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