The directional anti-derivative about a point: existence conditions and some applications

Sufficient conditions are proved to guarantee existence and uniqueness of the directional anti-derivative of a smooth (possibly, analytic) scalar function along a smooth (possibly, analytic) vector field. In particular, it is shown that smooth directional anti-derivatives always exist in the neighbourhood of regular points, whereas their existence in the neighbourhood of singular points is guaranteed just in the case that the eigenvalues of the corresponding linearised system are linearly independent over the set of non-negative integer numbers. If, additionally, the scalar function and the vector field are both analytic and the eigenvalues of the linearised system belong to the Poincaré domain, then the directional anti-derivative is analytic. Based on the theoretical results, techniques are proposed to either compute or approximate such a function. Several examples are proposed to illustrate how the directional anti-derivative can be used to solve some relevant control problems, such as determining a change of coordinates that recasts scalar and triangular nonlinear systems into linear form, and evaluating a performance index on the transient behaviour of an asymptotically stable closed loop system.