Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction

<p style='text-indent:20px;'>We study the asymptotic stability of periodic solutions for sweeping processes defined by a polyhedron with translationally moving faces. Previous results are improved by obtaining a stronger <inline-formula><tex-math id="M1">\begin{document}$ W^{1,2} $\end{document}</tex-math></inline-formula> convergence. Then we present an application to a model of crawling locomotion. Our stronger convergence allows us to prove the stabilization of the system to a running-periodic (or derivo-periodic, or relative-periodic) solution and the well-posedness of an average asymptotic velocity depending only on the gait adopted by the crawler. Finally, we discuss some examples of finite-time versus asymptotic-only convergence.</p>

On Asymptotic Dynamical Regimes of Manakov N-soliton Trains in Adiabatic Approximation

We analyze the dynamical behavior of the N-soliton train in the adiabatic approximation of the Manakov model. The evolution of Manakov N-soliton trains is described by the complex Toda chain (CTC) which is a completely integrable dynamical model. Calculating the eigenvalues of its Lax matrix allows us to determine the asymptotic velocity of each soliton. So we describe sets of soliton parameters that ensure one of the two main types of asymptotic regimes: the bound state regime (BSR) and the free asymptotic regime (FAR). In particular we find explicit description of special symmetric configurations of N solitons that ensure BSR and FAR. We find excellent matches between the trajectories of the solitons predicted by CTC with the ones calculated numerically from the Manakov system for wide classes of soliton parameters. This confirms the validity of our model

Synthesizing spacecraft orbits with high inclinations using Venusian gravity assists

An adaptive, semi‑analytical, and geometrically clear method for synthesis of sequences of Venusian gravity‑assist maneuvers setting the desired inclination of a spacecraft orbit is proposed. The geometric constraint on the maximum possible inclination of a spacecraft orbit, which depends on the asymptotic spacecraft velocity relative to Venus, and the dynamic constraint (arising when a gravity‑assist maneuver is performed) on the angle of rotation of the vector of asymptotic velocity relative to Venus are considered simultaneously.

Emergent behaviors of continuous and discrete thermomechanical Cucker–Smale models on general digraphs

We present emergent dynamics of continuous and discrete thermomechanical Cucker–Smale (TCS) models equipped with temperature as an extra observable on general digraph. In previous literature, the emergent behaviors of the TCS models were mainly studied on a complete graph, or symmetric connected graphs. Under this symmetric setting, the total momentum is a conserved quantity. This determines the asymptotic velocity and temperature a priori using the initial data only. Moreover, this conservation law plays a crucial role in the flocking analysis based on the elementary [Formula: see text] energy estimates. In this paper, we consider a more general connection topology which is registered by a general digraph, and the weights between particles are given to be inversely proportional to the metric distance between them. Due to this possible symmetry breaking in communication, the total momentum is not a conserved quantity, and this lack of conservation law makes the asymptotic velocity and temperature depend on the whole history of solutions. To circumvent this lack of conservation laws, we instead employ some tools from matrix theory on the scrambling matrices and some detailed analysis on the state-transition matrices. We present two sufficient frameworks for the emergence of mono-cluster flockings on a digraph for the continuous and discrete models. Our sufficient frameworks are given in terms of system parameters and initial data.

Non-interlacing peakon solutions of the Geng–Xue equation

The aim of the present article is to derive explicit formulas for arbitrary non-overlapping pure peakon solutions of the Geng–Xue (GX) equation, a two-component generalization of Novikov’s cubically non-linear Camassa–Holm type equation. By performing limiting procedures on the previously known formulas for so-called interlacing peakon solutions, where the peakons in the two component occur alternatingly, we turn some of the peakons into zero-amplitude ‘ghostpeakons’, in such a way that the remaining ordinary peakons occur in any desired configuration. A novel feature compared to the interlacing case is that the Lax pairs for the GX equation do not provide all the constants of motion necessary for the integration of the system. We also study the large-time asymptotics of the non-interlacing solutions. As in the interlacing case, the peakon amplitudes grow or decay exponentially, and their logarithms display phase shifts similar to those for the positions. Moreover, within a group of adjacent peakons in one component, all peakons but one have the same asymptotic velocity. A curious phenomenon occurs when the number of such peakon groups is odd, namely that the sets of incoming and outgoing velocities are unequal.