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Spectral analysis and stabilization of a chain of serially connected EulerBernoulli beams and strings
On the structure of the global attractor for nonautonomous dynamical systems with weak convergence
1.  Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Campus Reina Mercedes, Apdo. de Correos 1160, 41080 Sevilla 
2.  State University of Moldova, Department of Mathematics and Informatics, A. Mateevich Street 60, MD–2009 Chişinău 
References:
[1] 
N. P. Bhatia and G. P. Szegö, "Stability Theory of Dynamical Systems," Lecture Notes in Mathematics, Springer, BerlinHeidelbergNew York, 1970. Google Scholar 
[2] 
I. U. Bronsteyn, "Extensions of Minimal Transformation Group," Noordhoff, 1979. Google Scholar 
[3] 
B. F. Bylov, R. E. Vinograd, D. M. Grobman and V. V. Nemytskii, "Lyapunov Exponents Theory and Its Applications to Problems of Stabity," Moscow, Nauka, 1966, 576 pp. (in Russian). Google Scholar 
[4] 
T. Caraballo and D. N. Cheban, Levitan/Bohr almost periodic and almost automorphic solutions of secondorder monotone differential equations, J. Differ. Eqns., 251 (2011), 708727. Google Scholar 
[5] 
D. N. Cheban, Quasiperiodic solutions of the dissipative systems with quasiperiodic coefficients, Differential Equations, 22 (1986), 267278. Google Scholar 
[6] 
D. N. Cheban, $\mathbb C$analytic dissipative dynamical systems, Differential Equations, 22 (1986), 19151922. Google Scholar 
[7] 
D. N. Cheban, Boundedness, dissipativity and almost periodicity of the solutions of linear and weakly nonlinear systems of differential equations, Dynamical systems and boundary value problems, Kishinev, "Shtiintsa," (1987), 143159. Google Scholar 
[8] 
D. N. Cheban, Global pullback atttactors of Canalytic nonautonomous dynamical systems, Stochastics and Dynamics, 1 (2001), 511535. Google Scholar 
[9] 
D. N. Cheban, "Global Attractors of NonAutonomous Dissipative Dynamical Systems," Interdisciplinary Mathematical Sciences 1. River Edge, NJ: World Scientific, 2004, 528pp. Google Scholar 
[10] 
D. N. Cheban, Levitan almost periodic and almost automorphic solutions of $V$monotone differential equations, J. Dynamics and Differential Equations, 20 (2008), 669697. Google Scholar 
[11] 
D. N. Cheban, "Asymptotically Almost Periodic Solutions of Differential Equations," Hindawi Publishing Corporation, New York, 2009, 203 pp. Google Scholar 
[12] 
D. N. Cheban, "Global Attractors of SetValued Dynamical and Control Systems," Nova Science Publishers, New York, 2010. Google Scholar 
[13] 
D. N. Cheban and C. Mammana, Invariant manifolds, global attractors and almost periodic solutions of nonautonomous difference equations, Nonlinear Analysis TMA, 56 (2004), 465484. Google Scholar 
[14] 
D. N. Cheban and B. Schmalfuß, Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of nonautonomous differential equations, J. Math. Anal. Appl., 340 (2008), 374393. Google Scholar 
[15] 
C. Conley, "Isolated Invariant Sets and the Morse Index," Region. Conf. Ser. Math., No.38, 1978. Am. Math. Soc., Providence, RI. Google Scholar 
[16] 
B. P. Demidovich, On Dissipativity of Certain Nonlinear Systems of Differential Equations, I, Vestnik MGU, 6 (1961), 1927. Google Scholar 
[17] 
B. P. Demidovich, On Dissipativity of Certain Nonlinear Systems of Differential Equations, II, Vestnik MGU, 1 (1962), 38. Google Scholar 
[18] 
B. P. Demidovich, "Lectures on Mathematical Theory of Stability," Moscow, Nauka, 1967. (in Russian) Google Scholar 
[19] 
A. M. Fink and P. O. Fredericson, Ultimate boundedness does not imply almost periodicity, Journal of Differential Equations, 9 (1971), 280284. Google Scholar 
[20] 
J. K. Hale, "Asymptotic Behaviour of Dissipative Systems," Amer. Math. Soc., Providence, RI, 1988. Google Scholar 
[21] 
M. W. Hirsch, H. L. Smith and X.Q. Zhao, Chain transitivity, attractivity, and strong repellers for semidynamical systems, J. Dyn. Diff. Eqns, 13 (2001), 107131. Google Scholar 
[22] 
B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations," Cambridge Univ. Press, London, 1982. Google Scholar 
[23] 
A. Pavlov, A. Pogrowsky, N. van de Wouw and N. Nijmeijer, Convergent dynamics, a tribute to Boris Pavlovich Demidovich, Systems and Control Letters, 52 (2007), 257261. Google Scholar 
[24] 
V. A. Pliss, "Nonlocal Problems in the Theory of Oscillations," Nauka, Moscow, 1964 (in Russian). [English translation: Nonlocal Problems in the Theory of Oscillations, Academic Press, 1966.] Google Scholar 
[25] 
V. A. Pliss, "Integral Sets of Periodic Systems of Differential Equations," Nauka, Moscow, 1977 (in Russian). Google Scholar 
[26] 
G. R. Sell, "Topological Dynamics and Ordinary Differential Equations," Van NostrandReinhold, London, 1971. Google Scholar 
[27] 
B. A. Shcherbakov, The comparability of the motions of dynamical systems with regard to the nature of their recurrence, Differential Equations, 11 (1975), 12461255. Google Scholar 
[28] 
B. A. Shcherbakov, "Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations," Ştiinţa, Chişinău, 1985. (In Russian) Google Scholar 
[29] 
R. E. Vinograd, Inapplicability of the method of characteristic exponents to the study of nonlinear differential equations, Mat. Sb. N.S., 41 (1957), 431438. (in Russian) Google Scholar 
[30] 
T. Yoshizawa, "Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions," Applied Mathematical Sciences, Vol. 14, SpringerVerlag, New YorkHeidelberg, 1975. vii+233 pp. Google Scholar 
[31] 
V. V. Zhikov, On stability and unstability of Levinson's centre, Differentsial'nye Uravneniya, 8 (1972), 21672170. Google Scholar 
[32] 
V. V. Zhikov, Monotonicity in the theory of almost periodic solutions of nonlinear operator equations, Mat. Sbornik, 90 (1973), 214228; English transl., Math. USSRSb., 19 (1974), 209223. Google Scholar 
[33] 
V. I. Zubov, "The Methods of A. M. Lyapunov and Their Application," Noordhoof, Groningen, 1964. Google Scholar 
[34] 
V. I. Zubov, "Theory of Oscillations," Nauka, Moscow, 1979. (in Russian) Google Scholar 
show all references
References:
[1] 
N. P. Bhatia and G. P. Szegö, "Stability Theory of Dynamical Systems," Lecture Notes in Mathematics, Springer, BerlinHeidelbergNew York, 1970. Google Scholar 
[2] 
I. U. Bronsteyn, "Extensions of Minimal Transformation Group," Noordhoff, 1979. Google Scholar 
[3] 
B. F. Bylov, R. E. Vinograd, D. M. Grobman and V. V. Nemytskii, "Lyapunov Exponents Theory and Its Applications to Problems of Stabity," Moscow, Nauka, 1966, 576 pp. (in Russian). Google Scholar 
[4] 
T. Caraballo and D. N. Cheban, Levitan/Bohr almost periodic and almost automorphic solutions of secondorder monotone differential equations, J. Differ. Eqns., 251 (2011), 708727. Google Scholar 
[5] 
D. N. Cheban, Quasiperiodic solutions of the dissipative systems with quasiperiodic coefficients, Differential Equations, 22 (1986), 267278. Google Scholar 
[6] 
D. N. Cheban, $\mathbb C$analytic dissipative dynamical systems, Differential Equations, 22 (1986), 19151922. Google Scholar 
[7] 
D. N. Cheban, Boundedness, dissipativity and almost periodicity of the solutions of linear and weakly nonlinear systems of differential equations, Dynamical systems and boundary value problems, Kishinev, "Shtiintsa," (1987), 143159. Google Scholar 
[8] 
D. N. Cheban, Global pullback atttactors of Canalytic nonautonomous dynamical systems, Stochastics and Dynamics, 1 (2001), 511535. Google Scholar 
[9] 
D. N. Cheban, "Global Attractors of NonAutonomous Dissipative Dynamical Systems," Interdisciplinary Mathematical Sciences 1. River Edge, NJ: World Scientific, 2004, 528pp. Google Scholar 
[10] 
D. N. Cheban, Levitan almost periodic and almost automorphic solutions of $V$monotone differential equations, J. Dynamics and Differential Equations, 20 (2008), 669697. Google Scholar 
[11] 
D. N. Cheban, "Asymptotically Almost Periodic Solutions of Differential Equations," Hindawi Publishing Corporation, New York, 2009, 203 pp. Google Scholar 
[12] 
D. N. Cheban, "Global Attractors of SetValued Dynamical and Control Systems," Nova Science Publishers, New York, 2010. Google Scholar 
[13] 
D. N. Cheban and C. Mammana, Invariant manifolds, global attractors and almost periodic solutions of nonautonomous difference equations, Nonlinear Analysis TMA, 56 (2004), 465484. Google Scholar 
[14] 
D. N. Cheban and B. Schmalfuß, Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of nonautonomous differential equations, J. Math. Anal. Appl., 340 (2008), 374393. Google Scholar 
[15] 
C. Conley, "Isolated Invariant Sets and the Morse Index," Region. Conf. Ser. Math., No.38, 1978. Am. Math. Soc., Providence, RI. Google Scholar 
[16] 
B. P. Demidovich, On Dissipativity of Certain Nonlinear Systems of Differential Equations, I, Vestnik MGU, 6 (1961), 1927. Google Scholar 
[17] 
B. P. Demidovich, On Dissipativity of Certain Nonlinear Systems of Differential Equations, II, Vestnik MGU, 1 (1962), 38. Google Scholar 
[18] 
B. P. Demidovich, "Lectures on Mathematical Theory of Stability," Moscow, Nauka, 1967. (in Russian) Google Scholar 
[19] 
A. M. Fink and P. O. Fredericson, Ultimate boundedness does not imply almost periodicity, Journal of Differential Equations, 9 (1971), 280284. Google Scholar 
[20] 
J. K. Hale, "Asymptotic Behaviour of Dissipative Systems," Amer. Math. Soc., Providence, RI, 1988. Google Scholar 
[21] 
M. W. Hirsch, H. L. Smith and X.Q. Zhao, Chain transitivity, attractivity, and strong repellers for semidynamical systems, J. Dyn. Diff. Eqns, 13 (2001), 107131. Google Scholar 
[22] 
B. M. Levitan and V. V. Zhikov, "Almost Periodic Functions and Differential Equations," Cambridge Univ. Press, London, 1982. Google Scholar 
[23] 
A. Pavlov, A. Pogrowsky, N. van de Wouw and N. Nijmeijer, Convergent dynamics, a tribute to Boris Pavlovich Demidovich, Systems and Control Letters, 52 (2007), 257261. Google Scholar 
[24] 
V. A. Pliss, "Nonlocal Problems in the Theory of Oscillations," Nauka, Moscow, 1964 (in Russian). [English translation: Nonlocal Problems in the Theory of Oscillations, Academic Press, 1966.] Google Scholar 
[25] 
V. A. Pliss, "Integral Sets of Periodic Systems of Differential Equations," Nauka, Moscow, 1977 (in Russian). Google Scholar 
[26] 
G. R. Sell, "Topological Dynamics and Ordinary Differential Equations," Van NostrandReinhold, London, 1971. Google Scholar 
[27] 
B. A. Shcherbakov, The comparability of the motions of dynamical systems with regard to the nature of their recurrence, Differential Equations, 11 (1975), 12461255. Google Scholar 
[28] 
B. A. Shcherbakov, "Poisson Stability of Motions of Dynamical Systems and Solutions of Differential Equations," Ştiinţa, Chişinău, 1985. (In Russian) Google Scholar 
[29] 
R. E. Vinograd, Inapplicability of the method of characteristic exponents to the study of nonlinear differential equations, Mat. Sb. N.S., 41 (1957), 431438. (in Russian) Google Scholar 
[30] 
T. Yoshizawa, "Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions," Applied Mathematical Sciences, Vol. 14, SpringerVerlag, New YorkHeidelberg, 1975. vii+233 pp. Google Scholar 
[31] 
V. V. Zhikov, On stability and unstability of Levinson's centre, Differentsial'nye Uravneniya, 8 (1972), 21672170. Google Scholar 
[32] 
V. V. Zhikov, Monotonicity in the theory of almost periodic solutions of nonlinear operator equations, Mat. Sbornik, 90 (1973), 214228; English transl., Math. USSRSb., 19 (1974), 209223. Google Scholar 
[33] 
V. I. Zubov, "The Methods of A. M. Lyapunov and Their Application," Noordhoof, Groningen, 1964. Google Scholar 
[34] 
V. I. Zubov, "Theory of Oscillations," Nauka, Moscow, 1979. (in Russian) Google Scholar 
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