## International Journal of Differential Equations

### On the Complex Inversion Formula and Admissibility for a Class of Volterra Systems

#### Abstract

This paper studies Volterra integral evolution equations of convolution type from the point of view of complex inversion formula and the admissibility in the Salamon-Weiss sens. We first present results on the validity of the inverse formula of the Laplace transform for the resolvent families associated with scalar Volterra integral equations of convolution type in Banach spaces, which extends and improves the results in Hille and Philllips (1957) and Cioranescu and Lizama (2003, Lemma 5), respectively, including the stronger version for a class of scalar Volterra integrodifferential equations of convolution type on unconditional martingale differences UMD spaces, provided that the leading operator generates a ${C}_{0}$-semigroup. Next, a necessary and sufficient condition for ${L}^{p}$-admissibility $(p\in [1,\infty [)$ of the system's control operator is given in terms of the UMD-property of its underlying control space for a wider class of Volterra integrodifferential equations when the leading operator is not necessarily a generator, which provides a generalization of a result known to hold for the standard Cauchy problem (Bounit et al., 2010, Proposition 3.2).

#### Article information

Source
Int. J. Differ. Equ., Volume 2014 (2014), Article ID 948597, 13 pages.

Dates
Revised: 13 April 2014
Accepted: 26 April 2014
First available in Project Euclid: 20 January 2017

https://projecteuclid.org/euclid.ijde/1484881403

Digital Object Identifier
doi:10.1155/2014/948597

Mathematical Reviews number (MathSciNet)
MR3219411

Zentralblatt MATH identifier
1291.44001

#### Citation

Fadili, Ahmed; Bounit, Hamid. On the Complex Inversion Formula and Admissibility for a Class of Volterra Systems. Int. J. Differ. Equ. 2014 (2014), Article ID 948597, 13 pages. doi:10.1155/2014/948597. https://projecteuclid.org/euclid.ijde/1484881403

#### References

• G. Da Prato and M. Iannelli, “Linear abstract integro-differential equations of hyperbolic type in Hilbert spaces,” Rendiconti del Seminario Matematico dell'Università di Padova, vol. 62, pp. 191–206, 1980.MR582950
• G. Da Prato and M. Iannelli, “Existence and regularity for a class of integro-differential equations of parabolic type,” Journal of Mathematical Analysis and Applications, vol. 112, no. 1, pp. 36–55, 1985.MR812792
• R. C. Grimmer and A. J. Pritchard, “Analytic resolvent operators for integral equations in Banach space,” Journal of Differential Equations, vol. 50, no. 2, pp. 234–259, 1983.MR719448
• A. Lunardi, “Laplace transform methods in integro-differential equations,” Journal of Integral Equations, vol. 10, pp. 185–211, 1985.MR831244
• J. Prüss, “On linear Volterra equations of parabolic type in Banach spaces,” Transactions of the American Mathematical Society, vol. 301, no. 2, pp. 691–721, 1987.MR882711
• J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, Switzerland, 1993.MR2964432
• Ph. Clément and G. Da Prato, “Existence and regularity results for an integral equation with infinite delay in a Banach space,” Integral Equations and Operator Theory, vol. 11, no. 4, pp. 480–500, 1988.MR950513
• R. K. Miller, “Volterra integral equations in a Banach space,” Funkcialaj Ekvacioj, vol. 18, no. 2, pp. 163–193, 1975.MR0410312
• G. Chen and R. Grimmer, “Integral equations as evolution equations,” Journal of Differential Equations, vol. 45, no. 1, pp. 53–74, 1982.MR662486
• G. Chen and R. Grimmer, “Semigroups and integral equations,” Journal of Integral Equations, vol. 2, no. 2, pp. 133–154, 1980.MR572484
• W. Desch and R. C. Grimmer, “Initial-boundary value problems for integro-differential equations,” Journal of Integral Equations, vol. 10, pp. 73–97, 1985.MR831236
• W. Desch and W. Schappacher, “A semigroup approach to integro-differential equations in Banach spaces,” Journal of Integral Equations, vol. 10, pp. 99–110, 1985.MR831237
• G. Di Blasio, K. Kunisch, and E. Sinestrari, “Stability for abstract linear functional differential equations,” Israel Journal of Mathematics, vol. 50, no. 3, pp. 231–263, 1985.MR793856
• R. Nagel and E. Sinestrari, “Inhomogeneous Volterra integrodifferential equations for Hille-Yosida operators,” in Functional Analysis, vol. 150 of Lecture Notes in Pure and Applied Mathematics, pp. 51–70, Marcel Dekker, New York, NY, USA, 1993.MR1241671
• K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2000.MR1721989
• T. Bárta, “Smooth solutions of Volterra equations via semigroups,” Bulletin of the Australian Mathematical Society, vol. 78, no. 2, pp. 249–260, 2008.MR2466862
• E. Hille and R. S. Philllips, Functional Analysis and Semigroups, American Mathematical Society Translations Series, 1957.
• P. F. Yao, “On the inversion of the Laplace transform of ${C}_{0}$ semigroups and its applications,” SIAM Journal on Mathematical Analysis, vol. 26, no. 5, pp. 1331–1341, 1995.MR1347423
• A. Driouich and O. El-Mennaoui, “On the inverse Laplace transform for ${C}_{0}$-semigroups in UMD-spaces,” Archiv der Mathematik, vol. 72, no. 1, pp. 56–63, 1999.MR1657355
• W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-Valued, Laplace Transforms and Cauchy Problems, Birkhäauser, Ulm, Germany, 2010.MR1886588
• I. Cioranescu and V. Keyantuo, “On operator cosine functions in UMD spaces,” Semigroup Forum, vol. 63, no. 3, pp. 429–440, 2001.MR1851822
• I. Cioranescu and C. Lizama, “On the inversion of the Laplace transform for resolvent families in UMD spaces,” Archiv der Mathematik, vol. 81, no. 2, pp. 182–192, 2003.MR2009561
• M. Haase, “The complex inversion formula revisited,” Journal of the Australian Mathematical Society, vol. 84, no. 1, pp. 73–83, 2008.MR2469268
• C. Lizama and J. C. de Souza, “The complex inversion formula in UMD spaces for families of bounded operators,” Applicable Analysis, vol. 91, no. 5, pp. 937–946, 2012.MR2911242
• L. F. Ho and D. L. Russell, “Admissible input elements for systems in Hilbert space and a Carleson measure criterion,” SIAM Journal on Control and Optimization, vol. 21, no. 4, pp. 614–640, 1983.MR704478
• G. Weiss, “Admissibility of unbounded control operators,” SIAM Journal on Control and Optimization, vol. 27, no. 3, pp. 527–545, 1989.MR993285
• G. Weiss, “Two conjectures on the admissibility of control operators,” in Estimation and Control of Distributed Parameter Systems, F. Kappel and W. Desch, Eds., pp. 367–378, Birkhäuser, Basel, Switzerland, 1989.MR1155659
• B. Jacob and J. R. Partington, “The Weiss conjecture on admissibility of observation operators for contraction semigroups,” Integral Equations and Operator Theory, vol. 40, no. 2, pp. 231–243, 2001.MR1831828
• C. Le Merdy, “The Weiss conjecture for bounded analytic semigroups,” Journal of the London Mathematical Society, vol. 67, no. 3, pp. 715–738, 2003.MR1967702
• B. Jacob and J. R. Partington, “Admissibility of control and observation operators for semigroups: a survey,” in Current Trends in Operator Theory and Its Applications, vol. 149, pp. 199–221, Springer, New York, NY, USA, 2004.MR2063754
• O. Staffans, Well-Posed Linear Systems, vol. 103 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 2005.MR2154892
• F. Maragh, H. Bounit, A. Fadili, and H. Hammouri, “On the admissible control operators for linear and bilinear systems and the Favard spaces,” Bulletin of the Belgian Mathematical Society–-Simon Stevin, vol. 21, no. 4, 2014.
• M. Jung, “Admissibility of control operators for solution families to Volterra integral equations,” SIAM Journal on Control and Optimization, vol. 38, no. 5, pp. 1323–1333, 2000.MR1766417
• B. Jacob and J. R. Partington, “Admissible control and observation operators for Volterra integral equations,” Journal of Evolution Equations, vol. 4, no. 3, pp. 333–343, 2004.MR2120127
• B. Jacob and J. R. Partington, “A resolvent test for admissibility of Volterra observation operators,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 346–355, 2007.MR2319665
• B. H. Haak, B. Jacob, J. R. Partington, and S. Pott, “Admissibility and controllability of diagonal Volterra equations with scalar inputs,” Journal of Differential Equations, vol. 246, no. 11, pp. 4423–4440, 2009.MR2517779
• A. Fadili and H. Bounit, “On the Favard spaces and the admissibility for Volterra systems with scalar kernel,” Under review.
• H. Bounit, A. Driouich, and O. El-Mennaoui, “Admissibility of control operators in UMD spaces and the inverse Laplace transform,” Integral Equations and Operator Theory, vol. 68, no. 4, pp. 451–472, 2010.MR2745473
• E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, Technische Universiteit Eindhoven, Eindhoven, The Netherlands, 2001.MR2715439
• W. Arendt and J. Prüss, “Vector-valued Tauberian theorems and asymptotic behavior of linear Volterra equations,” SIAM Journal on Mathematical Analysis, vol. 23, no. 2, pp. 412–448, 1992.MR1147871
• W. Desch and J. Prüss, “Counterexamples for abstract linear Volterra equations,” Journal of Integral Equations and Applications, vol. 5, no. 1, pp. 29–45, 1993.MR1214709
• J. Bourgain, “Some remarks on Banach spaces in which martingale difference sequences are unconditional,” Arkiv för Matematik, vol. 21, no. 2, pp. 163–168, 1983.MR727340
• J. Bourgain, “Vector-valued singular integrals and the ${H}^{1}$-BMO duality,” in Probability Theory and Harmonic Analysis, vol. 98 of Textbooks Pure and Applied Mathematics, pp. 1–19, 1986.MR830227
• D. L. Burkholder, “Martingales and singular integrals in Banach spaces,” in Handbook of the Geometry of Banach Spaces, pp. 233–269, North-Holland, Amsterdam, The Netherlands, 2001.MR1863694
• M. Haase, New Tends in the Theory of Hyperbolic Equations, Birkhäauser, Boston, Mass, USA, 2000.
• H. Amann, Linear and Quasilinear Parabolic Problems, Birkhäauser, Boston, Mass, USA, 1995.
• A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983.MR710486
• R. Grimmer and J. Prüss, “On linear Volterra equations in Banach spaces,” Computers & Mathematics with Applications, vol. 11, no. 1–3, pp. 189–205, 1985.MR787436 \endinput