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Article Dans Une Revue IEEE Control Systems Letters Année : 2022

Stability and performance verification of dynamical systems controlled by neural networks: algorithms and complexity

Milan Korda

Résumé

This note makes several observations on stability and performance verification of nonlinear dynamical systems controlled by neural networks. First, we show that the stability and performance of a polynomial dynamical system controlled by a neural network with semialgebraically representable activation functions (e.g., ReLU) can be certified by convex semidefinite programming. The result is based on the fact that the semialgebraic representation of the activation functions and polynomial dynamics allows one to search for a Lyapunov function using polynomial sum-of-squares methods; the approach can be viewed as a special case of the general framework of [3]. Second, we remark that even in the case of a linear system controlled by a neural network with ReLU activation functions, the problem of verifying asymptotic stability is undecidable. Finally, under additional assumptions, we establish a converse result on the existence of a polynomial Lyapunov function for this class of systems. Numerical results on academic examples of dimension up to 50 and neural networks with several hundred neurons demonstrate the method.

Dates et versions

hal-03441657 , version 1 (22-11-2021)

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Milan Korda. Stability and performance verification of dynamical systems controlled by neural networks: algorithms and complexity. IEEE Control Systems Letters, 2022, 6, pp.3265 - 3270. ⟨10.1109/LCSYS.2022.3181806⟩. ⟨hal-03441657⟩
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