Analysis of Lurie systems with magnitude nonlinearities and connections to neural network stability analysis
Analysis of Lurie systems with magnitude nonlinearities and connections to neural network stability analysis
This paper considers the interconnection of a continuous time linear time-invariant system and a multivariable magnitude nonlinearity. A number of different quadratic constraints are established for the magnitude nonlinearity and then used to derive stability criteria based on quadratic and Lurie-type Lyapunov functions. The new stability criteria are cast as matrix inequalities and in some cases solved using semi-definite programming. Connections are made between the magnitude nonlinearity and neural network activation functions, such as the rectified linear unit (ReLU) and leaky ReLU, effectively allowing the stability criteria derived here to be used to analyse interconnections of dynamical systems and neural networks. Using the positive homogeneity property, shared by the (leaky) ReLU and magnitude functions, mild conditions are also established to show that the existence of a unique equilibrium point is sufficient for local and global stability to be equivalent. Finally, the new global stability criteria are tested on several numerical examples, including a Hopfield network with 100 states and neurons, and compare favourably with competing criteria from the literature.
Lyapunov methods, neural networks, semi-definite programming, stability of nonlinear systems
Richardson, Carl Robert
3406b6af-f00d-410b-8051-a0ecc27baba5
Gunn, Steve
306af9b3-a7fa-4381-baf9-5d6a6ec89868
Turner, Matthew
6befa01e-0045-4806-9c91-a107c53acba0
16 February 2026
Richardson, Carl Robert
3406b6af-f00d-410b-8051-a0ecc27baba5
Gunn, Steve
306af9b3-a7fa-4381-baf9-5d6a6ec89868
Turner, Matthew
6befa01e-0045-4806-9c91-a107c53acba0
Richardson, Carl Robert, Gunn, Steve and Turner, Matthew
(2026)
Analysis of Lurie systems with magnitude nonlinearities and connections to neural network stability analysis.
IEEE Transactions on Automatic Control.
(doi:10.1109/TAC.2026.3665137).
Abstract
This paper considers the interconnection of a continuous time linear time-invariant system and a multivariable magnitude nonlinearity. A number of different quadratic constraints are established for the magnitude nonlinearity and then used to derive stability criteria based on quadratic and Lurie-type Lyapunov functions. The new stability criteria are cast as matrix inequalities and in some cases solved using semi-definite programming. Connections are made between the magnitude nonlinearity and neural network activation functions, such as the rectified linear unit (ReLU) and leaky ReLU, effectively allowing the stability criteria derived here to be used to analyse interconnections of dynamical systems and neural networks. Using the positive homogeneity property, shared by the (leaky) ReLU and magnitude functions, mild conditions are also established to show that the existence of a unique equilibrium point is sufficient for local and global stability to be equivalent. Finally, the new global stability criteria are tested on several numerical examples, including a Hopfield network with 100 states and neurons, and compare favourably with competing criteria from the literature.
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Published date: 16 February 2026
Additional Information:
Funding Information:
This work was supported in part by the Defence Science and Technology Laboratory (DSTL) and in part by the U.K. Research and Innovation (UKRI) Centre of Machine Intelligence for Nano-Electronic Devices and Systems under Grant EP/S024298/1.
Publisher Copyright:
© 2017 IEEE.
Keywords:
Lyapunov methods, neural networks, semi-definite programming, stability of nonlinear systems
Identifiers
Local EPrints ID: 510819
URI: http://eprints.soton.ac.uk/id/eprint/510819
ISSN: 0018-9286
PURE UUID: 8b6e54ce-d724-41ed-906d-b2b8cd49f25b
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Date deposited: 22 Apr 2026 16:49
Last modified: 23 Apr 2026 02:09
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Contributors
Author:
Carl Robert Richardson
Author:
Steve Gunn
Author:
Matthew Turner
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