Abstract The current work proposes a new and constructive proof for the Caratheodory's theorem on existence and uniqueness of trajectories of dynamical systems. The key concern is the numerical uncertainty, i.e., the discrepancy between mathematical proofs, algorithms, and their implementations, which may affect the correct functioning of a control system. Due to growing demands on security and compliance with specifications, correctness of the control system functioning is becoming ever more important. Since in both dynamical systems and many control design approaches, one of the central notions is the system trajectory, it is important to address existence and uniqueness of system trajectories in a way which incorporates numerical uncertainty. Constructive analysis is a particular approach to formalizing numerical uncertainty and is used as the basis of the current work. The major difficulties of guaranteeing existence and uniqueness of system trajectories arise in the case of systems and controllers which possess discontinuities in time, since classical solutions to initial value problems do not exist. This issue is addressed in Caratheodory's theorem. A particular constructive variant of the theorem is proven which covers a large class of problems found in practice.

Pavel Osinenko, Grigory Devadze, Stefan Streif, "Analysis of the Caratheodory's Theorem on Dynamical System Trajectories Under Numerical Uncertainty," IEEE/CAA Journal of Automatica Sinica, vol. 5, no. 4, pp. 787-793, 2018.

[1] P. Jia, P. Hao, and H. Yu, "Function observer based event-triggered control for linear systems with guaranteed L_{∞}-gain, " IEEE/CAA J. of Autom. Sinica, vol. 2, no. 4, pp. 394-402, Oct. 2015. [2] X. X. Mi and S. Y. Li, "Event-triggered MPC design for distributed systems with network communications, " IEEE/CAA J. of Autom. Sinica, vol. 5, no. 1, pp. 240-250, Jan. 2018. [3] D. R. Liu, Y. C. Xu, Q. L. Wei, and X. L. Liu, "Residential energy scheduling for variable weather solar energy based on adaptive dynamic programming, " IEEE/CAA J. of Autom. Sinica, vol. 5, no. 1, pp. 36-46, Jan. 2018. [4] L. S. Pontryagin, Mathematical Theory of Optimal Processes. London: CRC Press, 1987. [5] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. New York: McGraw-Hill, 1984. [6] J. K. Hale, Ordinary Differential Equations. New York: Wiley, 1969. [7] W. Rudin, Real and Complex Analysis (3rd Edition). New York: McGraw-Hill Inc, 1986. [8] L. Q. Thuan and M. K. Camlibel, "On the existence, uniqueness and nature of caratheodory and filippov solutions for bimodal piecewise affine dynamical systems, " Syst. Contr. Lett., vol. 68, pp. 76-85, 2014. [9] J. A Cid, S. Heikkila, and R. Pouso, "Uniqueness and existence results for ordinary differential equations, " J. Math. Anal. Appl., vol. 316, no. 1, pp. 178-188, 2006. [10] W. P. M. H. Heemels, M. K. Camlibel, A. J. Van der Schaft, and J. M. Schumacher, "On the existence and uniqueness of solution trajectories to hybrid dynamical systems, " in Nonlinear and Hybrid Control in Automotive Applications, R. Johannson and A. Rantzer, Eds. London: Springer, 2002, pp. 391-422. [11] T. Ito, "A filippov solution of a system of differential equations with discontinuous right-hand sides, " Econom. Lett., vol. 4, no. 4, pp. 349-354, 1979. [12] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides: Control Systems. Netherlands: Springer, 2013. [13] S. C. Hu, "Differential equations with discontinuous right-hand sides, " J. Math. Anal. Appl., vol. 154, no. 2, pp. 377-390, 1991. [14] F. H. Clarke, Y. S. Ledyaev, E. D. Sontag, and A. I. Subbotin, "Asymptotic controllability implies feedback stabilization, " IEEE Trans. Autom. Contr., vol. 42, no. 10, pp. 1394-1407, 1997. [15] L. Fridland and A. Levant, "Higher Order Sliding Modes, " in Sliding Mode in Automatic Control, J. Barbot and W. Perruguetti, Eds. Ecole Central de Lille, 1999. [16] K. D. Young, V. I. Utkin, and U. Ozguner, "A control engineer guide to sliding mode control, " IEEE Trans. Contr. Syst. Technol., vol. 7, no. 3, pp. 328-342, May 1999. [17] F. Fontes, "Discontinuous feedbacks, discontinuous optimal controls, and continuous-time model predictive control, " Int. J. Robust Nonlin. Contr., vol. 13, no. 3-4, pp. 191-209, 2003. [18] J. Cortes, "Discontinuous dynamical systems, " IEEE Contr. Syst., vol. 28, no. 3, pp. Article No. 10010348, Jun. 2008. [19] N. Dunford, J. T. Schwartz, W. G. Bade, and R. G. Bartle, Linear Operators. New York: Wiley-interscience, 1971. [20] J. Schauder, "der Fixpunktsatz in Funktionalraumen, " Stud. Math., vol. 2, no. 1, pp. 171-180, 1930. [21] A. V. Surkov, "On functional-differential equations with discontinuous right-hand side, " Diff. Equat., vol. 44, no. 2, pp. 278-281, 2008. [22] M. J. Beeson, Foundations of Constructive Mathematics: Metamathematical Studies. Berlin Heidelberg: Springer Science & Business Media, 1980. [23] H. Schwichtenberg, Constructive Analysis with Witnesses. Munich: Mathernatisches Institut der LMU, 2016. [24] F. Ye, Strict Finitism and the Logic of Mathematical Applications. Netherlands: Springer, 2011. [25] E. Bishop, Foundations of Constructive Analysis. New York: McGraw-Hill, 1967. [26] L. E. J. Brouwer and D. van Dalen, Brouwer Cambridge Lectures on Intuitionism. Cambridge: Cambridge University Press, 2011. [27] P. Osinenko, G. Devadze, and S. Steif, "Constructive analysis of control system stability, " in Proc. 20th IFAC Congr., Quebec, Canada, 2016, pp. 7467-7474. [28] E. Bishop and D. S. Bridges, Constructive Analysis. Berlin Heidelberg: Springer Science & Business Media, 1985. [29] A. F. Filippov, "Differential equations with discontinuous right-hand side, " Matematicheskii Sbornik, vol. 93, no. 1, pp. 99-128, 1960. [30] M. L. Pinedo, Scheduling: Theory, Algorithms, and Systems. Boston, MA: Springer, 2012. [31] F. Clarke, "Nonsmooth Analysis in Systems and Control Theory, " in Encyclopedia of Complexity and Systems Science, R. Meyers, Ed. New York, NY: Springer, 2009, pp. 6271-6285. [32] D. Liberzon, Switching in Systems and Control, ser. Systems & Control: Foundations & Applications. Birkhauser Boston, 2012.