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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)