Safe and Robust Domains of Attraction for Discrete-Time Systems: A Set-Based Characterization and Certifiable Neural Network Estimation

Mohamed Serry; Maxwell Fitzsimmons; Jun Liu

eess.SYMar 3, 2026

Safe and Robust Domains of Attraction for Discrete-Time Systems: A Set-Based Characterization and Certifiable Neural Network Estimation

Authors: Mohamed Serry, Maxwell Fitzsimmons, Jun Liu

Organizations: department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada · Department of Mechanical and Mechatronics Engineering at the University of Waterloo

Abstract

Analyzing nonlinear systems with attracting robust invariant sets (RISs) requires estimating their domains of attraction (DOAs). Despite extensive research, accurately characterizing DOAs for general nonlinear systems remains challenging due to both theoretical and computational limitations, particularly in the presence of uncertainties and state constraints. In this paper, we propose a novel framework for the accurate estimation of safe (state-constrained) and robust DOAs for discrete-time nonlinear uncertain systems with continuous dynamics, open safe sets, compact disturbance sets, and uniformly locally $\ell_p$ -stable compact RISs. The notion of uniform $\ell_p$ stability is quite general and encompasses, as special cases, uniform exponential and polynomial stability. The DOAs are characterized via newly introduced value functions defined on metric spaces of compact sets. We establish their fundamental mathematical properties and derive the associated Bellman-type (Zubov-type) functional equations. Building on this characterization, we develop a physics-informed neural network (NN) framework to learn the corresponding value functions by embedding the derived Bellman-type equations directly into the training process. To obtain certifiable estimates of the safe robust DOAs from the learned neural approximations, we further introduce a verification procedure that leverages existing formal verification tools. The effectiveness and applicability of the proposed methodology are demonstrated through four numerical examples involving nonlinear uncertain systems subject to state constraints, and its performance is compared with existing methods from the literature.

Figures & tables

Parameters/Variables		$\mathbb{X}_{l}/\mathbb{X}_{v}$	$N_{s}$	$N_{\mathrm{d}}$ / $N_{\mathrm{pi}}$	$N_{\mathrm{traj}}$	$\mathrm{Loss}(\theta)$	$t_{\mathrm{cpu,ev}}$	$t_{\mathrm{t}}$	$t_{\mathrm{e}}$	$t_{\mathrm{v,gpu}}$
Ex. VII-B	$\mathrm{NN}_{\mathrm{nom}}$	0.7 $\llbracket-1_{2},1_{2}\rrbracket$	500	5000	1	9e-6	1.9	227.0	0.6/6.2	16.0/NA
Ex. VII-B	$\mathrm{NN}_{\mathrm{prop}}$	0.7 $\llbracket-1_{2},1_{2}\rrbracket$	500	5000	1000	4.0e-6	278.0	218.7	0.6/6.1	17.2/19.5
Ex. VII-C	$\mathrm{NN}_{\mathrm{nom}}$	$\llbracket-1_{2},1_{2}\rrbracket$	500	5000	1	4.3e-5	0.5	215.0	0.6/6.5	22.9/22.9
Ex. VII-C	$\mathrm{NN}_{\mathrm{prop}}$	$\llbracket-1_{2},1_{2}\rrbracket$	500	5000	1000	7.8e-5	245.4	211.0	0.6/7.8	15.7/38.6
Ex. VII-D	$\mathrm{NN}_{\mathrm{nom}}$	$[-\frac{\pi}{4},\frac{\pi}{4}]\times\llbracket-1_{2},1_{2}\rrbracket$	500	7000	1	1.4e-4	0.9	656.8	66.5/399.0	26.3/NA
Ex. VII-D	$\mathrm{NN}_{\mathrm{prop}}$		500	7000	3000	1.0e-4	683.7	622.2	65.5/385.5	23.4/43.0

TABLE I: Neural network architectures/parameters and corresponding CPU times for the four numerical examples. Here,

\mathrm{NN}_{\mathrm{nom}}

denotes the neural networks trained using the nominal dynamics (i.e., the framework in [ 3 ] ), whereas

\mathrm{NN}_{\mathrm{prop}}

denotes those trained using the proposed framework. The quantity

t_{\mathrm{ev}}

represents the CPU time required to evaluate

\bigcup_{i=1}^{N_{\mathrm{d}}}\{\tilde{\mathcal{W}}_{N_{s}}(\{z_{\mathrm{d}}^{(i)}\})\}

t_{\mathrm{t}}

is the CPU time required for neural network training,

t_{\mathrm{e}}

is the CPU time required to estimate the parameters

c_{2}

\omega_{1}

, and

\omega_{2}

(for both unperturbed and perturbed scenarios), and

t_{\mathrm{v,gpu}}

is the CPU time required to verify these parameters (for both unperturbed and perturbed scenarios) using a GPU. “NA” indicates verification failure. All reported CPU times are given in seconds.

Fig. 1: Certified ROAs for Example VII-B under Scenario 1 (uncertainty neglected, top) and Scenario 2 (full disturbance considered, bottom). Red circles indicate the state constraints. The nominally trained neural network fails to yield a certifiable ROA under Scenario 2.

Fig. 2: Certified ROAs for Example VII-C under Scenario 1 (uncertainty neglected, top) and Scenario 2 (full disturbance considered, bottom). Red lines indicate the state constraints.

Panel of Figure 4 — Fig. 3: Certified ROAs for Example VII-D under Scenario 1 (uncertainty neglected, top) and Scenario 2 (full disturbance considered, bottom). The nominally trained neural network fails to yield a certifiable ROA under Scenario 2.

Fig. 4: Certified ROAs for Example VII-E . The ROA based on parameter-dependent Lyapunov functions is digitized from [ 8 ] .

Robust Control

Robust control designs feedback that maintains stability, safety, and performance despite disturbances, uncertainty, model errors, and changing conditions.

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Set-Based Value Function Characterization and Neural Approximation of Stabilization Domains for Input-Constrained Discrete-Time Systems

Analyzing nonlinear systems with stabilizable controlled invariant sets (CISs) requires accurate estimation of their domains of stabilization (DOS) together with associated stabilizing controllers. Despite extensive research, estimating DOSs for general nonlinear systems remains challenging due to fundamental theoretical and computational limitations. In this paper, we propose a novel framework for estimating DOSs for controlled input-constrained discrete-time systems. The DOS is characterized via newly introduced value functions defined on metric spaces of compact sets. We establish the fundamental properties of these value functions and derive the associated Bellman-type (Zubov-type) functional equations. Building on this characterization, we develop a physics-informed neural network (NN) framework that learns the value functions by embedding the derived functional equations directly into the training process. The proposed methodology is demonstrated through two numerical examples, illustrating its ability to accurately estimate DOSs and synthesize stabilizing controllers from the learned value functions.

Mohamed Serry, S. Sivaranjani, Jun Liu

Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada. · Department of Mechanical and Mechatronics Engineering at the University of Waterloo. · School of Industrial Engineering, Purdue University, 15 N Grant Street, West Lafayette, 47907, Indiana.

Dynamical Systems Value Function Estimation

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