The concept is due to I. Michael Ross and named in honor of Constantin Carathéodory.
[1] Its practicality was demonstrated in 2008 by Ross et al.[2] in a laboratory implementation of the concept.
The concept is most useful for implementing feedback controls, particularly those generated by an application of Ross' pseudospectral optimal control theory.
[3] A Carathéodory-π solution addresses the fundamental problem of defining a solution to a differential equation, when g(x,t) is not differentiable with respect to x.
Such problems arise quite naturally[4] in defining the meaning of a solution to a controlled differential equation, when the control, u, is given by a feedback law, where the function k(x,t) may be non-smooth with respect to x. Non-smooth feedback controls arise quite often in the study of optimal feedback controls and have been the subject of extensive study going back to the 1960s.
[5] An ordinary differential equation, is equivalent to a controlled differential equation, with feedback control,
Then, given an initial value problem, Ross partitions the time interval
, generate a control trajectory, to the controlled differential equation, A Carathéodory solution exists for the above equation because
has discontinuities at most in t, the independent variable.
, Continuing in this manner, the Carathéodory segments are stitched together to form a Carathéodory-π solution.
A Carathéodory-π solution can be applied towards the practical stabilization of a control system.
[6][7] It has been used to stabilize an inverted pendulum,[6] control and optimize the motion of robots,[7][8] slew and control the NPSAT1 spacecraft[3] and produce guidance commands for low-thrust space missions.