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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...

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.

Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, II

Manabu Naito (2021)

Archivum Mathematicum

We consider the half-linear differential equation of the form ( p ( t ) | x ' | α sgn x ' ) ' + q ( t ) | x | α sgn x = 0 , t t 0 , under the assumption that p ( t ) - 1 / α is integrable on [ t 0 , ) . It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as t .

Remarks on the uniqueness of second order ODEs

Dalibor Pražák (2011)

Applications of Mathematics

We are concerned with the uniqueness problem for solutions to the second order ODE of the form x ' ' + f ( x , t ) = 0 , subject to appropriate initial conditions, under the sole assumption that f is non-decreasing with respect to x , for each t fixed. We show that there is non-uniqueness in general; on the other hand, several types of reasonable additional assumptions make the problem uniquely solvable. The interest in this problem comes, among other, from the study of oscillations of lumped parameter systems with implicit...

Resonance in Preisach systems

Pavel Krejčí (2000)

Applications of Mathematics

This paper deals with the asymptotic behavior as t of solutions u to the forced Preisach oscillator equation w ¨ ( t ) + u ( t ) = ψ ( t ) , w = u + 𝒫 [ u ] , where 𝒫 is a Preisach hysteresis operator, ψ L ( 0 , ) is a given function and t 0 is the time variable. We establish an explicit asymptotic relation between the Preisach measure and the function ψ (or, in a more physical terminology, a balance condition between the hysteresis dissipation and the external forcing) which guarantees that every solution remains bounded for all times. Examples show...

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