Electronic Journal of Differential Equations,
Vol. 2004(2004), No. 96, pp. 1-48.
Title: A class of nonlinear differential equations on the
space of symmetric matrices
Authors: Vasile Dragan (Romanian Academy, Romania)
Gerhard Freiling (Univ. Duisburg-Essen, Germany)
Andreas Hochhaus (Univ. Duisburg-Essen, Germany)
Toader Morozan (Romanian Academy, Romania)
Abstract:
In the first part of this paper we analyze the properties of
the evolution operators of linear differential equations generating
a positive evolution and provide a set of conditions which
characterize the exponential stability of the zero solution, which
extend the classical theory of Lyapunov.
In the main part of this work we prove a monotonicity and a
comparison theorem for the solutions of a class of time-varying
rational matrix differential equations arising from stochastic
control and derive existence and (in the periodic case) convergence
results for the solutions. The results obtained are similar to those
known for matrix Riccati differential equations. Moreover we provide
necessary and sufficient conditions which guarantee the existence
of some special solutions for the considered nonlinear differential
equations as: maximal solution, stabilizing solution, minimal positive
semi-definite solution. In particular it turns out that under the
assumption that the underlying system satisfies adequate generalized
stabilizability, detectability and definiteness conditions there
exists a unique stabilizing solution.
Submitted March 18, 2003. Published August 6, 2004.
Math Subject Classifications: 34A34, 34C11, 34C25, 93E03, 93E20.
Key Words: Rational matrix differential equations; generalized Riccati
differential equations; generalized stabilizability and detectability;
comparison theorem; existence and convergence results