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


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

Generalized Legendre–Clebsch
In optimal control, the situation is more complicated because of the possibility of a singular solution. The generalized Legendre–Clebsch condition, 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.,


 * $$\frac{\partial H}{\partial u} = 0$$

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.