Perturbation of nonnegative time scale quadratic functionals

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Authors

HILSCHER Roman RŮŽIČKOVÁ Viera

Year of publication 2007
Type Article in Proceedings
Conference Difference Equations, Special Functions, and Orthogonal Polynomials
MU Faculty or unit

Faculty of Science

Citation
Web http://www.worldscibooks.com/mathematics/6446.html
Field General mathematics
Keywords Quadratic functional; Nonnegativity; Positivity; Time scale; Time scale symplectic system; Hamiltonian system
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.
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