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A Corollary for Nonsmooth Systems
N. Fischer, R. Kamalapurkar, W. E. Dixon
TL;DR
The paper addresses the lack of a formal LaSalle-Yoshizawa result for nonsmooth nonautonomous systems with discontinuous state dynamics. It uses Filippov differential inclusions and generalized Lyapunov analysis to establish two corollaries for convergence under negative semi-definite derivative bounds. The results provide tools for analyzing discontinuous closed-loop error systems, with one corollary applying when an inequality holds for almost all time and the other when a stronger state-uniform inequality is available.
Problem
A formal extension of the LaSalle-Yoshizawa Theorem to systems with discontinuous right-hand sides and nonlocally Lipschitz state dynamics was missing.
Method
The note develops generalized Lyapunov analysis using Filippov solutions and differential inclusions for discontinuous nonautonomous systems.
Results
The two corollaries provide theoretical tools for discontinuous closed-loop error systems when the Lyapunov derivative is upper bounded by a negative semi-definite function.
Takeaways & Limitations
Corollary 1 applies when the derivative inequality holds for almost all time, while Corollary 2 uses a stronger inequality holding for all states.
Abstract
from arXiv · showhide
In this note, two generalized corollaries to the LaSalle-Yoshizawa Theorem are presented for nonautonomous systems described by nonlinear differential equations with discontinuous right-hand sides. Lyapunov-based analysis methods are developed using differential inclusions to achieve asymptotic convergence when the candidate Lyapunov derivative is upper bounded by a negative semi-definite function.
I. INTRODUCTION
The note addresses the unresolved extension of LaSalle-Yoshizawa analysis to nonsmooth systems whose state dynamics are not locally Lipschitz. It develops a Filippov-based approach for discontinuous nonautonomous systems with negative semi-definite Lyapunov derivative bounds.
- Extending LaSalle-Yoshizawa analysis to systems with nonlocally Lipschitz state derivatives remains an open problem.Its use for nonsmooth systems had previously been introduced only as a remark without a formal proof.
- The note considers Filippov solutions for nonautonomous nonlinear systems with discontinuous right-hand sides.The analysis uses Lipschitz continuous and regular Lyapunov functions whose Filippov time derivatives can be upper bounded by negative semi-definite functions.
- The proposed framework develops Lyapunov-based analysis for nonsmooth dynamics through differential inclusions.
II. PRELIMINARIES
The preliminaries replace classical solutions with generalized differential-inclusion solutions when discontinuities prevent classical existence or uniqueness. They introduce directional and generalized gradients, regularity, and a chain rule for Lyapunov analysis along Filippov trajectories.
- Discontinuous right-hand sides can prevent classical solutions from existing, motivating Filippov or Krasovskii generalized solutions.These solutions interpret derivatives through nearby-point behavior, but Filippov solutions are generally not unique.
- The system state lies in D ⊂ R^n, while f is Lebesgue measurable and essentially locally bounded.
- A regular function has matching right directional and generalized directional derivatives in every direction.The note also records that continuously differentiable functions are regular and sums of regular functions remain regular.
- Clarke’s generalized gradient is defined as the convex closure of limiting gradients at points where the gradient exists.The excluded set where the gradient is undefined has measure zero.
- The chain rule makes V(x(t),t) absolutely continuous and provides an almost-everywhere time derivative along Filippov solutions.The derivative is computed using Clarke’s generalized gradient and the set-valued map K[f](x,t).
III. MAIN RESULT
The note develops nonsmooth LaSalle-Yoshizawa corollaries using Filippov differential inclusions and Lyapunov derivatives bounded by negative semi-definite functions. The results establish boundedness and asymptotic convergence, with one corollary applying to a specific solution and the other to all Filippov solutions.
- Framework: Nonsmooth Lyapunov analysis replaces differential equations with inclusions and ordinary gradients with generalized gradients for discontinuous, non-Lipschitz systems.Filippov solutions and regular, locally Lipschitz Lyapunov functions provide the analytical framework.
- Corollary 1: If the Filippov Lyapunov derivative satisfies ˙V(x(t),t) ≤ 0 almost everywhere, V(x(t),t) is non-increasing along the solution.The chain rule and auxiliary lemma justify this conclusion for locally Lipschitz, regular Lyapunov functions.
- Corollary 1: Corollary 1 shows that solutions initialized in {x ∈ B_r | W2(x) ≤ c} remain bounded inside {x ∈ B_r | W1(x) ≤ c} for all future time.The time-dependent sublevel set Ω_t,c is used to establish forward containment and the bound ∥x(t)∥ < r.
- Corollary 1: Boundedness, integrability of W(x(t)), and uniform continuity support Barbalat’s Lemma, yielding the corollary’s asymptotic convergence conclusion.The argument uses the non-increasing Lyapunov function and continuity of W on the compact set B_r.
- Corollary 2: Corollary 2 extends the result from a specific Filippov solution to every Filippov solution when every element of the set-valued derivative satisfies ˙˜V(x,t) ≤ −W(x).The stronger set-valued inequality implies the trajectory condition required by Corollary 1 for arbitrary solution selections.
- Scope and applicability: Corollary 1 is useful when the stronger pointwise set-valued inequality for Corollary 2 is difficult or impossible to verify, but the trajectory inequality holds almost everywhere.Corollary 2 may be easier for some closed-loop error systems, including systems with sliding mode control laws.
IV. CONCLUSION
The note extends the LaSalle-Yoshizawa Theorem to systems with state-discontinuous, time-piecewise-continuous right-hand sides, developing Filippov-based Lyapunov tools for asymptotic convergence.
- The LaSalle-Yoshizawa Theorem is extended to differential systems with right-hand sides discontinuous in the state and piecewise continuous in time.
- Two theoretical tools are presented for nonautonomous systems with discontinuities in the closed-loop error system.
- Differential inclusions in the sense of Filippov support generalized Lyapunov-based analysis for these nonsmooth systems.
- The methods achieve asymptotic convergence when the candidate Lyapunov derivative is upper bounded by a negative semi-definite function.