Algebraic Structure and Poisson Integral Method of Snake-Like Robot Systems

The algebraic structure and Poisson's integral of snake-like robot systems are studied. The generalized momentum, Hamiltonian function, generalized Hamilton canonical equations, and their contravariant algebraic forms are obtained for snake-like robot systems. The Lie-admissible algebra structures of the snake-like robot systems are proved and partial Poisson integral methods are applied to the snake-like robot systems. The first integral methods of the snake-like robot systems are given. An example is given to illustrate the results.

EXACT EQUALITIES FOR APPROXIMATION OF FUNCTIONS FROM THE SOBOLEV CLASS BY THEIR GENERALIZED POISSON INTEGRALS

In most cases, solutions to problems of the motion of a system of interacting material points are reduced to either ordinary differential equations or partial differential equations. One of the solutions of this type of equations is the so-called generalized Poisson integrals, which in partial cases turn into the well-known Abel-Poisson integrals or biharmonic Poisson integrals. A number of results is known on the approximation of various classes of differentiable periodic and nonperiodic functions by the mentioned above integrals (the so-called Kolmogorov-Nikol’skii problem in the terminology of A.I. Stepanets). Nevertheless, there is a significant drawback practically in all of the solved Kolmogorov-Nikol’skii problems for both Abel-Poisson integrals and Poisson biharmonic integrals from the mathematical modeling (computational experiment) point of view. The core point here is that in most of the previously solved Kolmogorov-Nikol’skii problems for both Abel-Poisson integrals and Poisson biharmonic integrals only the leading and remainder terms of the approximation were obtained, that can significantly affect the accuracy of the computational experiment. In the present paper we obtain exact equalities for approximation of functions from the Sobolev classes by their generalized Poisson integrals. Consequently, the theorem proved in this paper is a generalization and refinement of previously known results characterizing the approximation properties of Abel-Poisson integrals and biharmonic Poisson integrals on the classes of differentiable periodic functions. A peculiarity of the solved in this work problem of approximation for the generalized Poisson integral on the classes of differentiable functions is that the result obtained is successfully written using the well-known Akhiezer-Krein-Favard constants. This fact substantially increases the accuracy of the mathematical modeling result (computational experiment) for a real process described using the generalized Poisson integral. These results can further significantly expand the scope of application of the Kolmogorov-Nikol’skii problems to mathematical modeling.

INTEGRAL REPRESENTATION OF SOLUTIONS OF HALF-SPACE HOMOGENEOUS DIRICHLET AND NEUMANN PROBLEMS FOR AN EQUATION OF FOKKER-PLANCK-KOLMOGOROV TYPE OF NORMAL MARKOV PROCESS

Solutions of a homogeneous model equation of the Fokker--Planck--Kolmogorov type of a normal Markov process are consider. They are defined in $\{(t,x_1,\dots,x_n)\in\mathbb{R}^{n+1}|0<t\le T, -\infty<x_j<\infty, j\in\{1,\dots,n-1\}, x_n>0\}$ and for $x_n=0$ satisfy the homogeneous Dirichlet or Neumann conditions and relate to special weighted Lebesgue $L_p$-spaces $L_p^{k(\cdot,a)}$. The representation of such solutions in the form of Poisson integrals is established. The kernels of these integrals are the homogeneous Green's functions of the considered problems, and their densities belong to specially constructed sets $\Phi_p^a$ of functions or generalized measures. The results obtained will be used to describe solutions of the problems from spaces $L_p^{k(\cdot,a)}$. Thus, the well-known Eidelman-Ivasyshen approach will be implemented for the considered problems. According to this approach, if the initial data are taken from the set $\Phi_p^a$, then there is only one solution to the problem from the space $L_p^{k(\cdot,a)}$. It is represented as a Poisson integral. Conversely, for any solution from the space $L_p^{k(\cdot,a)}$ there is only one element $\varphi \in\Phi_p^a$ such that this solution can be represented as a Poisson integral with density $\varphi$. In this case, it becomes clear in what sense the initial condition is satisfied.