Решения гамильтоновой системы с двумерным управлением в окрестности особой экстремали второго порядка

2021 ◽  
Vol 76 (5(461)) ◽  
pp. 201-202
Author(s):  
Мария Игоревна Ронжина ◽  
Mariya Igorevna Ronzhina ◽  
Лариса Анатольевна Манита ◽  
Larisa Anatol'evna Manita ◽  
Лев Вячеславович Локуциевский ◽  
...  
Keyword(s):  
Singular Point ◽  
Hamiltonian System ◽  
Infinite Number ◽  
Finite Time ◽  
Second Order ◽  
Two Dimensional ◽  
Optimal Solutions ◽  
Countable Number ◽  
Bounded Control ◽  
Singular Extremal

We consider a Hamiltonian system that is affine in two-dimensional bounded control that takes values in an ellipse. In the neighborhood of a singular extremal of the second order, we find two families of optimal solutions: chattering trajectories that attain the singular point in a finite time with a countable number of control switchings, and logarithmic-like spirals that reach the singular point in a finite time and undergo an infinite number of rotations.

scholarly journals Optimal spiral-like solutions near a singular extremal in a two-input control problem

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Larisa Manita ◽  
Mariya Ronzhina
Keyword(s):  
Optimal Control ◽  
Singular Point ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimal Controls ◽  
Two Dimensional ◽  
Spherical Pendulum ◽  
Bounded Control ◽  
Input Control ◽  
Singular Extremal

<p style='text-indent:20px;'>We study an optimal control problem affine in two-dimensional bounded control, in which there is a singular point of the second order. In the neighborhood of the singular point we find optimal spiral-like solutions that attain the singular point in finite time, wherein the corresponding optimal controls perform an infinite number of rotations along the circle <inline-formula><tex-math id="M1">\begin{document}$ S^{1} $\end{document}</tex-math></inline-formula>. The problem is related to the control of an inverted spherical pendulum in the neighborhood of the upper unstable equilibrium.</p>

Averages for polygons formed by random lines in Euclidean and hyperbolic planes

1972 ◽  
Vol 9 (1) ◽  
pp. 140-157 ◽  
Author(s):  
L. A. Santaló ◽  
I. Yañez
Keyword(s):  
Infinite Number ◽  
Stochastic Geometry ◽  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Second Order ◽  
Countable Number ◽  
Geometrical Probability ◽  
Starting Point ◽  
The Mean ◽  
Hyperbolic Planes

We consider a countable number of independent random uniform lines in the hyperbolic plane (in the sense of the theory of geometrical probability) which divide the plane into an infinite number of convex polygonal regions. The main purpose of the paper is to compute the mean number of sides, the mean perimeter, the mean area and the second order moments of these quantities of such polygonal regions. For the Euclidean plane the problem has been considered by several authors, mainly Miles [4]–[9] who has taken it as the starting point of a series of papers which are the basis of the so-called stochastic geometry.

Averages for polygons formed by random lines in Euclidean and hyperbolic planes

1972 ◽  
Vol 9 (01) ◽  
pp. 140-157 ◽  
Author(s):  
L. A. Santaló ◽  
I. Yañez
Keyword(s):  
Infinite Number ◽  
Stochastic Geometry ◽  
Hyperbolic Plane ◽  
Euclidean Plane ◽  
Second Order ◽  
Countable Number ◽  
Geometrical Probability ◽  
Starting Point ◽  
The Mean ◽  
Hyperbolic Planes

We consider a countable number of independent random uniform lines in the hyperbolic plane (in the sense of the theory of geometrical probability) which divide the plane into an infinite number of convex polygonal regions. The main purpose of the paper is to compute the mean number of sides, the mean perimeter, the mean area and the second order moments of these quantities of such polygonal regions. For the Euclidean plane the problem has been considered by several authors, mainly Miles [4]–[9] who has taken it as the starting point of a series of papers which are the basis of the so-called stochastic geometry.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

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