# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022027
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## Existence of complete Lyapunov functions with prescribed orbital derivative

 1 Department of Mathematics, University of Sussex, Falmer, Brighton, BN1 9QH, United Kingdom 2 Faculty of Physical Sciences, Dunhagi 5,107 Reykjavík, Iceland; Current address: Faculty of Physical Sciences, University of Iceland, Dunhagi 5,107 Reykjavík, Iceland 3 Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany

*Corresponding author: Stefan Suhr

Received  September 2021 Revised  January 2022 Early access February 2022

Fund Project: Suhr is partially supported by the SFB/TRR 191 "Symplectic Structures in Geometry, Algebra and Dynamics", funded by the Deutsche Forschungsgemeinschaft

Complete Lyapunov functions for a dynamical system, given by an autonomous ordinary differential equation, are scalar-valued functions that are strictly decreasing along orbits outside the chain-recurrent set. In this paper we show that we can prescribe the (negative) values of the derivative along orbits in any compact set, which is contained in the complement of the chain-recurrent set. Further, the complete Lyapunov function is as smooth as the vector field defining the dynamics. This delivers a theoretical foundation for numerical methods to construct complete Lyapunov functions and renders them accessible for further theoretical analysis and development.

Addendum: “Current address: Faculty of Physical Sciences, University of Iceland, Dunhagi 5,107 Reykjavík, Iceland” is added for the second author Sigurdur Freyr Hafstein. We apologize for any inconvenience this may cause.

Citation: Peter Giesl, Sigurdur Freyr Hafstein, Stefan Suhr. Existence of complete Lyapunov functions with prescribed orbital derivative. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022027
##### References:
 [1] C. Argáez, P. Giesl and S. Hafstein, Analysing dynamical systems towards computing complete Lyapunov functions, In Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, Madrid, Spain, (2017), 323–330. [2] C. Argáez, P. Giesl and S. Hafstein, Complete Lyapunov functions: Computation and applications, Simulation and Modeling Methodologies, Technologies and Applications, 873 (2017), 200-221. [3] J. Auslander, Generalized recurrence in dynamical systems, Contributions to Differential Equations, 3 (1964), 65-74. [4] H. Ban and W. Kalies, A computational approach to Conley's decomposition theorem, J. Comput. Nonlinear Dynam, 1 (2006), 312-319.  doi: 10.1115/1.2338651. [5] P. Bernhard and S. Suhr, Lyapounov functions of closed cone fields: From Conley theory to time functions, Comm. Math. Phys., 359 (2018), 467-498.  doi: 10.1007/s00220-018-3127-7. [6] C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38. American Mathematical Society, Providence, R.I., 1978. [7] M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, Handbook of Dynamical Systems, 2 (2002), 221-264.  doi: 10.1016/S1874-575X(02)80026-1. [8] A. Fathi and P. Pageault, Smoothing Lyapunov functions, Trans. Amer. Math. Soc., 371 (2019), 1677-1700.  doi: 10.1090/tran/7329. [9] A. Fathi and A. Siconolfi, On smooth time functions, Math. Proc. Cambridge Philos. Soc., 152 (2012), 303-339.  doi: 10.1017/S0305004111000661. [10] P. Giesl, C. Argáez, S. Hafstein and H. Wendland, Construction of a complete Lyapunov function using quadratic programming, Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics, 1 (2018), 560-568.  doi: 10.5220/0006944305600568. [11] P. Giesl, C. Argáez, S. Hafstein and H. Wendland, Minimization with differential inequality constraints applied to complete Lyapunov functions, Math. Comp., 90 (2021), 2137-2160.  doi: 10.1090/mcom/3629. [12] S. Hafstein and S. Suhr, Smooth complete Lyapunov functions for ODEs, J. Math. Anal. Appl., 499 (2021), Paper No. 125003, 15 pp. doi: 10.1016/j.jmaa.2021.125003. [13] S. W. Hawking, The existence of cosmic time functions, Proc. Roy. Soc. London Ser. A, 308 (1969), 433-435. [14] M. Hurley, Chain recurrence and attraction in non-compact spaces, Ergodic Theory Dynam. Systems, 11 (1991), 709-729.  doi: 10.1017/S014338570000643X. [15] M. Hurley, Chain recurrence, semiflows, and gradients, J. Dyn. Diff. Equat., 7 (1995), 437-456.  doi: 10.1007/BF02219371. [16] M. Hurley, Lyapunov functions and attractors in arbitrary metric spaces, Proc. Amer. Math. Soc., 126 (1998), 245-256.  doi: 10.1090/S0002-9939-98-04500-6. [17] W. Kalies, K. Mischaikow and R. VanderVorst, An algorithmic approach to chain recurrence, Found. Comput. Math, 5 (2005), 409-449.  doi: 10.1007/s10208-004-0163-9. [18] G. Osipenko, Dynamical Systems, Graphs, and Algorithms, Lecture Notes in Math. 1889, Springer, 2007. [19] M. Patrão, Existence of complete Lyapunov functions for semiflows on separable metric spaces, Far East J. Dyn. Syst., 17 (2011), 49-54.

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##### References:
 [1] C. Argáez, P. Giesl and S. Hafstein, Analysing dynamical systems towards computing complete Lyapunov functions, In Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, Madrid, Spain, (2017), 323–330. [2] C. Argáez, P. Giesl and S. Hafstein, Complete Lyapunov functions: Computation and applications, Simulation and Modeling Methodologies, Technologies and Applications, 873 (2017), 200-221. [3] J. Auslander, Generalized recurrence in dynamical systems, Contributions to Differential Equations, 3 (1964), 65-74. [4] H. Ban and W. Kalies, A computational approach to Conley's decomposition theorem, J. Comput. Nonlinear Dynam, 1 (2006), 312-319.  doi: 10.1115/1.2338651. [5] P. Bernhard and S. Suhr, Lyapounov functions of closed cone fields: From Conley theory to time functions, Comm. Math. Phys., 359 (2018), 467-498.  doi: 10.1007/s00220-018-3127-7. [6] C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Mathematics, 38. American Mathematical Society, Providence, R.I., 1978. [7] M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, Handbook of Dynamical Systems, 2 (2002), 221-264.  doi: 10.1016/S1874-575X(02)80026-1. [8] A. Fathi and P. Pageault, Smoothing Lyapunov functions, Trans. Amer. Math. Soc., 371 (2019), 1677-1700.  doi: 10.1090/tran/7329. [9] A. Fathi and A. Siconolfi, On smooth time functions, Math. Proc. Cambridge Philos. Soc., 152 (2012), 303-339.  doi: 10.1017/S0305004111000661. [10] P. Giesl, C. Argáez, S. Hafstein and H. Wendland, Construction of a complete Lyapunov function using quadratic programming, Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics, 1 (2018), 560-568.  doi: 10.5220/0006944305600568. [11] P. Giesl, C. Argáez, S. Hafstein and H. Wendland, Minimization with differential inequality constraints applied to complete Lyapunov functions, Math. Comp., 90 (2021), 2137-2160.  doi: 10.1090/mcom/3629. [12] S. Hafstein and S. Suhr, Smooth complete Lyapunov functions for ODEs, J. Math. Anal. Appl., 499 (2021), Paper No. 125003, 15 pp. doi: 10.1016/j.jmaa.2021.125003. [13] S. W. Hawking, The existence of cosmic time functions, Proc. Roy. Soc. London Ser. A, 308 (1969), 433-435. [14] M. Hurley, Chain recurrence and attraction in non-compact spaces, Ergodic Theory Dynam. Systems, 11 (1991), 709-729.  doi: 10.1017/S014338570000643X. [15] M. Hurley, Chain recurrence, semiflows, and gradients, J. Dyn. Diff. Equat., 7 (1995), 437-456.  doi: 10.1007/BF02219371. [16] M. Hurley, Lyapunov functions and attractors in arbitrary metric spaces, Proc. Amer. Math. Soc., 126 (1998), 245-256.  doi: 10.1090/S0002-9939-98-04500-6. [17] W. Kalies, K. Mischaikow and R. VanderVorst, An algorithmic approach to chain recurrence, Found. Comput. Math, 5 (2005), 409-449.  doi: 10.1007/s10208-004-0163-9. [18] G. Osipenko, Dynamical Systems, Graphs, and Algorithms, Lecture Notes in Math. 1889, Springer, 2007. [19] M. Patrão, Existence of complete Lyapunov functions for semiflows on separable metric spaces, Far East J. Dyn. Syst., 17 (2011), 49-54.
Schematic figure of a flow box $\mathcal{V}_{\tau', r, T}$
Schematic presentation of the sets $\mathcal{V}_{s_i, 1}$ in the construction of the functions $\tilde{\tau}_i$
The first step. Note that $M$ can intersect the boundary of $[-(k+1), k+1]\times W_s$ at $\{-(k+1)\}\times W_s$ and $[-(k+1), k+1)\times \partial W_s$, but not at $\{k+1\}\times W_s$ (right side)
The second step
The third step
The fourth step
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