Geodesic
A class of nonlinear differential equations on the space of symmetric matrices
Electronic journal of differential equations, Tome 2004 (2004)
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.
Classification : 34A34, 34C11, 34C25, 93E03, 93E20
Keywords: rational matrix differential equations, generalized Riccati differential equations, generalized stabilizability and detectability, comparison theorem, existence and convergence results 
@article{EJDE_2004__2004__a19,
     author = {Dragan,  Vasile and Freiling,  Gerhard and Hochhaus,  Andreas and Morozan,  Toader},
     title = {A class of nonlinear differential equations on the space of symmetric matrices},
     journal = {Electronic journal of differential equations},
     year = {2004},
     volume = {2004},
     zbl = {1053.34006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a19/}
}
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Dragan,  Vasile; Freiling,  Gerhard; Hochhaus,  Andreas; Morozan,  Toader. A class of nonlinear differential equations on the space of symmetric matrices. Electronic journal of differential equations, Tome 2004 (2004). http://geodesic.mathdoc.fr/item/EJDE_2004__2004__a19/