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.

Last change: August 21, 2006. (c) Roman Hilscher