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On the canonical development of Parseval formulas for singular differential operators

Robert W. Carroll (1982)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Per funzioni opportune f , g si ottiene una formula di Parseval 𝐑 Q , 𝒬 f 𝒬 g λ = Δ Q - 1 / 2 f , Δ Q - 1 / 2 g per operatori differenziali singolari di tipo dell'operatore radiale di Laplace-Beltrami. 𝐑 Q è una funzione spettrale generalizzata di tipo Marčenko e può essere rappresentata per mezzo di un certo nucleo della trasmutazione.

On the center of the generalized Liénard system

Cheng Dong Zhao, Qi-Min He (2002)

Czechoslovak Mathematical Journal

In this paper, we discuss the conditions for a center for the generalized Liénard system d x d t = ϕ ( y ) - F ( x ) , d y d t = - g ( x ) , or d x d t = ψ ( y ) , dy d t = - f ( x ) h ( y ) - g ( x ) , with f ( x ) , g ( x ) , ϕ ( y ) , ψ ( y ) , h ( y ) , F ( x ) = 0 x f ( x ) d x , and x g ( x ) > 0 for x 0 . By using a different technique, that is, by introducing auxiliary systems and using the differential inquality theorem, we are able to generalize and improve some results in [1], [2].

On the circle criterion for boundary control systems in factor form : Lyapunov stability and Lur’e equations

Piotr Grabowski, Frank M. Callier (2006)

ESAIM: Control, Optimisation and Calculus of Variations

A Lur’e feedback control system consisting of a linear, infinite-dimensional system of boundary control in factor form and a nonlinear static sector type controller is considered. A criterion of absolute strong asymptotic stability of the null equilibrium is obtained using a quadratic form Lyapunov functional. The construction of such a functional is reduced to solving a Lur’e system of equations. A sufficient strict circle criterion of solvability of the latter is found, which is based on results...

On the circle criterion for boundary control systems in factor form: Lyapunov stability and Lur'e equations

Piotr Grabowski, Frank M. Callier (2005)

ESAIM: Control, Optimisation and Calculus of Variations

A Lur'e feedback control system consisting of a linear, infinite-dimensional system of boundary control in factor form and a nonlinear static sector type controller is considered. A criterion of absolute strong asymptotic stability of the null equilibrium is obtained using a quadratic form Lyapunov functional. The construction of such a functional is reduced to solving a Lur'e system of equations. A sufficient strict circle criterion of solvability of the latter is found, which is based on...

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