About: Perturbation of nonnegative time scale quadratic functionals

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Description	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. 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. (en)
Title	Perturbation of nonnegative time scale quadratic functionals Perturbation of nonnegative time scale quadratic functionals (en)
skos:prefLabel	Perturbation of nonnegative time scale quadratic functionals Perturbation of nonnegative time scale quadratic functionals (en)
skos:notation	RIV/00216224:14310/07:00020419!RIV10-MSM-14310___
http://linked.open...avai/riv/aktivita	P
http://linked.open...avai/riv/aktivity	P(1K04001), P(GA201/04/0580)
http://linked.open...vai/riv/dodaniDat	2010
http://linked.open...aciTvurceVysledku	Šimon Hilscher, Roman Růžičková, Viera
http://linked.open.../riv/druhVysledku	D - Článek ve sborníku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Masarykova univerzita / Přírodovědecká fakulta
http://linked.open...dnocenehoVysledku	441073
http://linked.open...ai/riv/idVysledku	RIV/00216224:14310/07:00020419
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	Quadratic functional; Nonnegativity; Positivity; Time scale; Time scale symplectic system; Hamiltonian system (en)
http://linked.open.../riv/klicoveSlovo	Hamiltonian system Time scale Quadratic functional Time scale symplectic system Nonnegativity Positivity
http://linked.open...ontrolniKodProRIV	[16F5788B5597]
http://linked.open...v/mistoKonaniAkce	Mnichov
http://linked.open...i/riv/mistoVydani	Londýn
http://linked.open...i/riv/nazevZdroje	Difference Equations, Special Functions, and Orthogonal Polynomials
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	2 (xsd:int)
http://linked.open...cetTvurcuVysledku	2 (xsd:int)
http://linked.open...vavai/riv/projekt	Optimality conditions on time scales Difference equations and dynamic equations on time scales
http://linked.open...UplatneniVysledku	2007
http://linked.open...iv/tvurceVysledku	Šimon Hilscher, Roman Růžičková, Viera
http://linked.open...vavai/riv/typAkce	WRD - Světová
http://linked.open...ain/vavai/riv/wos	000261024200022
http://linked.open.../riv/zahajeniAkce	2005-01-01 (xsd:date)
number of pages	10 (xsd:int)
http://purl.org/ne...btex#hasPublisher	World Scientific
https://schema.org/isbn	978-981-270-643-0
http://localhost/t...ganizacniJednotka	14310
is http://linked.open...avai/riv/vysledek of	Perturbation of nonnegative time scale quadratic functionals Perturbation of nonnegative time scale quadratic functionals Perturbation of nonnegative time scale quadratic functionals

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