On the Existence of Approximately Coprime Factorizations for Retarded Systems Eduardo D. Sontag† Department of Mathematics Rutgers University New Brunswick, NJ 08903 U.S.A. and Yutaka Yamamoto‡ Department of Applied Systems Science Faculty of Engineering Kyoto University Kyoto 606 JAPAN. September 1988 Revised December 1988 Abstract This note establishes a result linking algebraically coprime factorizations of transfer matrices of delay systems to approximately coprime factorizations in the sense of distribu- tions. The latter have been employed by the second author in the study of function-space controllability for such systems. Keywords: delay systems, coprime factorizations, controllability, realization. † Author supported in part by grants AFOSR-85-0247 and AFOSR-88-0235. Electronic mail: sontag@fermat.rutgers.edu ‡ Author supported in part by Ministry of Education under Grant-in-Aid # 62302034 for Co- operative Research. Electronic mail: c51160@jpnkudpc.bitnet 1 Introduction This note deals with retarded delay-differential systems (with one commensurable delay) of the type: d dt x(t)= N X i=0 F i x(t - ih)+ N X i=0 Gu(t - ih). (1) 1