Caratheodory-π solution

A Carathéodory-π solution is a generalized solution to an ordinary differential equation. 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]

Mathematical background

A Carathéodory-π solution addresses the fundamental problem of defining a solution to a differential equation,

${\displaystyle {\dot {x}}=g(x,t)}$

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,

${\displaystyle {\dot {x}}=f(x,u)}$

when the control, u, is given by a feedback law,

${\displaystyle u=k(x,t)}$

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]

Ross' concept

An ordinary differential equation,

${\displaystyle {\dot {x}}=g(x,t)}$

is equivalent to a controlled differential equation,

${\displaystyle {\dot {x}}=u}$

with feedback control, ${\displaystyle u=g(x,t)}$. Then, given an initial value problem, Ross partitions the time interval ${\displaystyle [0,\infty )}$ to a grid, ${\displaystyle \pi =\{t_{i}\}_{i\geq 0}}$ with ${\displaystyle t_{i}\to \infty {\text{ as }}i\to \infty }$. From ${\displaystyle t_{0}}$ to ${\displaystyle t_{1}}$, generate a control trajectory,

${\displaystyle u(t)=g(x_{0},t),\quad x(t_{0})=x_{0},\quad t_{0}\leq t\leq t_{1}}$

to the controlled differential equation,

${\displaystyle {\dot {x}}=u(t),\quad x(t_{0})=x_{0}}$

A Carathéodory solution exists for the above equation because ${\displaystyle t\mapsto u}$ has discontinuities at most in t, the independent variable. At ${\displaystyle t=t_{1}}$, set ${\displaystyle x_{1}=x(t_{1})}$ and restart the system with ${\displaystyle u(t)=g(x_{1},t)}$,

${\displaystyle {\dot {x}}(t)=u(t),\quad x(t_{1})=x_{1},\quad t_{1}\leq t\leq t_{2}}$

Continuing in this manner, the Carathéodory segments are stitched together to form a Carathéodory-π solution.

Engineering applications

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.^[2]

See also

• Ross' π lemma