zbMATH — the first resource for mathematics

A survey of the recent results on characterizations of exponential stability and dichotomy over finite dimensional spaces. (English) Zbl 07238388
Summary: The main purpose of this article is the investigation of the recent advances on the exponential stability and dichotomy of autonomous and nonautonomous linear differential systems, in both continuous and discrete cases i.e. \( \dot x(t)=Ax(t)\), \(\dot x(t)=A(t)x(t)\), \(x_{n+1}=Ax_n\) and \(x_{n+1}=A_nx_n\) in terms of the boundedness of solutions of some Cauchy problems, where \(A\), \(A_n\), and \(A(t)\) are square matrices, for any \(n\in\mathbb{Z}_+\) and \(t\in\mathbb{R}_+\).
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34D09 Dichotomy, trichotomy of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
37C60 Nonautonomous smooth dynamical systems
39A12 Discrete version of topics in analysis
34A30 Linear ordinary differential equations and systems, general
Full Text: MNR
[1] W. Arendt, C. J. K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Monographs in Mathematics, 96, Birkhäuser, Basel, 2001 · Zbl 0978.34001
[2] S. Arshad, C. Buşe, A. Nosheen, A. Zada, “Connections between the stability of a Poincaré map and boundedness of certain associate sequences”, Electronic Journal of Qualitative Theory of Differential Equations, 16 (2011), 1-12 · Zbl 1281.39009
[3] S. Arshad, C. Buşe, O. Saierli, “Connections between exponential stability and boundedness of solutions of a couple of differential time depending and periodic systems”, Electronic Journal of Qualitative Theory of Differential Equations, 90 (2011), 1-16 · Zbl 1340.34178
[4] A. V. Balakrishnan, “Fractional powers of closed operators and semigroups generated by them”, Pacific J. Math., 10 (1960), 419-439 · Zbl 0103.33502
[5] S. Balint, “On the Perron-Bellman theorem for systems with constant coefficients”, Ann. Univ. Timisoara, 21 (1983), 3-8 · Zbl 0536.34038
[6] V. Barbu, Differential Equations, Ed. Junimea, Iasi, 1985
[7] D. Barbu, C. Buşe, “Some remarks about Perron condition for strongly continuous \(C_0\) semi groups”, Analele univ. Timisora, 35:1 (1997), 3-8 · Zbl 1003.93043
[8] B. Basit, Some problems concerning different types of vector-valued almost periodic functions, Dissertationes Math. (Rozprawy Mat.), 338, 1995 · Zbl 0828.43004
[9] A. G. Baskakov, “Some conditions for invertibility of linear differential and difference operators”, Russian Acad. Sci. Dokl. Math., 48 (1994), 498-501
[10] C. J. K. Batty, R. Chill, Y. Tomilov, “Strong stability of bounded evolution families and semigroups”, Journal of Functional Analysis, 193 (2002), 116-139 · Zbl 1040.47030
[11] C. J. K. Batty, M. D. Blake, “Convergence of Laplace integrals”, C. R. Acad. Sci. Paris Serie 1, 330 (2000), 71-75 · Zbl 0945.44001
[12] S. Braza, C. Buşe, J. Pecaric, “New characteriazation of asymptotic stability for evolution families on Banach Spaces”, Electronic Journal of Differential Equations, 38 (2004), 1-9
[13] C. Buşe, “On the Perron-Bellman theorem for evolutionary processes with exponential growth in Banach spaces”, New Zealand Journal of Mathematics, 27 (1998), 183-190 · Zbl 0972.47027
[14] C. Buşe, D. Barbu, “Some remarks about the Perron condition for strongly continious semigroups”, Analele Univ. Timisora, 35:1 (1997), 3-8 · Zbl 1003.93043
[15] C. Buşe, P. Cerone, S. S. Dragomir, A. Sofo, “Uniform stability of periodic discrete system in Banach spaces”, J. Difference Equ. Appl., 11 (2005), 1081-1088 · Zbl 1094.47040
[16] C. Buşe, A. Pogan, “Individual exponential stability for evolution families of bounded and linear operators”, New Zealand Journal of Mathematics, 30 (2001), 15-24 · Zbl 0990.35020
[17] C. Buşe, M. Reghiş, “On the Perron-Bellman theorem for strongly continious semigroups and periodic evolutionary processes in Banach spaces”, Italian Journal of Pure and Applied Mathematics, 4 (1998), 155-166 · Zbl 0981.47030
[18] C. Buşe, A. Zada, “Dichotomy and boundedness of solutions for some discrete Cauchy problems”, Proceedings of IWOTA-2008, Operator Theory, Advances and Applications, 203, eds. J. A. Ball, V. Bolotnikov, W. Helton, L. Rodman, T. Spitkovsky, Birkhäuser Verlag, Basel, 2010, 165-174 · Zbl 1193.39004
[19] C. Buşe, A. Zada, “Boundedness and exponential stability for periodic time dependent systems”, Electronic Journal of Qualitative Theory of Differential Equations, 37 (2009), 1-9 · Zbl 1180.34049
[20] C. Chicone, Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Math. Surv. and Monographs, 70, Amer. Math. Soc., 1999 · Zbl 0970.47027
[21] S. Clark, Y. Latushkin, S. Montgomery-Smith, T. Randolph, “Stability radius and internal versus external stability in Banach spaces: an evolution semigroup approach”, SIAM J. Control Optim., 38 (2000), 1757-1793 · Zbl 0978.47030
[22] C. Corduneanu, Almost Periodic Functions, 2nd edition, Chelsea, New-York, 1989 · Zbl 0672.42008
[23] Y. L. Daletkii, M. G. Krein, Stability of solutions of differential equations in Banach spaces, Izd. Nauka, Moskwa, 1970
[24] R. Datko, “Uniform asymptotic stability of evolutionary processes in Banach space”, SIAM J. Math. Analysis, 3 (1973), 428-445 · Zbl 0241.34071
[25] D. A. Halanay, Stabilization of linear systems, Ed. All, 1994
[26] S. Elaydi, An introduction to difference equations, Undergraduate Texts in Mathematics, Third edition, Springer, New-York, 2005 · Zbl 1071.39001
[27] A. Elkoutri, M. A. Taoudi, “Spectral inclusions and stability results for strongly continuous semigroups”, International Journal of Mathematics and Mathematical Sciences (IJMMS), 37 (2003), 2379-2387 · Zbl 1068.47045
[28] K. J. Engel, R. Nagel, One-Parameter semigroups for linear evolutions equations, Springer-Verlag, 2000 · Zbl 0952.47036
[29] H. O. Fattorini, The Cauchy Problem, Addison Wesley, 1983 · Zbl 1167.34300
[30] C. Foias, F. H. Vasilescu, “Non-analytic local functional calculus”, Czech. Math. J., 24:99 (1974), 270-283 · Zbl 0314.47008
[31] I. Gohberg, S. Goldberg, Basic Operator Theory, Birkhäuser, Boston-Basel, 1981 · Zbl 0458.47001
[32] J. A. Goldstein, “Periodic and pseudo-periodic solutions of evolution equations”, Semigroups, Theory and applications, v. I, eds. H. Brézis, M. G. Crandall, F. Kappel, Longman, 1986, 142-146
[33] E. Hille, R. S. Philips, Functional Analysis and Semi-groups, Amer. Math. Soc. Coll. Publ., 31, Providence, R.I., 1957
[34] A. Ichikawa, “Null controllability with vanishing energy for discrete-time systems in Hilbert space”, SIAM J. Control Optim., 46:2 (2007), 683-693 · Zbl 1330.93034
[35] D. Lassoued, “Exponential dichotomy of nonautonomous periodic systems in terms of the boundedness of certain periodic cauchy problems”, Electronic Journal of Differential Equations, 2013 (2013), 89, 7 pp. · Zbl 1294.34059
[36] Academic Press, 1966 · Zbl 0161.06303
[37] Yu. I. Lyubich, Quôc Phong Vū, “Asymptotic stability of linear differential equations in Banach Spaces”, Studia Math., 88 (1988), 37-42 · Zbl 0639.34050
[38] M. Megan, A. L. Sasu, B. Sasu, “On uniform exponential stability of periodic evolution operators in Banach spaces”, Acta Math. Univ. Comenian., 69 (2000), 97-106 · Zbl 0955.34037
[39] N. Van Minh, Frank Räbiger, R. Schnaubelt, “Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half line”, Integral Equations Operator Theory, 32 (1998), 332-353 · Zbl 0977.34056
[40] V. Müller, “Power bounded operators and supercyclic vectors”, Proceedings of the American Mathematical Society, 131:12 (2003), 3807-3812 · Zbl 1054.47010
[41] J. V. Neerven, The asymptotic behaviour of semigroups of linear operators, Birkhäuser, Basel, 1996 · Zbl 0905.47001
[42] J. V. Neerven, “Individual stability of strongly continuous semigroups with uniformly bounded local resolvent”, Semigroup Forum, 53 (1996), 155-161 · Zbl 0892.47040
[43] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, 1983 · Zbl 0516.47023
[44] G. Peano, “Integrazione per serie delle equazioni differenziali lineari”, Atti Reale Acc. Sci. Torino, 22 (1887), 293-302 · JFM 19.0308.02
[45] G. Peano, “Intégration par séries des équations différentielles linéaires”, Math. Ann., 32 (1888), 450-456 · JFM 20.0329.02
[46] O. Perron, “Die Stabilitätsfrage bei Differentialgleichungen”, Math. Z., 32 (1930), 703 -728 · JFM 56.1040.01
[47] V. Q. Phong, “On stability of \(C_0\)-semigroups”, Proceedings of the American Mathematical Society, 129:10 (2002), 2871-2879 · Zbl 0998.47022
[48] G. D. Prato, E. Sinestrari, “Non autonomous evolution operators of hyperbolic type”, Semigroup Forum, 45 (1992), 302-321 · Zbl 0791.47040
[49] J. Prüss, Evolutionary integral equations and applications, Monographs in Mathematics, 87, Birkhäuser Verlag, Basel, 1993 · Zbl 0784.45006
[50] P. Preda, M. Megan, “Exponential dichotomy of evolutionary processes in Banach spaces”, Czehoslovak Math. Journal, 35:2 (1985), 312-333 · Zbl 0609.47051
[51] C. I. Preda, P. Preda, “An extension of the admissibility-type conditions for the exponential dichotomy of \(C_0\)-semigroups”, Integral Equations and Operator Theory, 63 (2009), 249-261 · Zbl 1175.47038
[52] M. Reghis, “About exponential dichotomy of linear autonomous differential equations”, Qualitative problems for differential equations and control theory, World Sci. Publ., River Edge, NJ, 1995, 53-60 · Zbl 0839.34071
[53] A. L. Sasu, “Integral equations on function spaces and dichotomy on the real line”, Integral Equations and Operator Theory, 58:1 (2007), 133-152 · Zbl 1144.34039
[54] A. L. Sasu, B. Sasu, “Exponential Stability for linear skew-product flows”, Bull. Sci. Math., 128:9 (2004), 728-738 · Zbl 1064.34037
[55] R. Schnaubelt, “Well-posedness and asymptotic behavior of non-autonomous linear evolution equations”, Evolution equations, Semigroups and functional analysi (Milano, 2000), Progr. Nonlinear Differential Equations Appl., 50, Birkhäuser, Basel, 2002, 311-338 · Zbl 1044.34016
[56] Y. Wang, A. Zada, N. Ahmad, D. Lassoued, T. Li, “Uniform exponential stability of discrete evolution families on space of \(p\)-periodic sequences”, Abstract and Applied Analysis, 2014 (2014), Article ID 784289, 4 pp. · Zbl 07023063
[57] G. Weiss, “Weak \(L^p\)-stability of linear semigroup on a Hilbert space implies exponential stability”, J. Differential Equations, 76 (1988), 269-285 · Zbl 0675.47031
[58] A. Zada, “A characterization of dichotomy in terms of boundedness of solutions for some Cauchy Problems”, Electronic Journal of Qualitative Theory of Differential Equations, 94 (2008), 1-5 · Zbl 1168.47034
[59] A. Zada, Asymptotic behavior of solutions for a class of semi-linear differential systems in finite Dimensional Spaces, Dissertation, Abdus Salam school of mathematical sciences GC University, Lahore, Pakistan, 2010
[60] A. Zada, R. Amin, M. Asif, G. A. Khan, “On discrete characterization of dichotomy for autonomous systems”, Journal of Advanced Research in Dynamical Systems and control, 6:1 (2014), 48-55
[61] A. Zada, N. Ahmad, I. Khan, F. Khan, “On the exponential stability of discrete semigroups”, Qual. Theory Dyn. Syst., 14:1 (2015), 149-155 · Zbl 1336.47047
[62] A. Zada, S. Arshad, G. Rahmat, R. Amin, “Dichotomy of Poincar?e Maps and Boundedness of some Cauchy sequences”, Applied Mathematics E-Notes, 12 (2012), 14-22 · Zbl 1261.39003
[63] A. Zada, S. Arshad, G. Rahmat, A. Khan, “On the Dichotomy of Non-Autonomous Systems Over Finite Dimensional Spaces”, Appl. Math. Inf. Sci., 9:4 (2015), 1941-1946
[64] A. Zada, G. A. Khan, M. Asif, R. Amin, “On dichotomy of autonomous systems and boundedness of some Cauchy problem”, International Journal of Research and Reviews in Applied Sciences, 14:3 (2013), 533-538
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.