The Second Euler-Lagrange Equation of Variational Calculus on Time Scales

Abstract

The fundamental problem of the calculus of variations on time scales concerns the minimization of a delta-integral over all trajectories satisfying given boundary conditions. In this paper we prove the second Euler-Lagrange necessary optimality condition for optimal trajectories of variational problems on time scales. As an example of application of the main result, we give an alternative and simpler proof to the Noether theorem on time scales recently obtained in [J. Math. Anal. Appl. 342 (2008), no. 2, 1220-1226].

This work was partially presented at the Workshop in Control, Nonsmooth Analysis and Optimization, celebrating Francis Clarke's and Richard Vinter's 60th birthday, Porto, May 4-8, 2009. Submitted 26-May-2009; Revised 12-Jan-2010; Accepted 29-March-2010 in revised form; for publication in the European Journal of Control

References in corpus (7)

• Fractional conservation laws in optimal control theory
• Noether's Theorem on Time Scales
• Calculus of Variations on Time Scales with Nabla Derivatives
• Higher-Order Calculus of Variations on Time Scales
• Isoperimetric problems on time scales with nabla derivatives
• Necessary and sufficient conditions for local Pareto optimality on time scales
• Remarks on the calculus of variations on time scales

Cited by in corpus (2)

• Noether's symmetry theorem for nabla problems of the calculus of variations
• Generalizing the variational theory on time scales to include the delta indefinite integral