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

[math]\displaystyle{ \int_{a}^{b} L(t,x,x')\, dt . \, }[/math]

the condition is

[math]\displaystyle{ L_{x' x'}(t,x(t),x'(t)) \ge 0, \, \forall t \in[a,b] }[/math]

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.,

[math]\displaystyle{ \frac{\partial H}{\partial u} = 0 }[/math]

The Hessian of the Hamiltonian is positive definite along the trajectory of the solution:

[math]\displaystyle{ \frac{\partial^2 H}{\partial u^2} \gt 0 }[/math]

In words, the generalized LC condition guarantees that over a singular arc, the Hamiltonian is minimized.

See also

• Bang–bang control

Further reading

• Hestenes, Magnus R. (1966). "A General Fixed Endpoint Problem". Calculus of Variations and Optimal Control Theory. New York: John Wiley & Sons. pp. 250–295. 
• "Legendre condition". Springer. https://encyclopediaofmath.org/wiki/Legendre_condition.