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WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9876)

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A '''Carathéodory-{{pi}} 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]].<ref name ="biles">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.</ref> Its practicality was demonstrated in 2008 by Ross et al.<ref name ="ross-1">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.</ref> in a laboratory implementation of the concept. The concept is most useful for implementing [[feedback control]]s, particularly those generated by an application of Ross' [[pseudospectral optimal control]] theory.<ref name = "ross-2">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.</ref>

==Mathematical background==
A Carathéodory-{{pi}} solution addresses the fundamental problem of defining a solution to a differential equation,
:<math> \dot x = g(x,t) </math>
when ''g''(''x'',''t'') is not differentiable with respect to ''x''. Such problems arise quite naturally <ref name ="clarke-1">Clarke, F. H., Ledyaev, Y. S., Stern, R. J., and Wolenski, P. R.,
Nonsmooth Analysis and Control Theory, Springer–Verlag, New York,
1998.</ref> in defining the meaning of a solution to a controlled differential equation,
:<math> \dot x = f(x,u) </math>
when the control, ''u'', is given by a feedback law,
:<math> u = k(x,t) </math>
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.<ref name ="pontryagin-1">Pontryagin, L. S., Boltyanskii, V. G., Gramkrelidze, R. V., and Mishchenko, E. F., The Mathematical Theory of Optimal Processes,
Wiley, New York, 1962.</ref>

==Ross' concept==
An ordinary differential equation,
:<math> \dot x = g(x,t) </math>
is equivalent to a controlled differential equation,
:<math> \dot x = u </math>
with feedback control,
<math> u = g(x,t) </math>. Then, given an initial value problem, Ross partitions the time interval <math> [0, \infty) </math> to a grid, <math> \pi = \{t_i\}_{i\ge 0} </math> with <math> t_i \to \infty \text{ as } i \to \infty </math>. From <math>t_0</math> to <math> t_1</math>, generate a control trajectory,
:<math> u(t) = g(x_0, t), \quad x(t_0) = x_0, \quad t_0 \le t \le t_1 </math>
to the controlled differential equation,
:<math> \dot x = u(t), \quad x(t_0) = x_0 </math>
A [[Carathéodory existence theorem|Carathéodory solution]] exists for the above equation because <math> t \mapsto u </math> has discontinuities at most in ''t'', the independent variable. At <math> t = t_1</math>, set <math> x_1 = x(t_1) </math> and restart
the system with <math> u(t) = g(x_1, t)</math>,
:<math> \dot x(t) = u(t), \quad x(t_1) = x^1, \quad t_1 \le t \le t_2 </math>
Continuing in this manner, the Carathéodory segments are stitched together to form a Carathéodory-{{pi}} solution.

==Engineering applications==
A Carathéodory-{{pi}} solution can be applied towards the practical stabilization of a control system.<ref name ="ross-3"> 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.</ref> <ref name="martin"> 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. </ref> It has been used to stabilize an inverted pendulum,<ref name="ross-3"/> control and optimize the motion of robots,<ref name="martin"/> <ref name="robot-2"> 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 </ref> slew and control the NPSAT1 spacecraft<ref name="ross-2"/> and produce guidance commands for low-thrust space missions.<ref name="ross-1"/>

==See also==
*[[Ross' π lemma]]

==References==
{{Reflist}}

{{DEFAULTSORT:Caratheodory-pi solution}}
[[Category:Ordinary differential equations]]
[[Category:Optimal control]]

Ramsey interferometry - Revision history

en>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9876)