Literature cited
A. I. Lur'e, Some Nonlinear Problems in the Theory of Automatic Control [in Russian], Gostekhizdat, Moscow (1951).
V. A. Yakubovich, “The solution of some matrix inequalities occurring in the theory of automatic regulation,” Dokl. Akad. Nauk SSSR,143, No. 3, 1304–1307 (1962).
R. E. Kalman, “Lyapunov functions for the problem of Lur'e in automatic control,” Proc. Nat. Acad. Sci. USA,49, No. 2, 201–205 (1963).
V. M. Popov, Hyperstability of Automatic Systems [in Russian], Nauka, Moscow (1970).
V. A. Yakubovich, “The frequency theorem in control theory,” Sib. Mat. Zh.,14, No. 2, 384–420 (1973).
A. Kh. Gelig, G. A. Leonov, and V. A. Yakubovich, The Stability of Nonlinear Systems with Nonunique Equilibrium State [in Russian], Nauka, Moscow (1978).
A. N. Churilov, “The frequency theorem and Lur'e's equation,” Sib. Mat. Zh.,20, No. 4, 854–868 (1979).
A. N. Churilov, “The solubility of certain matrix inequalities,” Vestn. Mosk. Gos. Univ., Ser. Mat., Mekh., Astron., No. 7, 51–55 (1980).
A. A. Voronov, Stability, Controllability and Observability [in Russian], Nauka, Moscow (1979).
K. R. Meyer, “Lyapunov functions for the problems of Lur'e,” Proc. Nat. Acad. Sci. USA,53, No. 3, 501–503 (1965).
K. R. Meyer, “On the existence of Lyapunov functions for the problem of Lur'e,” SIAM J. Control, Ser. A,3, No. 3, 373–383 (1965).
M. D. Srinath, M. A. L. Thathacher, and H. K. Ramapriyan, “Absolute stability of systems with multiple nonlinearities,” Int. J. Control,7, No. 4, 365–375 (1968).
A. V. Zharkov, “The problem of the existence of a positive solution of Lur'e's equation,” Vestn. Mosk. Gos. Univ., Ser. Mat., Mekh., Astron., No. 13, 134–135 (1978).
W. A. Goppel, “Matrix quadratic equations,” Bull. Austral. Math. Soc,10, No. 3, 377–401 (1974).
P. Lancaster and L. Rodman, “Existence and uniqueness theorems for the algebraic Riccati equation,” Int. J. Control,32, No. 2, 285–310 (1980).
A. N. Churilov, “On solutions of a quadratic matrix equation,” in: Nonlinear Oscillations and Control Theory [in Russian], Vol. 2, Udmurtsk Univ., Izhevsk (1978), pp. 24–33.
F. R. Gantmakher, The Theory of Matrices, Chelsea Publ.
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Translated from Matematicheskie Zametki, Vol. 36, No. 5, pp. 725–732, November, 1984.
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Churilov, A.N. Solubility of matrix inequalities. Mathematical Notes of the Academy of Sciences of the USSR 36, 862–866 (1984). https://doi.org/10.1007/BF01139934
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DOI: https://doi.org/10.1007/BF01139934