1. Introduction.- 1. Stability as a property of a family of systems.- 2. The families of systems considered in the problem of absolute stability.- 3. Selecting the most natural families of systems.- 4. Introducing new families of systems.- 5. The concept of hyperstability.- 6. Indications on the use of the monograph.- 2. Classes of Equivalent Systems.- § 1. Equivalence classes for quadratic forms with relations between the variables.- § 2. Classes of single-input systems.- § 3. The characteristic polynomial of single-input systems.- § 4. Conditions under which all systems with the same characteristic polynomial belong to the same class.- § 5. Equivalence classes for multi-input systems.- § 6. Equivalence classes for discrete systems.- § 7. Equivalence classes for systems with time dependent coefficients.- 3. Positive Systems.- § 8. Single-input positive systems.- § 9. Multi-input positive systems.- § 10. Discrete positive systems.- § 11. Positive systems with time-dependent coefficients.- § 12. Nonlinear positive systems.- 4. Hyperstable Systems and Blocks.- § 13. General properties of the hyperstable systems.- § 14. Single-input hyperstable systems.- § 15. Simple hyperstable blocks.- § 16. Multi-input hyperstable systems.- § 17. Multi-input hyperstable blocks.- § 18. Discrete hyperstable systems and blocks.- § 19. Hyperstability of more general systems.- § 20. Integral hyperstable blocks.- § 21. Lemma of I. Barb?lat and its use in the study of asymptotic stability.- § 22. Other methods for studying asymptotic stability.- § 23. Conditions of asymptotic stability of single-input and multi-input systems with constant coefficients.- § 24. Characterization of the hyperstability property by the stability of systems with negative feedback.- 5.Applications.- § 25. Inclusion of the problem of absolute stability in a problem of hyperstability.- § 26. Determination of some Liapunov functions.- § 27. Stability in finite domains of the state space.- § 28. Stability of systems containing nuclear reactors.- § 29. Stability of some systems with non-linearities of a particular form.- § 30. Optimization of control systems for integral performance indices.- Appendix A. Controllability; Observability; Nondegeneration.- § 31. Controllability of single-input systems.- 1. Definition of the complete controllability of single-input systems.- 2. Theorem of complete controllability of single-input systems.- 3. Discussion.- 4. Proof of the theorem.- 5. Relations between single-input completely controllable systems.- § 32. Single-output completely observable systems.- § 33. Nondegenerate systems.- 1. Definition of the property of non degeneration and statement of the theorem of non degeneration.- 2. Remarks on the theorem of nondegeneration.- 3. Proof of the theorem of non degeneration.- 4. Bringing nondegenerate systems into the Jordan-Lur'e-Lefschetz form.- § 34. Controllability of multi-input systems.- 1. Definition and theorem of the complete controllability of multi-input systems.- 2. Proof of Theorem 1.- 3. Other properties of completely controllable multi-input systems.- § 35. Completely observable multi-output systems.- § 36. Special forms for multi-input blocks.- Appendix B. Factorization of Polynomial Matrices.- § 37. Auxiliary proposition.- § 38. Theorem of factorization on the unit circle.- 1. Statement of the theorem.- 2. Preliminary remarks.- 3. Some additional assumptions.- 5. A family of factorization relations.- 6. A special way of writing polynomial matrices.- 7. A nonsingular factorization.- 8.Properties of the nonsingular factorizations.- 9. Bringing the nonsingular factorization to the form required in Theorem 1.- 10. More about Assumption (e).- 11. Eliminating restrictions (C) and (e).- § 39. The theorem of factorization on the imaginary axis.- 1. Statement of the theorem.- 2. Definition of a matrix factorizable on the unit circle.- 3. Relations between ?(?) and ?(?).- 4. Factorization of the imaginary axis.- Appendix C. Positive Real Functions.- Appendix D. The Principal Hyperstable Blocks.- Appendix E. Notations.- Appendix F. Bibliography.