![](//s1.studylibfr.com/store/data-gzf/81f1bccc3902d1a470fb694aeda77337/1/010061908.htmlex.zip/bg2.jpg)
2I. EL HARRAKI AND A. BOUTOULOUT
around a constant reference trajectory in the space H3(0, 1)×H2(0, 1).InKhapalov (2010), the author
investigated the approximate controllability of the state equilibriums for the following control system:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
ytt =yxx +v(x,t)y+γ(t)yt−F(y)x∈(0, 1),t>0
y(x,0)=y0(x),yt(x,0)=y1(x)x∈(0, 1)
y(0, t)=y(1, t)=0,
(1.1)
in which the functions v∈L∞((0, +∞)×(0, 1)) and γ∈L∞(0, ∞)are regarded as bilinear or
multiplicative controls. If the system (1.1) models the oscillations of vibrating string with clapped ends,
then v(x,t)can be viewed as an axial load at point xand time t, and γ(t)as the gain of viscous (motion-
activated) damping, acting upon the string at time t. Such controllability problems may arise in the
context of smart materials, whose properties can be altered by applying various factors (temperature,
electric current, magnetic field). In Ouzahra (2014), the author considered the linear case of (1.1) with
bounded damping, and proved the approximate and the exact controllability for the equilibrium states.
The problem of optimal control of (1.1) without damping has been considered in Liang (1999), where a
bilinear control has been used to make the state solution close to a desired profile minimizing a quadratic
cost of control.
Our purpose in this paper is to study two cases for the multidimensional version of problem (1.1).
The first one deals with the linear wave with unbounded damping. The second one concerns the semi-
linear wave with bounded damping. The paper is organized as follows. The second section presents the
considered system and establishes an approximate controllability result of a set of target states in the
state space H1
0(Ω) ×L2(Ω). In the third section, we provide bilinear controls ensuring the approximate
controllability of the multi-dimensional semi-linear wave equation, and we give an extension result. The
rest of the section is devoted to exact controllability of the one-dimensional semi-linear wave equation
using the approximate controllability result, and the exact controllability of an equivalent system with
additive control.
2. Approximate controllability of the linear wave equation
2.1. Problem statement
Let Ωbe an open bounded subset of Rnwith regular boundary ∂Ω. We consider the space H=L2(Ω)
with its inner product and the corresponding norm denoted by
H. We define the unbounded operator
A=−Δwith the domain D(A)=H2(Ω) ∩H1
0(Ω). Let us also consider the space V=H1
0(Ω) with
the norm vV=A1
2vH,v∈V. For a time T1>0 large enough, we set Q=Ω×(0, T1)and
Σ=∂Ω ×(0, T1). We further consider a complex Hilbert space Uendowed with its inner product
., .U, and the induced norm
U. We denote by (D(A1/2))the dual space of (D(A1/2)) with respect
to the inner product in H. Let us consider the following initial and boundary value problem for the
n-dimensional wave equation:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
ytt =Δy+v(x,t)y−u(t)¯
Bytin Q
y(x,0)=y0(x),yt(x,0)=y1(x)in Ω
y(ξ,t)=0onΣ,
(2.1)
where ¯
B=BB∗for B∈L(U,(D(A1/2))with adjoint operator B∗.
at Ryerson University on October 19, 2016http://imamci.oxfordjournals.org/Downloaded from