Asymptotic solutions of a class of matrix differential equations arising in neutral network modelling

The following matrix differential equation initiated by research of dynamical network models is studied: φ=-P(φ t)a(t)a(t)TP(φT,t), where a is a vector-valued function of time t and P is a matrix-valued function of the solution matrix and time. A set of assumptions is given under which the solution, starting from a symmetric positive sotnidofinite initial matrix, tends asymptotically to a positive semidefinito matrix; if the initial matrix is a projection matrix and a(t) hua a special step-like structure, then the solution tends to a projection matrix on a specified subspace. Finally, a generalization of the basic class of equations is studied and an explicit expression is given for the limit matrix under certain conditions.