Have a personal or library account? Click to login

Comparison of direct and perturbation approaches to analysis of infinite–dimensional feedback control systems

Open Access
|Jul 2021

References

  1. Adler, M., Bombieri, M. and Engel, K.-J. (2017). Perturbation of analytic semigroups and applications to partial differential equations, Journal of Evolution Equations 17(4): 1183–1208.10.1007/s00028-016-0377-8
  2. Arendt, W., Batty, C., Hieber, M. and Neubrander, F. (2011). Vector-Valued Laplace Transforms and Cauchy Problems, Springer-Basel AG, Basel.10.1007/978-3-0348-0087-7
  3. Ball, J. (1977). Strongly continuous semigroups, weak solutions and the variation of constants formula, Proceedings of the American Mathematical Society 63(2): 370–373.10.1090/S0002-9939-1977-0442748-6
  4. Bouldin, R. (1971). The numerical range of a product, Journal of Mathematical Analysis and Applications 33(1): 243–263.10.1016/0022-247X(71)90193-4
  5. Crouzeix, M. (2008). A functional calculus based on the numerical range: Applications, Linear and Multilinear Algebra 56(1–2): 81–103.10.1080/03081080701336610
  6. Curtain, R. (1984). Spectral systems, International Journal of Control 39(4): 657–666.10.1080/00207178408933195
  7. Curtain, R. and Zwart, H. (1995). An Introduction to Infinite-Dimensional Linear Systems Theory, Springer, New York.10.1007/978-1-4612-4224-6
  8. Deckard, D., Foias, C., and Pearcy, C. (1979). Compact operators with root vectors that span, Proceedings of the American Mathematical Society 76(1): 101–106.10.1090/S0002-9939-1979-0534397-8
  9. DeLaubenfels, R. (1988). Inverses of generators, Proceedings of the American Mathematical Society 104(2): 443–448.10.1090/S0002-9939-1988-0962810-6
  10. Dunford, N. and Schwartz, J. (1971). Linear Operators. Part III: Spectral Operators, Wiley-Interscience, New York.
  11. Engel, K.-J. and Nagel, A. (2000). One-Parameter Semigroups, Springer, New York.
  12. Furuta, T. (1977). Relations between generalized growth conditions and several classes of convexoid operators, Canadian Journal of Mathematics 29(1–2): 1010–1030.10.4153/CJM-1977-099-0
  13. Furuta, T. (2001). Invitation to Linear Operators. From Matrices to Bounded Linear Operators on a Hilbert Space, CRC Press, London.10.1201/b16820
  14. Gohberg, I. and Krein, M. (1965). Introduction to the Theory of Linear Non-selfadjoint Operators, Nauka, Moscow, (in Russian).
  15. Grabowski, P. (1990). On spectral-Lyapunov approach to parametric optimization of distributed parameter systems, IMA Journal of Mathematical Control and Information 7(4): 317–338.10.1093/imamci/7.4.317
  16. Grabowski, P. (1995). Admissibility of observation functionals, International Journal of Control 62(5): 1161–1173.10.1080/00207179508921589
  17. Grabowski, P. (1999). Lecture Notes on Optimal Control Systems, AGH University Press, Kraków, http://home.agh.edu.pl/~pgrab/grabowski_files/lecturedition2/newlecture.xml.
  18. Grabowski, P. (2006). Well-posedness and stability analysis of hybrid feedback systems using Shkalikov’s theory, Opuscula Mathematica 26(1): 43–95.
  19. Grabowski, P. (2017). Some modifications of the Weiss–Staffans perturbation theorem, International Journal of Robust and Nonlinear Control 27(7): 1094–1121.10.1002/rnc.3617
  20. Grabowski, P. and Callier, F. (1999). Admissible observation operators. duality of observation and control using factorizations, Dynamics of Continuous, Discrete and Impulsive Systems 6(1): 87–119.
  21. Gustafson, K. and Rao, D. (1997). Numerical Range. The Field of Values of Linear Operators and Matrices, Springer, New York.
  22. Ionkin, N. (1977). Solutions of a boundary-value problem in heat conduction with a nonclassical boundary condition, Differentsial’nye Uravnieniya 13(2): 294–304.
  23. Janas, J. (1989). On unbounded hyponormal operators I, Archive för Matematik 27(1–2): 273–281.10.1007/BF02386376
  24. Kantorovitz, S. (2000). Topics in Operator Semigroups, Springer, New York.
  25. Kato, T. (1995). Perturbation Theory for Linear Operators, Springer, Berlin.10.1007/978-3-642-66282-9
  26. Katsnel’son, V. (1967). Conditions for a system of root vectors of certain classes of operators to be a basis, Fuktsjonal’nyj analiz i evo prilozhenija 1(2): 39–51.
  27. Kesel’man, G. (1964). On the unconditional convergence of eigenfunction expansions of some differential operators, Izvestya Vyshych Uchebnych Zavedenii: Matematika 39(2): 82–93, (in Russian).
  28. Lang, P. and Locker, J. (1989). Spectral theory of two-point differential operators determined by −D2. I: Spectral properities, Journal of Mathematical Analysis and Applications 141(2): 538–558.
  29. Lang, P. and Locker, J. (1990). Spectral theory of two-point differential operators determined by −D2. II: Analysis of cases, Journal of Mathematical Analysis and Applications 146(1): 148–191.
  30. Marchenko, V. (1977). Sturm–Liouville Operators and Applications, Naukova Dumka, Kiev.
  31. Mennicken, R. and Möller, M. (2003). Non-self-adjoint Boundary Eigenvalue Problems, Elsevier, Amsterdam.10.1016/S0304-0208(03)80005-1
  32. Mikhajlov, V. (1962). On Riesz bases in L2(0, 1), Doklady Akademii Nauk SSSR 144(5): 981–984, (in Russian).
  33. Orland, G. (1964). On a class of operators, Proceedings of the American Mathematical Society 15(1): 75–79.10.1090/S0002-9939-1964-0157244-4
  34. Paunonen, L. (2014). Robustness of strong stability of semigroups, Journal of Differential Equations 257(12): 403–436.10.1016/j.jde.2014.08.011
  35. Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York.10.1007/978-1-4612-5561-1
  36. Prüss, J. (1984). On the spectrum of C0-semigroups, Transactions of the American Mathematical Society 284(2): 847–857.10.1090/S0002-9947-1984-0743749-9
  37. Röh, H. (1982a). Dissipative operator with finite dimensional damping, Proceedings of the Royal Society of Edinburgh 91A(3–4): 243–263.10.1017/S0308210500017480
  38. Röh, H. (1982b). Spectral Analysis of Non Self-Adjoint C0-Semigroup Generators, PhD thesis, Hariot–Watt University, Edinburgh.
  39. Shapiro, J. (2017). Notes on the numerical range, Technical Report, May, 5, Michigan State University, East Lansing.
  40. Shkalikov, A. (1982). Basis property of eigenfunctions of ordinary differential operators with integral boundary conditions, Vestnik Moskovskogo Universiteta: Matematika i Mekhanika 6: 12–21, (in Russian).
  41. Shkalikov, A. (1986). Boundary problem for ordinary differential operators with parameter in the boundary conditions, Journal of Soviet Mathematics 33(6): 1311–1342.10.1007/BF01084754
  42. Triggiani, R. (1975). On the stabilizability problem in Banach space, SIAM Journal on Control 13(3): 383–403.10.1016/0022-247X(75)90067-0
  43. Weidmann, J. (1980). Linear Operators in Hilbert Spaces, Springer, Heidelberg.10.1007/978-1-4612-6027-1
DOI: https://doi.org/10.34768/amcs-2021-0014 | Journal eISSN: 2083-8492 | Journal ISSN: 1641-876X
Language: English
Page range: 195 - 218
Submitted on: Aug 25, 2020
Accepted on: Jan 19, 2021
Published on: Jul 8, 2021
Published by: University of Zielona Góra
In partnership with: Paradigm Publishing Services
Publication frequency: 4 issues per year

© 2021 Piotr Grabowski, published by University of Zielona Góra
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.