Global Exact Controllability of Semi-linear Time Reversible Systems in Infinite Dimensional Space
In this paper, we survey the updated available results on global exact controllability problem of some semi-linear time reversible systems in infinite dimensional space. The typical models we will consider are semi-linear wave equations and plate equation
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School of Mathematics, Sichuan University, Chengdu 610064, China Departamento de Matematica Aplicada, Universidad Complutense, 28040 Madrid, Spain
Summary. In this paper, we survey the updated available results on global exact controllability problem of some semi-linear time reversible systems in infinite dimensional space. The typical models we will consider are semi-linear wave equations and plate equations, where the nonlinearity is globally Lipschitz continuous, or more generally, satisfies some super-linear growth condition at infinity.
1 Introduction Time reversible systems in infinite dimensional space include the wave equation, the plate equation, Schrodinger equation, Maxwell's equations and so on. The heat equation is however a typical time irreversible system. It is wellknown that the controllability theories between the time reversible system and the irreversible one are quite different. In this paper, we are concerned with the global exact controllability problem of two classes of semi-linear time reversible systems in infinite dimensional space, i.e., semi-linear wave equations and plate equations. Let T > 0 and n c R n (n E N) be a bounded domain with a smooth boundary r. Let w be a proper sub-domain of D and denote the characteristic function of the set w by Xw' Fix a nonlinear function fECI (R). Let us consider the following controlled semi-linear wave equation
Ytt - i1y + f(y) = Xw(x)u(t,x) in (0, T) x D, { y=O on (0, T) x r, in D y(O) = Yo, Yt(O) = Y1
(1)
and controlled semi-linear plate equation
{
~t~+i1~2: y(O) = Yo,
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\7. (xw(x)\7u(t,x)) m (0, T) x D, on (0, T) x r, in D. Yt(O) = Y1
f(y)
=
(2)
* Supported by the Foundation for the Author of National Excellent Doctoral Dissertation of China (Project No: 200119), the Grant BFM2002-03345 of the Spanish MCYT, and NSF of China under Grant 19901024.
G. C. Cohen et al. (eds.), Mathematical and Numerical Aspects of Wave Propagation WAVES 2003 © Springer-Verlag Berlin Heidelberg 2003
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Xu Zhang
The (global) exact (internal) controllability of (1) (resp. (2)) in HJ(D) x L2(D) (resp. HJ(D) x H-1(D)) at time T is defined as follows: for any (Yo, yd and (zo, Zl) in this space, there is a control u E L2((0, T) x D) (resp. C([O, T]; HJ(D))) such that the solution y of (1) (resp. (2)) satisfies y(T) = Zo and Yt(T) = Zl in D. The exact (boundary) controllability of (1) (resp. (2)) can be defined similarly. In that case the control u enters on the system through the boundary conditions. However, in order to avoid unnecessary technical difficulties, we will concentrate on considering only the internal controllability problem. The study of controllability problem of the semi-linear time reversible systems began at the last 60's, we refer to [17] and [18] for a good list for some early references. The early studies are mainly devoted to the local controllability problem, i.e. the problem with some small datum. A breakthrough in this respect is Zuazua's works ([17] and [18]), which combine the so-called Hilbert Uniqueness Method, due to J.
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