Riccati equations for abnormal time scale quadratic functionals

This paper focuses on developing new Riccati type conditions for an abnormal time scale symplectic system ( ). These conditions provide characterizations of the nonnegativity (with and without a certain “image condition”) and positivity of the quadratic functionals associated with such a system. The novelty of these conditions rely on the natural conjoined basis (Xa,Ua) of ( ) in which Xa(t) is not necessarily invertible, and thus the system ( ) could be abnormal. These results are new even in the special case of continuous time, as are some of them in the discrete time setting.