226.-IEEE TRANSACTIONS ON AUTOXATIC CONTROL, URIL 1972 was established as a set of sufficient conditions for the ESL of system (9) (by means of the vector Lyapunov approach). Since cll 5 c12 and e l _< & it follows that (27) yields less conservative results t.han (28).
VI. CONCLUSIONSNew stability results for composite systems with linear and nonlinear interconnections were established by utilizing scalar Lyapunov functions. It was shown that for large classes of problems, the use of scalar Lyapunov functions leads t.0 results which are in general less conservat.ive than those obtained from t,he vector Lyapunov function approach.
ACKNOWLEDQMENTThe authors wish to thank one of t.he reviewers for his const,ructive suggestions. REFERENCES netted systems," SIA% J . Contr.. ser. A, vo?; 3, pp. 113-462. 1966. F. X. Bailey "The a plication of Lyapunov's second method to intercon-A . A. Piontkovskn and L. D. Rutkovskaya. Investigation of certain stability-theory problems by the vector Lyapunov function method," Automat. Remote Contr., v01. 28, pp.,l422-1429, 0,ct. 1965. H. K. Ramaqriyan and N. Bailey, A decomposition method for the sta-Space Congr., Oct. 1970. bility analysls of a spacecraft attitude control system." in Proc. 1st West. H. K. Ramapriyan. Study of coupling in interconnected systems," Ph.D. dissertation, Univ. hfj,nnesota, Minneapolis. BIar. 1970.Abstract-The concept of the so-called generalized polynomial operators is considered and applied especially to systems described by certain types of nonlinear dzerential equations. A theorem concerning local invertibility of polynomial operators is given. By an example it is shown how this theorem can be used to prove the existence of solutions, to construct these solutions, and to find a region of BIB0 stability of the aforementioned systems. The treatment is quite general, being based on functional analysis. In particular, it can be applied to the systems analyzed by using functional series of Volterra type.
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