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Rank-2 distributions satisfying the Goursat condition: all their local models in dimension 7 and 8

Mohamad Cheaito, Piotr Mormul (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We study the rank–2 distributions satisfying so-called Goursat condition (GC); that is to say, codimension–2 differential systems forming with their derived systems a flag. Firstly, we restate in a clear way the main result of[7] giving preliminary local forms of such systems. Secondly – and this is the main part of the paper – in dimension 7 and 8 we explain which constants in those local forms can be made 0, normalizing the remaining ones to 1. All constructed equivalences are explicit. ...

Rayleigh principle for linear Hamiltonian systems without controllability∗

Werner Kratz, Roman Šimon Hilscher (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we consider linear Hamiltonian differential systems without the controllability (or normality) assumption. We prove the Rayleigh principle for these systems with Dirichlet boundary conditions, which provides a variational characterization of the finite eigenvalues of the associated self-adjoint eigenvalue problem. This result generalizes the traditional Rayleigh principle to possibly abnormal linear Hamiltonian systems. The main tools...

Rayleigh principle for linear Hamiltonian systems without controllability∗

Werner Kratz, Roman Šimon Hilscher (2012)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we consider linear Hamiltonian differential systems without the controllability (or normality) assumption. We prove the Rayleigh principle for these systems with Dirichlet boundary conditions, which provides a variational characterization of the finite eigenvalues of the associated self-adjoint eigenvalue problem. This result generalizes the traditional Rayleigh principle to possibly abnormal linear Hamiltonian systems. The main tools...

Realization theory methods for the stability investigation of nonlinear infinite-dimensional input-output systems

Volker Reitmann (2011)

Mathematica Bohemica

Realization theory for linear input-output operators and frequency-domain methods for the solvability of Riccati operator equations are used for the stability and instability investigation of a class of nonlinear Volterra integral equations in a Hilbert space. The key idea is to consider, similar to the Volterra equation, a time-invariant control system generated by an abstract ODE in a weighted Sobolev space, which has the same stability properties as the Volterra equation.

Reconstruction of map projection, its inverse and re-projection

Tomáš Bayer, Milada Kočandrlová (2018)

Applications of Mathematics

This paper focuses on the automatic recognition of map projection, its inverse and re-projection. Our analysis leads to the unconstrained optimization solved by the hybrid BFGS nonlinear least squares technique. The objective function is represented by the squared sum of the residuals. For the map re-projection the partial differential equations of the inverse transformation are derived. They can be applied to any map projection. Illustrative examples of the stereographic and globular Nicolosi projections...

Regular half-linear second order differential equations

Ondřej Došlý, Jana Řezníčková (2003)

Archivum Mathematicum

We introduce the concept of the regular (nonoscillatory) half-linear second order differential equation r ( t ) Φ ( x ' ) ' + c ( t ) Φ ( x ) = 0 , Φ ( x ) : = | x | p - 2 x , p > 1 ( * ) and we show that if (*) is regular, a solution x of this equation such that x ' ( t ) ≠ 0 for large t is principal if and only if ∫ ∞ d t r ( t ) x 2 ( t ) | x ' ( t ) | p - 2 = ∞ . Conditions on the functions r , c are given which guarantee that (*) is regular.

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