In the calculus of variations the LegendreâClebsch condition is a second-order condition which a solution of the EulerâLagrange 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 LegendreâClebsch 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.
In the calculus of variations the LegendreâClebsch condition is a second-order condition which a solution of the EulerâLagrange 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 LegendreâClebsch 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.