The concept of a resonance set is used to state and prove sufficient conditions for the existence of a periodic solution of the nonlinear vector differential equation f(D)x + + BMg(D)x=p, in which D=d/dt and the functions M, p are periodic in the t variable.
S. Ahmad,An existence theorem for periodically perturbed conservative systems, Michigan Math. J.,20 (1973), pp. 385–392.
R. W. Brockett,Finite Dimensional Linear Systems, Wiley, New York, 1970.
K. J. Brown,Nonlinear boundary value problems and a global inverse function theorem, Ann. Mat. Pura Appl. (4)106 (1975), pp. 205–217.
V. A. Jakubovič,A class of nonlinear differential equations for which the problem of stability or instability in the large can be solved effectively (Russian), Dokl. Akad. Nauk SSSR,186 (1969), pp. 1027–1030, translated as Soviet Math. Dokl.,10 (1969), pp. 720–723.
V. A. Jakubovič (Yakubovich),Absolute stability of nonlinear control systems - I: General frequence criteria (Russian), Avtomat. i. Telemeh., n. 12 (1970), pp. 5–14, translated as Automat. Remote Control, n. 12 (1970), pp. 1903–1912.
A. C. Lazer -D. A. Sanchez,On periodically perturbed conservative systems, Michigan Math. J.,16 (1969), pp. 193–200.
D. E. Leach,On Poincaré's perturbation theorem and a theorem of W. S. Loud, J. Diff. Eqs.,7 (1970), pp. 34–53.
G. A. Leonov,A certain class of nonlinear differential equations for which questions of the existence of bounded and periodic solutions may be solved effectively (Russian), Vestnik Leningrad. Univ. No. 19 Mat. Meh. Astronom. Vyp,4 (1972), pp. 29–32.
W. S. Loud,Periodic solutions of nonlinear differential equations of Duffing type, Proc. U.S.-Japan Seminar on Differential and Functional Equations (Minneapolis, 1967), pp. 119–224, Benjamin, New York, 1967.
R. A. Smith,Absolute stability of certain differential equations, J. London Math. Soc., (2),7 (1973), pp. 203–210.
R. A. Smith,Forced oscillations of the feedback control equation, Proc. Roy Soc. Edinburgh,76A (1976), pp. 31–42,
Entrata in Redazione il 19 giugno 1976.
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Smith, R.A. Resonance sets of the feedback control equation. Annali di Matematica 114, 195–205 (1977). https://doi.org/10.1007/BF02413786
- Differential Equation
- Periodic Solution
- Feedback Control
- Control Equation
- Nonlinear Vector