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calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...

the Legendre–Clebsch condition is a second-order condition which a solution of the

Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...

must satisfy in order to be a minimum. For the problem of minimizing :

\int_^  L(t,x,x')\, dt . \,

the condition is :

L_(t,x(t),x'(t)) \ge 0, \, \forall t \in,b /math>

Generalized Legendre–Clebsch

optimal control Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations ...

, 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 Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...

to changes in u is zero, i.e., :

\frac = 0,

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

\frac > 0

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

Generalized Legendre–Clebsch

See also

References

Further reading