Publication details

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

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

Faculty of Science

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.
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