Legendre–Clebsch condition
Open-knowledge reference entry
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
∫
a
b
L
(
t
,
x
,
x
′
)
d
t
.
{\displaystyle \int _{a}^{b}L(t,x,x')\,dt.\,}
the condition is
L
x
′
x
′
(
t
,
x
(
t
)
,
x
′
(
t
)
)
≥
0
,
∀
t
∈
[
a
,
b
]
{\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, 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.,
∂
H
∂
u
=
0
,
{\displaystyle {\frac {\partial H}{\partial u}}=0,}
the Hessian of the Hamiltonian is positive definite along the trajectory of the solution:
∂
2
H
∂
u
2
>
0
{\displaystyle {\frac {\partial ^{2}H}{\partial u^{2}}}>0}
In words, the generalized LC condition gives ones more necessary condition for the Hamiltonian be minimized over a singular arc.
See also
Bang–bang control
References
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.
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