Converging outer approximations to global attractors using semidefinite programming
This paper develops a method for obtaining guaranteed outer approximations for global attractors of continuous and discrete time nonlinear dynamical systems. The method is based on a hierarchy of semidefinite programming problems of increasing size with guaranteed convergence to the global attractor...
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| Veröffentlicht in: | Automatica (Oxford) Jg. 134; S. 109900 |
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| Format: | Journal Article |
| Sprache: | Englisch |
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01.12.2021
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| ISSN: | 0005-1098, 1873-2836 |
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| Abstract | This paper develops a method for obtaining guaranteed outer approximations for global attractors of continuous and discrete time nonlinear dynamical systems. The method is based on a hierarchy of semidefinite programming problems of increasing size with guaranteed convergence to the global attractor. The approach taken follows an established line of reasoning, where we first characterize the global attractor via an infinite dimensional linear programming problem (LP) in the space of Borel measures. The dual to this LP is in the space of continuous functions and its feasible solutions provide guaranteed outer approximations to the global attractor. For systems with polynomial dynamics, a hierarchy of finite-dimensional sum-of-squares tightenings of the dual LP provides a sequence of outer approximations to the global attractor with guaranteed convergence in the sense of volume discrepancy tending to zero. The method is very simple to use and based purely on convex optimization. Numerical examples with the code available online demonstrate the method. |
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| AbstractList | This paper develops a method for obtaining guaranteed outer approximations for global attractors of continuous and discrete time nonlinear dynamical systems. The method is based on a hierarchy of semidefinite programming problems of increasing size with guaranteed convergence to the global attractor. The approach taken follows an established line of reasoning, where we first characterize the global attractor via an infinite dimensional linear programming problem (LP) in the space of Borel measures. The dual to this LP is in the space of continuous functions and its feasible solutions provide guaranteed outer approximations to the global attractor. For systems with polynomial dynamics, a hierarchy of finite-dimensional sum-of-squares tightenings of the dual LP provides a sequence of outer approximations to the global attractor with guaranteed convergence in the sense of volume discrepancy tending to zero. The method is very simple to use and based purely on convex optimization. Numerical examples with the code available online demonstrate the method. |
| ArticleNumber | 109900 |
| Author | Schlosser, Corbinian Korda, Milan |
| Author_xml | – sequence: 1 givenname: Corbinian surname: Schlosser fullname: Schlosser, Corbinian email: cschlosser@laas.fr organization: CNRS-LAAS, 7 avenue du colonel Roche, F-31400 Toulouse, France – sequence: 2 givenname: Milan surname: Korda fullname: Korda, Milan email: korda@laas.fr organization: CNRS-LAAS, 7 avenue du colonel Roche, F-31400 Toulouse, France |
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| Cites_doi | 10.1137/070685051 10.1109/TAC.2013.2283095 10.1512/iumj.1993.42.42045 10.1080/10556780802699201 10.3934/dcdsb.2015.20.2291 10.1137/17M1121044 10.1088/1361-6544/ab018b 10.1137/130914565 10.1137/19M1305835 10.1109/CACSD.2004.1393890 10.1137/S1052623400366802 10.1137/0313003 10.1145/3282678.3282681 10.1109/TAC.2003.823000 10.1016/j.automatica.2015.05.015 10.1016/j.tcs.2008.09.025 10.1016/j.sysconle.2016.11.010 |
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| Keywords | Occupation measures Semidefinite programming Sum-of-squares Infinite-dimensional linear programming Outer approximations Dynamical systems Global attractor |
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| SubjectTerms | Dynamical Systems Global attractor Infinite-dimensional linear programming Mathematics Occupation measures Optimization and Control Outer approximations Semidefinite programming Sum-of-squares |
| Title | Converging outer approximations to global attractors using semidefinite programming |
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