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

Optimal Control of Nonlocal Thermistor Equations

We are concerned with the optimal control problem of the well known nonlocal thermistor problem, i.e., in studying the heat transfer in the resistor device whose electrical conductivity is strongly dependent on the temperature. Existence of an optimal control is proved. The optimality system consisting of the state system coupled with adjoint equations is derived, together with a characterization of the optimal control. Uniqueness of solution to the optimality system, and therefore the uniqueness of the optimal control, is established. The last part is devoted to numerical simulations.Comment: Submitted 21-March-2012; revised 11-June-2012; accepted 13-June-2012; for publication in the International Journal of Contro

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

Galerkin spectral method for the fractional nonlocal thermistor problem

We develop and analyse a numerical method for the time-fractional nonlocal thermistor problem. By rigorous proofs, some error estimates in different contexts are derived, showing that the combination of the backward differentiation in time and the Galerkin spectral method in space leads, for an enough smooth solution, to an approximation of exponential convergence in space. © 2016 Elsevier Lt

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