Nonemptiness of viability kernels for infinite-dimensional differential inclusions

We investigate infinite-dimensional differential inclusions. Sufficient conditions for nonemptiness of the viability kernels of such systems are obtained.     

Computing the viability kernel in large state dimension

T-viable states in a closed set K under a certain set-valued dynamic are states from which there exists at least one solution remaining in K until a given time horizon T. Minimizing the cost to constraints lets us determine whether a given state is T-viable or not, and this is implementable in large dimension for the state-space. Minimizing on the initial condition itself lets find viable states. Quincampoix's semi-permeability property helps find other states located close to the viability boundary, which is then gradually delineated. The algorithm is particularly suited to the identification of specific trajectories, such as the heavy viable solution, or to the computation of viability kernels associated with delayed dynamics. The volume of the viability kernel and its confidence interval can be estimated by randomly drawing states and checking their viability status. Examples are given.     

Approximation of kernel matrices by circulant matrices and its application in kernel selection methods

This paper focuses on developing fast numerical algorithms for selection of a kernel optimal for a given training data set. The optimal kernel is obtained by minimizing a cost functional over a prescribed set of kernels. The cost functional is defined in terms of a positive semi-definite matrix determined completely by a given kernel and the given sampled input data. Fast computational algorithms are developed by approximating the positive semi-definite matrix by a related circulant matrix so that the fast Fourier transform can apply to achieve a linear or quasi-linear computational complexity for finding the optimal kernel. We establish convergence of the approximation method. Numerical examples are presented to demonstrate the approximation accuracy and computational efficiency of the proposed methods.     

控制系统生存核的逼近算法

生存核的计算是控制理论中的一个重要研究方向．给出了一种计算一般离散控制系统生存核的新算法．基于机器学习的方法,给出了逼近生存核的算法．并在一定条件下,证明了此算法的收敛性．此算法在一定程度上避免了计算量随控制空间的维数增长而指数增长的问题．最后,给出具体的实际例子来说明算法的有效性．     

Robustness of viability controllers under small perturbations

In this note, a perturbed control problem with state constraints depending on a parameteru is considered. Assuming that, for a certain value ofu, there exists a viability controller, we explicitly estimate the range of variations ofu for which the same controller gives viable solutions.This work was supported by the Academy of Sciences of the Czech Republic and by EC, Cooperation in Science and Technology with Central and Eastern European Countries.     

φ-单调型集值变分包含解的Ishikawa迭代逼近

在实自反的Banach空间中引入和研究了一类新的非线性φ-单调型集值变分包含,并证明了此类变分包含解的唯一性及其具误差项的Ishikawa迭代程序的收敛性.本文结果在多方面改进和推广了Huang和Fang等人的一些相关结果.     

Solving integro-differential equations with Cauchy kernel

A simple method for solving the Fredholm singular integro-differential equations with Cauchy kernel is proposed based on a new reproducing kernel space. Using a transformation and modifying the traditional reproducing kernel method, the singular term is removed and the analytical representation of the exact solution is obtained in the form of series in the new reproducing kernel space. The advantage of the approach lies in the fact that, on the one hand, by improving the definition of traditional inner product, the representation of new reproducing kernel function becomes simple and requirement for image space of operator is weakened comparing with traditional reproducing kernel method; on the other hand, the approximate solution and its derivatives converge uniformly to the exact solution and its derivatives. Some examples are displayed to demonstrate the validity and applicability of the proposed method.     

Systems with distributions and viability theorem

The approach to the consideration of the ordinary differential equations with distributions in the classical space D^′ of distributions with continuous test functions has certain insufficiencies: the notations are incorrect from the point of view of distribution theory, the right-hand side has to satisfy the restrictive conditions of equality type. In the present paper we consider an initial value problem for the ordinary differential equation with distributions in the space of distributions with dynamic test functions T^′, where the continuous operation of multiplication of distributions by discontinuous functions is defined [V. Derr, D. Kinzebulatov, Distributions with dynamic test functions and multiplication by discontinuous functions, preprint, arXiv: math.CA/0603351, 2006], and show that this approach does not have the aforementioned insufficiencies. We provide the sufficient conditions for viability of solutions of the ordinary differential equations with distributions (a generalization of the Nagumo Theorem), and show that the consideration of the distributional (impulse) controls in the problem of avoidance of encounters with the set (the maximal viability time problem) allows us to provide for the existence of solution, which may not exist for the ordinary controls.     

Discrete Approximation of Impulsive Differential Inclusions

The article deals with the approximation of the solution set and the reachable sets of an impulsive differential inclusion with variable times of impulses. It is strongly connected to [11 T. Donchev ( 2008 ). Approximation of the solution set of impulsive systems . In Lecture Notes in Comput. Sci . ( et al. . eds.). Vol. 4818 , Springer-Verlag , Berlin , pp. 309 – 316 . [Google Scholar]] and is its continuation. We achieve order of convergence 1 for the Euler approximation under Lipschitz assumptions on the set-valued right-hand side and on the functions describing the jump surfaces and jumps themselves. Another criterion prevents the beating phenomena and generalizes available conditions. Several test examples illustrate the conditions and the practical evaluation of the jump conditions.     

Approximation theorems for set-valued stochastic integrals

The article is devoted to new properties of Aumann, Lebesgue, and Itô set-valued stochastic integrals considered in papers [1 Kisielewicz, M. (2014). Properties of generalized set-valued stochastic integrals. Discuss. Math. (DICO) 34:131–147. [Google Scholar],2 Kisielewicz, M., Michta, M. (2017). Integrably bounded set-valued stochastic integrals. J. Math. Anal. Appl. 449:1893–1910.[Crossref], [Web of Science ®] , [Google Scholar]]. In particular, it contains some approximation theorems for Aumann and Itô set-valued stochastic integrals. Hence, in particular, it follows that Aumann and Lebesgue set-valued stochastic integrals cover a.s., both for measurable and IF-nonanticipative integrably bounded set-valued stochastic processes.     

Approximation of the reproducing kernel

One discusses the asymptotic behavior of the reproducing kernel of the space.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 97, pp. 195–198, 1980.     

Shape evolutions under state constraints: A viability theorem

The aim of this paper is to adapt the Viability Theorem from differential inclusions (governing the evolution of vectors in a finite-dimensional space) to so-called morphological inclusions (governing the evolution of nonempty compact subsets of the Euclidean space).In this morphological framework, the evolution of compact subsets of is described by means of flows along differential inclusions with bounded and Lipschitz continuous right-hand side. This approach is a generalization of using flows along bounded Lipschitz vector fields introduced in the so-called velocity method alias speed method in shape analysis.Now for each compact subset, more than just one differential inclusion is admitted for prescribing the future evolution (up to first order)—correspondingly to the step from ordinary differential equations to differential inclusions for vectors in the Euclidean space.We specify sufficient conditions on the given data such that for every initial compact set, at least one of these compact-valued evolutions satisfies fixed state constraints in addition. The proofs follow an approximative track similar to the standard approach for differential inclusions in , but they use tools about weak compactness and weak convergence of Banach-valued functions. Finally the viability condition is applied to constraints of nonempty intersection and inclusion, respectively, in regard to a fixed closed set .     

Reproducing kernel method of solving singular integral equation with cosecant kernel

In the paper, a reproducing kernel method of solving singular integral equations (SIE) with cosecant kernel is proposed. For solving SIE, difficulties lie in its singular term. In order to remove singular term of SIE, an equivalent transformation is made. Compared with known investigations, its advantages are that the representation of exact solution is obtained in a reproducing kernel Hilbert space and accuracy in numerical computation is higher. On the other hand, the representation of reproducing kernel becomes simple by improving the definition of traditional inner product and requirements for image space of operators are weakened comparing with traditional reproducing kernel method. The final numerical experiments illustrate the method is efficient.     

Simplified reproducing kernel method for impulsive delay differential equations

In this paper, a new algorithm is presented to solve the impulsive delay initial value problems. This is the first time to propose the simplified reproducing kernel method (SRKM for short) to solve the impulsive delay differential equations. Then the uniform convergence of the numerical solution is proved, and the time consuming Schmidt orthogonalization process is avoided. The proposed method is proved to be stable and is not less than second order convergence. The algorithm is employed successfully in some numerical examples.     

A Multivalued Approach to the Planar Isosceles Three Body Problem

The isosceles three body problem consists of three point masses located on the vertices of an isosceles triangle on the plane. The two masses on the asymmetric edge are equal. This problem has been extensively studied but not as a perturbation of the Kepler problem. In this case we arrive at a differential inclusion as a natural formulation when we regularize the problem. We also derive an extension of the vectorfield that allows us to consider orbits across singular sets.      

Stochastic viability of convex sets

We investigate necessary and sufficient conditions for viability of a closed convex set K under weak solutions of a stochastic differential equation. These conditions are expressed in terms of the distance function to K. When in addition the boundary of K is smooth, then our necessary and sufficient conditions reduce to two relations that have to be verified just on the boundary of K.     

Subexponential behaviour of the Dirichlet heat kernel

We obtain a formula for the asymptotic behaviour of the Dirichlet heat kernel for large time in terms of the survival probability of a Brownian motion, under the assumption that the latter decays subexponentially for large time. We also obtain a lower bound for the Dirichlet heat kernel for arbitrary open and connected sets in Euclidean space.     

Calculation and estimation of the Poisson kernel

We provide a simple method for obtaining boundary asymptotics of the Poisson kernel on a domain in RN.