Leitmann's direct method of optimization for absolute extrema of certain problems of the calculus of variations 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. This includes the discrete-time, the quantum, and the continuous/classical calculus of variations as particular cases. In this note we follow Leitmann’s direct method to give explicit solutions for some concrete optimal control problems on an arbitrary time scale.

[1]  Delfim F. M. Torres,et al.  Automatic Computation of Conservation Laws in the Calculus of Variations and Optimal Control , 2005 .

[2]  Wei Wei,et al.  Hamilton-Jacobi-Bellman equations on time scales , 2009, Math. Comput. Model..

[3]  George Leitmann,et al.  Some Extensions to a Direct Optimization Method , 2001 .

[4]  A. Peterson,et al.  Advances in Dynamic Equations on Time Scales , 2012 .

[5]  Georges Zaccour,et al.  Dynamic games : theory and applications , 2005 .

[6]  Ferhan Merdivenci Atici,et al.  An application of time scales to economics , 2006, Math. Comput. Model..

[7]  Agnieszka B. Malinowska,et al.  Natural boundary conditions in the calculus of variations , 2008, 0812.0705.

[8]  Delfim F. M. Torres,et al.  Contrasting Two Transformation-based Methods for Obtaining Absolute Extrema , 2008 .

[9]  Florian Wagener,et al.  On the Leitmann Equivalent Problem Approach , 2009, J. Optimization Theory and Applications.

[10]  Delfim F. M. Torres,et al.  Absolute Extrema of Invariant Optimal Control Problems , 2006 .

[11]  Delfim F. M. Torres,et al.  Higher-Order Calculus of Variations on Time Scales , 2007, 0706.3141.

[12]  Martin Bohner.  CALCULUS OF VARIATIONS ON TIME SCALES , 2004 .

[13]  George Leitmann,et al.  Coordinate Transformation Method for the Extremization of Multiple Integrals , 2005 .

[14]  Ferhan Merdivenci Atici,et al.  A production-inventory model of HMMS on time scales , 2008, Appl. Math. Lett..

[15]  George Leitmann.  On a class of direct optimization problems 1,2 , 2002 .

[16]  Delfim F. M. Torres,et al.  Remarks on the calculus of variations on time scales , 2007 .

[17]  Delfim F. M. Torres,et al.  Isoperimetric Problems on Time Scales with Nabla Derivatives , 2008, 0811.3650.

[18]  Agnieszka B. Malinowska,et al.  Necessary and sufficient conditions for local Pareto optimality on time scales , 2008 .

[19]  George Leitmann,et al.  A Direct Method for Open-Loop Dynamic Games for Affine Control Systems , 2005 .

[20]  George Leitmann,et al.  Fields of extremals and sufficient conditions for the simplest problem of the calculus of variations , 2008, J. Glob. Optim..

[21]  A. Peterson,et al.  Dynamic Equations on Time Scales , 2001 .

[22]  D. A. Carlson.  An Observation on Two Methods of Obtaining Solutions to Variational Problems , 2002 .

[23]  Delfim F. M. Torres,et al.  Noether's theorem on time scales , 2008 .

[24]  G. Leitmann,et al.  On a Class of Direct Optimization Problems , 2001 .

[25]  Delfim F. M. Torres,et al.  Calculus of variations on time scales with nabla derivatives , 2008, 0807.2596.

[26]  George Leitmann,et al.  A note on absolute extrema of certain integrals , 1967 .