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 maximum (and not a minimum or another kind of extremal).

For the problem of maximizing

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. ↑ H.M. Robbins, A generalized Legendre-Clebsch condition for the singular cases of optimal control, IBM Journal of Research and Development, 1967
  2. ↑ Choset, H.M. (2005). Principles of Robot Motion: Theory, Algorithms, and Implementation. The MIT Press. ISBN 0-262-03327-5.
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