Research and Exposition in Mathematics -- Volume 22
Optimal Control of Undamped Linear Vibrations
188 p., soft cover, ISBN 3-88538-222-9, EUR 40.00, 1995
Considered are vibrating systems with finitely many
and infinitely many degrees of freedom which are undamped and linear. The first can be
described by linear systems of ordinary differential equations of second order without
first order terms. The control of the systems is performed by exterior forces on the
right hand side of the equations and is called lumped control. In the case of infinitely
many degrees of freedom the vibrating systems are described by linear partial differential
equations of second order with respect to the time under linear boundary conditions. There
are two kinds of control: distributed control where the control is performed by an exterior
force on the right hand side of the differential equation and boundary control which is
applied to one or more boundary conditions.
The main purpose of this book is to investigate the vibrating systems under the aspect of
null-controllability. In particular the relationship between time-minimal null-controllability
with norm bounded controls and minimum norm null-controls on fixed time intervals is studied
because it is the key for a numerical solution.
Distributed control of partial differential equations is uniformly treated within the framework
of abstract wave equations for Hilbert space valued functions which also comprises the case of
lumped control of vibrating systems with finitely many degrees of freedom. The corresponding
control problems are approximately solved with the aid of Galerkin's method and by applying
duality results for certain minimum norm problems.