Boundary energy control of a system governed by the nonlinear Klein–Gordon equation

  • Maksim DolgopolikEmail author
  • Alexander L. Fradkov
  • Boris Andrievsky
Original Article


The boundary energy control problem for the sine-Gordon and the nonlinear Klein–Gordon equations is posed. Two control laws solving this problem based on the speed-gradient method with smooth and nonsmooth goal functions are proposed. The control law obtained via a nonsmooth goal function is proved to steer the system to any required nonzero energy level in finite time. The results of the numerical evaluation of the proposed algorithm for an undamped nonlinear elastic string demonstrate attainability of the control goal for the cases of both decreasing and increasing systems’ energy and show high rate of vanishing of the control error.


Sine-Gordon equation Klein–Gordon equation Energy control Speed-gradient method 



The authors are grateful to the anonymous referees for thoughtful and stimulating comments that helped to improve the quality of the article.


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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Problems of Mechanical EngineeringRussian Academy of SciencesSaint PetersburgRussia
  2. 2.ITMO UniversitySaint PetersburgRussia
  3. 3.Saint Petersburg State UniversitySaint PetersburgRussia

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