Abstract
This work studies the controllability of a class of delay differential equations. Instead of \(C_0\)-semigroup associated with the mild solution of the system, we use the concept of fundamental solution. Approximate controllability of the system is shown using sequence method. Finally, an illustrative example has been provided.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
R.F. Curtain, H. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory, Texts in Applied Mathematics, vol. 21 (Springer, New York, 1995)
K. Naito, Controllability of semilinear control systems dominated by the linear part. SIAM J. Control Optim. 25, 715–722 (1987)
K. Balachandran, J.P. Dauer, Controllability of nonlinear systems in Banach spaces: a survey. J Optim. Theory Appl. 115(1), 7–28 (2002)
J. Klamka, Relative controllability of minimum energy control of linear systems with distributed delays in control. IEEE T. Automat. Contr. 21, 594–595 (1976)
J. Klamka, Schauder’s fixed point theorem in nonlinear controllability problems. Control Cybern. 29, 153–165 (2000)
N.I. Mahmudov, N. Semi, Approximate controllability of semilinear control systems in Hilbert spaces. TWMS J. App. Eng. Math. 2, 67–74 (2012)
C. Wang, R. Du, Approximate controllability of a class of semilinear degenerate systems with convection term. J. Differ. Equ. 254(9), 3665–3689 (2013)
L. Wang, Approximate controllability of integrodifferential equations with multiple delays. J. Optim Theory Appl. 143, 185–206 (2009)
N. Sukavanam, Approximate controllability of semilinear control systems with growing nonlinearity, in Mathematical theory of control proceedings of international conference (Marcel Dekker, New York, 1993), pp. 353–357
J. Klamka, Stochastic controllability of systems with variable delay in control. Bull. Pol. Ac. Tech. 56, 279–284 (2008)
J. Klamka, Stochastic controllability and minimum energy control of systems with multiple delays in control. Appl. Math. Comput. 206, 704–715 (2008)
I. Davies, P. Jackreece, Controllability and null controllability of linear systems. J. Appl. Sci. Environ. Manag. 9, 31–36 (2005)
A. Shukla, N. Sukavanam, D.N. Pandey, Approximate controllability of semilinear system with state delay using sequence method. J. Frankl. Inst. 352, 5380–5392 (2015)
N. Sukavanam, S. Tafesse, Approximate controllability of a delayed semilinear control system with growing nonlinear term. Nonlinear Anal. 74, 6868–6875 (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Haq, A., Sukavanam, N. (2020). Controllability of Semilinear Control Systems with Fixed Delay in State. In: Deo, N., Gupta, V., Acu, A., Agrawal, P. (eds) Mathematical Analysis II: Optimisation, Differential Equations and Graph Theory. ICRAPAM 2018. Springer Proceedings in Mathematics & Statistics, vol 307. Springer, Singapore. https://doi.org/10.1007/978-981-15-1157-8_4
Download citation
DOI: https://doi.org/10.1007/978-981-15-1157-8_4
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-15-1156-1
Online ISBN: 978-981-15-1157-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)