scholarly journals The Planar Inverse Problem for Autonomous Systems

1983 ◽  
Vol 74 ◽  
pp. 353-367 ◽  
Author(s):  
Basilis C. Xanthopoulos ◽  
George Bozis
Keyword(s):  
Inverse Problem ◽  
Dynamical Systems ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
Parametric Family ◽  
General Version ◽  
Conic Sections ◽  
Family Of Curves ◽  
Necessary And Sufficient ◽  
Autonomous Dynamical Systems

AbstractWe study the general version of the inverse problem for planar trajectories and for autonomous dynamical systems possessing three integrals, i.e., for a given three-parametric family of curves f(x,y,a,b)=c we find the potential V(x,y) for which these curves are orbits of a unit mass. All possible cases, depending on the preassigned function f, are classified and in each case the necessary and sufficient conditions for the.existence of a solution are established. Among the examples is the case of the Keplerian conic sections which is studied in detail.

scholarly journals An inverse problem for the non-self-adjoint matrix Sturm-Liouville operator

2018 ◽  
Vol 50 (1) ◽  
pp. 71-102 ◽  
Author(s):  
Natalia Pavlovna Bondarenko
Keyword(s):  
Inverse Problem ◽  
Liouville Operator ◽  
Finite Interval ◽  
Spectral Characteristics ◽  
Sufficient Conditions ◽  
Adjoint Matrix ◽  
Method Of Spectral Mappings ◽  
Sturm Liouville ◽  
Necessary And Sufficient

The inverse problem of spectral analysis for the non-self-adjoint matrix Sturm-Liouville operator on a finite interval is investigated. We study properties of the spectral characteristics for the considered operator, and provide necessary and sufficient conditions for the solvability of the inverse problem. Our approach is based on the constructive solution of the inverse problem by the method of spectral mappings. The characterization of the spectral data in the self-adjoint case is given as a corollary of the main result.

Reconstruction of a convolution operator from the right-hand side on the semiaxis

2015 ◽  
Vol 23 (5) ◽  
Author(s):  
Anatoly F. Voronin
Keyword(s):  
Inverse Problem ◽  
Integral Operator ◽  
Sufficient Conditions ◽  
Stability Theorem ◽  
Explicit Formulas ◽  
Right Hand ◽  
The Right ◽  
Necessary And Sufficient ◽  
Operator Kernel ◽  
Problem Solvability

AbstractIn this paper, a Volterra integral equation of the first kind in convolutions on the semiaxis when the integral operator kernel and the right-hand side of the equation have a bounded support is considered. An inverse problem of reconstructing the solution to the equation and the integral operator kernel from values of the right-hand side is formulated. Necessary and sufficient conditions for the inverse problem solvability are obtained. A uniqueness and stability theorem is proved. Explicit formulas for reconstruction of the solution and kernel are obtained.

On Solutions of Inverse Problem for Hermitian Generalized Hamiltonian Matrices

2013 ◽  
Vol 860-863 ◽  
pp. 2727-2731
Author(s):  
Kai Fu Liang ◽  
Ming Jun Li ◽  
Ze Lin Zhu
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Design Automation ◽  
Sufficient Conditions ◽  
Matrix Equations ◽  
Complex Matrix ◽  
Linear Quadratic ◽  
Real Representation ◽  
The Matrix ◽  
Necessary And Sufficient

Hamiltonian matrices have many applications to design automation and autocontrol, in particular in the linear-quadratic autocontrol problem. This paper studies the inverse problems of generalized Hamiltonian matrices for matrix equations. By real representation of complex matrix, we give the necessary and sufficient conditions for the existence of a Hermitian generalized Hamiltonian solutions to the matrix equations, and then derive the representation of the general solutions.

scholarly journals Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft

1996 ◽  
Vol 2 (4) ◽  
pp. 277-299 ◽  
Author(s):  
Xinzhi Liu ◽  
Allan R. Willms
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Dynamical Systems ◽  
Novel Approach ◽  
Necessary And Sufficient

Necessary and sufficient conditions for impulsive controllability of linear dynamical systems are obtained, which provide a novel approach to problems that are basically defined by continuous dynamical systems, but on which only discrete-time actions are exercised. As an application, impulsive maneuvering of a spacecraft is discussed.

Inverse problem about two-spectra for finite Jacobi matrices with zero diagonal

2016 ◽  
Vol 24 (6) ◽  
Author(s):  
Adil Huseynov
Keyword(s):  
Inverse Problem ◽  
Finite Order ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Jacobi Matrices ◽  
The Matrix ◽  
Explicit Procedure ◽  
Necessary And Sufficient

AbstractThe necessary and sufficient conditions for solvability of the inverse problem about two-spectra for finite order real Jacobi matrices with zero-diagonal elements are established. An explicit procedure of reconstruction of the matrix from the two-spectra is given.

Switchability in Discontinuous, Discrete Dynamical Systems

2013 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Dynamical Systems ◽  
Sufficient Conditions ◽  
Discrete Dynamical Systems ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Tin this paper, a theory for switchability and singularity of discontinuous, discrete dynamical systems. G-functions for the discrete dynamical systems are introduced through the boundary, and the necessary and sufficient conditions for the switchability of discrete mappings are presented.

Characterization of isochronous foci for planar analytic differential systems

2005 ◽  
Vol 135 (5) ◽  
pp. 985-998 ◽  
Author(s):  
Jaume Giné ◽  
Maite Grau
Keyword(s):  
Critical Point ◽  
Analytic Functions ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Differential Systems ◽  
Systems Of Differential Equations ◽  
Image Position ◽  
Necessary And Sufficient

We consider the two-dimensional autonomous systems of differential equations of the form where P(x,y) and Q(x,y) are analytic functions of order greater than or equal to 2. These systems have a focus at the origin if λ ≠ 0, and have either a centre or a weak focus if λ = 0. In this work we study the necessary and sufficient conditions for the existence of an isochronous critical point at the origin. Our result is, to the best of our knowledge, original when applied to weak foci and gives known results when applied to strong foci or to centres.

Metrizability problem for spray on Lie algebroids

2016 ◽  
Vol 13 (09) ◽  
pp. 1650115
Author(s):  
Zahra Pirbodaghi ◽  
Morteza Mir Mohammad Rezaii
Keyword(s):  
Inverse Problem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Lie Algebroids ◽  
Jacobi Endomorphism ◽  
Necessary And Sufficient

In this paper, we study inverse problem for sprays on Lie algebroids. We obtain necessary and sufficient conditions, based on semi-basic forms, for a spray to be Lagrangian. Then we discuss the Finsler metrizability of a spray and obtain some equations on the Jacobi endomorphism.

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