Noether's Theorem for Control Problems on Time Scales

We prove a generalization of Noether's theorem for optimal control problems defined on time scales. Particularly, our results can be used for discrete-time, quantum, and continuous-time optimal control problems. The generalization involves a one-parameter family of maps which depend also on the control and a Lagrangian which is invariant up to an addition of an exact delta differential. We apply our results to some concrete optimal control problems on an arbitrary time scale.Comment: This is a preprint of a paper whose final and definite form is published in International Journal of Difference Equations ISSN 0973-6069, Vol. 9 (2014), no. 1, 87--10

Holder's and Hardy's Two Dimensional Diamond-alpha Inequalities on Time Scales

We prove a two dimensional Holder and reverse-Holder inequality on time scales via the diamond-alpha integral. Other integral inequalities are established as well, which have as corollaries some recent proved Hardy-type inequalities on time scales.Comment: Accepted for publication (October 21, 2009) in the journal "Annals of the University of Craiova, Mathematics and Computer Science Series" (http://inf.ucv.ro/~ami

Existence and uniqueness of a positive solution to generalized nonlocal thermistor problems with fractional-order derivatives

In this work we study a generalized nonlocal thermistor problem with fractional-order Riemann-Liouville derivative. Making use of fixed-point theory, we obtain existence and uniqueness of a positive solution.Comment: Submitted 17-Jul-2011; revised 09-Oct-2011; accepted 21-Oct-2011; for publication in the journal 'Differential Equations & Applications' (http://dea.ele-math.com

A Dual Mesh Method for a Non-Local Thermistor Problem

We use a dual mesh numerical method to study a non-local parabolic problem arising from the well-known thermistor problem.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Optimal Control for a Steady State Dead Oil Isotherm Problem

We study the optimal control of a steady-state dead oil isotherm problem. The problem is described by a system of nonlinear partial differential equations resulting from the traditional modelling of oil engineering within the framework of mechanics of a continuous medium. Existence and regularity results of the optimal control are proved, as well as necessary optimality conditions.Comment: This is a preprint of a paper whose final and definitive form will appear in Control and Cybernetics. Paper submitted 24-Sept-2012; revised 21-March-2013; accepted for publication 17-April-2013. arXiv admin note: text overlap with arXiv:math/061237