# Masaryk University

Publication details

# Perturbation of nonnegative time scale quadratic functionals

Authors RŮŽIČKOVÁ Viera 2007 Article in Proceedings Difference Equations, Special Functions, and Orthogonal Polynomials http://www.worldscibooks.com/mathematics/6446.html General mathematics Quadratic functional; Nonnegativity; Positivity; Time scale; Time scale symplectic system; Hamiltonian system 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.