Sufficient lognorm conditions beyond μ1 and μ∞ for continuous-time reservoir contraction

Determine analytic sufficient conditions, in terms of logarithmic norms μℓ(·) with ℓ other than 1 and ∞, that ensure global exponential stability (contraction), the Echo State Property, and Generalized Synchronization for the continuous-time reservoir system ẋ = −Cx + σ(Ax + Bu), where σ is an element-wise saturating activation function, A ∈ R^{N×N} is the connectivity matrix, B ∈ R^{N×M} is the input gain, and C ∈ R^{N×N} is a symmetric positive-definite leak matrix.

Background

The paper derives sufficient conditions for contraction, ESP, and GS in continuous-time reservoirs using weak pairings and logarithmic norms. Closed-form conditions are obtained for the 1-lognorm and ∞-lognorm by bounding the Jacobian of the activation function and leveraging properties of the leak matrix C.

The authors note that while these two cases are covered, generalizing the approach to other lognorms remains unresolved. Establishing broader lognorm-based conditions would provide practical, computable guarantees for a wider class of norms and reservoir designs.