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The mathematical properties and data-driven learning of the Koopman operator, which represents nonlinear dynamics as a linear mapping on a properly defined functional spaces, have become key problems in nonlinear system identification and control. However, Koopman operators that are approximately learned from snapshot data may not always accurately predict the system evolution on long horizons. In this work, by defining the Koopman operator on a space of weighted continuous functions and learning it on a weighted reproducing kernel Hilbert space, the Koopman operator is guaranteed to be contractive and the accumulation learning error is bounded. The weighting function, assumed to be known a priori, has an exponential decay with the flow or decays exponentially when compensated by an exponential factor. Under such a construction, the Koopman operator learned from data is used to estimate (i) Lyapunov functions for globally asymptotically stable dynamics, and (ii) Zubov-Lyapunov functions that characterize the domain of attraction. For these estimations, probabilistic bounds on the errors are derived., Comment: 8 pages, 3 figures, submitted to 2025 American Control Conference

This work proposes a data-driven approach for bifurcation analysis in nonlinear systems when the governing differential equations are not available. Specifically, regularized regression with barrier terms is used to learn a homeomorphism that transforms the underlying system to a reference linear dynamics -- either an explicit reference model with desired qualitative behavior, or Koopman eigenfunctions that are identified from some system data under a reference parameter value. When such a homeomorphism fails to be constructed with low error, bifurcation phenomenon is detected. A case study is performed on a planar numerical example where a pitchfork bifurcation exists., Comment: 12 pages, 4 figures, submitted to the 6th Annual Learning for Dynamics and Control (L4DC) Conference

Topological defects play a key role in the structures and dynamics of liquid crystals (LCs) and other ordered systems. There is a recent interest in studying defects in different biological systems with distinct textures. However, a robust method to directly recognize defects and extract their structural features from various traditional and nontraditional nematic systems remains challenging to date. Here we present a machine learning solution, termed Machine Eye for Defects (MED), for automated defect analysis in images with diverse nematic textures. MED seamlessly integrates state-of-the-art object detection networks, Segment Anything Model, and vision transformer algorithms with tailored computer vision techniques. We show that MED can accurately identify the positions, winding numbers, and orientations of $\pm 1/2$ defects across distinct cellular contours, sparse vector fields of nematic directors, actin filaments, microtubules, and simulation images of Gay--Berne particles. MED performs faster than conventional defect detection method and can achieve over 90\% accuracy on recognizing $\pm1/2$ defects and their orientations from vector fields and experimental tissue images. We further demonstrate that MED can identify defect types that are not included in the training data, such as giant-core defects and defects with higher winding number. Remarkably, MED provides correct structural information about $\pm 1$ defects, i.e., the phase angle for $+1$ defects and the orientation angle for $-1$ defects. As such, MED stands poised to transform studies of diverse ordered systems by providing automated, rapid, accurate, and insightful defect analysis.

This paper focuses on the model-free synthesis of state observers for nonlinear autonomous systems without knowing the governing equations. Specifically, the Kazantzis-Kravaris/Luenberger (KKL) observer structure is leveraged, where the outputs are fed into a linear time-invariant (LTI) system to obtain the observer states, which can be viewed as the states nonlinearly transformed by an immersion mapping, and a neural network is used to approximate the inverse of the nonlinear immersion and estimate the states. In view of the possible existence of noises in output measurements, this work proposes to impose an upper bound on the Lipschitz constant of the neural network for robust and safe observation. A relation that bounds the generalization loss of state observation according to the Lipschitz constant, as well as the $H_2$-norm of the LTI part in the KKL observer, is established, thus reducing the model-free observer synthesis problem to that of Lipschitz-bounded neural network training, for which a direct parameterization technique is used. The proposed approach is demonstrated on a chaotic Lorenz system., Comment: submitted to American Control Conference 2024

Topological spin and polar textures have fascinated people in different areas of physics and technologies. However, the observations are limited in magnetic and solid-state ferroelectric systems. Ferroelectric nematic is the first liquid-state ferroelectric that would carry many possibilities of spatially distributed polarization fields. Contrary to traditional magnetic or crystalline systems, anisotropic liquid crystal interactions can compete with the polarization counterparts, thereby setting a challenge in understating their interplays and the resultant topologies. Here, we discover chiral polarization meron-like structures during the emergence and growth of quasi-2D ferroelectric nematic domains, which are visualized by fluorescence confocal polarizing microscopy and second harmonic generation microscopies. Such micrometre-scale polarization textures are the modified electric variants of the magnetic merons. Unlike the conventional liquid crystal textures driven solely by the elasticity, the polarization field puts additional topological constraints, e.g., head-to-tail asymmetry, to the systems and results in a variety of previously unidentified polar topological patterns. The chirality can emerge spontaneously in polar textures and can be additionally biased by introducing chiral dopants. An extended mean-field modelling for the ferroelectric nematics reveals that the polarization strength of systems plays a dedicated role in determining polarization topology, providing a guide for exploring diverse polar textures in strongly-polarized liquid crystals.

Super-high-\k{appa} materials that exhibit exceptionally high dielectric permittivity are recognized as potential candidates for a wide range of next-generation photonic and electronic devices. Generally, the high dielectricity for achieving a high-\k{appa} state requires a low symmetry of materials so that most of the discovered high-\k{appa} materials are symmetry-broken crystals. There are scarce reports on fluidic high-\k{appa} dielectrics. Here we demonstrate a rational molecular design, supported by machine-learning analyses, that introduces high polarity to asymmetric molecules, successfully realizing super-high-\k{appa} fluid materials (dielectric permittivity, {\epsilon} > 104) and strong second harmonic generation with macroscopic spontaneous polar ordering. The polar structures are confirmed to be identical for all the synthesized materials. Our experiments and computational calculation reveal the unique orientational structures coupled with the emerging polarity. Furthermore, adopting this strategy to high-molecular-weight systems additionally extends the novel material category from monomer to polar polymer materials, creating polar soft matters with spontaneous symmetry breaking.

Distributed optimization, where the computations are performed in a localized and coordinated manner using multiple agents, is a promising approach for solving large-scale optimization problems, e.g., those arising in model predictive control (MPC) of large-scale plants. However, a distributed optimization algorithm that is computationally efficient, globally convergent, amenable to nonconvex constraints and general inter-subsystem interactions remains an open problem. In this paper, we combine three important modifications to the classical alternating direction method of multipliers (ADMM) for distributed optimization. Specifically, (i) an extra-layer architecture is adopted to accommodate nonconvexity and handle inequality constraints, (ii) equality-constrained nonlinear programming (NLP) problems are allowed to be solved approximately, and (iii) a modified Anderson acceleration is employed for reducing the number of iterations. Theoretical convergence towards stationary solutions and computational complexity of the proposed algorithm, named ELLADA, is established. Its application to distributed nonlinear MPC is also described and illustrated through a benchmark process system., Comment: 18 pages, 5 figures