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Discrete Filippov-Type Stability for One-Sided Lipschitzian Difference Inclusions

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Control Systems and Mathematical Methods in Economics

Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 687))

Abstract

We state and prove Filippov-type stability theorems for discrete difference inclusions obtained by the Euler discretization of a differential inclusion with perturbations in the set of initial points, in the right-hand side and in the state variable. We study the cases in which the right-hand side of the inclusion is not necessarily Lipschitz, but satisfies a weaker one-sided Lipschitz (OSL) or strengthened one-sided Lipschitz (SOSL) condition. The obtained estimates imply stability of the discrete solutions for infinite number of fixed time steps if the OSL constant is negative and the perturbations are bounded in certain norms. We show a better order of stability for SOSL right-hand sides and apply our theorems to estimate the distance from the solutions of other difference methods, as for the implicit Euler scheme to the set of solutions of the Euler scheme. We also prove a discrete relaxation stability theorem for the considered difference inclusion, which also extends a theorem of Grammel (Set-Valued Anal. 11(1):1–8, 2003) from the class of Lipschitz maps to the wider class of OSL ones.

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References

  • J.-P. Aubin, A. Cellina, Differential Inclusions. Grundlehren der Mathematischen Wissenschaften, vol. 264 (Springer, Berlin-Heidelberg-New York-Tokyo, 1984)

    Book  Google Scholar 

  • J.-P. Aubin, H. Frankowska, Set-Valued Analysis. Systems & Control: Foundations & Applications, vol. 2 (Birkhäuser, Boston, MA, 1990)

    Google Scholar 

  • W. Auzinger, R. Frank, F. Macsek, Asymptotic error expansions for stiff equations: the implicit Euler scheme. SIAM J. Numer. Anal. 27(1), 67–104 (1990)

    Article  Google Scholar 

  • R. Baier, I.A. Chahma, F. Lempio, Stability and convergence of Euler’s method for state-constrained differential inclusions. SIAM J. Optim. 18(3), 1004–1026 (2007). (electronic). D. Dentcheva, J. Revalski (eds.), special issue on “Variational Analysis and Optimization”

    Article  Google Scholar 

  • R. Baier, E. Farkhi, Regularity of set-valued maps and their selections through set differences. Part 2: one-sided Lipschitz properties. Serdica Math. J. 39(3–4), 391–422 (2013). Special issue dedicated to the 65th anniversary of Professor Asen L. Dontchev and to the 60th anniversary of Professor Vladimir M. Veliov

    Google Scholar 

  • W.-J. Beyn, J. Rieger, The implicit Euler scheme for one-sided Lipschitz differential inclusions. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 14(2), 409–428 (2010)

    Article  Google Scholar 

  • F.E. Browder, Nonlinear accretive operators in Banach spaces. Bull. Am. Math. Soc. 73, 470–476 (1967a)

    Article  Google Scholar 

  • F.E. Browder, Nonlinear mappings of nonexpansive and accretive type in Banach spaces. Bull. Am. Math. Soc. 73, 875–882 (1967b)

    Article  Google Scholar 

  • I.A. Chahma, Set-valued discrete approximation of state-constrained differential inclusions. Bayreuth. Math. Schr. 67, 3–162 (2003)

    Google Scholar 

  • K. Deimling, Multivalued Differential Equations. de Gruyter Series in Nonlinear Analysis and Applications, vol. 1 (de Gruyter, Berlin-New York, 1992)

    Google Scholar 

  • K. Dekker, J.G. Verwer, Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. CWI Monographs, vol. 2 (North-Holland, Amsterdam, 1984)

    Google Scholar 

  • T.D. Donchev, Functional-differential inclusion with monotone right-hand side. Nonlinear Anal. 16(6), 533–542 (1991)

    Article  Google Scholar 

  • T.D. Donchev, Properties of one-sided Lipschitz multivalued maps. Nonlinear Anal. 49(1), 13–20 (2002)

    Article  Google Scholar 

  • T.D. Donchev, One sided Lipschitz multifunctions and applications, in Optimal Control, Stabilization and Nonsmooth Analysis. Lecture Notes in Control and Information Science (Springer, Berlin, 2004), pp. 333–341

    Chapter  Google Scholar 

  • T.D. Donchev, E. Farkhi, Stability and Euler approximation of one-sided Lipschitz differential inclusions. SIAM J. Control Optim. 36(2), 780–796 (1998) (electronic)

    Article  Google Scholar 

  • T.D. Donchev, E. Farkhi, Approximations of one-sided Lipschitz differential inclusions with discontinuous right-hand sides, in Calculus of Variations and Differential Equations (Haifa, 1998). Chapman & Hall/CRC Research Notes in Mathematics, vol. 410 (Chapman & Hall/CRC, Boca Raton, FL, 2000), pp. 101–118

    Google Scholar 

  • T.D. Donchev, E. Farkhi, On the theorem of Filippov-Pliś and some applications. Control and Cybern. 38(4A), 1251–1271 (2009)

    Google Scholar 

  • T.D. Donchev, R. Ivanov, On the existence of solutions of differential inclusions in uniformly convex Banach space. Math. Balkanica (N.S.) 6(1), 13–24 (1992)

    Google Scholar 

  • E. Farkhi, T.D. Donchev, R. Baier, Existence of solutions for nonconvex differential inclusions of monotone type. C. R. Acad. Bulg. Sci. 67(3), 323–330 (2014)

    Google Scholar 

  • A.F. Filippov, Classical solutions of differential equations with multi-valued right-hand side. SIAM J. Control 5, 609–621 (1967)

    Article  Google Scholar 

  • H. Frankowska, F. Rampazzo, Filippov’s and Filippov-Wazewski’s theorems on closed domains. J. Differ. Equ. 161(2), 449–478 (2000)

    Article  Google Scholar 

  • G. Grammel, Towards fully discretized differential inclusions. Set-Valued Anal. 11(1), 1–8 (2003)

    Article  Google Scholar 

  • E. Hairer, G. Wanner, Solving Ordinary Differential Equations. II Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics, vol. 14, 2nd edn. (Springer, Berlin, 1996)

    Google Scholar 

  • J.L. Haunschmied, A. Pietrus, V.M. Veliov, The Euler method for linear control systems revisited, in Large-Scale Scientific Computing. Revised Selected Papers from the 9th International Conference (LSSC ‘13), Sozopol, June 3–7, 2013, ed. by I. Lirkov, S.D. Margenov, J. Waśniewski. Lecture Notes in Computer Science, vol. 8353 (Springer, Heidelberg, 2014), pp. 90–97

    Google Scholar 

  • A. Kastner-Maresch, Diskretisierungsverfahren zur Lösung von Differentialinklusionen [Discretization Methods for the Solution of Differential Inclusions]. Ph.D. thesis, Department of Mathematics, University of Bayreuth, Bayreuth, (1990a)

    Google Scholar 

  • A. Kastner-Maresch, Implicit Runge-Kutta methods for differential inclusions. Numer. Funct. Anal. Optim. 11(9–10), 937–958 (1990b)

    Article  Google Scholar 

  • A. Kastner-Maresch, The implicit midpoint rule applied to discontinuous differential equations. Computing 49(1), 45–62 (1992)

    Article  Google Scholar 

  • A. Kastner-Maresch, F. Lempio, Difference methods with selection strategies for differential inclusions. Numer. Funct. Anal. Optim. 14(5–6), 555–572 (1993)

    Article  Google Scholar 

  • F. Lempio, Difference methods for differential inclusions, in Modern Methods of Optimization. Proceedings of a Summer School at the Schloß Thurnau of the University of Bayreuth (Germany), FRG, October 1–6, 1990. Lecture Notes in Economics and Mathematical Systems, vol. 378 (Springer, Berlin-Heidelberg-New York, 1992), pp. 236–273

    Google Scholar 

  • F. Lempio, Modified Euler methods for differential inclusions, in Set-Valued Analysis and Differential Inclusions. A Collection of Papers resulting from a Workshop held in Pamporovo, September 17–21, 1990. Progress in Systems and Control Theory, vol. 16 (Birkhäuser, Boston, MA-Basel-Berlin, 1993), pp. 131–148

    Google Scholar 

  • F. Lempio, Euler’s method revisited. Proc. Steklov Inst. Math. 211, 429–449 (1995)

    Google Scholar 

  • F. Lempio, D.B. Silin, Differential inclusions with strongly one-sided-Lipschitz right-hand sides. Differ. Equ. 32(11), 1485–1491 (1997)

    Google Scholar 

  • F. Lempio, V.M. Veliov, Discrete approximations of differential inclusions. Bayreuth. Math. Schr. 54, 149–232 (1998)

    Google Scholar 

  • G. Lumer, R.S. Phillips, Dissipative operators in a Banach space. Pac. J. Math. 11, 679–698 (1961)

    Article  Google Scholar 

  • R. Mannshardt, One-step methods of any order for ordinary differential equations with discontinuous right-hand sides. Numer. Math. 31(2), 131–152 (1978/1979)

    Article  Google Scholar 

  • J.T. Marti, Konvexe Analysis. Lehrbücher und Monographien aus dem Gebiet der Exakten Wissenschaften, Mathematische Reihe, vol. 54 (Birkhäuser, Basel-Stuttgart, 1977)

    Google Scholar 

  • R.H. Martin Jr., A global existence theorem for autonomous differential equations in a Banach space. Proc. Am. Math. Soc. 26, 307–314 (1970)

    Article  Google Scholar 

  • R. Model, Zur Integration über Unstetigkeiten in gewöhnlichen Differentialgleichungen. Z. Angew. Math. Mech. 68(3), 161–169 (1988)

    Article  Google Scholar 

  • A. Pietrus, V.M. Veliov, On the discretization of switched linear systems. Syst. Control Lett. 58(6), 395–399 (2009)

    Article  Google Scholar 

  • V.M. Veliov, Second order discrete approximations to strongly convex differential inclusions. Syst. Control Lett. 13(3), 263–269 (1989)

    Article  Google Scholar 

  • V.M. Veliov, Second order discrete approximation to linear differential inclusions. SIAM J. Numer. Anal. 29(2), 439–451 (1992)

    Article  Google Scholar 

  • V.M. Veliov, On the time-discretization of control systems. SIAM J. Control Optim. 35(5), 1470–1486 (1997)

    Article  Google Scholar 

  • V.M. Veliov, Error analysis of discrete approximations to bang-bang optimal control problems: the linear case. Control Cybern. 34(3), 967–982 (2005)

    Google Scholar 

  • V.M. Veliov, On the relationship between continuous- and discrete-time control systems. Cent. Eur. J. Oper. Res. 18(4), 511–523 (2010)

    Article  Google Scholar 

  • V.M. Veliov, Relaxation of Euler-type discrete-time control system, in Large-Scale Scientific Computing. Revised Selected Papers from the 10th International Conference (LSSC ‘15), Sozopol, June 8–12, 2015, ed. by I. Lirkov, S. D. Margenov, and J. Waśniewski. Lecture Notes in Computer Science, vol. 9374 (Springer, Cham, 2015), pp. 134–141

    Chapter  Google Scholar 

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Acknowledgements

The authors would like to thank Frank Lempio and Janosch Rieger for the helpful discussions on the explicit and implicit Euler method. The authors also acknowledge the partial support by The Hermann Minkowski Center for Geometry at Tel Aviv University, Israel.

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Authors and Affiliations

  1. University of Bayreuth, Bayreuth, Germany

    Robert Baier

  2. School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel

    Elza Farkhi

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  1. Robert Baier
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Correspondence to Elza Farkhi .

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Editors and Affiliations

  1. Operations Research and Control Systems, Vienna University of Technology , Vienna, Austria

    Gustav Feichtinger

  2. Operations Research and Control Systems, Vienna University of Technology, Vienna, Austria

    Raimund M. Kovacevic

  3. Operations Research and Control Systems, Vienna University of Technology, Vienna, Austria

    Gernot Tragler

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Baier, R., Farkhi, E. (2018). Discrete Filippov-Type Stability for One-Sided Lipschitzian Difference Inclusions. In: Feichtinger, G., Kovacevic, R., Tragler, G. (eds) Control Systems and Mathematical Methods in Economics. Lecture Notes in Economics and Mathematical Systems, vol 687. Springer, Cham. https://doi.org/10.1007/978-3-319-75169-6_3

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