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

$\int_{a}^{b} L(t,x,x')\, dt . \,$

the condition is

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

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.

See also

• Singular control
• Bang–bang control

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.