Legendre, Jacobi, and Riccati type conditions for time scale variational problem with application

Investor logo

Warning

This publication doesn't include Faculty of Education. It includes Faculty of Science. Official publication website can be found on muni.cz.
Authors

HILSCHER Roman ZEIDAN Vera

Year of publication 2007
Type Article in Periodical
Magazine / Source Dynamic Systems and Applications
MU Faculty or unit

Faculty of Science

Citation
Field General mathematics
Keywords Time scale quadratic functional; Nonnegativity; Positivity; Jacobi equation; Linear Hamiltonian system; Conjugate point; Conjoined basis; Riccati matrix equation; Strengthened Legendre condition; Time-dependent impulsive dynamical system
Description A time scale quadratic problem J with piecewise right-dense continuous coefficients and one varying endpoint is considered. Such problems are ``hybrid'', since they include mixing of continuous- and discrete-time problems. A new notion of a generalized conjugate point involving ``dynamic'' (hybrid) systems and comprising as special cases those known for the continuous- and discrete-time settings is introduced. A type of a strengthened Legendre condition is identified and used to establish characterizations of the nonnegativity and positivity of J in terms of (i) the nonexistence of such conjugate points, (ii) the natural conjoined basis of the associated time scale Jacobi equation, and (iii) a solution of the corresponding time scale Riccati equation. These results furnish second order necessary optimality conditions for a nonlinear time scale variational problem.
Related projects:

You are running an old browser version. We recommend updating your browser to its latest version.