Time scale embedding theorem and coercivity of quadratic functionals
|Year of publication
|Article in Periodical
|Magazine / Source
|MU Faculty or unit
|Time scale; Time scale embedding theorem; Quadratic functional; Positivity; Coercivity; Riccati equation; Legendre condition; Calculus of variations; Weak local extremum
|In this paper we study the relation between the coercivity and positivity of a time scale quadratic functional J, which could be a second variation for a nonlinear time scale calculus of variations problem (P). We prove for the case of general jointly varying endpoints that J is coercive if and only if it is positive definite and the time scale version of the strengthened Legendre condition holds. In order to prove this, we establish a time scale embedding theorem and apply it to the Riccati matrix equation associated with the quadratic functional J. Consequently, we obtain sufficiency criteria for the nonlinear problem (P) in terms of the positivity of J or in terms of the time scale Riccati equation. This result is new even for the continuous time case when the endpoints are jointly varying .