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Professor J. P. LaSalle died on July 7, 1983 at the age of 67. The present book is being published posthumously with the careful assistance of Kenneth Meyer, one of the students of Professor LaSalle. It is appropriate that the last publi- cation of Professor LaSalle should be on a subject which con- tains many interesting ideas, is very useful in applications and can be understood at an undergraduate level. In addition to making many significant contributions at the research level to differential equations and control theory, he was an excel- lent teacher and had the ability to make sophisticated con- cepts appear to be very elementary. Two examples of this are his books with N. Hasser and J. Sullivan on analysis published by Ginn and Co. , 1949 and 1964, and the book with S. Lefschetz on stability by Liapunov's second method published by Academic Press, 1961. Thus, it is very fitting that the present volume could be completed. Jack K. Hale Kenneth R. Meyer TABLE OF CONTENTS page 1. Introduction 1 2. Liapunov's direct method 7 3. Linear systems Xl = Ax. 13 4. An algorithm for computing An. 19 5. Acharacterization of stable matrices. Computational criteria. 24 6. Liapunovls characterization of stable matrices. A Liapunov function for Xl = Ax. 32 7. Stability by the linear approximation. 38 8. The general solution of Xl = Ax. The Jordan Canonical Form. 40 9. Higher order equations. The general solution of ~(z)y = O.