### Bernd Thaller

Institut f. Mathematik

Universität Graz

A-8010 Graz, Austria

e-mail: bernd.thaller@kfunigraz.ac.at

## Some Considerations on Degenerate Control Systems: The LQR problem

### Abstract:

We investigate linear inhomogeneous Cauchy problems in a Hilbert space which are degenerate in the sense that there is a non-invertible operator M at the time derivative. Under certain conditions the degenerate system is replaced with an ``effective'' non-degenerate system in the subspace given by the orthogonal complement of the kernel of M. The connection between the solutions of the nondegenerate system and the solutions of the degenerate system is in general given by an unbounded and sometimes not closable linear operator. The previously developed theory is extended and criteria are obtained when the solutions can be described in terms of a strongly continuous semigroup. We describe some examples and introduce the concept of a mild solution in order to deal with the LQR-problem in degenerate control theory.

(Bericht Nr. 45 des Spezialforschungsbereichs F003: Optimierung und Kontrolle - Oktober 1995)

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Bernd Thaller

Institute of Mathematics

University of Graz.

Page design of course by: Bernd Thaller.

last changed: 03-21-1997