[Paper Review] Non-Euclidean Contraction Theory via Semi-Inner Products.

A. S. Davydov, Saber Jafarpour+1 more|arXiv (Cornell University)|Mar 23, 2021

Control and Stability of Dynamical Systems48 references4 citations

TL;DR

This paper introduces weak semi-inner products (WSIPs) to extend contraction theory to non-Euclidean norms, such as ℓ₁ and ℓ∞, providing five equivalent characterizations for contraction, including one-sided Lipschitz conditions and generalized Demidovich criteria. It establishes incremental stability and input-to-state properties for contracting systems, enabling hierarchical analysis of interconnected systems.

ABSTRACT

In this paper, we study necessary and sufficient conditions for contraction and incremental stability of dynamical systems with respect to non-Euclidean norms. First, generalizing Lumer semi-inner products, we introduce weak semi-inner products (WSIPs), and characterize their properties. We introduce and study the sign and max WSIPs for the $\ell_{1}$ and $\ell_{\infty}$ norms, respectively. Using WSIPs, we then establish five equivalent characterizations for contraction, including one-sided Lipschitz continuity of the vector field as well as matrix measure and generalized Demidovich conditions for the corresponding Jacobian. Third, we extend our contraction framework in two directions: we prove equivalences for contraction of continuous vector fields and we formalize the weaker notion of equilibrium contraction, which ensures exponential convergence to a given equilibrium. Finally, as an application of the general theory, we provide (i) incremental input-to-state stability and finite input-state gain properties for contracting systems, and (ii) a hierarchical analysis approach to establish contraction properties for interconnected systems.

Motivation & Objective

• To develop a contraction theory framework applicable to non-Euclidean norms, such as ℓ₁ and ℓ∞, beyond standard Euclidean settings.
• To generalize Lumer semi-inner products by introducing weak semi-inner products (WSIPs) with well-characterized properties.
• To establish equivalent conditions for contraction, including one-sided Lipschitz continuity and matrix measure-based criteria for Jacobians.
• To formalize equilibrium contraction, ensuring exponential convergence to a given equilibrium, and extend results to continuous vector fields.
• To apply the framework to prove incremental input-to-state stability and finite input-state gain for contracting systems, and to enable hierarchical analysis of interconnected systems.

Proposed method

• Introduce weak semi-inner products (WSIPs) as a generalization of Lumer semi-inner products, tailored to non-Euclidean norms.
• Define and analyze sign and max WSIPs specifically for ℓ₁ and ℓ∞ norms, respectively, to capture directional derivatives in non-smooth settings.
• Derive five equivalent characterizations of contraction, including one-sided Lipschitz conditions on the vector field and generalized Demidovich conditions on the Jacobian matrix.
• Establish equivalence between contraction and exponential convergence to equilibrium under the new WSIP framework.
• Extend the theory to continuous vector fields and define equilibrium contraction as a weaker, yet stabilizing, condition.
• Apply the framework to prove incremental input-to-state stability and finite input-state gain for contracting systems, and develop a hierarchical analysis method for interconnected systems.

Experimental results

Research questions

• RQ1 What conditions ensure contraction of dynamical systems under non-Euclidean norms such as ℓ₁ and ℓ∞?
• RQ2 How can semi-inner products be generalized to enable contraction analysis beyond Euclidean geometry?
• RQ3 What are the equivalent characterizations of contraction in terms of vector field properties and Jacobian conditions under non-Euclidean norms?
• RQ4 Can equilibrium contraction be formally defined and characterized to ensure exponential convergence to a specific equilibrium?
• RQ5 How can the theory be extended to interconnected systems and used to establish input-to-state stability with finite gain?

Key findings

• The introduction of weak semi-inner products (WSIPs) enables a rigorous contraction analysis for non-Euclidean norms, including ℓ₁ and ℓ∞.
• Five equivalent conditions for contraction are established, including one-sided Lipschitz continuity of the vector field and generalized Demidovich conditions on the Jacobian.
• Equilibrium contraction is formalized as a weaker but sufficient condition for exponential convergence to a given equilibrium, extending the scope of contraction theory.
• The framework proves incremental input-to-state stability and finite input-state gain for contracting systems under non-Euclidean norms.
• A hierarchical analysis method is developed to verify contraction in interconnected systems using the proposed WSIP-based criteria.
• The theory is extended to continuous vector fields, broadening applicability beyond smooth systems.