Discrete Approximations of a Controlled Sweeping Process

The paper is devoted to the study of a new class of optimal control problems governed by the classical Moreau sweeping process with the new feature that the polyhe- dral moving set is not fixed while controlled by time-dependent functions. The dynamics of such problems is described by dissipative non-Lipschitzian differential inclusions with state constraints of equality and inequality types. It makes challenging and difficult their anal- ysis and optimization. In this paper we establish some existence results for the sweeping process under consideration and develop the method of discrete approximations that allows us to strongly approximate, in the W^{1,2} topology, optimal solutions of the continuous-type sweeping process by their discrete counterparts

Accelerating an adiabatic process by nonlinear sweeping

We investigate the acceleration of an adiabatic process with the same survival probability of the ground state by sweeping a parameter nonlinearly, fast in the wide gap region and slow in the narrow gap region, as contrast to the usual linear sweeping. We find the expected acceleration in the Laudau-Zener tunneling model and in the adiabatic quantum computing model for factorizing the number N=21.Comment: 4 pages, 3 figure

Lattice Model of Sweeping Interface for Drying Process in Water-Granule Mixture

Based on the invasion percolation model, a lattice model for the sweeping interface dynamics is constructed to describe the pattern forming process by a sweeping interface upon drying the water-granule mixture. The model is shown to produce labyrinthine patterns similar to those found in the experiment[Yamazaki and Mizuguchi, J. Phys. Soc. Jpn. \textbf{69} (2000) 2387]. Upon changing the initial granular density, resulting patterns undergo the percolation transition, but estimated critical exponents are different from those of the conventional percolation. Loopless structure of clusters in the patterns produced by the sweeping dynamics seems to influence the nature of the transition.Comment: 6 pages, 7 figure

Sweeping process by prox-regular sets in Riemannian Hilbert manifolds

In this paper, we deal with sweeping processes on (possibly infinite-dimensional) Riemannian Hilbert manifolds. We extend the useful notions (proximal normal cone, prox-regularity) already defined in the setting of a Hilbert space to the framework of such manifolds. Especially we introduce the concept of local prox-regularity of a closed subset in accordance with the geometrical features of the ambient manifold and we check that this regularity implies a property of hypomonotonicity for the proximal normal cone. Moreover we show that the metric projection onto a locally prox-regular set is single-valued in its neighborhood. Then under some assumptions, we prove the well-posedness of perturbed sweeping processes by locally prox-regular sets.Comment: 27 page