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Quadratic functionals: positivity, oscillation, Rayleigh's principle

Werner Kratz (1998)

Archivum Mathematicum

In this paper we give a survey on the theory of quadratic functionals. Particularly the relationships between positive definiteness and the asymptotic behaviour of Riccati matrix differential equations, and between the oscillation properties of linear Hamiltonian systems and Rayleigh’s principle are demonstrated. Moreover, the main tools form control theory (as e.g. characterization of strong observability), from the calculus of variations (as e.g. field theory and Picone’s identity), and from matrix...

Quadratic functionals with a variable singular end point

Zuzana Došlá, PierLuigi Zezza (1992)

Commentationes Mathematicae Universitatis Carolinae

In this paper we introduce the definition of coupled point with respect to a (scalar) quadratic functional on a noncompact interval. In terms of coupled points we prove necessary (and sufficient) conditions for the nonnegativity of these functionals.

Qualitative theory of half-linear second order differential equations

Ondřej Došlý (2002)

Mathematica Bohemica

Some recent results concerning properties of solutions of the half-linear second order differential equation ( r ( t ) Φ ( x ' ) ) ' + c ( t ) Φ ( x ) = 0 , Φ ( x ) : = | x | p - 2 x , p > 1 , ( * ) are presented. A particular attention is paid to the oscillation theory of ( * ) . Related problems are also discussed.

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...

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