ
0
µ
0
ρ
0
∂
t
u
∂
x
u
+
1
2
ρ(L)(1 + α
2
) − 2αµ(L)
Z
τ
0
|∂
t
u(L, t)|
2
dt
−
ρ(0)
2
Z
τ
0
|∂
t
u(0, t)|
2
+ |∂
x
u(0, t)|
2
dt
+µ(0)
Z
τ
0
F (∂
t
u(0, t))∂
t
u(0, t) dt.
(21)
Equation (21) is obtained by multiplying the wave equation
∂
tt
u(x, t) − ∂
xx
u(x, t) = 0 by φ(x, y) = µ(x)∂
t
u(x, t) +
ρ(x)∂
x
u(x, t) ∈ H
1
(Q
τ
), integrating over Q
τ
, and perform-
ing a few integrations by parts. If ρ and µ are nonnegative,
writing |F (s)| ≤ q|s| where q is the Lipschitz constant in
Assumption 3, we have the following inequality:
Γ
u
ρ,µ
(t)
τ
0
≤ −
1
2
ZZ
Q
τ
∂
t
u
∂
x
u
>
ρ
0
µ
0
µ
0
ρ
0
∂
t
u
∂
x
u
+
1
2
ρ(L)(1 + α
2
) − 2αµ(L)
Z
τ
0
|∂
t
u(L, t)|
2
dt
+
1
2
[2qµ(0) − ρ(0)]
Z
τ
0
|∂
t
u(0, t)|
2
dt.
(22)
Take µ(x) = 1. For (16) to hold, it suffices to have ρ(x) ≤
1 − for some > 0. Now, we derive some sufficient
conditions for the energy to decay exponentially. It suffices
to have ρ(0) ≥ 2q + , and ρ
0
(x) ≥ a.e. as well as
ρ(L) ≤
2α
1 + α
2
− (23)
for some > 0. Since q < 1/2, there exists an increasing
affine function ρ such that ρ(0) > 2q and ρ(L) < 1. Let
α = 1 so that (23) holds. As a result, by Gr
¨
onwall’s lemma,
we obtain the following: with this particular choice of ρ and
µ, there exists a positive constant (solution independent) η
such that
∀t ≥ 0, Γ
u
ρ,µ
(t) ≤ Γ
u
ρ,µ
(0) exp(−ηt). (24)
By a density-continuity argument, the uniform estimate (24)
holds for weak solutions as well – see Proposition 1. Since
(16) holds, then there exists M > 0 such that (20) is
verified by any solution. As for the second statement of
Theorem 2, let u be a solution. Take an increasing sequence
of nonnegative real numbers t
n
such that t
n
→ +∞ when
n → +∞. Then {u(t
n
)}
n≥0
is a Cauchy sequence in V .
Indeed, for any m ≥ n, writing the variation of u(t) between
t
n
and t
m
, we have
ku(t
m
) − u(t
n
)k
H
≤
Z
t
m
t
n
ku
0
(s)k
H
ds
≤ M
0
Z
+∞
t
n
exp(−ηs/2) ds,
(25)
which converges to 0 when n → +∞; also, we already
know that a(u(t), u(t)) → 0 when t → +∞, thus ku(t
m
) −
u(t
n
)k
V
can be arbitrarily small. Using a similar argument,
one verifies that the limit does not depend on the sequence
{t
n
}. Therefore, u(t) converges to some u
∞
∈ V and u
∞
is constant by (18).
IV. NUMERICAL SIMULATIONS
We provide some numerical computations for illustrative
purposes.
Fig. 1. Evolution of the boundary position u(0, t) over time. It obeys a
second-order differential equation coupled with the wave equation.
Fig. 2. Evolution of the mechanical energy E(u(t), u
0
(t)) over time. The
uncontrolled system would be unstable due to the boundary anti-damping.
We discretize (1) using finite elements over space and
finite differences over time, on the basis of the functional
formulation of the problem. Figures 1 and 2 are obtained with
the following parameters: we take α = 1, F(x) = qx with
q = 0.1 and L = 1. The domain Ω is discretized into 100
points and the time step is set to 0.001. Further computations
suggest that the condition on q derived in Theorem 2 is nearly
sharp as taking q = 0.5 leads to an exponentially unstable
system. Computations also suggest that the proportional
feedback is not robust to (numerical) errors: taking values
of q slightly below 0.5 induces unclear situations where an
exponential decay is not easily identifiable.
V. PROOF OF THE HADAMARD THEOREM
The proof of Theorem 1 relies on nonlinear semigroup
techniques and appropriate energy estimates. We begin this
section with some remarks on the functional settings of the
problem. The following result states some useful properties
of the spaces H and V .
Lemma 1. V is a separable Hilbert space isomorphic to
H
1
(Ω). Moreover, V is a dense subset of H.
Since kuk
2
V
= kuk
2
H
+ a(u, u), we see that V is con-
tinuously embedded into H. We denote H
0
the topological
dual of H. By Riesz representation theorem, we make the
identification H ' H
0
. Because V is a dense subset of H, the
latter can be identified as a dense subset of V
0
, leading to the
classical injection chain V → H ' H
0
→ V
0
, where each
space is dense and continuously embedded into the following
one. We denote by A the continuous operator from V into
V
0
defined by hAu
1
, u
2
i
V
, a(u
1
, u
2
) for all u
1
, u
2
in V .
Next, define an unbounded (nonlinear) operator A
g
on X by
D(A
g
) , {[u, v] ∈ W × V : ∂
x
u(L) = −g(v(L))}
∀X = [u, v] ∈ D(A
g
), A
g
(X) , −
v
(∂
xx
u, ∂
x
u(0))
.
(26)
Note that the domain D(A
g
) need not be a subspace. We
start with the following first-order abstract Cauchy problem:
(
˙
X(t) + A
g
(X(t)) = F(X(t))
X(0) = X
0
,
(27)
where F is the nonlinear perturbation operator on X defined
by F(X) , [0, B(v)] for all X = [u, v] ∈ X . We see that
F is Lipschitz, since the H-valued mapping B is Lipschitz.
We wish to prove that (27) is a Lipschitz perturbation of an
evolution equation with maximal monotone generator, hence
the following result.
Proposition 2. The unbounded operator A
g
+id is maximal
monotone.
Proof. We start with the monotonicity, and then we shall
prove the surjectivity of [A
g
+ id] + id.
Step 1: Monotonicity. Let X
1
= [u
1
, v
1
] and X
2
=
[u
2
, u
2
] in D(A
g
). We denote u
1
− u
2
(resp. v
1
− v
2
) by u
(resp. v), and also X
1
− X
2
by X. We have
(A
g
(X
1
)−A
g
(X
2
), X)
X
= −(u, v)
V
− (∂
xx
u, v)
L
2
(Ω)
− ∂
x
u(0)v(0)
(28)
Recall that (u, v)
V
= (u, v)
H
+ a(u, v). Moreover, by
integration by parts, we have a(u, v) = −(∂
xx
u, v)
L
2
(Ω)
+
∂
x
u(L)v(L) − ∂
x
u(0)v(0). Thus, from (28) we obtain
(A
g
(X
1
)−A
g
(X
2
), X)
X
= −(u, v)
H
− ∂
x
u(L)v(L).
(29)
By definition of D(A
g
) and Assumption 2, −∂
x
u(L)v(L) =
[g(v
1
(L)) − g(v
2
(L))]v(L) ≥ 0, hence
(A
g
(X
1
) − A
g
(X
2
), X)
X
≥ −(u, v)
H
. (30)
From (29) we then obtain
(A
g
(X
1
)−A
g
(X
2
) + X, X)
X
≥ −(u, v)
H
+ kuk
2
V
+ kvk
2
H
≥ −(u, v)
H
+
1
2
kuk
2
H
+
1
2
kvk
2
H
≥ 0,
(31)