Legendre–Clebsch condition

In the calculus of variations the Legendre–Clebsch condition is a second-order condition which a solution of the Euler–Lagrange equation must satisfy in order to be a minimum.

For the problem of minimizing

the condition is

Generalized Legendre–Clebsch

In optimal control, the situation is more complicated because of the possibility of a singular solution. The generalized Legendre–Clebsch condition,[1] also known as convexity,[2] is a sufficient condition for local optimality such that when the linear sensitivity of the Hamiltonian to changes in u is zero, i.e.,

The Hessian of the Hamiltonian is positive definite along the trajectory of the solution:

In words, the generalized LC condition guarantees that over a singular arc, the Hamiltonian is minimized.

See also

References

  1. Robbins, H. M. (1967). "A Generalized Legendre–Clebsch Condition for the Singular Cases of Optimal Control". IBM Journal of Research and Development. 11 (4): 361–372. doi:10.1147/rd.114.0361.
  2. Choset, H.M. (2005). Principles of Robot Motion: Theory, Algorithms, and Implementation. The MIT Press. ISBN 0-262-03327-5.

Further reading

  • Hestenes, Magnus R. (1966). "A General Fixed Endpoint Problem". Calculus of Variations and Optimal Control Theory. New York: John Wiley & Sons. pp. 250–295.
  • "Legendre condition". Encyclopedia of Mathematics. Springer.
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