Fully continuous vector fields for mobile robot navigation on sequences of discrete triangular regions

We study the solvability and removable singularities of the equation divF=µ , with measure data µ , in the class of continuous or Lp vector fields F , where 1=p=8 . In particular, we show that, for a signed measure µ , the equation divF=µ has a solution F?L8(Rn) if and only if |µ(U)|=CHn-1(?U) for any open set U with smooth boundary. For non-negative measures µ , we obtain explicit characterizations of the solvability of divF=µ in terms of potential energies of µ for p?8 , and in terms of densities of µ for continuous vector fields. These existence results allow us to characterize the removable singularities of the corresponding equation divF=µ with signed measures µ .

In this paper we present an n-dimensional canonical piecewise-linear electrical circuit. It contains 2n two-terminal elements: n linear dynamic elements (capacitors and inductors), n - 1 linear resistors and one nonlinear (piecewise-linear) resistor. This circuit can realize any prescribed eigenvalue pattern, except for a set of measure zero, associated with (i) any n-dimensional two-region continuous piecewise-linear vector fields and (ii) any n-dimensional three-region symmetric (with respect to the origin) piecewise-linear continuous vector fields. We also proved a theorem that specifies the conditions for a vector field, realized with our canonical circuit, to have two or three equilibrium points.

The well-known Cauchy integral decomposition in the plane is a representation of a continuous function on a curve as a sum of two functions analytic inside and outside the curve. These analytic components inherit smoothness of the original function almost independently of the smoothness of the curve itself. Let G be a domain in homeomorphic to a ball with C 1-smooth boundary S. Let F be a continuous vector field on S. When it is possible to represent it as a sumF = F + + F −, where vector fields F ± are harmonic in G and respectively, and smooth of prescribed order s up to the boundary? The following two conditions on F turn out to be necessary and sufficient: 1. The circulation of F along any closed path on S vanishes. 2. For any ball B with radius r centered on S there exists a polynomial harmonic vector field H of degree m = [s] such that on B∩S. This is a multidimensional analog of the Cauchy integral decomposition. The result does not presuppose any extra smoothness of S.

In this work a theorical framework to apply the Poincare compactification technique to locally Lipschitz continuous vector fields is developed. It is proved that these vectors fields are compactifiable in the n-dimensional sphere, though the compactified vector field can be identically null in the equator. Moreover, for a fixed projection to the hemisphere, all the compactifications of a vector field, which are not identically null on the equator are equivalent. Also, the conditions determining the invariance of the equator for the compactified vector field are obtained. Up to the knowledge of the authors, this is the first time that the Poincare compactification of locally Lipschitz continuous vector fields is studied. These results are illustrated applying them to some families of vector fields, like polynomial vector fields, vector fields defined as a sum of homogeneous functions and vector fields defined by piecewise linear functions.

Stabilizability of Bimodal Piecewise Linear Systems With Continuous Vector Field

When Brouwer continued his investigations into Hilbert 5, he discovered that his main topology source, Schoenflies Bericht, was far from correct. He set himself to straighten out the defective parts; the best known fall out of this research was his work on indecomposable continua, with the spectacular example of three domains with one common boundary. The chapter also contains the story of Brouwer’s research on fixed points on the sphere and his translation theorem (on fixed point free continuous maps of the plane onto itself). He simultaneously produced a number of papers on vector field on surfaces. The best known result was the hairy ball theorem: a continuous vector field on a sphere must be zero or infinite at at least one point.

A problem of great interest in the control of hybrid systems is the design of least restrictive controllers for reachability specifications. Controller design typically uses game theoretic methods to compute the region of the state space for which there exists a control such that for all disturbances, an unsafe set is not reached. In general, the computation of the controllers requires the steady state solution of a Hamilton-Jacobi partial differential equation which is very difficult to compute, if it exists. In this paper, we show that for special classes of hybrid systems where the continuous vec-tor fields are linear, the controller synthesis problem is semi-decidable: There exists a computational algo-rithm which, if it terminates in a finite number of steps, will exactly compute the least restrictive con-troller. This result is achieved by a very interesting interaction of results from mathematical logic and o p timal control. 1

Nontangential a.e. vanishing of the oblique derivative of a harmonic function with respect to a Hölder continuous vector field on a Lipschitz boundary is shown to imply that the harmonic function is constant.

The methods for navigation and control of a space robot during the approach to a geostationary information satellite under conditions of uncertainty and incompleteness of measurement are considered.

The Hairy Ball Theorem states that every continuous tangent vector field on an even-dimensional sphere must have a zero. We prove that the associated computational problem of computing an approximate zero is PPAD-complete. We also give a FIXP-hardness result for the general exact computation problem. In order to show that this problem lies in PPAD, we provide new results on multiple-source variants of End-of-Line, the canonical PPAD-complete problem. In particular, finding an approximate zero of a Hairy Ball vector field on an even-dimensional sphere reduces to a 2-source End-of-Line problem. If the domain is changed to be the torus of genus g >= 2 instead (where the Hairy Ball Theorem also holds), then the problem reduces to a 2(g-1)-source End-of-Line problem. These multiple-source End-of-Line results are of independent interest and provide new tools for showing membership in PPAD. In particular, we use them to provide the first full proof of PPAD-completeness for the Imbalance problem defined by Beame et al. in 1998.

In the present paper we are concerned with the existence of Γ-periodic solutions for the differential equation x(t) = f(t, .x( r)), t e R, where / is a continuous time dependent Γ-periodic tangent vector field defined on an ^-dimensional differentiable manifold M possibly with boundary. We prove that if the Euler characteristic of the average vector field w(p) = (l/T)fίff(t,p)dt is defined and nonzero and if all the possible orbits of the parametrized equation x(t) = λf(ί,x(t)), /eR and λ £ (0,1], lie in a compact set and do not hit the boundary of M, then the given equation admits a Γ-periodic solution. 0. Introduction. Let M be an w-dimensional differentiable manifold, possibly with boundary, and let /: RxM->Γ(M) be a time dependent Γ-periodic tangent vector field on M. In this paper we give a topological result concerning the existence of Γ-periodic solutions for the differential equation

We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two nonequivalent distances. The two distances under consideration are the Euclidean distance and, roughly speaking, the geodesic distance along integral curves of a (possibly multi-valued) flow of a continuous vector field. The Lipschitz constant for the geodesic distance of the extension can be estimated in terms of the Lipschitz constant for the geodesic distance of the original function. This Lipschitz extension lemma allows us to remove the high integrability assumption on the solution needed for the uniqueness within the DiPerna-Lions theory of continuity equations in the case of vector fields in the Sobolev space where p is larger than the space dimension, under the assumption that the so-called “forward-backward integral curves” associated to the vector field are trivial for almost every starting point. More precisely, for such vector fields we prove uniqueness and Lagrangianity for weak solutions of the continuity equation that are just locally integrable. Additionally, for such vector fields it is possible to prove almost everywhere uniqueness of (standard) integral curves, which also implies uniqueness of positive measure solutions to the continuity equation with absolutely continuous initial datum.

The problem of vector field approximation emerges in the wide range of fields such as motion control, computer vision and so on. The paper discusses an approximation method for reconstructing an entire continuous vector field from a sparse set of sample data by neural networks. In order to improve approximation accuracy and efficiency, we incorporate the inherent property of vector fields into the learning problem of neural networks and derive a new learning algorithm. It is shown through numerical experiments that the proposed method makes it possible to reconstruct vector fields accurately and efficiently.

Many algorithms for scientific visualization and image analysis are rooted in the world of continuous scalar, vector, and tensor fields, but are programmed in low-level languages and libraries that obscure their mathematical foundations. Diderot is a parallel domain-specific language that is designed to bridge this semantic gap by providing the programmer with a high-level, mathematical programming notation that allows direct expression of mathematical concepts in code. Furthermore, Diderot provides parallel performance that takes advantage of modern multicore processors and GPUs. The high-level notation allows a concise and natural expression of the algorithms and the parallelism allows efficient execution on real-world datasets.

A new radar echo tracking algorithm known as multi-scale tracking radar echoes by cross-correlation (MTREC) was developed in this study to analyze movements of radar echoes at different spatial scales. Movement of radar echoes, particularly associated with convective storms, exhibits different characteristics at various spatial scales as a result of complex interactions among meteorological systems leading to the formation of convective storms. For the null echo region, the usual correlation technique produces zero or a very small magnitude of motion vectors. To mitigate these constraints, MTREC uses the tracking radar echoes by correlation (TREC) technique with a large “box” to determine the systematic movement driven by steering wind, and MTREC applies the TREC technique with a small “box” to estimate small-scale internal motion vectors. Eventually, the MTREC vectors are obtained by synthesizing the systematic motion and the small-scale internal motion. Performance of the MTREC technique was compared with TREC technique using case studies: the Khanun typhoon on 11 September 2005 observed by Wenzhou radar and a squall-line system on 23 June 2011 detected by Beijing radar. The results demonstrate that more spatially smoothed and continuous vector fields can be generated by the MTREC technique, which leads to improvements in tracking the entire radar reflectivity pattern. The new multi-scale tracking scheme was applied to study its impact on the performance of quantitative precipitation nowcasting. The location and intensity of heavy precipitation at a 1-h lead time was more consistent with quantitative precipitation estimates using radar and rain gauges.