An infinite-time relaxation theorem for differential inclusions
Authors: Brian Ingalls, Eduardo D. Sontag and Yuan Wang
Journal: Proc. Amer. Math. Soc. 131 (2003), 487-499
MSC (2000): Primary 34A60; Secondary 34D23
Published electronically: May 22, 2002
MathSciNet review: 1933340
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Abstract: The fundamental relaxation result for Lipschitz differential inclusions is the Filippov-Wazewski Relaxation Theorem, which provides approximations of trajectories of a relaxed inclusion on finite intervals. A complementary result is presented, which provides approximations on infinite intervals, but does not guarantee that the approximation and the reference trajectory satisfy the same initial condition.
-  D. Angeli, B. Ingalls, E. D. Sontag, and Y. Wang, A Relaxation Theorem for Asymptotically Stable Differential Inclusions, in preparation.
-  Jean-Pierre Aubin and Arrigo Cellina, Differential inclusions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 264, Springer-Verlag, Berlin, 1984. Set-valued maps and viability theory. MR 755330
-  Jean-Pierre Aubin and Hélène Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1048347
-  R. M. Colombo, A. Fryszkowski, T. Rzeżuchowski, and V. Staicu, Continuous selections of solution sets of Lipschitzean differential inclusions, Funkcial. Ekvac. 34 (1991), no. 2, 321–330. MR 1130468
-  Klaus Deimling, Multivalued differential equations, De Gruyter Series in Nonlinear Analysis and Applications, vol. 1, Walter de Gruyter & Co., Berlin, 1992. MR 1189795
-  A. F. Filippov, Differential equations with discontinuous righthand sides, Mathematics and its Applications (Soviet Series), vol. 18, Kluwer Academic Publishers Group, Dordrecht, 1988. Translated from the Russian. MR 1028776
-  Andrzej Fryszowski and Tadeusz Rzeżuchowski, Continuous version of Filippov-Ważewski relaxation theorem, J. Differential Equations 94 (1991), no. 2, 254–265. MR 1137615, https://doi.org/10.1016/0022-0396(91)90092-N
-  Eduardo D. Sontag and Yuan Wang, New characterizations of input-to-state stability, IEEE Trans. Automat. Control 41 (1996), no. 9, 1283–1294. MR 1409473, https://doi.org/10.1109/9.536498
- D. Angeli, B. Ingalls, E. D. Sontag, and Y. Wang, A Relaxation Theorem for Asymptotically Stable Differential Inclusions, in preparation.
- J.-P. Aubin and A. Cellina, Differential Inclusions, Spring-Verlag, Berlin, 1984. MR 85j:49010
- J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990. MR 91d:49001
- R. M. Colombo, A. Fryszkowski, T. Rzezuchowski, and V. Staicu, Continuous Selections of Solution Sets of Lipschitzean Differential Inclusions, Funkcialaj Ekvacioj, 34 (1991), pp. 321-330. MR 93i:34022
- K. Deimling, Multivalued Differential Equations, Walter De Gruyter & Co., Berlin, 1992. MR 94b:34026
- A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Kluwer Academic, Dordrecht, The Netherlands, 1988. MR 90i:34002
- A. Fryszkowski and T. Rzezuchowski, Continuous Version of Filippov-Wazewski Relaxation Theorem, Journal of Differential Equations, 94 (1991), pp. 254-265. MR 92j:34031
- E. D. Sontag and Y. Wang, New characterizations of the input to state stability property, IEEE Transactions on Automatic Control, 41 (1996), pp. 1283-1294. MR 97g:93069
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903-2101
Eduardo D. Sontag
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903- 2101
Affiliation: Department of Mathematics, Florida Atlantic University, Boca Raton, Florida 33431-0991
Received by editor(s): May 28, 2001
Received by editor(s) in revised form: September 19, 2001
Published electronically: May 22, 2002
Additional Notes: The second author was supported in part by US Air Force Grant F49620-98-1-0242.
The third author’s research was supported in part by NSF Grant DMS-9457826.
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2002 American Mathematical Society