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

${\displaystyle \int _{a}^{b}L(t,x,x')\,dt.\,}$

the condition is

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

Generalized Legendre–Clebsch

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

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

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

${\displaystyle {\frac {\partial ^{2}H}{\partial u^{2}}}>0}$

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

See also

• Bang–bang control

References

1. Robbins, H. M. (1967). "A Generalized Legendre–Clebsch Condition for the Singular Cases of Optimal Control". IBM Journal of Research and Development. 11 (4): 361–372. doi:10.1147/rd.114.0361.
2. Choset, H.M. (2005). Principles of Robot Motion: Theory, Algorithms, and Implementation. The MIT Press. ISBN 0-262-03327-5.

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". Encyclopedia of Mathematics. Springer.