Perturbation of nonnegative time scale quadratic functionals
| Authors | |
|---|---|
| Year of publication | 2007 |
| Type | Article in Proceedings |
| Conference | Difference Equations, Special Functions, and Orthogonal Polynomials |
| MU Faculty or unit | |
| 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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