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

$\underset{a}{\overset{b}{\int }}L(t,x,x^{\prime })dt.$

the condition is

$L_{x^{\prime }{x}^{\prime }}(t,x(t),x^{\prime }(t))\ge 0,\mathrm{\forall }t\in [a,b]$

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.

See also

• Bang–bang control

Further reading

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