Examples for separable control Lyapunov functions and their neural network approximation

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Abstract

In this paper, we consider nonlinear control systems and discuss the existence of a separable control Lyapunov function. To this end, we assume that the system can be decomposed into subsystems and formulate conditions such that a weighted sum of Lyapunov functions of the subsystems yields a control Lyapunov function of the overall system. Since deep neural networks are capable of approximating separable functions without suffering from the curse of dimensionality, we can thus identify systems where an efficient approximation of a control Lyapunov function via a deep neural network is possible. A corresponding network architecture and training algorithm are proposed. Further, numerical examples illustrate the behavior of the algorithm.

Keywords

deep neural network
curse of dimensionality
separable function
control Lyapunov function
nonlinear control system
small-gain theory

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This research has been supported by the German Research Foundation (DFG) under project GR 1569/23-1 within the priority program 2298 “Theoretical Foundations of Deep Learning”.

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