Construction of a contraction metric for non-smooth time-periodic systems

Jordan Boote; Peter Giesl; Stefan Suhr

doi:10.3934/dcdsb.2024147

Article Contents

2025, Volume 30, Issue 6: 1903-1929. Doi: 10.3934/dcdsb.2024147

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Construction of a contraction metric for non-smooth time-periodic systems

1.
Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom
2.
Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany

^*Corresponding author: Peter Giesl
^*Corresponding author: Peter Giesl

Received: January 2024

Revised: October 2024

Early access: October 17, 2024

Published online: October 17, 2024

The third author is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 281071066 - TRR 191.

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Abstract

We consider a general non-smooth ordinary differential equation of the form $ \dot{x} = f(t, x) $, where $ x\in\mathbb{R} $, $ f $ is $ t $-periodic and smooth except at $ x = 0 $. The existence, uniqueness and exponential stability of a periodic orbit is equivalent to the existence of a contraction metric. In this paper, we develop a method to numerically construct a contraction metric. In more detail, we discretise the conditions and use quadratic programming to determine a solution. We show that the method always succeeds if it employs sufficiently many discretisation points, and we apply the method to several examples.

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Mathematics Subject Classification: Primary: 37M22, 34A36, 90C20; Secondary: 35A23, 34C25.

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Figure 1. Example 4.1, equation (37): Points in $ \Omega $, which are used in the algorithm

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Figure 2. Example 4.1, equation (37). Left: $ \dot{W}^{+}(t, x) $, which is negative, and thus shows that $ f_x^+(t, x)+\dot{W}^+(t, x)<0 $ holds for $ (t, x)\in \Omega^+ $ (assumption 1. in Theorem 2.2 for $ \Omega^+ $). Right: $ \dot{W}^{-}(t, x) $, which is negative, and thus shows that $ f_x^-(t, x)+\dot{W}^-(t, x)<0 $ holds for $ (t, x)\in \Omega^- $ (assumption 1. in Theorem 2.2 for $ \Omega^- $)

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Figure 3. Example 4.1, equation (37). Left: $ \dot{W}^{+}(t, 0) $, which is negative, and thus shows that $ f_x^+(t, x)+\dot{W}^+(t, x)<0 $ holds for $ (t, x)\in \Omega^+ $ with $ x\to 0 $ (condition 1. in Theorem 2.2 for $ \Omega^+ $). Right: $ \dot{W}^{-}(t, 0) $, which is negative, and thus shows that $ f_x^-(t, x)+\dot{W}^-(t, x)<0 $ holds for $ (t, x)\in \Omega^- $ with $ x\to 0 $ (condition 1. in Theorem 2.2 for $ \Omega^- $)

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Figure 4. Example 4.1, equation (37). Left: The function $ \frac{f^-(t, 0)}{f^+(t, 0)}\exp(W^-(t, 0)-W^+(t, 0)) $ from condition 2. in Theorem 2.2 is plotted for all $ (t, 0)\in \Gamma^- $. As all values are $ <1 $, this shows that 2. is true for all $ (t, 0)\in \Gamma^- $. Right: The function $ \frac{f^+(t, 0)}{f^-(t, 0)}\exp(W^+(t, 0)-W^-(t, 0)) $ from condition 3. in Theorem 2.2 is plotted for all $ (t, 0)\in \Gamma^+ $. As all values are $ <1 $, this shows that 3. is true for all $ (t, 0)\in \Gamma^+ $

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Figure 5. Example 4.2, equation (40). Left: $ f_x^{+}(t, x)+\dot{W}^{+}(t, x) $, which is negative, and thus shows that $ f_x^+(t, x)+\dot{W}^+(t, x)<0 $ holds for $ (t, x)\in \Omega^+ $ (assumption 1. in Theorem 2.2 for $ \Omega^+ $). Right: $ f_x^-(t, x)+\dot{W}^{-}(t, x) $, which is negative, and thus shows that $ f_x^-(t, x)+\dot{W}^-(t, x)<0 $ holds for $ (t, x)\in \Omega^- $ (assumption 1. in Theorem 2.2 for $ \Omega^- $)

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Figure 6. Example 4.2, equation (40). Left: $ f_x^+(t, 0)+\dot{W}^{+}(t, 0) $, which is negative, and thus shows that $ f_x^+(t, x)+\dot{W}^+(t, x)<0 $ holds for $ (t, x)\in \Omega^+ $ with $ x\to 0 $ (condition 1. in Theorem 2.2 for $ \Omega^+ $). Right: $ f_x^-(t, 0)+\dot{W}^{-}(t, 0) $, which is negative, and thus shows that $ f_x^-(t, x)+\dot{W}^-(t, x)<0 $ holds for $ (t, x)\in \Omega^- $ with $ x\to 0 $ (condition 1. in Theorem 2.2 for $ \Omega^- $)

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Figure 7. Example 4.2, equation (40). Left: The function $ \frac{f^-(t, 0)}{f^+(t, 0)}\exp(W^-(t, 0)-W^+(t, 0)) $ from condition 2. in Theorem 2.2 is plotted for all $ (t, 0)\in \Gamma^- $. As all values are $ <1 $, this shows that 2. is true for all $ (t, 0)\in \Gamma^- $. Right: The function $ \frac{f^+(t, 0)}{f^-(t, 0)}\exp(W^+(t, 0)-W^-(t, 0)) $ from condition 3. in Theorem 2.2 is plotted for all $ (t, 0)\in \Gamma^+ $. As all values are $ <1 $, this shows that 3. is true for all $ (t, 0)\in \Gamma^+ $

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