Existence and Optimal Control Results for Second-Order Semilinear System in Hilbert Spaces

Next, we discuss the existence and the uniqueness of mild solutions for the above proposed problem using Banach fixed point theorem. The stated Lagrange’s problem admits at least one optimal control pair under certain assumptions. Finally, the validation of theoretical results is provided through an example.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic €32.70 /Month

Buy Now

Price includes VAT (France)

Instant access to the full article PDF.

Rent this article via DeepDyve

Similar content being viewed by others

HJB Equations and Stochastic Control on Half-Spaces of Hilbert Spaces

Article Open access 27 May 2023

Results on exact controllability of second-order semilinear control system in Hilbert spaces

Article Open access 16 October 2021

Tikhonov solutions of approximately controllable second-order semilinear control systems

Article 26 August 2022

Data Availability Statement

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

References

  1. A.V. Balakrishnan, Optimal control problems in Banach spaces. J. Soc. Indust. Appl. Math. Ser. A Control 3, 152–180 (1965) ArticleMathSciNetGoogle Scholar
  2. E.J. Balder, Necessary and sufficient conditions for \(L_1\) -strong-weak lower semicontinuity of integral functionals. Nonlinear Anal. 11(12), 1399–1404 (1987) ArticleMathSciNetGoogle Scholar
  3. J.-M. Jeong, H.-J. Hwang, Optimal control problems for semilinear retarded functional differential equations. J. Optim. Theory Appl. 167(1), 49–67 (2015) ArticleMathSciNetGoogle Scholar
  4. J.-M. Jeong, J.-R. Kim, H.-H. Roh, Optimal control problems for semilinear evolution equations. J. Korean Math. Soc. 45(3), 757–769 (2008)
  5. J.-M. Jeong, S.-J. Son, Time optimal control of semilinear control systems involving time delays. J. Optim. Theory Appl. 165(3), 793–811 (2015) ArticleMathSciNetGoogle Scholar
  6. S. Kumar, Mild solution and fractional optimal control of semilinear system with fixed delay. J. Optim. Theory Appl. 174(1), 108–121 (2017) ArticleMathSciNetGoogle Scholar
  7. S. Kumar, The solvability and fractional optimal control for semilinear stochastic systems. Cubo 19(3), 1–14 (2017) ArticleMathSciNetGoogle Scholar
  8. N.S. Papageorgiou, Existence of optimal controls for nonlinear systems in Banach spaces. J. Optim. Theory Appl. 53(3), 451–459 (1987) ArticleMathSciNetGoogle Scholar
  9. N.S. Papageorgiou, On the optimal control of strongly nonlinear evolution equations. J. Math. Anal. Appl. 164(1), 83–103 (1992) ArticleMathSciNetGoogle Scholar
  10. N.S. Papageorgiou, Optimal control of nonlinear evolution equations with nonmonotone nonlinearities. J. Optim. Theory Appl. 77(3), 643–660 (1993) ArticleMathSciNetGoogle Scholar
  11. D.-G. Park, J.-M. Jeong, W.-K. Kang, Optimal problem for retarded semilinear differential equations. J. Korean Math. Soc. 36(2), 317–332 (1999) MathSciNetMATHGoogle Scholar
  12. J.-Y. Park, J.-M. Jeong, Y.-C. Kwun, Optimal control for retarded control system. Nihonkai Math. J. 8(1), 59–70 (1997) MathSciNetMATHGoogle Scholar
  13. A. Shukla et al., Approximate controllability of second-order semilinear control system. Circuits Syst. Signal Process. 35(9), 3339–3354 (2016) ArticleMathSciNetGoogle Scholar
  14. C.C. Travis, G.F. Webb, Cosine families and abstract nonlinear second order differential equations. Acta Math. Acad. Sci. Hungar. 32(1–2), 75–96 (1978). (58 #17404) ArticleMathSciNetGoogle Scholar
  15. J. Wang, Y. Zhou, W. Wei, Optimal feedback control for semilinear fractional evolution equations in Banach spaces. Syst. Control Lett. 61(4), 472–476 (2012) ArticleMathSciNetGoogle Scholar
  16. J. Wang, Y. Zhou, M. Medved, On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay. J. Optim. Theory Appl. 152(1), 31–50 (2012) ArticleMathSciNetGoogle Scholar
  17. E. Zeidler, Nonlinear functional analysis and its applications. I, translated from the German by Peter R. Wadsack, Springer-Verlag, New York, (1986)

Acknowledgements

Authors are thankful to the reviewers for the valuable suggestions. Authors are also thankful to Dr. Chhaya Singh (Assistant Professor, English, Rajkiya Engineering College Kannauj, India-209732), who helped us to improve the language of the article.

Author information

Authors and Affiliations

  1. Department of Applied Science, Rajkiya Engineering College Kannauj, Kannauj, 209732, India Anurag Shukla & Rohit Patel
  1. Anurag Shukla