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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,

{\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,

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

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

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,

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

is equivalent to a controlled differential equation,

{\dot {x}}=u

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

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,

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

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

{\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]

References

^ Biles, D. C., and Binding, P. A., “On Carathéodory’s Conditions for the Initial Value Problem," Proceedings of the American Mathematical Society, Vol. 125, No. 5, May 1997, pp. 1371–1376.
^ ^a ^b Ross, I. M., Sekhavat, P., Fleming, A. and Gong, Q., "Optimal Feedback Control: Foundations, Examples and Experimental Results for a New Approach," Journal of Guidance, Control and Dynamics, Vol. 31, No. 2, pp. 307–321, 2008.
^ ^a ^b Ross, I. M. and Karpenko, M. "A Review of Pseudospectral Optimal Control: From Theory to Flight," Annual Reviews in Control, Vol.36, No.2, pp. 182–197, 2012.
^ Clarke, F. H., Ledyaev, Y. S., Stern, R. J., and Wolenski, P. R., Nonsmooth Analysis and Control Theory, Springer–Verlag, New York, 1998.
^ Pontryagin, L. S., Boltyanskii, V. G., Gramkrelidze, R. V., and Mishchenko, E. F., The Mathematical Theory of Optimal Processes, Wiley, New York, 1962.
^ ^a ^b Ross, I. M., Gong, Q., Fahroo, F. and Kang, W., "Practical Stabilization Through Real-Time Optimal Control," 2006 American Control Conference, Minneapolis, MN, June 14-16 2006.
^ ^a ^b Martin, S. C., Hillier, N. and Corke, P., "Practical Application of Pseudospectral Optimization to Robot Path Planning," Proceedings of the 2010 Australasian Conference on Robotics and Automation, Brisbane, Australia, December 1-3, 2010.
^ Björkenstam, S., Gleeson, D., Bohlin, R. "Energy Efficient and Collision Free Motion of Industrial Robots using Optimal Control," Proceedings of the 9th IEEE International Conference on Automation Science and Engineering (CASE 2013), Madison, Wisconsin, August, 2013