FACTORIZATION OF DEGENERATE CAUCHY PROBLEMS: THE LINEAR CASE

• Published in: Journal of operator theory Vol. 36; no. 1; pp. 121 - 146
• Main Authors: THALLER, B., THALLER, S.
• Format: Journal Article
• Language: English
• Published: Theta Foundation 01-07-1996
• Subjects: Cauchy problem; Control systems; Control theory; Dirac equation; Evolution equations; Factorization; Hilbert spaces; Linear transformations; Mathematics; Semigroups
• ISSN: 0379-4024; 1841-7744

We investigate an abstract degenerate Cauchy problem with a non-invertible operator M at the derivative. The problem is formulated in a Hilbert space ℌ which can be written as an orthogonal direct sum of Ker M and Ran M*. Under certain conditions it is possible to reduce the problem to an equivalent non-degenerate Cauchy problem in the factor space ℌ/Ker M. The explicit form of the generator for the restricted problem is investigated. As an example we discuss the Dirac equation, where our theory leads to a new interpretation of the nonrelativistic limit. We show that this limit can be understood in terms of a degenerate Cauchy problem where the generator of the restricted problem is a Schrödinger operator. Finally, we describe some consequences for the treatment of degenerate control systems. In particular, we introduce dual methods of factorization in order to investigate the compatibility of factorization and transition to the dual system, which is essential for the definition of basic notions like observability and detectability.