Jacobi Equation

  • Andrei A. Agrachev
  • Yuri L. Sachkov
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 87)


In Chap. 20 we established that the sign of the quadratic form λ t Hess ũ F t is related to optimality of the extremal control ũ. Under natural assumptions, the second variation is negative on short segments. Now we wish to catch the instant of time where this quadratic form fails to be negative. We derive an ODE (Jacobi equation) that allows to find such instants (conjugate times). Moreover, we give necessary and sufficient optimality conditions in these terms.


Quadratic Form Optimal Control Problem Hamiltonian Function Jacobi Equation Conjugate Point 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Andrei A. Agrachev
    • 1
  • Yuri L. Sachkov
    • 2
  1. 1.SISSA-ISASTriesteItaly
  2. 2.Program Systems InstitutePereslavl-ZalesskyRussia

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