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 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,[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.,
- [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
References
- ↑ 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.
- ↑ 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". Springer. https://encyclopediaofmath.org/wiki/Legendre_condition.
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