22,345 research outputs found
Continuous Uniform Finite Time Stabilization of Planar Controllable Systems
Continuous homogeneous controllers are utilized in a full state feedback setting for the uniform finite time stabilization of a perturbed double integrator in the presence of uniformly decaying piecewise continuous disturbances. Semiglobal strong Lyapunov functions are identified to establish uniform asymptotic stability of the closed-loop planar system. Uniform finite time stability is then proved by extending the homogeneity principle of discontinuous systems to the continuous case with uniformly decaying piecewise continuous nonhomogeneous disturbances. A finite upper bound on the settling time is also computed. The results extend the existing literature on homogeneity and finite time stability by both presenting uniform finite time stabilization and dealing with a broader class of nonhomogeneous disturbances for planar controllable systems while also proposing a new class of homogeneous continuous controllers
Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps
We study the positive recurrence of multi-dimensional birth-and-death
processes describing the evolution of a large class of stochastic systems, a
typical example being the randomly varying number of flow-level transfers in a
telecommunication wire-line or wireless network.
We first provide a generic method to construct a Lyapunov function when the
drift can be extended to a smooth function on , using an
associated deterministic dynamical system. This approach gives an elementary
proof of ergodicity without needing to establish the convergence of the scaled
version of the process towards a fluid limit and then proving that the
stability of the fluid limit implies the stability of the process. We also
provide a counterpart result proving instability conditions.
We then show how discontinuous drifts change the nature of the stability
conditions and we provide generic sufficient stability conditions having a
simple geometric interpretation. These conditions turn out to be necessary
(outside a negligible set of the parameter space) for piece-wise constant
drifts in dimension 2.Comment: 18 pages, 4 figure
Generalized Voronoi Tessellation as a Model of Two-dimensional Cell Tissue Dynamics
Voronoi tessellations have been used to model the geometric arrangement of
cells in morphogenetic or cancerous tissues, however so far only with flat
hypersurfaces as cell-cell contact borders. In order to reproduce the
experimentally observed piecewise spherical boundary shapes, we develop a
consistent theoretical framework of multiplicatively weighted distance
functions, defining generalized finite Voronoi neighborhoods around cell bodies
of varying radius, which serve as heterogeneous generators of the resulting
model tissue. The interactions between cells are represented by adhesive and
repelling force densities on the cell contact borders. In addition, protrusive
locomotion forces are implemented along the cell boundaries at the tissue
margin, and stochastic perturbations allow for non-deterministic motility
effects. Simulations of the emerging system of stochastic differential
equations for position and velocity of cell centers show the feasibility of
this Voronoi method generating realistic cell shapes. In the limiting case of a
single cell pair in brief contact, the dynamical nonlinear Ornstein-Uhlenbeck
process is analytically investigated. In general, topologically distinct tissue
conformations are observed, exhibiting stability on different time scales, and
tissue coherence is quantified by suitable characteristics. Finally, an
argument is derived pointing to a tradeoff in natural tissues between cell size
heterogeneity and the extension of cellular lamellae.Comment: v1: 34 pages, 19 figures v2: reformatted 43 pages, 21 figures, 1
table; minor clarifications, extended supplementary materia
Inversion and Symmetries of the Star Transform
The star transform is a generalized Radon transform mapping a function of two
variables to its integrals along "star-shaped" trajectories, which consist of a
finite number of rays emanating from a common vertex. Such operators appear in
mathematical models of various imaging modalities based on scattering of
elementary particles. The paper presents a comprehensive study of the inversion
of the star transform. We describe the necessary and sufficient conditions for
invertibility of the star transform, introduce a new inversion formula and
discuss its stability properties. As an unexpected bonus of our approach, we
prove a conjecture from algebraic geometry about the zero sets of elementary
symmetric polynomials
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