Discrete Approximate Solution of Differential Inclusion with Normal Cone and Prox-Regular Set.

Our aim is to calculate the discrete approximate solution of di⁄erential inclusion with normal cone and prox-regular set, the question is how to calculate this solution? We use the discrete approximation property of a new variant of nonconvex sweeping processes involving normal cone and a nite element method. Knowing that The majority of mathematicians have proved only the existence and uniqueness of the solution for this type of inclusions, like: Mordukhovich, Thibault, Aubin, Messaoud, ...etc.

Explicit and Implicit Non-convex Sweeping Processes in the Space of Absolutely Continuous Functions

We show that sweeping processes with possibly non-convex prox-regular constraints generate a strongly continuous input-output mapping in the space of absolutely continuous functions. Under additional smoothness assumptions on the constraint we prove the local Lipschitz continuity of the input-output mapping. Using the Banach contraction principle, we subsequently prove that also the solution mapping associated with the state-dependent problem is locally Lipschitz continuous.

Rural Life in Late Socialism

Late socialist countries are transforming faster than ever. Across China, Laos and Vietnam, where market economies coexist with socialist political rhetoric and the Communist party state’s rule, sweeping processes of change open up new vistas of imaginaries of the future alongside uncertainty and anxiety. These countries are three of very few living examples that combine capitalist economics with party state politics. Consequently, societal transformations in these contexts are subject to pressures and agendas not found elsewhere, and yet they are no less subject to global forces than elsewhere. As all three countries maintain substantial rural populations, and because those rural areas are themselves places of change, how rural people across these changing contexts undertake future making is a timely and significant question. The contributions in the issue address this question by engaging with lived experiences and government agendas across Laos, China and Vietnam, showing a politics of development in which desire and hope are entangled with the contradictions and struggles of late socialism.

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>

Modeling of crowds in regions with moving obstacles

<p style='text-indent:20px;'>We present a model of crowd motion in regions with moving obstacles, which is based on the notion of measure sweeping process. The obstacle is modeled by a set-valued map, whose values are complements to <inline-formula><tex-math id="M1">\begin{document}$ r $\end{document}</tex-math></inline-formula>-prox-regular sets. The crowd motion obeys a nonlinear transport equation outside the obstacle and a normal cone condition (similar to that of the classical sweeping processes theory) on the boundary. We prove the well-posedness of the model, give an application to environment optimization problems, and provide some results of numerical computations.</p>