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
- 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.
- 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.