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 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.