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, also known as convexity, 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 gives ones more necessary condition for the Hamiltonian be minimized over a singular arc.