A class of nonlinear differential equations on the space of symmetric matrices
Date
2004-08-06
Authors
Dragan, Vasile
Freiling, Gerhard
Hochhaus, Andreas
Morozan, Toader
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
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.
Description
Keywords
Rational matrix differential equations, Generalized Riccati differential equations, Generalized stabilizability and detectability, Comparison theorem, Existence and convergence results
Citation
Dragan, V., Freiling, G., Hochhaus, A., & Morozan, T. (2004). A class of nonlinear differential equations on the space of symmetric matrices. <i>Electronic Journal of Differential Equations, 2004</i>(96), pp. 1-48.
Rights
Attribution 4.0 International