Integrability

In some cases, it is possible that the state transition equation is integrable. This implies that is can be expressed purely as a function of x and u, and not of ${\dot x}$. In the case of an integrable state transition equation, the motions is actually restricted to a lower-dimensional subset of X, which is a global constraints as opposed to a local constraint.

CIRCLE EXAMPLE

The Frobenius theorem gives an interesting condition that may be applied to determine whether the state transition equation is integrable.

Theorem 763 ((Frobenius))

The state transition equation is integrable if and only if all vectors fields that can be obtained by Lie bracket operations are contained in $\triangle$.

Intuitively, if the Lie bracket operation is unable to produce any new (linearly-independent) vector fields that lie outside of $\triangle$,then the state transition equation can be integrated. Thus, the equation is not needed, and the problem can be reformulated without using ${\dot x}$. This is, however, a theoretical result; it may be a difficult or impossible task in general to actually integrate the state transition equation.

The Frobenius theorem can also be expressed in terms of dimensions. If $dim(CLA(\triangle)) = dim(\triangle)$, then the state transition equation is integrable. Note that the dimension of $CLA(\triangle)$can never be greater than n.

If the state transition equation is not integrable, then it is called nonholonomic. These equations are of greatest interest.