2003, Vol.6, No.3, pp.705-716

The main purpose of the paper is to summarize some fundamental results concerning the matrix impulsive differential equations with impulses at fixed points. The theory of impulsive matrix differential equations is interesting in itself and it will assume greater importance in the near future since the application of the theory to various fields of science is also increasing, especially in the theory of polyphase transmission line and surge phenomena. The calculations in this theory are based on the concept of polyphase reflection factor K(x). The matrix K(x) satisfies the matrix Riccati equation, which may be considered as a simplest nonlinear matrix equation with bilinear main part. Local nonhomogeneities of the transmission line at some points lead to the case when the matrix function of local reflection K(x) becomes matrix -function (Dirac function), that is equivalent the pulse condition at some fixed points. The conditions of existence of solutions of homogeneous and nonhomogeneous impulsive matrix equation have been determined. The T-periodic matrix impulsive equation has been considered. The condition of existence the unique T-periodic solution has been obtained. The invariant set of impulsive matrix equation has been studied. The condition of its existence for nonlinear matrix equation has been determined.
Key words: impulsive matrix equation, transmission line

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