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@inproceedings{722452, author = {Hilscher, Roman and Růžičková, Viera}, address = {Londýn}, booktitle = {Difference Equations, Special Functions, and Orthogonal Polynomials}, keywords = {Quadratic functional; Nonnegativity; Positivity; Time scale; Time scale symplectic system; Hamiltonian system}, language = {eng}, location = {Londýn}, isbn = {978-981-270-643-0}, pages = {266-275}, publisher = {World Scientific}, title = {Perturbation of nonnegative time scale quadratic functionals}, url = {http://www.worldscibooks.com/mathematics/6446.html}, year = {2007} }
TY - JOUR ID - 722452 AU - Hilscher, Roman - Růžičková, Viera PY - 2007 TI - Perturbation of nonnegative time scale quadratic functionals PB - World Scientific CY - Londýn SN - 9789812706430 KW - Quadratic functional KW - Nonnegativity KW - Positivity KW - Time scale KW - Time scale symplectic system KW - Hamiltonian system UR - http://www.worldscibooks.com/mathematics/6446.html N2 - 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. ER -
HILSCHER, Roman a Viera RŮŽIČKOVÁ. Perturbation of nonnegative time scale quadratic functionals. In \textit{Difference Equations, Special Functions, and Orthogonal Polynomials}. Londýn: World Scientific, 2007, s.~266-275. ISBN~978-981-270-643-0.
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