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

\int _{a}^{b}L(t,x,x')\,dt.\,

the condition is

L_{x'x'}(t,x(t),x'(t))\geq 0,\,\forall t\in [a,b]

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.,

{\frac {\partial H}{\partial u}}=0,

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

{\frac {\partial ^{2}H}{\partial u^{2}}}>0

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

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.

Legendre–Clebsch condition

Generalized Legendre–Clebsch

See also

References

Further reading