View source for Legendre–Clebsch condition

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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 maximum (and not a minimum or another kind of extremal).

For the problem of maximizing

:<math>  \int_{a}^{b}  L(t,x,x')\, dt . \,</math>

the condition is

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

==Generalized Legendre–Clebsch==

In [[optimal control]], the situation is more complicated because of the possibility of a [[singular control|singular solution]].  The '''generalized Legendre–Clebsch condition''',<ref>{{cite journal |first=H. M. |last=Robbins |title=A Generalized Legendre–Clebsch Condition for the Singular Cases of Optimal Control |journal=IBM Journal of Research and Development |year=1967 }}</ref> also known as convexity,<ref name=Choset2005>{{cite book
 | author = Choset, H.M.
 | year = 2005
 | title = Principles of Robot Motion: Theory, Algorithms, and Implementation
 | publisher = The MIT Press
 | isbn = 0-262-03327-5
}}</ref> is a sufficient condition for local optimality such that when the linear sensitivity of the [[Hamiltonian (control theory)|Hamiltonian]] to changes in u is zero, i.e.,

: <math>\frac{\partial H}{\partial u} = 0</math>

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

: <math>\frac{\partial^2 H}{\partial u^2} > 0</math>

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

==See also==
* [[Bang–bang control]]

==References==
{{reflist}}

==Further reading==
*{{cite book |first=Magnus R. |last=Hestenes |authorlink=Magnus Hestenes |title=Calculus of Variations and Optimal Control Theory |location=New York |publisher=John Wiley & Sons |year=1966 |chapter=A General Fixed Endpoint Problem |pages=250–295 }}

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[[Category:Optimal control]]
[[Category:Calculus of variations]]