Perturbace nezáporných kvadratických funkcionálů na time scales.

Perturbation of nonnegative time scale quadratic functionals

conference paper

en

In this paper we consider a bounded time scale T=[a,b] , a quadratic functional F(x,u) defined over such time scale, and its perturbation G(x,u)=F(x,u)+\alpha\,|x(a)|2 , where the endpoints of F are zero, while the initial endpoint x(a) of G can vary and x(b) is zero. It is known that there is no restriction on x(a) in G when studying the positivity of these functionals. We prove that, when studying the nonnegativity, the initial state x(a) in G must be restricted to a certain subspace, which is the kernel of a specific conjoined basis of the associated time scale symplectic system. This result generalizes a known discrete-time special case, but it is new for the corresponding continuous-time case. We provide several examples which illustrate the theory.

V tomto článku uvažujeme time scale T=[a,b] , kvadratický functional F(x,u), definovaný na této množině, a jeho perturbaci G(x,u)=F(x,u)+\alpha\,|x(a)|2 , kde F má nulové konce, počáteční konec x(a) u G se mění a x(b) je nula. Je známo, že v případě pozitivity není omezení na x(a) u G. Dokážeme, že v případě nezápornosti, počáteční hodnota x(a) u G musí být omezená na jistý podprostor, který je jádrem specifické konjugované báze asociovaného symplektického systému na time scale. Tento výsledek zobecňuje známý speciální diskrétní případ, a je nový pro odpovídající spojitý případ. Uvedeme několik příkladů ilustrujících teorii.

In this paper we consider a bounded time scale T=[a,b] , a quadratic functional F(x,u) defined over such time scale, and its perturbation G(x,u)=F(x,u)+\alpha\,|x(a)|2 , where the endpoints of F are zero, while the initial endpoint x(a) of G can vary and x(b) is zero. It is known that there is no restriction on x(a) in G when studying the positivity of these functionals. We prove that, when studying the nonnegativity, the initial state x(a) in G must be restricted to a certain subspace, which is the kernel of a specific conjoined basis of the associated time scale symplectic system. This result generalizes a known discrete-time special case, but it is new for the corresponding continuous-time case. We provide several examples which illustrate the theory.

Quadratic functional, Nonnegativity, Positivity, Time scale, Time scale symplectic system, Hamiltonian system

01.05.2007

978-981-270-643-0

DIFFERENCE EQUATIONS, SPECIAL FUNCTIONS AND ORTHOGONAL POLYNOMIALS Proceedings of the International Conference

266–275

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