Publication Cover
Optimization
A Journal of Mathematical Programming and Operations Research
Volume 71, 2022 - Issue 12
224
Views
0
CrossRef citations to date
0
Altmetric
Research Article

Fundamental solutions for semi-linear neutral retarded integro-differential systems and applications to control problems

&
Pages 3439-3483 | Received 25 Nov 2020, Accepted 04 Mar 2021, Published online: 23 Mar 2021

References

  • Desch W, Grimmer R, Schappacher W. Well-posedness and wave propagation for a class of integrodifferential equations in Banach space. J Differ Equ. 1988;74:391–411.
  • Miller RK. An integro-differential equation for rigid heat conductors with memory. J Math Anal Appl. 1978;66:313–332.
  • Nunziato JW. On heat conduction in materials with memory. Quart Appl Math. 1971;29:187–304.
  • Cannarsa P, Sforza D. Global solutions of abstract semilinear parabolic equations with memory terms. NoDEA Nonlinear Differ Equ Appl. 2003;10:399–430.
  • Clément P, Nohel JA. Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels. SIAM J Math Anal. 1981;12:514–535.
  • Clément P, Prüss J. Global existence for a semilinear parabolic Volterra equation. Math Z. 1992;209:17–26.
  • Lunardi A. On the linear heat equation with fading memory. SIAM J Math Anal. 1990;21:1213–1224.
  • Prüss J. Evolutionary integral equations and applications. Basel: Birkhäuser-Verlag; 1993. (Monographs in mathematics; 87).
  • Grimmer R. Resolvent operator for integral equations in a Banach space. Trans Am Math Soc. 1982;273:333–349.
  • Grimmer R, Kappel F. Series expansions of Volterra integrodifferential equations in Banach space. SIAM J Math Anal. 1984;15:595–604.
  • Grimmer R, Pritchard AJ. Analytic resolvent operators for integral equations in a Banach space. J Differ Equ. 1983;50:234–259.
  • Caraballo T, Ogouyandjou C, Allognissode FK, et al. Existence and exponential stability for neutral stochastic integro-differential equations with impulses driven by a Rosenblatt process. Discrete Contin Dyn Syst Ser B. 2020;25:507–528.
  • Diallo MA, Ezzinbi K, Sène A. Optimal control problem for some integrodifferential equations in Banach spaces. Optim Control Appl Meth. 2018;39:563–574.
  • Ezzinbi K, Ghnimi S. Existence and regularity of solutions for some partial integrodifferential equations involving the nonlocal conditions. Numer Funct Anal Optim. 2019;40:1532–1549.
  • Fu X, Gao Y, Zhang Y. Existence of solutions for neutral integrodifferential equations with nonlocal conditions. Taiwan J Math. 2012;16:1879–1909.
  • Fu X, Huang R. Existence of solutions for neutral integro-differential equations with state-dependent delay. Appl Math Comput. 2013;224:743–759.
  • Hernández E, O'Regan D. On a new class of abstract neutral integro-differential equations and applications. Acta Appl Math. 2017;149:125–137.
  • Dos Santos JPC, Henríquez H, Hernández E. Existence results for neutral integrodifferential equations with unbounded delay. J Integral Equ Appl. 2011;23:289–330.
  • Dos Santos JPC, Henríquez H. Existence of S-asymptotically ω-periodic solutions to abstract integro-differential equations. Appl Math Comput. 2015;256:109–118.
  • Jeet K, Sukavanam N. Approximate controllability of nonlocal and impulsive neutral integro-differential equations using the resolvent operator theory and an approximating technique. Appl Math Comput. 2020;364:124690, 15 pp.
  • Vijayakumar V. Approximate controllability results for abstract neutral integro-differential inclusions with infinite delay in Hilbert spaces. IMA J Math Control Inf. 2018;35:297–314.
  • Liu K. The fundamental solution and its role in the optimal control of infinite dimensional neutral systems. Appl Math Optim. 2009;60:1–38.
  • Huang H, Fu X. Approximate controllability of semi-linear stochastic integro-differential equations with infinite delay. IMA J Math Control Inf. 2020;37:1133–1167.
  • Mokkedem FZ, Fu X. Approximate controllability of a semi-linear neutral evolution system with infinite delay. Int J Robust Nonlinear Control. 2017;27:1122–1146.
  • Sukavanam N, Tafesse S. Approximate controllability of a delayed semilinear control system with growing nonlinear term. Nonlinear Anal. 2011;74:6868–6875.
  • Jeong JM, Hwang HJ. Optimal control problems for semilinear retarded functional differential equations. J Optim Theory Appl. 2015;157:49–67.
  • Jeong JM, Son SJ. Time optimal control of semilinear control systems involving time delays. J Optim Theory Appl. 2015;165:793–811.
  • Mokkedem FZ, Fu X. Optimal control problems for a semilinear evolution system with infinite delay. Appl Math Optim. 2019;79:41–67.
  • Mordukhovich BS, Wang D, Wang L. Optimal control of delay-differential inclusions with functional endpoint constraints in infinite dimensions. Nonlinear Anal. 2009;71:2740–2749.
  • Nakagiri S. Optimal control of linear retarded systems in Banach spaces. J Math Anal Appl. 1986;120:169–210.
  • Chen P, Zhang X, Li Y. Approximate controllability of non-autonomous evolution system with nonlocal conditions. J Dyn Control Syst. 2020;26:1–16.
  • Mokkedem FZ, Fu X. Approximate controllability of semi-linear neutral integro-differential systems with finite delay. Appl Math Comput. 2014;242:202–215.
  • Mourad K. Approximate controllability of fractional neutral stochastic evolution equations in Hilbert spaces with fractional Brownian motion. Stoch Anal Appl. 2018;36:209–223.
  • Zhou Y, Suganya S, Mallika AM, et al. Approximate controllability of impulsive fractional integro-differential equation with state-dependent delay in Hilbert spaces. IMA J Math Control Inf. 2019;36:603–622.
  • Bashirov AE, Mahmudov NI. On concepts of controllability for linear deterministic and stochastic systems. SIAM J Control Optim. 1999;37:1808–1821.
  • Li X, Yong J. Optimal control theory for infinite dimensional systems. Boston: Birkhäuser; 1995.
  • Lions JL. Optimal control of systems governed by partial differential equations. New York (NY): Springer; 1971.
  • Balasubramaniam P, Tamilalagan P. The solvability and optimal controls for impulsive fractional stochastic integro-differential equations via resolvent operators. J Optim Theory Appl. 2017;174:139–155.
  • Chang Y, Pei Y. Degenerate type fractional evolution hemivariational inequalities and optimal controls via fractional resolvent operators. Int J Control. 2020;93:528–540.
  • Harrat A, Nieto JJ, Debbouche A. Solvability and optimal controls of impulsive Hilfer fractional delay evolution inclusions with Clarke subdifferential. J Comput Appl Math. 2018;344:725–737.
  • Kumar S. Mild solution and fractional optimal control of semilinear system with fixed delay. J Optim Theory Appl. 2017;174:108–121.
  • Engel KJ, Nagel R. One-Parameter semigroups for linear evolution equations. New York (NY): Springer; 2000.
  • Pazy A. Semigroups of linear operators and applications to partial differential equations. New York (NY): Springer-Verlag; 1983.
  • Desch W, Grimmer R, Schappacher W. Some considerations for linear integro-differential equations. J Math Anal Appl. 1984;104:219–234.
  • Grimmer R, Schappacher W. Weak solutions of integro-differential equations and resolvent operators. J Integral Equ. 1984;6:205–229.
  • Curtain R, Zwart HJ. An introduction to infinite dimensional linear systems theory. New York (NY): Springer-Verlag; 1995.
  • Travis CC, Webb GF. Existence, stability and compactness with α-norm for partial functional differential equations. Trans Am Math Soc. 1978;240:129–143.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.