Skip to main content

Existence of Admissible Controls and Optimal Controls

  • Chapter
  • First Online:
Book cover Time Optimal Control of Evolution Equations

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE-SC,volume 92))

  • 795 Accesses

Abstract

In this chapter, we will study the existence of admissible controls and optimal controls for Problem (TP) (given by (2.4) in Chapter 1) for some special cases.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. J.P. Aubin, Optima and Equilibria, An Introduction to Nonlinear Analysis, Translated from the French by Stephen Wilson. Graduate Texts in Mathematics, vol. 140 (Springer, Berlin, 1993)

    Google Scholar 

  2. V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems. Mathematics in Science and Engineering, vol. 190 (Academic, Boston, MA, 1993)

    Google Scholar 

  3. V. Barbu, The time optimal control of Navier-Stokes equations. Syst. Control Lett. 30, 93–100 (1997)

    Article  MathSciNet  Google Scholar 

  4. V. Barbu, Exact controllability of the superlinear heat equation. Appl. Math. Optim. 42, 73–89 (2000)

    Article  MathSciNet  Google Scholar 

  5. R. Conti, Teoria del controllo e del controllo ottimo, UTET, Torino (1974)

    Google Scholar 

  6. J.M. Coron, A.V. Fursikov, Global exact controllability of the 2D Navier-Stokes equations on a manifold without boundary. Russian J. Math. Phys. 4, 429–446 (1996)

    MathSciNet  MATH  Google Scholar 

  7. T. Duyckaerts, X. Zhang, E. Zuazua, On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials. Ann. Inst. H. Poincaré, Anal. Non Linéaire 25, 1–41 (2008)

    Google Scholar 

  8. C. Fabre, J.P. Puel, E. Zuazua, Approximate controllability of the semilinear heat equation. Proc. R. Soc. Edinb. A 125, 31–61 (1995)

    Article  MathSciNet  Google Scholar 

  9. H.O. Fattorini, Infinite Dimensional Linear Control Systems, the Time Optimal and Norm Optimal Problems. North-Holland Mathematics Studies, vol. 201 (Elsevier Science B.V., Amsterdam, 2005)

    Google Scholar 

  10. F. Gozzi, P. Loreti, Regularity of the minimum time function and minimum energy problems: the linear case. SIAM J. Control Optim. 37, 1195–1221 (1999)

    Article  MathSciNet  Google Scholar 

  11. J.B. Hiriart-Urruty, C. Lemaréchal, Fundamentals of Convex Analysis (Springer, Berlin, 2001)

    Book  Google Scholar 

  12. R.E. Kalman, Mathematical description of linear dynamical system. J. SIAM Control A 1, 152–192 (1963)

    MathSciNet  MATH  Google Scholar 

  13. K. Kunisch, L. Wang, The bang-bang property of time optimal controls for the Burgers equation. Discrete Contin. Dyn. Syst. 34, 3611–3637 (2014)

    Article  MathSciNet  Google Scholar 

  14. K. Kunisch, L. Wang, Bang-bang property of time optimal controls of semilinear parabolic equation. Discrete Contin. Dyn. Syst. 36, 279–302 (2016)

    Article  MathSciNet  Google Scholar 

  15. X. Li, J. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Systems & Control: Foundations & Applications (Birkhäuser Boston, Boston, MA, 1995)

    Google Scholar 

  16. P. Lin, Quenching time optimal control for some ordinary differential equations. J. Appl. Math. Art. ID 127809, 13 pp. (2014)

    Google Scholar 

  17. P. Lin, S. Luan, Time optimal control for some ordinary differential equations with multiple solutions. J. Optim. Theory Appl. 173, 78–90 (2017)

    Article  MathSciNet  Google Scholar 

  18. P. Lin, G. Wang, Blowup time optimal control for ordinary differential equations. SIAM J. Control Optim. 49, 73–105 (2011)

    Article  MathSciNet  Google Scholar 

  19. H. Lou, J. Wen, Y. Xu, Time optimal control problems for some non-smooth systems. Math. Control Relat. Fields 4, 289–314 (2014)

    Article  MathSciNet  Google Scholar 

  20. Q. Lü, G. Wang, On the existence of time optimal controls with constraints of the rectangular type for heat equations. SIAM J. Control Optim. 49, 1124–1149 (2011)

    Article  MathSciNet  Google Scholar 

  21. S. Micu, L.E. Temereancă, A time-optimal boundary controllability problem for the heat equation in a ball. Proc. R. Soc. Edinb. A 144, 1171–1189 (2014)

    Google Scholar 

  22. K.D. Phung, G. Wang, X. Zhang, On the existence of time optimal controls for linear evolution equations. Discrete Contin. Dyn. Syst. Ser. B 8, 925–941 (2007)

    Article  MathSciNet  Google Scholar 

  23. K.D. Phung, L. Wang, C. Zhang, Bang-bang property for time optimal control of semilinear heat equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, 477–499 (2014)

    Article  MathSciNet  Google Scholar 

  24. W.E. Schmitendorf, B.R. Barmish, Null controllability of linear system with constrained controls. SIAM J. Control Optim. 18, 327–345 (1980)

    Article  MathSciNet  Google Scholar 

  25. E.D. Sontag, Mathematical Control Theory: Deterministic Finite-Dimensional Systems. Texts in Applied Mathematics, 2nd edn., vol. 6 (Springer, New York, 1998)

    Google Scholar 

  26. M. Tucsnak, G. Wang, C. Wu, Perturbations of time optimal control problems for a class of abstract parabolic systems. SIAM J. Control Optim. 54, 2965–2991 (2016)

    Article  MathSciNet  Google Scholar 

  27. L.J. A-Vázquez, F.J. Fernández, A. Martínez, Analysis of a time optimal control problem related to the management of a bioreactor. ESAIM Control Optim. Calc. Var. 17, 722–748 (2011)

    Article  MathSciNet  Google Scholar 

  28. G. Wang, The existence of time optimal control of semilinear parabolic equations. Syst. Control Lett. 53, 171–175 (2004)

    Article  MathSciNet  Google Scholar 

  29. L. Wang, G. Wang, The optimal time control of a phase-field system. SIAM J. Control Optim. 42, 1483–1508 (2003)

    Article  MathSciNet  Google Scholar 

  30. L. Wang, Q. Yan, Bang-bang property of time optimal null controls for some semilinear heat equation. SIAM J. Control Optim. 54, 2949–2964 (2016)

    Article  MathSciNet  Google Scholar 

  31. G. Wang, Y. Zhang, Decompositions and bang-bang problems. Math. Control Relat. Fields 7, 73–170 (2017)

    Article  MathSciNet  Google Scholar 

  32. G. Wang, Y. Xu, Y. Zhang, Attainable subspaces and the bang-bang property of time optimal controls for heat equations. SIAM J. Control Optim. 53, 592–621 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Wang, G., Wang, L., Xu, Y., Zhang, Y. (2018). Existence of Admissible Controls and Optimal Controls. In: Time Optimal Control of Evolution Equations. Progress in Nonlinear Differential Equations and Their Applications(), vol 92. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-95363-2_3

Download citation

Publish with us

Policies and ethics