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, 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 guarantees that over a singular arc, the Hamiltonian is minimized.