Bounded confidence models of opinion formation are paradigmatic examples of coevolutionary networks, where agents update their opinions by averaging those of peers deemed sufficiently similar. In this paper, we study a multidimensional extension of the Hegselmann–Krause (HK) model with stubborn agents (MHK-S), in which the confidence sets are defined as balls in a general (not necessarily Euclidean) norm and some agents remain stubborn, being resistant to social influence. We establish a behavioral dichotomy: an agent’s opinion either terminates in finite time (if not influenced by stubborn agents) or converges asymptotically to the convex hull of the stubborn agents’ opinions. In the special case where the stubborn agents’ opinions are sufficiently close, the influenced agents converge to their barycenter; we further extend this result to scenarios featuring multiple clusters of stubborn opinions that are sufficiently distant from each other.

While monotone operator theory is often studied on Hilbert spaces, many interesting problems in machine learning and optimization arise naturally in finite-dimensional vector spaces endowed with non-Euclidean norms, such as diagonally-weighted $\ell_{1}$ or $\ell_{\infty}$ norms. This paper provides a natural generalization of monotone operator theory to finite-dimensional non-Euclidean spaces. The key tools are weak pairings and logarithmic norms. We show that the resolvent and reflected resolvent operators of non-Euclidean monotone mappings exhibit similar properties to their counterparts in Hilbert spaces. Furthermore, classical iterative methods and splitting methods for finding zeros of monotone operators are shown to converge in the non-Euclidean case. We apply our theory to equilibrium computation and Lipschitz constant estimation of recurrent neural networks, obtaining novel iterations and tighter upper bounds via forward-backward splitting.

Implicit neural networks, a.k.a., deep equilibrium networks, are a class of implicit-depth learning models where function evaluation is performed by solving a fixed point equation. They generalize classic feedforward models and are equivalent to infinite-depth weight-tied feedforward networks. While implicit models show improved accuracy and significant reduction in memory consumption, they can suffer from ill-posedness and convergence instability.This paper provides a new framework, which we call Non-Euclidean Monotone Operator Network (NEMON), to design well-posed and robust implicit neural networks based upon contraction theory for the non-Euclidean norm $\ell_\infty$. Our framework includes (i) a novel condition for well-posedness based on one-sided Lipschitz constants, (ii) an average iteration for computing fixed-points, and (iii) explicit estimates on input-output Lipschitz constants. Additionally, we design a training problem with the well-posedness condition and the average iteration as constraints and, to achieve robust models, with the input-output Lipschitz constant as a regularizer. Our $\ell_\infty$ well-posedness condition leads to a larger polytopic training search space than existing conditions and our average iteration enjoys accelerated convergence. Finally, we evaluate our framework in image classification through the MNIST and the CIFAR-10 datasets. Our numerical results demonstrate improved accuracy and robustness of the implicit models with smaller input-output Lipschitz bounds. Code is available at https://github.com/davydovalexander/Non-Euclidean_Mon_Op_Net.

Whereas classical control theory provides many methods for designing continuous-time feedback controllers, nowadays control algorithms are implemented on digital platforms and have to be designed in sampled time. Approaches to sampled-time control design are based on either discretization of the plant enabling discrete-time controller synthesis, or various redesign methods converting a continuous-time controller into a sampled-time approximation, providing comparable closed-loop system properties. The simplest of redesign approaches, typically used in practice, is the emulation of continuous-time feedback by sufficiently fast sampling. In spite of its simplicity, emulation gives rise to an important problem: does emulation at a sufficiently high rate (or, equivalently, with a small sampling time) preserve the stability of the closed-loop system? In this paper, we address this problem for the case of exponential stability (local or global). Even for linear systems, the problem of stability preservation becomes non-trivial when sampling is aperiodic. For nonlinear systems, viability of emulation approach is usually proved only under quite restrictive assumptions on the plant and the controller, which, as will be shown, in fact be discarded.

The pull-in range, known also as the acquisition or capture range, is an important characteristics of synchronization circuits such as e.g. phase-, frequency- and delay-locked loops (PLL/FLL/DLL). For PLLs, the pull-in range characterizes the maximal frequency detuning under which the system provides phase locking (mathematically, every solution of the system converges to one of the equilibria). The presence of periodic nonlinearities (characteristics of phase detectors) and infinite sequences of equilibria makes rigorous analysis of PLLs very difficult in spite of their seeming simplicity. The models of PLLs can be featured by multi-stability, hidden attractors and even chaotic trajectories. For this reason, the pull-in range is typically estimated numerically by e.g. using harmonic balance or Galerkin approximations. Analytic results presented in the literature are not numerous and primarily deal with ordinary differential equations. In this paper, we propose an analytic method for pull-in range estimation, applicable to synchronization systems with infinite-dimensional linear part, in particular, for PLLs with delays. The results are illustrated by analysis of a PLL described by second-order delay equations.

Control Lyapunov Functions (CLF) method gives a constructive tool for stabilization of nonlinear systems. To find a CLF, many methods have been proposed in the literature, e.g. backstepping for cascaded systems and sum of squares (SOS) programming for polynomial systems. Dealing with continuous-time systems, the CLF-based controller is also continuous-time, whereas practical implementation on a digital platform requires sampled-time control. In this paper, we show that if the continuous-time controller provides exponential stabilization, then an exponentially stabilizing event-triggered control strategy exists with the convergence rate arbitrarily close to the rate of the continuous-time system.

Phase-locked loops (PLL), Costas loops and other synchronizing circuits are featured by the presence of a nonlinear phase detector, described by a periodic nonlinearity. In general, nonlinearities can cause complex behavior of the system such as multi-stability and chaos. Even if the phase locking is guaranteed for any initial conditions, the transient behavior of the circuit can still be unsatisfactory due to the cycle slipping. Growth of the phase error caused by cycle slipping is undesirable, leading e.g. to demodulation and decoding errors. This makes the problem of estimating the phase error oscillations and number of slipped cycles in nonlinear PLL-based circuits extremely important for modern telecommunications. Most mathematical results in this direction, available in the literature, focus on the phase jitter and cycle slipping under random noise and examine the relations between the probabilistic characteristics of the noise and of the phase error, e.g. the expected number of slipped cycles. At the same time, cycle slipping occurs also in deterministic systems with periodic nonlinearities, depending on the initial conditions, properties of the linear part and the periodic nonlinearity and other factors such as delays in the loop. In the present paper we give analytic estimates for the number of slipped cycles in PLL-based systems, governed by integro-differential equations, allowing to capture effects of high-order dynamics, discrete and distributed delays. We also consider the effects of singular small-parameter perturbations on the cycle slipping behavior.

This paper addresses consensus problems in nonlinearly coupled networks of dynamic agents described by a common and arbitrary linear model. Interagent interaction rules are uncertain but satisfy the standard sector condition with known sector bounds; both the agent's model and interaction topology are time-invariant. A novel frequency-domain criterion for consensus is offered that is similar to and extends the classical Popov's absolute stability criterion.