Mathematics in Applied Sciences and Engineering https://ojs.lib.uwo.ca/mase
Online First, pp.1-24 https://doi.org/10.5206/mase/15949

OPTIMAL ACTUATOR PLACEMENT FOR CONTROL OF VIBRATIONS

INDUCED BY PEDESTRIAN-BRIDGE INTERACTIONS

MARTIN DEOSBORNS AROP, HENRY KASUMBA, JUMA KASOZI, AND FREDRIK BERNTSSON

Abstract. In this paper, an optimal actuator placement problem with a linear wave equation as the

constraint is considered. In particular, this work presents the frameworks for finding the best location

of actuators depending upon the given initial conditions, and where the dependence on the initial

conditions is averaged out. The problem is motivated by the need to control vibrations induced by

pedestrian-bridge interactions. An approach based on shape optimization techniques is used to solve

the problem. Specifically, the shape sensitivities involving a cost functional are determined using the

averaged adjoint approach. A numerical algorithm based on these sensitivities is used as a solution

strategy. Numerical results are consistent with the theoretical results, in the two examples considered.

1. Introduction

An actuator is a device that introduces or prevents motion in a control system [10]. In this work, an

actuator is defined as a device that prevents motion in a control system.

Optimal actuator placement problems involve the question of finding the optimal location of the

subdomain [23]. They arise naturally in many practical applications, for example, in seismic inversion

[20], placement of loudspeakers for ideal acoustics [11], and medical applications [2].

There are extensive works on the optimal actuator placement problems governed by linear ordinary

differential equations in the literature, see [9, 22] and the references therein. From among the earlier

publications in this direction, we quote the work in [9], where the optimal placement of actuators

and sensors for gyroelastic bodies is studied based on controllability and observability criteria. Another

important study is by Van de Wal and de Jager [22], where a linear system is solved using controllability

and observability Gramians.

The optimal placement of actuators in dynamical systems governed by heat, advection, and wave

equations has also received a growing amount of attention. In [21], an actuator and sensor placement

problem is considered using an advection equation with an application in building systems. The authors

proposed a Gramian criterion, where the degree of controllability and observability is maximized with

respect to the least controllable and observable states.

An optimal actuator design and placement problem for a linear heat equation is investigated in

[10] using a shape and topology optimization approach. The authors parametrized the actuators by

considering controls over some subsets of the domains using indicator functions.

In [7] and [8], optimal stabilizations of the one-dimensional wave equation are investigated using a

genetic algorithm and frequential analysis approach, respectively. Furthermore, the optimal location of

controllers for the one-dimensional wave equation is studied in [16] as an exact controllability problem

Received by the editors 31 January 2023; accepted 7 August 2023; published online 20 August 2023.

2020 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20.

Key words and phrases. Functionals, wave equation, optimal actuator placement, shape optimization, finite difference.

M. D. Arop was supported in part by SIDA bilateral programme (2015–2022) with Makerere University; Project 316.

1

https://ojs.lib.uwo.ca/mase
https://dx.doi.org/https://doi.org/10.5206/mase/15949


2 M. D. AROP, H. KASUMBA, J. KASOZI, AND F. BERNTSSON

using the frequential analysis approach. In addition, the optimal location of the support of the control

for the one-dimensional wave equation as an exact controllability problem is studied in [14].

Inspired by the work in [14], we study an optimal actuator placement problem for linear wave

dynamics by using shape optimization techniques. In particular, we extend the techniques presented in

[10] to a dynamic system governed by the linear wave equation. Numerical realization of the problem

is achieved by using a finite difference method, see e.g., [13].

In this paper, we determine the optimal actuator placement for the stabilization of pedestrian-bridge

vibrations. More precisely, we use a shape optimization approach to find the optimal actuator location

so that the vibrations induced by pedestrian-bridge interactions are controlled.

The remainder of this paper is organized as follows. In Section 2, we fix the notations utilized in

the sequel and formulate the state and optimization problems. Section 3 is devoted to proving well-

posedness and deriving the optimality system for our optimization problems. In Section 4, we derive the

shape derivatives of the optimization problems. Numerical tests that illustrate the theoretical results

are given in Section 5. The paper ends with concluding remarks and future work.

2. Formulation of the Problem

2.1. Notations. Let G be either the domain Ω or its boundary ∂Ω. Then, we define L2(G) as a linear
space of all measurable functions y : G → R such that

‖y‖L2(G) :=
(∫
G
|y|2 dx

)1
2

< ∞.

The standard Sobolev space of order m ∈ R+ ∪{0}, denoted by Hm(G), is defined as

Hm(G) := {y ∈ L2(G)|Dγy ∈ L2(G), for all 0 ≤ |γ| ≤ m},

where Dγ is the weak partial derivative and γ is a multi-index. The norm ‖ · ‖Hm(G) associated with
Hm(G) is given by

‖y‖Hm(G) :=

√√√√ ∑
|γ|≤m

∫
G
|Dγy|2 dx.

For a functional space X, we denote by Lp(0,T; X) (1 ≤ p < ∞) the space of measurable functions
y : [0,T] → X such that

‖y‖Lp(0,T;X) :=
(∫ T

0

‖y(·, t)‖pX dt
)1
p

< ∞,

where T is the final time. The space of essentially bounded functions from [0,T] into X is denoted by

L∞(0,T; X) and is equipped with the norm ess supt∈[0,T]‖y(·, t)‖X, where ess sup denotes the essential
supremum. The duality pairing between H10 (Ω) and H

−1(Ω) will be denoted by 〈·, ·〉H−1(Ω),H10 (Ω) while
the inner product in R2 will be denoted by (·, ·). We denote the control space by U := L2(0,T; L2(Ω))
and the collection of measurable subdomains of Ω by E(Ω). We shall use L2(L2(Ω)),L2(H10 (Ω)) and

L∞(H10 (Ω)) as the short forms for L
2(0,T; L2(Ω)),L2(0,T; H10 (Ω)) and L

∞(0,T; H10 (Ω)), respectively.

2.2. Setup of the Problem. In this work, we consider the problem of controlling vibrations induced

by pedestrian-bridge interactions, see Figure 1.

The vibrations y(x,t) at position x and time t are governed by the wave equation:



OPTIMAL ACTUATOR PLACEMENT FOR CONTROL OF VIBRATIONS 3

ω1 ω2

Ω

Figure 1. Control of vibrations on the domain Ω using the supports at ω := ω1 ∪ω2.

∂2y

∂t2
−4y = χωu, (x,t) ∈ Ω × (0,T],

y = 0, (x,t) ∈ ∂Ω × (0,T], (2.1)

y(x, 0) = f(x),
∂y

∂t
(x, 0) = g(x), x ∈ Ω,

where u = u(x,t) denotes the control variable, χω the characteristic function for the domain ω ⊂ Ω, and
x ∈ R2. The domain ω represents the location of the actuators. It is not known where these supports
should be placed in order to control the vibrations on the bridge. The goal is to determine the optimal

location of these supports. The vibrations may depend on the initial conditions f and g, control variable

u, and subdomain ω. This leads to the cost functional J : E(Ω)×Uad×H10 (Ω)×L2(Ω) → R defined by

J(ω,u,f,g) :=

∫ T
0

1

2
‖yu,f,g,ω(·, t)‖2L2(Ω) +

1

2

∥∥∥∥dyu,f,g,ωdt (·, t)
∥∥∥∥2
L2(Ω)

+
α

2
‖χωu(·, t)‖2L2(Ω)dt, (2.2)

where α > 0 is a given parameter and Uad is the admissible set of controls consisting of a closed and

convex subset of U. The first and second terms in (2.2) suggest that we minimize the vibrations and

speed, respectively while the third term is the control cost.

Remark 2.1. The notation χωu(x,t) is used to stress the fact that u(x,t) is zero outside of ω.

Let ω,f and g be fixed. Then by taking the infimum of the cost J over all controls u ∈ Uad, we
obtain the functional J1 : E(Ω) ×H10 (Ω) ×L2(Ω) → R defined by

J1(ω,f,g) := inf
u∈Uad

J(ω,u,f,g). (2.3)

Note that the shape functional J1 depends on the initial conditions f and g. To overcome such a

dependence, we introduce a functional J2 : E(Ω) → R defined by

J2(ω) := sup
f∈K1,g∈K2

J1(ω,f,g), (2.4)

where K1 and K2 denote weakly compact subsets of H
1
0 (Ω) and L

2(Ω) defined by

K1 := {f : ‖f‖H10 (Ω)) ≤ 1} and K2 := {g : ‖g‖L2(Ω)) ≤ 1},

respectively. These conditions are used to average out the dependence of J1 on the initial conditions,

and overcome overflow for large values of f and g.

After introducing the two functionals in (2.3) and (2.4), we now study the problems of finding a

minimum cost functional for a fixed ω ⊂ Ω and a Lipschitz vector field X.



4 M. D. AROP, H. KASUMBA, J. KASOZI, AND F. BERNTSSON

Definition 2.1. The optimal actuator placement problems related to J1 and J2 are defined by the

minimization problems:

inf
X∈R2

J1((id + X)(ω),f,g) (2.5)

and

inf
X∈R2

J2((id + X)(ω)), (2.6)

where f ∈ K1, g ∈ K2 and (id + X)(ω) := {x + X : x ∈ ω}, respectively.

3. Well-Posedness of the Functionals

To simplify the analysis, we reformulate the wave equation as a system. Note that by setting

∂yu,f,g,ω

∂t
= vu,f,g,ω,

we can rewrite (2.1) as the following first-order system:

∂yu,f,g,ω

∂t
−vu,f,g,ω = 0, (x,t) ∈ Ω × (0,T],

∂vu,f,g,ω

∂t
−4yu,f,g,ω −χωu = 0, (x,t) ∈ Ω × (0,T],

yu,f,g,ω(x, 0) = f(x), vu,f,g,ω(x, 0) = g(x), x ∈ Ω,
yu,f,g,ω = 0, (x,t) ∈ ∂Ω × (0,T].

(3.1)

This reformulation is useful in the derivation of the optimality system and the discretization of the

optimization problems.

The well-posedness of (3.1) and hence, (2.1) is guaranteed by the following Lemma:

Lemma 3.1. Let f ∈ H10 (Ω),g ∈ L2(Ω) and χωu ∈ L2(L2(Ω)). Then the problem


〈
∂vu,f,g,ω

∂t
,φ

〉
H−1(Ω),H10 (Ω)

+
∫

Ω
Oyu,f,g,ω ·Oφ dx =

∫
Ω
χωuφ dx,

(
∂yu,f,g,ω

∂t
,ψ

)
=

(
vu,f,g,ω,ψ

)
,

(3.2)

for all φ ∈ L2(H10 (Ω)) and ψ ∈ L2(L2(Ω)) for a.e. t ∈ (0,T] with yu,f,g,ω(x, 0) = f(x), vu,f,g,ω(x, 0) =
g(x), has a unique weak solution yu,f,g,ω ∈ L2(H10 (Ω)) and vu,f,g,ω ∈ L2(L2(Ω)) with

∂vu,f,g,ω

∂t
∈ L2(H−1(Ω)).

Moreover, yu,f,g,ω ∈ L∞(H2∩H10 (Ω)) and vu,f,g,ω ∈ L∞(H10 ∩L2(Ω)), and there exists a constant c > 0
that depends on Ω and T such that

‖yu,f,g,ω‖L∞(H10 (Ω)) + ‖v
u,f,g,ω‖L∞(L2(Ω)) ≤ c

(
‖χωu‖L2(L2(Ω)) + ‖f‖H10 (Ω) + ‖g‖L2(Ω)

)
. (3.3)

Proof. It is well known that problem (3.2) has a unique and stable weak solution yu,f,g,ω ∈ L∞(H10 (Ω))∩
L2(H10 (Ω)) and v

u,f,g,ω ∈ L∞(L2(Ω)) ∩L2(L2(Ω)), see e.g., [6, Chap. 7] . �

Now, we establish the convergence of the sequence of solutions to (3.1).



OPTIMAL ACTUATOR PLACEMENT FOR CONTROL OF VIBRATIONS 5

Lemma 3.2. Suppose that {fn} is a sequence in K1 that converges weakly in H10 (Ω) to f ∈ K1, {gn}
is a sequence in K2 that converges weakly in L

2(Ω) to g ∈ K2 and {un} is a sequence in Uad that
converges weakly to a function u ∈ Uad. Then:

yun,fn,gn,ω → yu,f,g,ω in L2(H10 (Ω)) as n →∞,

vun,fn,gn,ω → vu,f,g,ω in L2(L2(Ω)) as n →∞.

Proof. Note that inequality (3.3) implies that the sequences {yun,fn,gn,ω} and {vun,fn,gn,ω} are bounded
in L2(H2(Ω) ∩ H10 (Ω)) and L2(H10 (Ω) ∩ L2(Ω)), respectively. By Rellich-Kondrachov theorem (see
e.g.,[1]), we can extract the subsequences again denoted by {yun,fn,gn,ω} and {vun,fn,gn,ω} such that
{yun,fn,gn,ω} converges weakly to yu,f,g,ω in L2(H2(Ω)∩H10 (Ω)) and strongly to yu,f,g,ω in L2(H10 (Ω)),
and {vun,fn,gn,ω} converges weakly to vu,f,g,ω in L2(H10 (Ω)) and strongly to vu,f,g,ω in L2(L2(Ω)). Thus,
replacing (u,f,g,ω) by (un,fn,gn,ω) in problem (3.2), we may pass to the limits and obtain by the

uniqueness that y = yu,f,g,ω and v = vu,f,g,ω. �

In the following lemma, we check that the optimization problem (2.3) is well-posed.

Lemma 3.3. Problem (2.3) admits a unique optimal solution u.

Proof. We refer to [19, Chap. 1]. �

The notation uf,g,ω will be used to indicate that u depends on f,g,ω.

Lemma 3.4. Suppose that {fn} is a sequence in H10 (Ω) that converges weakly to f in H10 (Ω) and {gn}
is a sequence in L2(Ω) that converges weakly to g in L2(Ω). Then we have

ufn,gn,ω → uf,g,ω in Uad as n →∞,

where uf,g,ω solves (2.3).

Proof. Since ufn,gn,ω minimizes J with (ω,f,g) replaced by (ω,fn,gn), for all u ∈ Uad and n ≥ 0, it
follows from (3.3) that we must have

1

2

∫ T
0

‖yu
fn,gn,ω,fn,gn,ω(·, t)‖2L2(Ω) + ‖v

ufn,gn,ω,fn,gn,ω(·, t)‖2L2(Ω) + α‖χωu
fn,gn,ω(·, t)‖2L2(Ω)dt

≤
1

2

∫ T
0

‖yu,fn,gn,ω(·, t)‖2L2(Ω) + ‖v
u,fn,gn,ω(·, t)‖2L2(Ω) + α‖χωu(·, t)‖

2
L2(Ω)dt, (3.4)

≤ c(‖χωu‖2L2(L2(Ω)) + ‖fn‖
2
H10 (Ω)

+ ‖gn‖2L2(Ω)).

This implies that {un} := {ufn,gn,ω} is bounded in Uad. By Rellich-Kondrachov theorem, we can extract
a subsequence {unk} such that unk ⇀ u in Uad as k →∞. Since u is a unique solution of J(ω, ·,f,g),
the whole sequence {un} converges weakly to u in Uad as n →∞. Thus, using Lemma 3.2 and by weak
lower semicontinuity of norms, we may pass to the limit infimum in (3.4) to obtain∫ T

0

‖yu,f,g,ω(·, t)‖2L2(Ω) + ‖v
u,f,g,ω(·, t)‖2L2(Ω) + α‖χωu(·, t)‖

2
L2(Ω)dt

≤
∫ T

0

‖yu,f,g,ω(·, t)‖2L2(Ω) + ‖v
u,f,g,ω(·, t)‖2L2(Ω) + α‖χωu(·, t)‖

2
L2(Ω)dt, (3.5)

for all u ∈ Uad. So, we must have u = uf,g,ω and since uf,g,ω is the minimizer of J(ω, ·,f,g) (see e.g.,
Lemma 3.3), the whole sequence {un} converges weakly to uf,g,ω. Therefore, un ⇀ uf,g,ω in Uad. As a
consequence of weak lower semicontinuity, we must have

‖uf,g,ω‖L2(L2(Ω)) ≤ limk→∞ inf ‖unk‖L2(L2(Ω)) = ‖u
f,g,ω‖L2(L2(Ω)).



6 M. D. AROP, H. KASUMBA, J. KASOZI, AND F. BERNTSSON

Thus, it follows from (3.5) that the norm ‖ufn,gn,ω‖L2(L2(Ω)) converges to ‖uf,g,ω‖L2(L2(Ω)). The weak
convergence and norm convergence of (un) imply that u

fn,gn,ω → uf,g,ω in Uad as n →∞. �

The following result will be used to characterize the optimal solution u.

Theorem 3.5. Suppose that Uad = U. Then we have the following optimality system:

∂yu,f,g,ω

∂t
−vu,f,g,ω = 0, (x,t) ∈ Ω × (0,T],

∂vu,f,g,ω

∂t
−4yu,f,g,ω −χωu = 0, (x,t) ∈ Ω × (0,T],

yu,f,g,ω(x, 0) = f, vu,f,g,ω(x, 0) = g, x ∈ Ω,
yu,f,g,ω = 0, (x,t) ∈ ∂Ω × (0,T],

(3.6)

∂pu,f,g,ω

∂t
−wu,f,g,ω = −vu,f,g,ω, (x,t) ∈ Ω × (0,T],

∂wu,f,g,ω

∂t
−4pu,f,g,ω = −yu,f,g,ω, (x,t) ∈ Ω × (0,T],

pu,f,g,ω(x,T) = 0, wu,f,g,ω(x,T) = 0, x ∈ Ω,
pu,f,g,ω = 0, (x,t) ∈ ∂Ω × (0,T]

(3.7)

and

αχωu−χωpu,f,g,ω = 0, (x,t) ∈ Ω × (0,T], (3.8)

where pu,f,g,ω ∈ L2(H10 (Ω)),wu,f,g,ω ∈ L2(L2(Ω)) and (yu,f,g,ω,vu,f,g,ω,u,pu,f,g,ω,wu,f,g,ω) solves
(3.6)–(3.8).

Proof. The optimality system (3.6)–(3.8) can be easily proved using standard techniques, see e.g.,

[12, Theorem 2.1], [19, Chap. 3]. �

Remark 3.1. Let Uad ( U. Then, instead of (3.8), we find the variational inequality∫
Ω×[0,T]

(αχωu−χωpu,f,g,ω)(u−u) dxdt ≥ 0, for all u ∈ Uad. (3.9)

The optimal solution u is now characterized using (3.9).

In the following lemma, the well-posedness of J2 is checked.

Lemma 3.6. Let K1 and K2 be two weakly compact sets containing the respective origins. Then for

every ω ∈ E(Ω), we can find f ∈ K1 and g ∈ K2 satisfying

‖f‖H10 (Ω) ≤ 1, ‖g‖L2(Ω) ≤ 1 and J2(ω) = J1(ω,f,g).

Proof. Note that 0 ∈ Uad. Let f ∈ K1 and g ∈ K2 with fixed ω ∈ E(Ω). Then in the absence of control,
using (3.3) we have

J1(ω,f,g) = min
u∈Uad

J(ω,u,f,g) ≤
∫ T

0

1

2
‖y0,f,g,ω(·, t)‖2L2(Ω) +

1

2
‖v0,f,g,ω(·, t)‖2L2(Ω) dt,

≤ c(‖f‖2H10 (Ω) + ‖g‖
2
L2(Ω)) ≤ cR

2,

(3.10)

where R =
√

2. Since f ∈ K1 and g ∈ K2, it follows that fR ∈ K1 and
g
R
∈ K2 with ‖fR‖H10 (Ω) ≤

1, ‖ g
R
‖L2(Ω) ≤ 1. Next, we show that J2(ω) = J1(ω,f,g). From (2.4), we have

J2(ω) = sup
f∈K1,g∈K2

∫ T
0

1

2
‖yu

f,g,ω,f,g,ω(·, t)‖2L2(Ω) +
1

2
‖vu

f,g,ω,f,g,ω(·, t)‖2L2(Ω)

+
α

2
‖χωuf,g,ω(·, t)‖2L2(Ω) dt. (3.11)



OPTIMAL ACTUATOR PLACEMENT FOR CONTROL OF VIBRATIONS 7

Let {fn} ⊂ K1, ‖fn‖H10 (Ω) ≤ 1 and {gn} ⊂ K2, ‖gn‖L2(Ω) ≤ 1 be maximizing sequences. Then, (3.11)
can be written as

J2(ω) = lim
n→∞

∫ T
0

1

2
‖yu

fn,gn,ω,fn,gn,ω(·, t)‖2L2(Ω) +
1

2
‖vu

fn,gn,ω,fn,gn,ω(·, t)‖2L2(Ω)

+
α

2
‖χωufn,gn,ω(·, t)‖2L2(Ω) dt. (3.12)

Since {fn} and {gn} are bounded in K1 and K2, respectively, a subsequence {fnk} converges weakly to
f ∈ K1; {gnk} converges weakly to g ∈ K2. Since {fn}⊂ K1 and {gn}⊂ K2, the limit elements satisfy

‖f‖H10 (Ω) ≤ limk→∞
inf ‖fnk‖H10 (Ω) ≤ 1, ‖g‖L2(Ω) ≤ limk→∞

inf ‖gnk‖L2(Ω) ≤ 1,

by lower semicontinuity of norms. Thus, ‖f‖H10 (Ω) ≤ 1 and ‖g‖L2(Ω) ≤ 1. Since {fnk} and {gnk} are
bounded in H10 (Ω) and L

2(Ω), respectively, fnk ⇀ f ∈ H
1
0 (Ω) and gnk ⇀ g in L

2(Ω). From Lemma

3.2, we note that {yun,fn,gn,ω} converges strongly to yu,f,g,ω in L2(H10 (Ω)) and {vun,fn,gn,ω} converges
strongly to vu,f,g,ω in L2(L2(Ω)), and from Lemma 3.4, ufn,gn,ω → uf,g,ω in Uad as n → ∞. Thus, by
lower semicontinuity, we have norm convergence. Hence, we may pass to the limit in (3.12) and obtain

J2(ω) =
1

2

∫ T
0

‖yu
f,g,ω,f,g,ω(·, t)‖2L2(Ω) + ‖v

uf,g,ω,f,g,ω(·, t)‖2L2(Ω) + α‖χωu
f,g,ω(·, t)‖2L2(Ω)dt

= J1(ω,f,g).

Since f ∈ K1 and g ∈ K2 satisfy

‖f‖H10 (Ω) ≤ 1, ‖g‖L2(Ω) ≤ 1, maxf∈K1,g∈K2
J1(ω,f,g) =: J2(ω) = J1(ω,f,g),

it follows that the map ω 7→ J2(ω) is well-posed. �

4. Sensitivity Analysis of the Functionals

4.1. Shape Derivative. In order to compute the shape derivatives of J1 and J2, we introduce a

perturbation of the identity. Consider the space C̊0,1(Ω,R2) of Lipschitz vector fields vanishing on ∂Ω.
We define a perturbation of the identity Tτ (x) by Tτ (x) := x + τX(x), where x ∈ Ω, X ∈ C̊0,1(Ω,R2)
and τ is the perturbation parameter [5, p.175]. In view of the perturbation of the identity, we give the

definition of a shape derivative of J as follows.

Definition 4.1. The directional derivative of J at ω ∈ E(Ω) in the direction X ∈ C̊0,1(Ω,R2) is defined
by

DJ(ω)(X) := lim
τ↘0

J(Tτ (ω)) −J(ω)
τ

,

provided the limit exists.

Remark 4.1. The cost functional J is shape differentiable at ω if X 7→ DJ(ω)(X) is linear and continuous
for all X ∈ C̊0,1(Ω,R2), see e.g., [5] and [4].

4.2. Sensitivity of the State Equation. The space-time cylinder and its boundary will be denoted

by ΩT := Ω × (0,T] and ΓT := Γ × (0,T], respectively. The sensitivity of the solution of (3.1) is given
in the following lemma.



8 M. D. AROP, H. KASUMBA, J. KASOZI, AND F. BERNTSSON

Lemma 4.1. Let Tτ = id + τX,τ ≥ 0. Suppose that ω is perturbed such that ωτ := Tτ (ω),ω ∈ E(Ω).
Then on the perturbed domain Ωτ × (0,T] with Ωτ := Tτ (Ω),τ ≥ 0, we have

∂yu,f,g,τ

∂t
−vu,f,g,τ = 0 in ΩT , (4.1)

∂vu,f,g,τ

∂t
−

1

ζ(τ)
div(A(τ)∇yu,f,g,τ ) = χωu in ΩT , (4.2)

yu,f,g,τ (x, 0) = f(x) ◦ Tτ, vu,f,g,τ (x, 0) = g(x) ◦ Tτ in Ω, (4.3)

yu,f,g,τ = 0 on ΓT , (4.4)

where

A(τ) := ζ(τ)(∂Tτ )
−1(∂Tτ )

−>, ζ(τ) := |det(∂Tτ )|. (4.5)

Proof. In view of (3.1) with ωτ := Tτ (ω),ω ∈ E(Ω), we have
∂yu,f,g,ωτ

∂t
−vu,f,g,ωτ = 0 in ΩT , (4.6)

∂vu,f,g,ωτ

∂t
−4yu,f,g,ωτ = χωτu in ΩT , (4.7)

yu,f,g,ωτ (x, 0) = f(x),vu,f,g,ωτ (x, 0) = g(x) in Ω, (4.8)

yu,f,g,ωτ = 0 on ΓT , (4.9)

where ωτ ⊂ Ω. Thus, considering (4.7) on the perturbed domain Ωτ × (0,T] with Ωτ = Tτ (Ω),τ ≥ 0,
we get the perturbed weak formulation:∫

Ωτ×(0,T]

∂vu,f,g,ωτ

∂t
ϕ dxτdt +

∫
Ωτ×(0,T]

∇yu,f,g,ωτ ·∇ϕ dxτdt =
∫

Ωτ×(0,T]
χωτuϕ dxτ dt, (4.10)

for all ϕ ∈ L2(H10 (Ωτ )) with (yu,f,g,ωτ ,vu,f,g,ωτ ) satisfying (4.6)–(4.9). Next, employing a change of
variables induced by Ωτ := Tτ (Ω) in (4.10) gives∫

ΩT

ζ(τ)
∂(vu◦T

−1
τ ,f,g,ωτ ◦ Tτ )
∂t

(ϕ◦ Tτ ) dxdt +
∫

ΩT

ζ(τ)O(yu◦T
−1
τ ,f,g,ωτ ◦ Tτ ) ·O(ϕ◦ Tτ ) dxdt

=

∫
ΩT

ζ(τ)(χωτu◦ Tτ )(ϕ◦ Tτ ) dxdt, for all ϕ ∈ L
2(H10 (Ωτ )). (4.11)

Applying the chain rule (see e.g., [17, p.63]) in (4.11) together with χωτ = χω◦T−1τ and the perturbed
variables (see e.g., [5, p.523])

yu,f,g,τ = yu◦T
−1
τ ,f,g,ωτ ◦ Tτ, vu,f,g,τ = vu◦T

−1
τ ,f,g,ωτ ◦ Tτ, (4.12)

yield ∫
ΩT

ζ(τ)
∂vu,f,g,τ

∂t
(ϕ◦ Tτ ) + ζ(τ)(∂Tτ )−>∇yu,f,g,τ · (∂Tτ )−>∇(ϕ◦ Tτ ) dxdt

=

∫
ΩT

ζ(τ)(χωu)(ϕ◦ Tτ ) dxdt, for all ϕ ∈ L2(H10 (Ωτ )). (4.13)

From (4.5), equality (4.13) simplifies to∫
ΩT

ζ(τ)
∂vu,f,g,τ

∂t
(ϕ◦ Tτ ) dxdt +

∫
ΩT

A(τ)∇yu,f,g,τ ·∇(ϕ◦ Tτ ) dxdt

=

∫
ΩT

ζ(τ)(χωu)(ϕ◦ Tτ ) dxdt, for all ϕ ∈ L2(H10 (Ωτ )). (4.14)



OPTIMAL ACTUATOR PLACEMENT FOR CONTROL OF VIBRATIONS 9

Since (4.14) is true for all ϕ ∈ L2(H10 (Ωτ )), it follows that for all φ ∈ L2(H10 (Ω)) the function φ◦ T−1τ
belongs to L2(H10 (Ωτ )). So, testing (4.14) with ϕ := φ◦T−1τ for an arbitrary φ ∈ L2(H10 (Ω)), we obtain∫

ΩT

ζ(τ)
∂vu,f,g,τ

∂t
(φ◦ T−1τ ◦ Tτ ) + A(τ)∇y

u,f,g,τ ·∇(φ◦ T−1τ ◦ Tτ ) dxdt

=

∫
ΩT

ζ(τ)(χωu)(φ◦ T−1τ ◦ Tτ ) dxdt, for all φ ∈ L
2(H10 (Ω)). (4.15)

Rewriting (4.15), we have∫
ΩT

ζ(τ)
∂vu,f,g,τ

∂t
φ dxdt +

∫
ΩT

A(τ)∇yu,f,g,τ ·∇φ dxdt

=

∫
ΩT

ζ(τ)χωuφ dxdt, for all φ ∈ L2(H10 (Ω)). (4.16)

Similarly, considering (4.6) on Ωτ × (0,T], it can be shown that:∫
ΩT

ζ(τ)
∂yu,f,g,τ

∂t
ψ − ζ(τ)vu,f,g,τψ dxdt = 0, for all ψ ∈ L2(L2(Ω)). (4.17)

Thus, after mapping back (4.16) and (4.17), and using (4.8)–(4.9) in (4.12), we have (4.1)–(4.4). �

In the following essential lemma, the sequence {τn}∞n=1 will be necessary.

Lemma 4.2. Let X ∈ C̊0,1(Ω,R2).
(a) Then as τn → 0+, we have

ζ(τn) − 1
τn

→ div(X) strongly in L∞(Ω), (4.18)

A(τn) − I
τn

→ div(X)I −∂X −∂X> strongly in L∞(Ω,R2×2), (4.19)

where I is the 2-dimensional identity matrix.

(b) Suppose that {Ψn} is a sequence in H10 (Ω) converging weakly to Ψ ∈ H10 (Ω).
(i) Then for all Ψ ∈ H10 (Ω), we have as τ → 0+,

Ψn ◦ Tτ → Ψ strongly in H10 (Ω). (4.20)

(ii) If {τn} is a null sequence, then as n →∞ we have
Ψn ◦ Tτn − Ψn

τn
⇀ ∇Ψ · X weakly in H10 (Ω). (4.21)

Proof. The results of the convergence (4.18), (4.19), and (4.21) are proved in [17]: Lemma 2.31, p.107

and proposition 2.72, respectively while (4.20) is proved in [5, p.527]. �

Remark 4.2. There are constants c1,c2 > 0 such that for all x ∈ Ω and τ ∈ [0,τX],τX ≥ 0,

c1 ≤ ζ(τ)(x), c2|ζ|2 ≤ A(τ)(x)ζ · ζ, (4.22)

for all ζ ∈ R2, see e.g., [5, p.559].

The following lemma gives the a-priori estimates for yu,f,g,ωτ , yu,f,g,τ , vu,f,g,ωτ and vu,f,g,τ .

Lemma 4.3. For all (u,f,g,ω) ∈ Uad × H10 (Ω) × L2(Ω) × E(Ω), there exists a constant c > 0, such
that

‖yu,f,g,ωτ‖L2(H10 (Ω)) + ‖v
u,f,g,ωτ‖L2(L2(Ω)) ≤ c

(
‖χωτu‖L2(L2(Ω)) + ‖f‖H10 (Ω) + ‖g‖L2(Ω)

)
, (4.23)



10 M. D. AROP, H. KASUMBA, J. KASOZI, AND F. BERNTSSON

‖yu,f,g,τ‖L2(H10 (Ω)) + ‖v
u,f,g,τ‖L2(L2(Ω)) ≤ c

(
‖χωu‖L2(L2(Ω)) + ‖f‖H10 (Ω) + ‖g‖L2(Ω)

)
. (4.24)

Proof. Note that (4.23) is a consequence of (3.3) and the proof is omitted here. We prove (4.24) as

follows. By a change of variables, we have∫ T
0

‖yu,f,g,τ (·, t)‖2L2(Ω) + ‖∇y
u,f,g,τ (·, t)‖2L2(Ω) dt

=

∫
ΩT

|yu,f,g,τ|2 + |∇yu,f,g,τ|2 dxdt,

=

∫
ΩT

ζ−1(τ)|yu,f,g,τ ◦T−1τ |
2 + ζ−1(τ)∇yu,f,g,τ ◦T−1τ ·∇y

u,f,g,τ ◦T−1τ dxdt, (4.25)

=

∫
ΩT

ζ−1(τ)|yu◦T
−1
τ ,f,g,ωτ |2 + A−1(τ)∇yu◦T

−1
τ ,f,g,ωτ ·∇yu◦T

−1
τ ,f,g,ωτ dxdt,

(4.22)

≤ c
∫

ΩT

|yu◦T
−1
τ ,f,g,ωτ |2 + ∇yu◦T

−1
τ ,f,g,ωτ ·∇yu◦T

−1
τ ,f,g,ωτ dxdt,

(4.23)

≤ c
(
‖χωτu◦ T

−1
τ ‖

2
L2(L2(Ω)) + ‖f‖

2
H10 (Ω)

+ ‖g‖2L2(Ω)
)
.

Using χωτ = χω ◦T−1τ and the natural norm on H1(Ω), i.e.,∫ T
0

‖yu,f,g,τ (·, t)‖2L2(Ω) + ‖∇y
u,f,g,τ (·, t)‖2L2(Ω) dt = ‖y

u,f,g,τ‖2L2(H1(Ω)),

in (4.25) (see e.g., [3, p.39]), we obtain the desired inequality. �

For the continuity results of (u,f,g,τ) 7→ yu,f,g,τ and (u,f,g,τ) 7→ vu,f,g,τ , we prove the lemma that
follows.

Lemma 4.4. For every (ω1,u1,f1,g1), (ω2,u2,f2,g2) ∈ E(Ω) × Uad × H10 (Ω) × L2(Ω), with (y1,v1)
and (y2,v2) being the corresponding solutions to (4.6)–(4.9), there is a constant c > 0, independent of

(ω1,u1,f1,g1) and (ω2,u2,f2,g2), such that

‖y1 −y2‖L2(H10 (Ω)) + ‖v1 −v2‖L2(L2(Ω))

≤ c
(
‖χω1u1 −χω2u2‖L2(L2(Ω)) + ‖f1 −f2‖H10 (Ω) + ‖g1 −g2‖L2(Ω)

)
. (4.26)

Proof. Since (y1,v1) and (y2,v2) solve (4.6)–(4.9), it follows that they satisfy

∂yk
∂t
−vk = 0 in ΩT ,

∂vk
∂t
−4yk = χωkuk in ΩT ,

yk(x, 0) = fk(x),vk(x, 0) = gk(x) in Ω,

yk = 0 on ΓT ,

for all k = 1, 2. Let y12 := y1 −y2 and v12 := v1 −v2. Then (y12,v12) satisfies
∂y12
∂t
−v12 = 0 in ΩT ,

∂v12
∂t
−4y12 = χω1u1 −χω2u2 in ΩT ,

y12(x, 0) = f1(x) −f2(x),v12(x, 0) = g1(x) −g2(x) in Ω,
y12 = 0 on ΓT .



OPTIMAL ACTUATOR PLACEMENT FOR CONTROL OF VIBRATIONS 11

Hence, (4.26) follows from (3.3). �

The following lemma is an immediate consequence of Lemma 4.4.

Lemma 4.5. Let ω ∈ E(Ω) be given. Suppose that for all τn ∈ (0,τX], un,u ∈ Uad, fn,f ∈ H10 (Ω) and
gn,g ∈ L2(Ω),

un ⇀ u in Uad, fn ⇀ f in H
1
0 (Ω),gn ⇀ g in L

2(Ω), τn → 0, as n →∞.

Then:

yun,fn,gn,τn → yu,f,g,ω in L2(H10 (Ω)) as n →∞,

vun,fn,gn,τn → vu,f,g,ω in L2(L2(Ω)) as n →∞.

Proof. Using inequality (4.24), we see that the sequences {yun,fn,gn,τn} and {vun,fn,gn,τn} are bounded
in L2(H2(Ω) ∩H10 (Ω)) and L2(H10 (Ω) ∩L2(Ω)), respectively. By Rellich-Kondrachov theorem, we can
extract subsequences {yunk,fnk,gnk,τnk} and {vunk,fnk,gnk,τnk} such that {yunk,fnk,gnk,τnk} converges
weakly to yu,f,g,ω in L2(H2(Ω) ∩H10 (Ω)) and strongly to yu,f,g,ω in L2(H10 (Ω)), and {v

unk,fnk,gnk,τnk}
converges weakly to vu,f,g,ω in L2(H10 (Ω)) and strongly to v

u,f,g,ω in L2(L2(Ω)). From (4.16) and (4.17),

it is known that (yk,vk) with yk := y
unk,fnk,gnk,τnk and vk := v

unk,fnk,gnk,τnk , k ∈{0}∪N satisfies the
variational formulations∫

ΩT

ζ(τnk)
∂vk
∂t

ϕ + A(τnk)∇yk ·∇ϕ dxdt =
∫

ΩT

ζ(τnk)χωunkϕ dxdt,∫
ΩT

ζ(τnk)
∂yk
∂t

ψ dxdt−
∫

ΩT

ζ(τnk)vkψ dxdt = 0, (4.27)

for all ϕ ∈ L2(H10 (Ω)) and ψ ∈ L2(L2(Ω)) with yk(x, 0) = fnk(x) ◦ Tτnk and vk(x, 0) = gnk(x) ◦ Tτnk
in Ω. From Lemma 4.2, it follows that fnk(x) ◦ Tτnk → f(x) in H

1
0 (Ω) and gnk(x) ◦ Tτnk → g(x) in

L2(Ω) as k → ∞. Thus, we have y(x, 0) = f(x) and v(x, 0) = g(x). Using the weak convergence of
{unk}, {yk}, {vk} and the strong convergence in Lemma 4.2, i.e., ζ(τnk) → 1 in L

∞(Ω), A(τnk) → I
in L∞(Ω,R2×2) as k →∞, we pass to the limits in (4.27) and obtain∫

ΩT

∂v

∂t
ϕ + ∇y ·∇ϕ dxdt =

∫
ΩT

χωuϕ dxdt,∫
ΩT

∂y

∂t
ψ dxdt−

∫
ΩT

vψ dxdt = 0, (4.28)

for all ϕ ∈ L2(H10 (Ω)) and ψ ∈ L2(L2(Ω)) with y(x, 0) = f(x),v(x, 0) = g(x). Furthermore, since
(4.28) with y(x, 0) = f(x),v(x, 0) = g(x) admits a unique solution, we must have y = yu,f,g,ω and

v = vu,f,g,ω. Thus, the sequences {yn} and {vn} converge to y = yu,f,g,ω in L2(H10 (Ω)) and v = vu,f,g,ω
in L2(L2(Ω)), respectively. This finishes the proof. �

The following lemmas will be employed in the proof of the theorem that follows.

Lemma 4.6. For every null-sequence {τn} in [0,τX], every sequence {fn} in K1 converging weakly in
H10 (Ω) to f ∈ K1 and for every sequence {gn} in K2 converging weakly in L2(Ω) to g ∈ K2, we have

ufn,gn,τn → uf,g,ω in Uad as n →∞.

Proof. We proceed as follows. Note that ωτn, u
fn,gn,ωτn and ufn,gn,τn represent the perturbed domain,

optimal control solution, and perturbed optimal control, respectively. Since ufn,gn,τn = ufn,gn,ωτn ◦Tτn
(see e.g., [5, p.523]) and ufn,gn,ωτn → uf,g,ω in Uad by Lemma 3.4, the desired result follows from
Lemma 4.2. �



12 M. D. AROP, H. KASUMBA, J. KASOZI, AND F. BERNTSSON

In the sequel, we denote the set of maximizers by X2(ω).

Lemma 4.7. For every null sequence {τn} in [0,τX] and every sequence {fn,gn} with (fn,gn) ∈
X2(ωτn), we can find a subsequence {fnk,gnk}, such that fnk ⇀ f in H

1
0 (Ω) and gnk ⇀ g in L

2(Ω) as

k →∞, where (f,g) ∈ X2(ω).

Proof. It is easy to prove this from (2.4) and the proof is left out. �

4.3. Averaged Adjoint Equations. Let τ ∈ [0,τX] be fixed. Then the mapping T−1τ : Uad →
Uad, u 7→ T−1τ ◦ u is a bijection between Uad and Uad that preserves the binary operations. As a
consequence and using the change of variables Tτ , it is easy to show that

inf
u∈Uad

J(ωτ,u,f,g) =
1

2
inf

u∈Uad

∫
ΩT

ζ(τ)

(
|yu,f,g,τ|2 + |vu,f,g,τ|2 + α|u|2

)
dxdt.

Note that p ∈ L2(H10 (Ω)) and w ∈ L2(L2(Ω)). By choosing Lagrange multipliers φ = p and ψ = w,
we can incorporate (4.1)–(4.4) in the formulation of the following Lagrangian functional.

Definition 4.2. Define the parametrized Lagrangian

H̃ : [0,τX] ×Uad ×K1 ×K2 ×H10 (Ω) ×L2(Ω) ×H10 (Ω) ×L2(Ω) → R by

H(τ,u,f,g) :=

∫
ΩT

1

2
ζ(τ)

(
(yu,f,g,τ )2 + (vu,f,g,τ )2 + α(u)2

)
dxdt

+

∫
ΩT

ζ(τ)
∂vu,f,g,τ

∂t
pu,f,g,τ + A(τ)∇yu,f,g,τ ·∇pu,f,g,τ − ζ(τ)χωupu,f,g,τ

+ ζ(τ)
∂yu,f,g,τ

∂t
wu,f,g,τ − ζ(τ)vu,f,g,τwu,f,g,τ dxdt (4.29)

+

∫
Ω

ζ(τ)(yu,f,g,τ (x, 0) −f ◦ Tτ )wu,f,g,τ (x, 0) + ζ(τ)(vu,f,g,τ (x, 0) −g ◦ Tτ )pu,f,g,τ (x, 0)dx,

where H(τ,u,f,g) := H̃(τ,u,f,g,yu,f,g,τ,vu,f,g,τ,pu,f,g,τ,wu,f,g,τ ).

In the sequel, the following definition is used.

Definition 4.3. Given τ ∈ [0,τX], 0 ≤ s ≤ 1 and (u,f,g) ∈ Uad ×K1 ×K2. We define the averaged
adjoint equations associated with yu,f,g,τ and yu,f,g,ω; vu,f,g,τ and vu,f,g,ω as: find pu,f,g,τ ∈ L2(H10 (Ω))
and wu,f,g,τ ∈ L2(L2(Ω)) such that∫ 1

0

∂yH̃(τ,u,f,g,sy
u,f,g,τ + (1 −s)yu,f,g,ω,vu,f,g,τ,pu,f,g,τ,wu,f,g,τ )(φ)ds = 0, (4.30)

for all φ ∈ L2(H10 (Ω)), and∫ 1
0

∂vH̃(τ,u,f,g,y
u,f,g,τ,svu,f,g,y,τ + (1 −s)vu,f,g,ω,pu,f,g,τ,wu,f,g,τ )(ψ)ds = 0, (4.31)

for all ψ ∈ L2(L2(Ω)), where ∂yH̃ and ∂vH̃ denote the partial derivatives of H̃ with respect to y and
v, respectively.

The following lemmas will be important in the proof of the theorem that follows.



OPTIMAL ACTUATOR PLACEMENT FOR CONTROL OF VIBRATIONS 13

Lemma 4.8. The averaged adjoint equations (4.30) and (4.31), associated with yu,f,g,τ and yu,f,g,ω;

vu,f,g,τ and vu,f,g,ω are given by∫
ΩT

−ζ(τ)φ
∂wu,f,g,τ

∂t
dxdt +

∫
ΩT

A(τ)∇φ ·∇pu,f,g,τ dxdt

= −
∫

ΩT

1

2
ζ(τ)(yu,f,g,τ + yu,f,g,ω)φ dxdt, for all φ ∈ L2(H10 (Ω)) (4.32)

and ∫
ΩT

−ζ(τ)ψ
(
∂pu,f,g,τ

∂t
+ wu,f,g,τ

)
dxdt = −

∫
ΩT

1

2
ζ(τ)(vu,f,g,τ + vu,f,g,ω)ψ dxdt, (4.33)

for all ψ ∈ L2(L2(Ω)), respectively.

Proof. Since yu,f,g,τ 7→ H̃(τ,u,f,g,yu,f,g,τ,vu,f,g,τ,pu,f,g,τ,wu,f,g,τ ) is affine, H̃ is Gâteaux differen-
tiable with respect to y, see e.g., [19, p.200]. Thus, it is easy to see that (4.32) and (4.33) hold. �

The lemma that follows is a direct consequence of Lemmas 4.5 and 4.8.

Lemma 4.9. For all τn ∈ (0,τX], un ∈ Uad, fn ∈ K1 and gn ∈ K2, such that

un ⇀ u in Uad, fn ⇀ f in H
1
0 (Ω),gn ⇀ g in L

2(Ω), τn → 0, as n →∞,

where u ∈ Uad, f ∈ K1 and g ∈ K2, we have

pun,fn,gn,τn → pu,f,g,ω in L2(H10 (Ω)) as n →∞,

wun,fn,gn,τn → wu,f,g,ω in L2(L2(Ω)) as n →∞,

with pu,f,g,ω ∈ L2(H10 (Ω)) and wu,f,g,ω ∈ L2(L2(Ω)) satisfying the adjoint equations∫
ΩT

−φ
∂wu,f,g,ω

∂t
dxdt +

∫
ΩT

∇φ ·∇pu,f,g,ω dxdt = −
∫

ΩT

yu,f,g,ωφ dxdt,∫
ΩT

−ψ
∂pu,f,g,ω

∂t
−ψwu,f,g,ω dxdt = −

∫
ΩT

vu,f,g,ωψ dxdt,

for all φ ∈ L2(H10 (Ω)) and ψ ∈ L2(L2(Ω)) with pu,f,g,ω(x,T) = 0 and wu,f,g,ω(x,T) = 0 a.e. in Ω.

Proof. Using (3.6)–(3.7) and the estimate in [6, p.391-393, Theorem 6] , we have the a-priori bound for

the adjoint given by

‖pu,f,g,ω‖L2(H10 (Ω)) +
∥∥∥∥∂pu,f,g,ω∂t

∥∥∥∥
L2(L2(Ω))

≤ c‖vu,f,g,τ + vu,f,g,ω‖L2(L2(Ω)). (4.34)

Using similar arguments as in Lemma 4.5 and replacing (u,f,g,τ) by (un,fn,gn,τn) in (4.32) and

(4.33), and passing to the limits as n →∞, we have the desired result. �

4.4. Directional Derivative of Max-Min Functions. Let H : [0,τX] ×Uad ×K1 ×K2 → R be a
function. Then, we define the max-min function h : [0,τX] → R by

h(τ) := sup
f∈K1,g∈K2

inf
u∈Uad

H(τ,u,f,g).

In the following lemma, we seek to find out sufficient conditions for the existence of the limit

d

d`
h(0+) := lim

τ↘0+
h(τ) −h(0)

`(0)
,

for any function ` : [0,τX] → R such that `(τ) > 0 for τ ∈ (0,τX], and `(0) = 0.

Lemma 4.10. Assume that the following hypotheses hold.



14 M. D. AROP, H. KASUMBA, J. KASOZI, AND F. BERNTSSON

(H0) The problem

inf
u∈Uad

H(τ,u,f,g)

admits a unique optimal solution u.

(H1) The set of maximizers

X2(ω) := {(f,g) : sup
f∈K1,g∈K2

inf
u∈Uad

H(τ,u,f,g) = inf
u∈Uad

H(τ,uτ,f,g,f,g)}

is nonempty for all τ ∈ [0,τX].
(H2) For all f ∈ K1,g ∈ K2 and τ ∈ [0,τX], the partial derivatives

lim
τ↘0

H(τ,uτ,f,g,f,g) −H(0,uτ,f,g,f,g)
`(τ)

and

lim
τ↘0

H(τ,u0,f,g,f,g) −H(0,u0,f,g,f,g)
`(τ)

exist and are equal.

(H3) For all τn ∈ [0,τX] and (fn,gn) ∈ X2(ωn), there exist subsequences {τnk} and {fnk,gnk} with
fnk ⇀ f in H

1
0 (Ω) and gnk ⇀ g in L

2(Ω) as k →∞ and (f,g) ∈ X2(ω), such that

lim
k→∞

H(τnk,unk,fnk,gnk) −H(0,unk,fnk,gnk)
`(τnk)

= ∂`H(0
+,u0,f,g,f,g)

and

lim
k→∞

H(τnk,u
fnk,gnk,0,fnk,gnk) −H(0,u

fnk,gnk,0,fnk,gnk)

`(τnk)
= ∂`H(0

+,uf,g,0,f,g).

Then, we have

d

d`
h(τ)|τ=0+ = max

(f,g)∈X2(ω)
∂`H(0

+,u0,f,g,f,g).

Proof. We refer to [5, p.524] and [18]. �

In the following theorem, we derive the directional derivative of J2 for `(τ) = τ.

Theorem 4.11. The directional derivative of J2(ω) at ω in the direction X ∈ C̊0,1(Ω,R2) is given by

DJ2(ω)(X) = max
(f,g)∈X2(ω)

∫
ΩT

S1

(
yf,g,ω,vf,g,ω,pf,g,ω,wf,g,ω,uf,g,ω

)
: ∂X + S0(f,g) · X dxdt, (4.35)

where

S1

(
yf,g,ω,vf,g,ω,pf,g,ω,wf,g,ω,uf,g,ω

)
:=

(
1

2
|yf,g,ω|2 +

1

2
|vf,g,ω|2 +

α

2
|uf,g,ω|2 −vf,g,ω

∂pf,g,ω

∂t

−yf,g,ω
∂wf,g,ω

∂t
+ ∇yf,g,ω ·∇pf,g,ω −χωuf,g,ωpf,g,ω −vf,g,ωwf,g,ω

−
1

T
gpf,g,ω(x, 0) −

1

T
fwf,g,ω(x, 0)

)
I −∇yf,g,ω ⊗∇pf,g,ω −∇pf,g,ω ⊗∇yf,g,ω,

S0(f,g) := −
1

T

(
∇fwf,g,ω(x, 0) + ∇gpf,g,ω(x, 0)

)
, (4.36)

and the adjoint (pf,g,ω,wf,g,ω) satisfies (3.6)–(3.7).



OPTIMAL ACTUATOR PLACEMENT FOR CONTROL OF VIBRATIONS 15

Proof. Since J1 and J2 are well-posed, it follows that (H0) and (H1) are satisfied. Next, we check

that (H2) and (H3) hold. Using the fundamental theorem of calculus on averaged adjoint equations

(4.30)–(4.31), it is easy to see that

H̃(τ,u,f,g,yu,f,g,τ,vu,f,g,τ,pu,f,g,τ,wu,f,g,τ ) = H̃(τ,u,f,g,yu,f,g,ω,vu,f,g,ω,pu,f,g,τ,wu,f,g,τ ). (4.37)

Since J(ωτ,u◦ T−1τ ,f,g) = H̃(τ,u,f,g,yu,f,g,τ,vu,f,g,τ,pu,f,g,τ,wu,f,g,τ ), it follows from (4.37) that

J(ωτ,u◦ T−1τ ,f,g) = H̃(τ,u,f,g,y
u,f,g,ω,vu,f,g,ω,pu,f,g,τ,wu,f,g,τ ).

Hence,

J1(ωτ,f,g) = inf
u∈Uad

H̃(τ,u,f,g,yu,f,g,ω,vu,f,g,ω,pu,f,g,τ,wu,f,g,τ ). (4.38)

Choosing u := uf,g,τ in (4.38) with

(τ,u,f,g,yu,f,g,ω,vu,f,g,ω,pu,f,g,τ,wu,f,g,τ )

replaced by

(τn,un,fn,gn,y
un,fn,gn,ω,vun,fn,gn,ω,pun,fn,gn,τn,wun,fn,gn,τn)

and substituting in (4.29), we have

H(τn,un,fn,gn) =

∫
ΩT

1

2
ζ(τn)(|yun,fn,gn,ω|2 + |vun,fn,gn,ω|2 + α|un|2) dxdt

+

∫
ΩT

ζ(τn)
∂vun,fn,gn,ω

∂t
pun,fn,gn,τn + A(τn)∇yun,fn,gn,ω ·∇pun,fn,gn,τn dxdt

+

∫
ΩT

(
− ζ(τn)χωunpun,fn,gn,τn + ζ(τn)

∂yun,fn,gn,ω

∂t
wun,fn,gn,τn

− ζ(τn)vun,fn,gn,ωwun,fn,gn,τn
)
dxdt +

∫
Ω

[
ζ(τn)

(
yun,fn,gn,ω(x, 0)

−fn ◦ Tτn
)
wun,fn,gn,τn(x, 0) + ζ(τn)

(
vun,fn,gn,ω(x, 0) −gn ◦ Tτn

)
pun,fn,gn,τn(x, 0)

]
dx. (4.39)

From (4.5) as τn → 0+, we have ζ(τn) → 1,A(τn) → I. Utilizing this result in (4.39), and re-arranging
the terms, we obtain

H(τn,un,fn,gn) −H(0,un,fn,gn)
τn

=

∫
ΩT

ζ(τn) − 1
τn

·
1

2

(
|yun,fn,gn,ω|2 + |vun,fn,gn,ω|2 + α|un|2

)
dxdt

+

∫
ΩT

ζ(τn) − 1
τn

∂vun,fn,gn,ω

∂t
pun,fn,gn,τn +

A(τn) − I
τn

∇yun,fn,gn,ω ·∇pun,fn,gn,τn

−
ζ(τn) − 1

τn
χωunp

un,fn,gn,τn dxdt +

∫
ΩT

ζ(τn) − 1
τn

(
∂yun,fn,gn,ω

∂t
wun,fn,gn,τn

−vun,fn,gn,ωwun,fn,gn,τn
)
dxdt +

∫
Ω

ζ(τn) − 1
τn

((
yun,fn,gn,ω(x, 0)

−fn ◦ Tτn
)
wun,fn,gn,τn(x, 0) +

(
vun,fn,gn,ω(x, 0) −gn ◦ Tτn

)
pun,fn,gn,τn(x, 0)

)
dx

−
∫

Ω

(
fn ◦ Tτn −fn

τn
wun,fn,gn,τn(x, 0) +

gn ◦ Tτn −gn
τn

pun,fn,gn,τn(x, 0)

)
dx. (4.40)



16 M. D. AROP, H. KASUMBA, J. KASOZI, AND F. BERNTSSON

Note that

gn ◦ Tτn =
∂

∂t

(
yu◦T

−1
τn
,fn,gn,ω(x, 0) ◦ Tτn

)
and

gn ◦ Tτn −gn
τn

=
∂

∂t

(
yu◦T

−1
τn
,fn,gn,ω(x, 0) ◦ Tτn −y

u◦T−1τn ,fn,gn,ω(x, 0)

τn

)
since Tτ is independent of t. Using these results, Lemmas 4.2 and 4.9, the right-hand side of (4.40)

converges to ∫
ΩT

div(X)

(
1

2
|yf,g,ω|2 +

1

2
|vf,g,ω|2 +

α

2
|uf,g,ω|2 +

∂vf,g,ω

∂t
pf,g,ω

+
∂yf,g,ω

∂t
wf,g,ω −vf,g,ωwf,g,ω + ∇yf,g,ω ·∇pf,g,ω −χωuf,g,ωpf,g,ω

)
dxdt

−
∫

ΩT

(
∂X∇yf,g,ω ·∇pf,g,ω + ∂XT∇yf,g,ω ·∇pf,g,ω +

1

T
∇f · Xwf,g,ω(x, 0)

+
1

T
∇g · Xpf,g,ω(x, 0)

)
dxdt. (4.41)

Integrating the fourth and fifth terms of (4.41) by partial integration in time t, and using the facts that

pf,g,ω(x,T) = 0,wf,g,ω(x,T) = 0, A: B =
∑2
i,l=1 ailbil and a⊗ b: A = a ·Ab, a,b ∈ R

2, A,B ∈ R2×2,
we have

∫
ΩT

((
1
2
|yf,g,ω|2 + 1

2
|vf,g,ω|2 + α

2
|uf,g,ω|2 −vf,g,ω ∂p

f,g,ω

∂t

−yf,g,ω ∂w
f,g,ω

∂t
+ ∇yf,g,ω ·∇pf,g,ω −χωuf,g,ωpf,g,ω −vf,g,ωwf,g,ω − 1T gp

f,g,ω(x, 0)

− 1
T
fwf,g,ω(x, 0)

)
I −∇yf,g,ω ⊗∇pf,g,ω −∇pf,g,ω ⊗∇yf,g,ω

)
: ∂X

− 1
T

(
∇fwf,g,ω(x, 0) + ∇gpf,g,ω(x, 0)

)
· X dxdt.

Thus, we have the tensor representation (4.35)–(4.36). Hence,

lim
n→∞

H(τn,un,fn,gn) −H(0,un,fn,gn)
τn

=

∫
ΩT

S1

(
yf,g,ω,vf,g,ω,pf,g,ω,wf,g,ω,uf,g,ω

)
: ∂X + S0(f,g) · X dxdt. (4.42)

Suppose that un,0 := u
fn,gn,0. Then similarly, modifying un as un,0, we obtain

lim
n→∞

H(τn,un,0,fn,gn) −H(0,un,0,fn,gn)
τn

=

∫
ΩT

S1

(
yf,g,ω,vf,g,ω,pf,g,ω,wf,g,ω,uf,g,ω

)
: ∂X + S0(f,g) · X dxdt. (4.43)

Let {fn} and {gn} be constant sequences. Then, it is clearly seen that H(τn,un,fn,gn)−H(0,un,fn,gn)
in (4.42) and H(τn,un,0,fn,gn) − H(0,un,0,fn,gn) in (4.43) are equal. Hence, (H2) is satisfied. Uti-
lizing Lemma 4.7, we obtain LHS of (4.42) and (4.43) as ∂τH(0

+,u0,f,g,f,g) and ∂τH(0
+,uf,g,0,f,g),

respectively. Hence, (H3) is satisfied. �

As a consequence of Theorem 4.11, we obtain the directional derivative of J1.



OPTIMAL ACTUATOR PLACEMENT FOR CONTROL OF VIBRATIONS 17

Corollary 4.12. Let the hypotheses of Theorem 4.11 hold. Let (f,g) ∈ H10 (Ω) ×L2(Ω) := V be given.
Then the directional derivative of J1(ω,f,g) at ω in the direction X ∈ C̊0,1(Ω,R2) is given by

DJ1(ω,f,g)(X) =

∫
ΩT

S1

(
yf,g,ω,vf,g,ω,pf,g,ω,wf,g,ω,uf,g,ω

)
: ∂X + S0(f,g) · Xdxdt, (4.44)

where S1

(
yf,g,ω,vf,g,ω,pf,g,ω,wf,g,ω,uf,g,ω

)
and S0(f,g) are defined by (4.36).

Proof. For a constant R > 0, we note that

max
f∈K1,g∈K2

‖f‖
H1

0
(Ω)
≤R,‖g‖

L2(Ω)
≤R

J1(ω,f,g) = R
2 max

f∈ 1
R
K1,g∈ 1RK2

‖f‖
H1

0
(Ω)
≤1,‖g‖

L2(Ω)
≤1

J1(ω,f,g). (4.45)

From (4.45) and by the hypotheses of Theorem 4.11, we deduce that f
R
∈ K1 and gR ∈ K2 with

‖f
R
‖H10 (Ω) ≤ 1 and ‖

g
R
‖L2(Ω) ≤ 1. Thus, we have the singleton {K1,K2} := {(f,g)}. So, for all

ω ∈ E(Ω), we have

J2(ω) = max
f∈K1,g∈K2

J1(ω,f,g) = J1(ω,f,g).

Hence, we deduce that X2(ω) = {(f,g)}. Since X2(ω) is a singleton, (4.44) follows by Theorem 4.11. �

As a further consequence of Theorem 4.11, we write (4.35) as an integral over ∂ω. To this end, we

require that ω and Ω are C2 domains. Additionally, for any two sets ω and Ω, the notation ω b Ω will
be used to mean that ω is compactly contained in Ω. In other words, ω b Ω if ω ⊂ Ω and ω is compact.

Corollary 4.13. Let f ∈ K1,g ∈ K2 and X ∈ C̊0,1(Ω,R2) be given. Assume that ω b Ω and Ω are C2
domains.

(a) Given (f,g) ∈ X2(ω), define Ŝ1(f,g) and Ŝ0(f,g) by Ŝ1(f,g) :=
∫T

0
S1(f,g)(s) ds and

Ŝ0(f,g) :=
∫T

0
S0(f,g)(s) ds, respectively. Then we have

Ŝ1(f,g)|ω∈ W 1,1(ω,R2×2), Ŝ1(f,g)|Ω\ω∈ W 1,1(Ω \ω,R2×2), Ŝ0(f,g)|ω∈ L2(ω,R2), (4.46)

−div(Ŝ1(f,g)) + Ŝ0(f,g) = 0 a.e. in ω ∪ (Ω \ω). (4.47)

Moreover, (4.35) can be written as

DJ2(ω)(X) = max
(f,g)∈X2(ω)

−
∫
∂ω

∫ T
0

uf,g,ω(t)pf,g,ω(t)(X ·ν) dtds, (4.48)

for X ∈ C̊0,1(Ω,R2), with ν the outer normal to ω. We denote the jump of Ŝ1(f,g)ν across ∂ω
by [Ŝ1(f,g)ν] := Ŝ1(f,g)|ων − Ŝ1(f,g)|Ω\ων.

(b) We have that (4.44) can be written as

DJ1(ω,f,g)(X) = −
∫
∂ω

∫ T
0

uf,g,ω(t)pf,g,ω(t)(X ·ν) dtds, (4.49)

for X ∈ C̊0,1(Ω,R2).

We begin by stating an important lemma, the so-called Nagumo’s lemma (see e.g., [15]) before

proving Corollary 4.13. The outer normal to ∂R2 will be denoted by ν.



18 M. D. AROP, H. KASUMBA, J. KASOZI, AND F. BERNTSSON

Lemma 4.14. Let Ω ⊂ R2 be a bounded domain of class Ck, k ≥ 1. Suppose that X ∈ C̊0,1(R2,R2) is
a vector field satisfying

X(x) ·ν(x) = 0, for all x ∈ ∂R2.

Then the flow Φτ of X satisfies

Φτ (Ω) = Ω and Φτ (∂Ω) = ∂Ω, for all τ.

Proof of Corollary 4.13. We prove (4.46)–(4.48) as follows. By Nagumo’s lemma, we have

DJ2(ω)(X) = 0, for all X ∈ C1c (Ω,R2). Using this condition and definitions of Ŝ1(f,g) and Ŝ0(f,g) in
(4.35), we see that ∫

Ω

Ŝ1(f,g) : ∂X + Ŝ0(f,g) · X dx = 0, (4.50)

for all X ∈ C1c (Ω,R2). Integrating the first term in (4.50) by partial integration and using X|∂Ω= 0,
we have ∫

Ω

(−div(Ŝ1(f,g)) + Ŝ0(f,g)) · X dx = 0, (4.51)

for all X ∈ C1c (Ω,R2). Since X ∈ C1c (Ω,R2), applying the fundamental lemma of calculus of variations
on (4.51) gives (4.47). Further, since y,p ∈ H2(Ω) ∩H10 (Ω) follows from elliptic regularity theory (see
e.g., [6, p.317] ), we have that (4.46) holds. Thus, noting that Ω = ω∪(Ω\ω) and by partial integration,
we have for all X ∈ C1c (Ω,R2),

DJ2(ω)(X) = max
(f,g)∈X2(ω)

∫
Ω

Ŝ1(f,g) : ∂X + Ŝ0(f,g) · X dx,

= max
(f,g)∈X2(ω)

(∫
ω

(
− div(Ŝ1(f,g)) + Ŝ0(f,g)

)
· X dx

+

∫
Ω\ω

(
− div(Ŝ1(f,g)) + Ŝ0(f,g)

)
· X dx +

∫
∂ω

[Ŝ1(f,g)ν] · X ds
)
,

(4.47)
= max

(f,g)∈X2(ω)

∫
∂ω

[Ŝ1(f,g)ν] · X ds.

(4.52)

Since (4.46) holds, it follows that

Tτ (f,g) := Ŝ1(f,g) +

∫ T
0

χωu
f,g,ω(t)pf,g,ω(t) dt ∈ W 1,1(ω,R2×2). (4.53)

So, Tτ (f,g)ν = 0 on ∂ω. Hence, it is easy to see from (4.53) that

[Ŝ1(f,g)ν] = −
(∫ T

0

χωu
f,g,ω(t)pf,g,ω(t) dt

)
ν. (4.54)

Since X and ν are independent of time t, substituting (4.54) in (4.52) gives

DJ2(ω)(X) = max
(f,g)∈X2(ω)

−
∫
∂ω

∫ T
0

uf,g,ω(t)pf,g,ω(t)(X ·ν) dtds,

as was to be proved.

The proof of (4.49) is similar to the proof of Corollary 4.12. �

4.5. Gradient Algorithm for Optimal Actuator Placement. Here, we present the steps of a

gradient-based algorithm for optimal actuator placement. The version of the algorithm is summarized

in Algorithm 1. It is important to note that we can also use J2 in this algorithm to investigate the

optimal actuator placement by replacing J1 with J2.



OPTIMAL ACTUATOR PLACEMENT FOR CONTROL OF VIBRATIONS 19

Algorithm 1 Shape derivative-based gradient algorithm for optimal actuator placement

Require: ω0 ∈ E(Ω),f,g, tolerance ε > 0,k = 0,λ,d0 := −∇J1(ω0,f,g).
while |dk| ≥ ε do

if J1((id + λdk)(ωk),f,g) < J1(ωk,f,g) then

dk = −∇J1(ωk,f,g)
ωk+1 = (id + λdk)(ωk)

k := k + 1

end if

end while

return optimal actuator placement ωk+1

5. Numerical Examples

5.1. Discretization. Let step sizes be h in space and 4t in time, i.e., 4x1 = 4x2 = h and tk = k4t.
Then, discretizing (3.6) and (3.7) using finite differences, we have for k = 1, 2, . . . ,M − 1

yk+1h = y
k
h + 4tv

k
h,

vk+1h = v
k
h + Ary

k
h + 4tχωu

k
h, (5.1)

y1h = fh,

v1h = gh,

and for k = M,M − 1, . . . , 2

pk−1h = p
k
h + 4t(w

k
h − v

k
h),

wk−1h = w
k
h + Arp

k
h −4ty

k
h, (5.2)

pMh = 0,

wMh = 0,

respectively, where

yh = (y11,y12, . . . ,y(N−1)2(N−1)2 )
>, vh = (v11,v12, . . . ,v(N−1)2(N−1)2 )

>,

uh = (u11,u12, . . . ,u(N−1)2(N−1)2 )
>, fh = (f11,f12, . . . ,f(N−1)2(N−1)2 )

>,

gh = (g11,g12, . . . ,g(N−1)2(N−1)2 )
>, ph = (p11,p12, . . . ,p(N−1)2(N−1)2 )

>,

wh = (w11,w12, . . . ,w(N−1)2(N−1)2 )
>,r =

4t
h2

and

Ar =




B I 0 . . . 0

I B I
. . .

...

0 I
. . .

. . . 0
...

. . .
. . . B I

0 . . . 0 I B




with B =




−4 1 0 . . . 0

1 −4 1
. . .

...

0
. . .

. . .
. . . 0

...
. . . 1 −4 1

0 . . . 0 1 −4




and I is the identity matrix. The matrix Ar is of size (N − 1)2 × (N − 1)2 while matrices B, I and 0
are of size (N − 1) × (N − 1). The discrete functionals of J1 and J2 are

J1,h(ω, fh, gh) =
1

2
min

uh∈Uad

∫ T
0

yh(t)
>yh(t) + vh(t)

>vh(t) + αχωuh(t)
>χωuh(t)dt (5.3)



20 M. D. AROP, H. KASUMBA, J. KASOZI, AND F. BERNTSSON

and

J2,h(ω) = max
fh,gh

J1,h(ω, fh, gh), (5.4)

respectively. The discrete derivatives J1,h and J2,h are given by

DJ1,h(ω, fh, gh)(X) = −
∫
∂ω

∫ T
0

uh(s,t)
>ph(s,t)(X ·ν) dtds (5.5)

and

DJ2,h(ω)(X) = max
fh,gh

J1,h(ω, fh, gh),

for X ∈ C̊0,1(Ω,R2), respectively. The vector b ∈ R2 has components

bj := −
∫
∂ω

∫ T
0

uh(s,t)
>ph(s,t)(ej ·ν) dtds,j = 1, 2,

where ej is the jth element of the standard basis of R2.

5.2. Examples. We illustrate the actuator placement optimizations for two cases of initial conditions

f and g. In all the experiments, the actuators ω1, and ω2 each of fixed size 0.2 × 0.2 are placed on the
domain and moved along the descent direction x1 = x2. We consider two actuators without overlap

such that they move into their optimal locations. We set the tolerance ε to 10−4 and N to 8.

Example 5.21. We consider the case

y(x1,x2, 0) = sin πx1 sin πx2, 0 ≤ x1,x2 ≤ 1,

v(x1,x2, 0) =
πc

20
sin πx1 sin πx2, 0 ≤ x1,x2 ≤ 1,

so that the initial speed v(x1,x2, 0) varies with the speed of wave 1 ≤ c ≤ 20π . First, we start by
investigating the optimal actuator placement using J1,h. For initial actuators ω1,ω2 centered at (0.4, 0.4)

and (0.825, 0.825), respectively, a shape optimization Algorithm 1 is utilized. The results are presented

in Figure 2. It is observed from Figure 2 that as the actuators move toward the optimal locations in the

subsequent iterations (see Figure 2(a)), the functional J1,h decays until a stationary point is reached,

see Figure 2(b). The optimization algorithm converges after 120 iterations. The optimal actuators are

centered at (0.325, 0.325) and (0.75, 0.75), respectively.

Next, we perform numerical experiments using J2,h but with initial actuators ω1,ω2 centered at

(0.2, 0.2) and (0.825, 0.825), respectively. Algorithm 1 is run until the set criterion is achieved. The

results are depicted in Figure 3. From this figure, we see that as the actuators move toward the optimal

locations, see Figure 3(a), the functional J2,h decays until a stationary point is reached, see Figure 3(b).

The convergence of the optimization algorithm occurs after 73 iterations. The final actuator locations

are found at (0.325, 0.325) and (0.75, 0.75), respectively. This is consistent with the result obtained by

using J1,h.

Lastly, the results of the experiments to investigate the influence of the wave speed are shown in

Figure 4 and Table 1.

From Table 1, we see that when ω1 is placed at (0.4, 0.4), and ω2 at (0.825, 0.825) (see Figure 4(a)),

the least values of both J1,h and J2,h are obtained. Furthermore, it is observed that J1,h increases with

an increase in the wave speed c, see Figure 4 and Table 1. We also note from Table 1 that the least

values of J2,h(ω) after 120 iterations are the same. This confirms the fact that the dependence of J1,h
on the initial conditions is averaged out.



OPTIMAL ACTUATOR PLACEMENT FOR CONTROL OF VIBRATIONS 21

(a) (b)

Figure 2. (a) The initial actuator center locations: (0.4, 0.4), (0.825, 0.825) (red) and

final actuator center locations: (0.325, 0.325), (0.75, 0.75) (blue). (b) The history of

cost functional J1,h, as the actuators move from the initial to the optimal actuator

locations. The speed of the wave is set to c = 1.

(a) (b)

Figure 3. (a) The initial actuator center locations: (0.2, 0.2), (0.825, 0.825) (red) and

final actuator center locations: (0.325, 0.325), (0.75, 0.75) (blue). (b) The history of

cost functional J2,h, as the actuators move from the initial to the optimal actuator

locations. The speed of the wave is set to c = 1.

Example 5.22. In this example, we set

y(x1,x2, 0) = x1x2(1 −x1)(1 −x2), 0 ≤ x1,x2 ≤ 1,

v(x1,x2, 0) =
1

2
sin(x1(1 −x1)x2(1 −x2)), 0 ≤ x1,x2 ≤ 1,

so that the initial conditions of the dynamics satisfy Dirichlet boundary conditions. Therefore, the

optimal actuator center locations are expected at points different from the boundary of the domain.

First, we start by investigating the optimal actuator placement using J1,h. With initial actuator center

locations ω1,ω2 at (0.2, 0.2) and (0.825, 0.825), respectively, the results are shown in Figure 5.

The minimum value of J1,h occurs when the actuators are placed at (0.3, 0.3) and (0.7, 0.7).



22 M. D. AROP, H. KASUMBA, J. KASOZI, AND F. BERNTSSON

(a) (b)

Figure 4. (a) The initial actuator center locations: (0.4, 0.4), (0.825, 0.825) (red) and

final actuator center locations: (0.325, 0.325), (0.75, 0.75) (blue). (b) Demonstration

of the influence of wave speed on the history of cost functional J1,h, as the actuators

move from the initial to the final actuator locations.

Table 1. The minimum values of functionals J1,h and J2,h after 120 iterations for the

given speed of wave c. The measure of the cost of control is set to α = 10−4.

c J1,h(ω,f,g) J2,h(ω)

1 78.8529 0.4307

3 80.2176 0.4307

5 83.4128 0.4307

Next, we perform a numerical experiment using J2,h. With initial actuator center locations at

(0.2, 0.2) and (0.825, 0.825), we run Algorithm 1 until the set criterion is achieved. The results are given

in Figure 6. From this figure, we see that the functional J2,h decays until a stationary point is reached.

The convergence of the optimization algorithm occurs after 50 iterations. The final actuator locations

are found at (0.3, 0.3) and (0.7, 0.7), see Figure 6(a). This is in agreement with the result obtained by

using J1,h.

6. Conclusion

In this paper, we proved important results for the differentiability of functionals J1 and J2. The shape

derivative is derived using the averaged adjoint approach. We also developed a shape derivative-based-

gradient algorithm for determining the optimal actuator placement for the control of vibrations induced

by pedestrian-bridge interactions. The algorithm is constructed by embedding the shape sensitivities

in a gradient-based method. The numerical results presented illustrate the potential of the shape

sensitivities for solving the optimal actuator placement problem whenever the actuator’s width is fixed

in advance. The optimal actuator design for the wave equation is under our next research study

plan. The term ”design” here means picking the best domain ω by parametrizing the set of admissible

domains.



OPTIMAL ACTUATOR PLACEMENT FOR CONTROL OF VIBRATIONS 23

(a) (b)

Figure 5. (a) The initial actuator center locations: (0.2, 0.2), (0.825, 0.825) (red) and

final actuator center locations: (0.3, 0.3), (0.7, 0.7) (blue). (b) Demonstration of the

history of cost functional J1,h, as the actuators move from the initial to the optimal

actuator locations.

(a) (b)

Figure 6. (a) The initial actuator center locations: (0.2, 0.2), (0.825, 0.825) (red) and

final actuator center locations: (0.3, 0.3), (0.7, 0.7) (blue). (b) History of cost functional

J2,h, as the actuators move from the initial to the optimal actuator locations.

Acknowledgements. This study was supported by the SIDA bilateral programme (2015–2022) with

Makerere University; Project 316: Capacity building in Mathematics. The authors sincerely thank the

anonymous referees for valuable comments and suggestions that led to the final version of the paper.

Finally, the authors thank Prof. Karl Kunisch for his readership.

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M. D. Arop, corresponding author, Department of Mathematics, Makerere University, Kampala, Uganda

Current address: Department of Mathematics, Muni University, Arua, Uganda

Email address: d.arop@muni.ac.ug

H. Kasumba, Department of Mathematics, Makerere University, Kampala, Uganda

Email address: henry.kasumba@mak.ac.ug

J. Kasozi, Department of Mathematics, Makerere University, Kampala, Uganda

Email address: juma.kasozi@mak.ac.ug

F. Berntsson, Department of Mathematics, Linköping University, Linköping, Sweden

Email address: fredrik.berntsson@liu.se


	1. Introduction
	2. Formulation of the Problem
	2.1. Notations
	2.2. Setup of the Problem

	3. Well-Posedness of the Functionals
	4. Sensitivity Analysis of the Functionals
	4.1. Shape Derivative
	4.2. Sensitivity of the State Equation
	4.3. Averaged Adjoint Equations
	4.4. Directional Derivative of Max-Min Functions
	4.5. Gradient Algorithm for Optimal Actuator Placement

	5. Numerical Examples
	5.1. Discretization
	5.2. Examples

	6. Conclusion
	Acknowledgements

	References