scholarly journals Differential Galois theory of infinite dimension

1996 ◽  
Vol 144 ◽  
pp. 59-135 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
19Th Century ◽  
Galois Theory ◽  
Infinite Dimension ◽  
Differential Galois Theory ◽  
Dimensional Theory ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
The 19Th Century

This paper is the second part of our work on differential Galois theory as we promised in [U3]. Differential Galois theory has a long history since Lie tried to apply the idea of Abel and Galois to differential equations in the 19th century (cf. [U3], Introduction). When we consider Galois theory of differential equation, we have to separate the finite dimensional theory from the infinite dimensional theory. As Kolchin theory shows, the first is constructed on a rigorous foundation. The latter, however, seems inachieved despite of several important contributions of Drach, Vessiot,…. We propose in this paper a differential Galois theory of infinite dimension in a rigorous and transparent framework. We explain the idea of the classical authors by one of the simplest examples and point out the problems.

scholarly journals Second proof of the irreducibility of the first differential equation of painlevé

1990 ◽  
Vol 117 ◽  
pp. 125-171 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
General Theory ◽  
Galois Theory ◽  
Rigorous Proof ◽  
Mathematical Foundation ◽  
Differential Galois Theory ◽  
Infinite Dimensional ◽  
Finite Dimensional

In our paper [U2], we proved the irreducibility of the first differential equation y″ = 6y2 + x of Painlevé. In that paper we explained the origin of the problem and the importance of giving a rigorous proof. We can say that our method in [U2] is algebraic and finite dimensional in contrast to a prediction of Painlevé who expected a proof depending on the infinite dimensional differential Galois theory. Even nowadays the latter remains to be established. It seems that Painlevé needed an armament with the general theory (the infinite dimensional differential Galois theory) in the controversy with R. Liouville on the mathematical foundation of the proof of the irreducibility of the first differential equation (1902-03).

scholarly journals Birational automorphism groups and differential equations

1990 ◽  
Vol 119 ◽  
pp. 1-80 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Critical Point ◽  
Galois Theory ◽  
Automorphism Groups ◽  
Rigorous Proof ◽  
Differential Galois Theory ◽  
General Attitude ◽  
Transcendental Functions ◽  
Birational Automorphism

Painlevé studied the differential equations y″ = R(y′ y, x) without moving critical point, where R is a rational function of y′ y, x. Most of them are integrated by the so far known functions. There are 6 equations called Painlevé’s equations which seem to be irreducible or seem to define new transcendental functions. The simplest one among them is y″ = 6y2 + x. Painlevé declared on Comptes Rendus in 1902-03 that y″ = 6y2 + x is irreducible. It seems that R. Liouville pointed out an error in his argument. In fact there are discussions on this subject between Painlevé and Liouville on Comptes Rendus in 1902-03. In 1915 J. Drach published a new proof of the irreducibility of the differential equation y″ = 6y2 + x. The both proofs depend on the differential Galois theory developed by Drach. But the differential Galois theory of Drach contains errors and gaps and it is not easy to understand their proofs. One of our contemporaries writes in his book: the differential equation y″ = 6y2 + x seems to be irreducible dans un sens que on ne peut pas songer à préciser. This opinion illustrates well the general attitude of the nowadays mathematicians toward the irreducibility of the differential equation y″ = 6y2 + x. Therefore the irreducibility of the differential equation y″ = 6y2 + x remains to be proved. We consider that to give a rigorous proof of the irreducibility of the differential equation y″ = 6y2 + x is one of the most important problem in the theory of differential equations.

scholarly journals Galois theory of aigebraic and differential equations

1996 ◽  
Vol 144 ◽  
pp. 1-58 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Galois Theory ◽  
Linear Partial Differential Equation ◽  
Linear Algebraic Group ◽  
Partial Differential ◽  
Linear Ordinary Differential Equations ◽  
Infinite Dimensional ◽  
Linear Partial Differential Equations

This paper will be the first part of our works on differential Galois theory which we plan to write. Our goal is to establish a Galois Theory of ordinary differential equations. The theory is infinite dimensional by nature and has a long history. The pioneer of this field is S. Lie who tried to apply the idea of Abel and Galois to differential equations. Picard [P] realized Galois Theory of linear ordinary differential equations, which is called nowadays Picard-Vessiot Theory. Picard-Vessiot Theory is finite dimensional and the Galois group is a linear algebraic group. The first attempt of Galois theory of a general ordinary differential equations which is infinite dimensional, is done by the thesis of Drach [D]. He replaced an ordinary differential equation by a linear partial differential equation satisfied by the first integrals and looked for a Galois Theory of linear partial differential equations. It is widely admitted that the work of Drach is full of imcomplete definitions and gaps in proofs. In fact in a few months after Drach had got his degree, Vessiot was aware of the defects of Drach’s thesis. Vessiot took the matter serious and devoted all his life to make the Drach theory complete. Vessiot got the grand prix of the academy of Paris in Mathematics in 1903 by a series of articles.

Basic Predictor Feedback for Single-Input Systems

2020 ◽  
pp. 19-34
Author(s):  
Yang Zhu ◽  
Miroslav Krstic
Keyword(s):  
Differential Equation ◽  
Target System ◽  
Alternative View ◽  
Cascade System ◽  
Actuator Dynamics ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Backstepping Approach ◽  
Plant Dynamics ◽  
Single Input

This chapter discusses the basic idea of a partial differential equation (PDE) backstepping approach for single-input LTI ordinary differential equation (ODE) systems with discrete input delay. The key point of the backstepping approach lies in it providing a systematic construction of an infinite-dimensional transformation of the actuator state, which yields a cascade system of transformed stable actuator dynamics and stabilized plant dynamics. The cascade system consisting of such infinite-dimensional stable actuator dynamics and finite-dimensional stabilized plant dynamics is referred to as the closed-loop “target system.” The chapter first presents an alternative view of the backstepping transformation based purely on standard ODE delay notation. Then the backstepping transformation is described in PDE and rescaled unity-interval transport PDE notation.

Finding abstract Lie symmetry algebras of differential equations without integrating determining equations

1991 ◽  
Vol 2 (4) ◽  
pp. 319-340 ◽  
Author(s):  
Gregory J. Reid
Keyword(s):  
Differential Equations ◽  
Standard Form ◽  
Structure Constant ◽  
Variable Coefficients ◽  
Lie Symmetry ◽  
Symbolic Language ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Structure Constants ◽  
Symmetry Algebras

There are symbolic programs based on heuristics that sometimes, but not always, explicitly integrate the determining equations for the infinitesimal Lie symmetries admitted by systems of differential equations. We present a heuristic-free algorithm ‘Structure constant’, which can always determine whether the Lie symmetry group of a given system of PDEs is finite- or infinite-dimensional. If the group is finite-dimensional then ‘Structure constant’ can determine the dimension and structure constants of its associated Lie algebra without the heuristics of integration involved in other methods. If the group is infinite-dimensional, then ‘Structure constant’ computes the number of arbitrary functions which determine the infinite-dimensional component of its Lie symmetry algebra and also calculates the dimension and structure of its associated finite-dimensional subalgebra. ‘Structure constant’ employs the algorithms ‘Standard form’ and ‘Taylor’, described elsewhere. ‘Standard form’ is a heuristic-free algorithm which brings any system of determining equations to a standard form by including all integrability conditions in the system. ‘Taylor’ uses the standard form of a system of differential equations to calculate its Taylor series solution. These algorithms have been implemented in the symbolic language MAPLE. ‘Structure constant’ can also automatically determine the dimension and structure constants of the Lie symmetry algebras of entire classes of differential equations dependent on variable coefficients. In particular, we obtain new group classification results for some physically interesting classes of nonlinear telegraph equations depending on two variable coefficients, one representing a nonlinear wave speed and the other representing a nonlinear dispersion.

Nonintegrability of an Infinite-Degree-of-Freedom Model for Unforced and Undamped, Straight Beams

2003 ◽  
Vol 70 (5) ◽  
pp. 732-738
Author(s):  
K. Yagasaki
Keyword(s):  
Differential Equation ◽  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Galois Theory ◽  
Degree Of Freedom ◽  
Differential Galois Theory ◽  
Finite Degree ◽  
Chaotic Motions ◽  
Infinite Degree ◽  
Simulation Results

We study a mathematical model for unforced and undamped, initially straight beams. This system is governed by an integro-partial differential equation, and its energy is conserved: It is an infinite-degree-of-freedom Hamiltonian system. We can derive “exact” finite-degree-of-freedom mode truncations for it. Using the differential Galois theory for Hamiltonian systems, we prove that any two or more modal truncations for the model are nonintegrable in the following sense: The Hamiltonian systems do not have the same number of “meromorphic” first complex integrals which are independent and in involution, as the number of their degrees of freedom, when they are regarded as Hamiltonian systems with complex time and coordinates. This also means the nonintegrability of the infinite-degree-of-freedom model for the beams. We present numerical simulation results and observe that chaotic motions occur as in typical nonintegrable Hamiltonian systems.

scholarly journals Optimization of functionals under uncertainties for Ito-Skorokhod stochastic differential equations in Hilbert spaces

2018 ◽  
pp. 65-70
Author(s):  
A. Nikitin ◽  
O. Baliasnikova
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Hilbert Spaces ◽  
Liquid Fuel ◽  
Degrees Of Freedom ◽  
Applied Mathematics ◽  
Optimal Controls ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Carry Over

In the article for the stochastic differential equations of Ito-Skorokhod, problems of optimization of functionals under conditions of uncertainty in Hilbert spaces are investigated. Purpose of the article is to investigate some properties of stochastic differential equations in Hilbert spaces. These objects arise in diverse areas of applied mathematics as models for various natural phenomena, in particular, the evolution of complex systems with infinitely many degrees of freedom. For instance, one may think of the liquid fuel motion in the tank of a spacecraft. Spacecraft constructors should take into account this motion, for it influences heavily the path of a spacecraft. Also, optimization of the motion is an issue of principal importance. It is not trivial to carry over the results concerning stochastic differential equations in finite-dimensional spaces to the infinite dimensional case. We give some statements, in which the existence, uniqueness is proved and the explicit form μ-optimal controls for such equations is constructed, in particular, μ-optimal control is found as a linear inverse relationship.

scholarly journals Adapting the range of validity for the Carleman linearization

2016 ◽  
Vol 14 ◽  
pp. 51-54 ◽  
Author(s):  
Harry Weber ◽  
Wolfgang Mathis
Keyword(s):  
Differential Equation ◽  
Linear System ◽  
Nonlinear Differential Equation ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Liouville Problem ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Linear Differential

Abstract. In this contribution, the limitations of the Carleman linearization approach are presented and discussed. The Carleman linearization transforms an ordinary nonlinear differential equation into an infinite system of linear differential equations. In order to transform the nonlinear differential equation, orthogonal polynomials which represent solutions of a Sturm–Liouville problem are used as basis. The determination of the time derivate of this basis yields an infinite dimensional linear system that depends on the considered nonlinear differential equation. The infinite linear system has the same properties as the nonlinear differential equation such as limit cycles or chaotic behavior. In general, the infinite dimensional linear system cannot be solved. Therefore, the infinite dimensional linear system has to be approximated by a finite dimensional linear system. Due to limitation of dimension the solution of the finite dimensional linear system does not represent the global behavior of the nonlinear differential equation. In fact, the accuracy of the approximation depends on the considered nonlinear system and the initial value. The idea of this contribution is to adapt the range of validity for the Carleman linearization in order to increase the accuracy of the approximation for different ranges of initial values. Instead of truncating the infinite dimensional system after a certain order a Taylor series approach is used to approximate the behavior of the nonlinear differential equation about different equilibrium points. Thus, the adapted finite linear system describes the local behavior of the solution of the nonlinear differential equation.

scholarly journals Approximation of solutions to retarded differential equations with applications to population dynamics

2005 ◽  
Vol 2005 (1) ◽  
pp. 1-11 ◽  
Author(s):  
D. Bahuguna ◽  
M. Muslim
Keyword(s):  
Differential Equation ◽  
Population Dynamics ◽  
Differential Equations ◽  
Unique Solution ◽  
Existence And Uniqueness ◽  
Finite Dimensional ◽  
Retarded Differential Equation ◽  
Retarded Differential Equations

We consider a retarded differential equation with applications to population dynamics. We establish the convergence of a finite-dimensional approximations of a unique solution, the existence and uniqueness of which are also proved in the process.

scholarly journals $ L^p $-exact controllability of partial differential equations with nonlocal terms

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Luisa Malaguti ◽  
Stefania Perrotta ◽  
Valentina Taddei
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Controllability ◽  
Partial Differential ◽  
State Spaces ◽  
Strongly Continuous Semigroups ◽  
Infinite Dimensional ◽  
Semilinear Equations ◽  
Finite Dimensional ◽  
Nonlocal Terms

<p style='text-indent:20px;'>The paper deals with the exact controllability of partial differential equations by linear controls. The discussion takes place in infinite dimensional state spaces since these equations are considered in their abstract formulation as semilinear equations. The linear parts are densely defined and generate strongly continuous semigroups. The nonlinear terms may also include a nonlocal part. The solutions satisfy nonlocal properties, which are possibly nonlinear. The states belong to Banach spaces with a Schauder basis and the results exploit topological methods. The novelty of this investigation is in the use of an approximation solvability method which involves a sequence of controllability problems in finite-dimensional spaces. The exact controllability of nonlocal solutions can be proved, with controls in <inline-formula><tex-math id="M2">\begin{document}$ L^p $\end{document}</tex-math></inline-formula> spaces, <inline-formula><tex-math id="M3">\begin{document}$ 1&lt;p&lt;\infty $\end{document}</tex-math></inline-formula>. The results apply to the study of the exact controllability for the transport equation in arbitrary Euclidean spaces and for the equation of the nonlinear wave equation.</p>

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