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Time Optimal Control for Some Ordinary Differential Equations with Multiple Solutions

  • Ping Lin
  • Shu Luan
Article

Abstract

This paper studies a time optimal control problem for a class of ordinary differential equations. The control systems may have multiple solutions. Based on the properties fulfilled by the solutions of the concerned equations, we get both the existence and the Pontryagin maximum principle for optimal controls.

Keywords

Nonlinear ordinary differential equations Multiple solution Time optimal control 

Mathematics Subject Classification

49J15 34A34 

Notes

Acknowledgements

This work was partially supported by the National Natural Science Foundation of China under Grant 11471070, 11301472, the Natural Science Foundation of Guangdong Province under Grant 2014A030307011, the Training Program Project for Outstanding Young Teachers of Colleges and Universities in Guangdong Province under Grant Yq2014116 and the Characteristic Innovation Project of Common Colleges and Universities in Guangdong Province under Grant 2014KTSCX158.

References

  1. 1.
    Li, X., Yong, J.: Optimal Control Theory for Infinite-Dimensional Systems. Birkhäuser, Boston (1995)CrossRefGoogle Scholar
  2. 2.
    Lions, J.L.: Some Methods in the Mathematical Analysis of System and Their Control. Science Press, Gordon and Breach, Scinece Publishers Inc, Beijing, New York (1981)zbMATHGoogle Scholar
  3. 3.
    Bonnans, J.F., Casas, E.: Optimal control of semilinear multistate systems with state constraints. SIAM J. Control Optim. 27(2), 446–455 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Casas, E., Kavian, O., Puel, J.P.: Optimal control of an ill-posed elliptic semilinear equation with an exponential non linearity. ESAIM: COCV 3, 361–380 (1998)CrossRefzbMATHGoogle Scholar
  5. 5.
    Luan, S., Gao, H., Li, X.: Optimal control problem for an elliptic equation which has exactly two solutions. Optim. Control Appl. Methods 32(6), 734–747 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Wang, G., Wang, L.: Maximum principle of optimal control of non-well posed elliptic differential equations. Nonlinear Anal. 52, 41–67 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Lin, P.: Extendability and optimal control after quenching for some ordinary differential equations. J. Optim. Theory Appl. 168, 769–784 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hale, J.K.: Ordinary Differential Equations. Robert E. Krieger Publishing Company Inc, New York (1980)zbMATHGoogle Scholar
  9. 9.
    Lin, P.: Quenching time optimal control for some ordinary differential equations. J. Appl. Math. 2014, 127809 (2014). doi: 10.1155/2014/127809 MathSciNetGoogle Scholar
  10. 10.
    Lin, P., Wang, G.: Blowup time optimal control for ordinary differential equations. SIAM J. Control Optim. 49, 73–105 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lou, H., Wang, W.: Optimal blowup time for controlled ordinary differential equations. ESAIM: COCV 21(3), 815–834 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lou, H., Wen, J., Xu, Y.: Time optimal control problems for some non-smooth systems. Mathematical Control and Related Fields 4, 289–314 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Agarwal, R.P., Lakshmikantham, V.: Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. World Scientific, Singapore (1993)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsNortheast Normal UniversityChangchunChina
  2. 2.School of Mathematics and Computation ScienceLingnan Normal UniversityZhanjiangChina

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