Abstract
We present a novel approach to address integro-differential systems incorporating state, input, and output delays. Our approach leverages product spaces and employs a boundary perturbation technique. Initially, we focus on state-delay equations, wherein we introduce a variation of constants formula for the mild solution. Additionally, we establish spectral properties using a characteristic equation. Subsequently, we extend our analysis to integro-differential systems affected by state, input, and output delays. Notably, we demonstrate the equivalence between a such delay system and a regular free-delay system within the Salamon–Weiss framework. This equivalence sheds valuable insights on the nature of the integro-differential system under consideration.
Similar content being viewed by others
References
Bátkai, A., Piazzera, S.: Semigroups and linear partial differential equations with delay. J. Math. Anal. Appl. 264, 1–20 (2001)
Fadili, A., Bounit, H.: On the complex inversion formula and admissibility for a class of Volterra systems. Int. J. Differ. Equ. 42, 1–13 (2014)
Bounit, H., Hadd, S.: Regular linear systems governed by neutral FDEs. J. Math. Anal. Appl. 320, 836–858 (2006)
Chen, J.H., Liang, J., Xiao, T.J.: Stability of solutions to integro-differential equations in Hibert spaces. Bull. Belg. Math. Soc. Simon Stevin 18, 781–792 (2011)
Desch, W., Grimmer, R., Schappacher, W.: Wellposedness and wave propagation for a class of integrodifferential equations in Banach space. J. Differ. Equ. 74, 391–411 (1988)
Desch, W., Schappacher, W.: A semigroup approach to integro-diffrential equations in Banach spaces. J. Integral Equ. 10, 99–110 (1985)
Engel, K.J., Nagel, R.: One-parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)
Greiner, G.: Perturbing the boundary conditions of a generator. Houston J. Math. 13, 213–229 (1987)
Grimmer, R., Prüss, J.: On linear Volterra equations in Banach spaces. Hyperbolic partial differential equations II. Comput. Math. Appl. 11(1), 189–205 (1985)
Hadd, S.: Unbounded perturbations of \(C_0\)-semigroups on Banach spaces and applications. In: Semigroup Forum, vol. 70, pp. 451–465. Springer, Berlin (2005)
Hadd, S., Idrissi, A.: Regular linear systems governed by systems with state, input and output delays. IMA J. Math. Control Inform. 22(4), 423–439 (2005)
Hadd, S., Idriss, A.: On the admissibility of observation for perturbed \(C_0\)-semigroups on Banach spaces. Systems Control Lett. 55(1), 1–7 (2006)
Hadd, S., Idrissi, A., Rhandi, A.: The regular linear systems associated with the shift semigroups and application to control linear systems with delay. Math. Control Signals Syst. 18(3), 272–291 (2006)
Hadd, S., Manzo, R., Rhandi, A.: Unbounded perturbations of the generator domain. Discret. Contin. Dyn. Syst. A 35(2), 703–723 (2015)
Jacob, B., Partington, J.R.: Admissible control and observation operators for Volterra integral equations. J. Evol. Equ. 4, 333–343 (2004)
Jacob, B., Partington, J.R.: A resolvent test for admissibility of Volterra observation operators. J. Math. Anal. Appl. 332, 346–355 (2007)
Jung, M.: Admissibility of control operators for solution families to Volterra integral equations. SIAM J. Control Optim. 38, 1323–1333 (2000)
Miller, R.K.: Linear Volterra integrodifferential equations as semigroups. Funkcialaj Ekvacioj 17, 39–55 (1974)
Miller, R.K.: Volterra integral equations in a Banach space. Funkcialaj Ekvacioj 18, 163–194 (1975)
Nagel, R., Sinestrari, E.: Inhomogeneous Volterra integrodifferential equations for Hille-Yosida operators. Functional Analysis (Proceeding Essen 1991), K.D. Bierstedt, A. Pietsch, W.M. Ruess, D. Vogt (eds.), Lect. Notes Pure Appl. Math., vol. 150, pp. 391–399. Marcel Dekker, New York (1993)
Prüss, J.: Evolutionary Integral Equations and Applications. Birkhäuser, Basel (1993)
Staffans, O.J.: Well-Posed Linear Systems. Cambridge Univ. Press, Cambridge (2005)
Tucsnak, M., Weiss, G.: Observation and Control for Operator Semigroups. Birkhäuser, Basel (2009)
Weiss, G.: Admissible observation operators for linear semigroups. Israel J. Math. 65, 17–43 (1989)
Weiss, G.: Admissibility of unbounded control operators. SIAM J. Control Optim. 27, 527–545 (1989)
Weiss, G.: Regular linear systems with feedback. Math. Control Signals Sys. 7, 23–57 (1994)
Acknowledgements
The authors would like to thank the editor and the referee whose detailed comments helped us to improve the organization and the content of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Abdelaziz Rhandi.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
El Kadiri, Y., Hadd, S. & Bounit, H. Analysis and control of integro-differential Volterra equations with delays. Semigroup Forum 107, 375–401 (2023). https://doi.org/10.1007/s00233-023-10378-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00233-023-10378-7