We consider dynamics of an oscillatory mechanical system with 1.5 degrees of freedom. The system consists of two linear subsystems, a leader and a follower, which are coupled via a kinetic friction interface, and additionally driven by a proportional control term. The kinetic friction model accounts for the Coulomb friction and the Stribeck effect (as negative damping) at the passage from zero to nonzero relative velocity. The equations of motion are equivalent to a Lur'e system with a single sign nonlinearity. We show that at a critical value of the parameter the zero steady state blows up into an attracting continuum (segment) of equilibrium states. Moreover, two antipodal smallamplitude periodic orbits bifurcate from the end points of the segment of equilibrium states, creating stick-slip vibrations. We consider other possible attractors consisting of stick-slip solutions of the Lur'e system such as a Z2-symmetric stick-slip periodic orbit and an attractor composed of a continuum of heteroclinic loops representing stick-slip motions towards a steady state.
In this paper, we investigate the spectral stability of two biharmonic Steklov problems under domain perturbation. We provide optimal conditions on the boundary perturbations ensuring the stability of both eigenvalues and eigenfunctions. To highlight the optimality of those conditions, we present alternative assumptions on the boundary perturbations that lead to either a degeneration of the spectrum or to the appearance of a strange term in the limiting problem. In particular, these phenomena are discussed for a boundary homogenization problem exhibiting a trichotomy in the asymptotic behaviour.
Deep learning has become a cornerstone in numerous applications, such as robotics, augmented reality, and autonomous systems, where accurate 6D pose estimation-determining an object's position and orientation in 3D space-is critical. Despite its success, designing optimal neural architectures and tuning hyperparameters remain computationally expensive and challenging, especially in high-variance and data-intensive domains like 6D pose estimation. To address this, we investigate the application of a Neural Architecture Search (NAS) technique guided by classical Machine Learning-based performance predictors. As these predictors estimate final model performance using early-stage training data, we demonstrate that leveraging such a strategy allows for efficient exploration of the hyperparameter space, significantly reducing the computational burden of exhaustive search while still achieving improved performance. Building on an existing NAS method, we introduce a novel modification that enhances the efficiency of the search process, further accelerating convergence by nearly 71% without sacrificing accuracy. Through extensive experimentation on the LineMOD dataset, we demonstrate that our method consistently discovers high-performing configurations of an Augmented Autoencoder for 6D pose estimation, outperforming benchmark models by almost 15% in pose accuracy and 42% in reconstruction loss. These results underscore the potential of predictor-based NAS as a powerful and computationally efficient tool for neural architecture optimization in complex, real-world tasks.
This paper investigates lump waves in a generalized (2+1)-dimensional Bogoyavlensky-Konopelchenko model with spatially balanced derivatives. Using a sum-of-squares ansatz, symbolic computation in Maple is employed to construct lump wave solutions of the nonlinear model from positive quadratic functions. The interplay of four sets of nonlinear terms and five dispersion terms gives rise to the resulting lump waves. The critical points of these quadratic functions are determined, and they travel at constant velocities along a straight line in the spatial plane. Along this characteristic line, the constructed lump waves remain invariant. Concluding remarks are provided in the final section.
This paper studies the far field refraction problem in negative refractive index material with loss of energy, which is a remaining problem in E. Stachura, Nonlinear Anal. 2017;157:76-103. The analysis is divided into two cases according to the relative refractive index kappa, that is, kappa < -1 and-1 < kappa < 0. For each case, we use the Minkowski method to establish the existence of the weak solution when the target measure is either discrete or a finite Radon measure. Eventually, the inequality involving a Monge-Ampe`re type operator satisfied by the solution of the problem is derived, which is useful to understand this complex optical phenomenon.
We consider a class of nonlinear integro-differential equations whose leading operator is modeled on a superposition of (-triangle(p))(s )and (-triangle(p))(t,) where 0 < s < t < 1 < p < infinity, weighted via two possibly degenerate coefficients a(& centerdot;,& centerdot;) >= 0 and b(& centerdot;,& centerdot;) >= 0, respectively. We prove local boundedness and Holder regularity of its weak solutions under natural assumptions on the coefficients a(& centerdot;,& centerdot;), b(& centerdot;,& centerdot;) and the powers s, t, p. Moreover, when a(& centerdot;,& centerdot;) equivalent to 1, we also prove a Harnack inequality for weak solutions.
We study optimization problems for partially hinged rectangular plates, modeling bridge roadways, in the presence of real and artificial obstacles. Real obstacles represent structural constraints to avoid, while artificial ones are introduced to enhance stability. For the former, aiming to prevent collisions, we set up a worst-case optimization problem in which we minimize the amplitude of oscillations with respect to the density distribution; for the latter, aiming to improve the torsional stability, we minimize, with respect to the obstacles, the maximum of a gap function quantifying the displacement between the long edges of the plate. For both problems, existence results are provided, along with a discussion about qualitative properties of optimal density distributions and obstacles.
We pursue a computational analysis of the biomedical problem on the identification of cancerous tumors at an early stage of development based on the Electrical Impedance Tomography (EIT) and optimal control of elliptic partial differential equations. Relying on the fact that the electrical conductivity of the cancerous tumor is significantly higher than that of healthy tissue, we consider an inverse EIT problem for identifying the conductivity map in the complete electrode model based on m current-to-voltage measurements on the boundary electrodes. A variational formulation as a PDE-constrained optimal control problem is introduced based on the novel idea of increasing the size of the input data by adding "voltage-to-current" measurements through various permutations of the single "current-to-voltage" measurement. The idea of permutation preserves the size of the unknown parameters at the expense of an increase in the number of PDE constraints. We apply a gradient projection method (GPM) based on the Frechet differentiability in Besov-Hilbert spaces. Numerical simulations of 2D and 3D model examples demonstrate the sharp increase in the resolution of the cancerous tumor by increasing the number of measurements from m to m2.
We propose a reduced-order modeling approach for nonlinear, parameter-dependent ordinary differential equations (ODE). Dimensionality reduction is achieved using nonlinear maps represented by autoencoders. The resulting low-dimensional ODE is then solved using standard integration in time schemes, and the high-dimensional solution is reconstructed from the low-dimensional one. We investigate the architecture of neural networks for constructing effective autoencoders that hold necessary properties to reconstruct the input manifold with exact representation capabilities. We study the convergence of the reduced-order model to the high-fidelity one. Numerical experiments show the robustness and accuracy of our approach in different scenarios, highlighting its effectiveness in highly complex and nonlinear settings without sacrificing accuracy. Moreover, we examine how the reduction influences the stability properties of the reconstructed high-dimensional solution.
Plasticity with softening and fracture mechanics lead to ill-posed mathematical problems due to the loss of monotonicity. Multiple co-existing solutions are possible when softening elements are coupled together, and solutions cannot be continued beyond the point of complete degradation of the set of admissible stresses. We present a state-dependent sweeping process which solves the evolution of elasto-plastic Lattice Spring Models with arbitrary placement of softening, hardening and perfectly plastic springs. Using numerical simulations of regular grid lattices with softening we demonstrate the emergence of non-symmetric shear bands with strain localization. At the same time, in toy examples it is easy to analytically derive multiple co-existing solutions. These solutions correspond to fixed points in the implicit catch-up algorithm and we observe a discontinuous bifurcation with the exchange of stability of those fixed points.
We study simultaneous homogenization and dimensional reduction of integral functionals for maps in manifold-valued Sobolev spaces. Due to the superlinear growth regime, we prove that the density of the Γ-limit is a tangential quasiconvex integrand represented by a cell formula.
We provide sharp boundary regularity estimates for solutions to elliptic equations driven by an integro-differential operator obtained as the sum of a Laplacian with a nonlocal operator generalizing a fractional Laplacian. Our approach makes use of weighted Hölder spaces as well as regularity estimates for the Laplacian in this context and a fixed-point argument. We show the optimality of the obtained estimates by means of a counterexample that we have striven to keep as explicit as possible.
We present three equivalent definitions of the fractional p-Laplacian (-Delta(n)(H))(s)(p), 0 < s < 1, p > 1, with normalizing constants, on hyperbolic spaces. The explicit values of the constants enable us to study the convergence of the fractional p-Laplacian to the p-Laplacian as s -> 1(-).
In this paper, we aim to establish a new class of weak Harnack inequalities for weak supersolutions to the nonhomogeneous nonlocal equations with general growth. Our approach mainly relies on the expansion of positivity in the spirit of De Giorgi classes, along with a refined energy estimate.
We introduce and analyse a variant of the two-dimensional XY-model energy which is suited to detect both topological defects and geometric defects in form of fractional vortices and domain walls, respectively. In contrast to previously introduced variants, the energies we consider here are defined without using an angular lifting of the S1-valued spin variables. Moreover, they combine in an explicit way the features of the XY-model energy on the one hand and weak-membrane energies on the other hand. This leads to simplified proofs of compactness and lower bound in the Gamma-convergence analysis.
This paper considers a class of hysteresis systems consisting of a linear part with an external input and feedback with a backlash nonlinearity. Assuming that the latter is specified by a strongly convex set, we establish estimates for the Lyapunov exponents which quantify the rate of convergence of the system state trajectories to a forced periodic regime when the input is a periodic function of time with a sufficiently large "amplitude". These results employ enhanced dissipation inequalities, arising from differential inclusions with strongly convex sets which were used previously for the Moreau sweeping process.
In this paper, we address a system of two ODEs including a hysteresis nonlinearity of generalized play type. Our system is subject to a composed perturbation, giving under particular choices of composants various types of common multivalued perturbations. We prove the existence of a solution to this system. The theoretical analysis is complemented by a discussion of a mechanical model illustrating potential applicability of our results and the physical meaning of the underlying assumptions.
In this paper, we propose a method for identifying the measure of the stochastic Preisach operator using machine learning techniques. The classical Preisach operator model is widely used to describe hysteresis phenomena in various fields such as physics, chemistry, economics, and biology. However, it does not account for uncontrolled fluctuations in the parameters of elementary hysteresis carriers - hysterons, which limits its applicability in real systems where these parameters can be stochastic variables. The proposed method is based on sequential reconstruction of the operator's measure on the plane of hysteron threshold parameters (alpha,beta). The identification is carried out through specially constructed input signals that trigger switching of only specific hysterons in given local regions of this plane. Multiple repetitions of such input actions and application of the law of large numbers allow for estimating the mathematical expectations of hysteron state changes. These estimates are used in the algorithm to minimize a loss functional, leading to the reconstruction of the operator's measure from experimental data. The results open new possibilities for modeling and analyzing complex systems with hysteretic properties, taking into account the stochastic nature of parameters.
In this paper we study the controllability problem for systems exhibiting hysteresis represented by play-type operators. To this end we first formalize and study in a functional setting the approximation of the Play operators by a finite weighted sum of delayed relay. Then we prove the controllability for the case with the Play operator, by the controllability result for the case with the weighted sum of delayed relay. Finally, we discuss potential applications of our approach to the sweeping process.
Microbially influenced corrosion (MIC) refers to any corrosion process caused or fostered by microbial activity, and represents a global concern, impacting infrastructure, economies, and the environment worldwide. MIC affects a wide range of materials and is particularly common in wastewater concrete pipes, where it is associated with the proliferation of biofilm colonies of sulfur-oxidizing bacteria (SOBs). SOBs oxidize hydrogen sulfide produced within wastewater effluents and generate corrosive sulfuric acid that triggers the degradation of concrete. We propose here a onedimensional, two-layer diffusion model with double free boundaries to investigate the proliferation of SOB biofilms and the related corrosion process in wastewater concrete pipes. The domain is composed of two free boundary regions: a monospecies SOB biofilm in contact with the sewer atmosphere, which grows towards the interior cavity of the pipe, sitting on a gypsum layer formed from corrosion, that penetrates the concrete pipe. Diffusion-reaction equations govern the transport and metabolic production or consumption of hydrogen sulfide, oxygen, and sulfuric acid within the biofilm layer. The biofilm free boundary tracks the growth of the microbial community, regulated by metabolic activity of SOBs and detachment phenomena. The corrosion process is incorporated in the model through a Stefan-type condition, which drives the advancement of the gypsum free boundary into the concrete pipe, governed by microbial production of sulfuric acid. Numerical simulations are carried out to investigate the model behavior, encompassing the development and progression of the biofilm as well as the corrosion advancement, with the aim of elucidating the influence of key factors such as hydrogen sulfide level in the sewer, calcium carbonate concentration in concrete, detachment phenomena, and acid diffusivity in the gypsum layer. Interestingly, the model suggests that, under specific conditions, biofilms may impose limitations on sulfuric acid diffusion and act as a partial protective barrier for the underlying concrete.
