EXISTENCE FOR CALCULUS OF VARIATIONS AND OPTIMAL CONTROL PROBLEMS ON TIME SCALES

1. Introduction. Continuous-time modelling and discrete-time modelling are two main approaches which dominate the methodology of mathematical modelling. System dynamics can be analyzed by using either approach. For example, both continuous-time recurrent neural networks and their discrete-time analogue have been studied in the literature (see, for example, [1] and the references therein). However, in practice, some processes consist of both continuous and discrete elements. A simple example of this kind of hybrid continuous-discrete time system is seasonally breeding population whose generations do not overlap. Temperate zone insects (including many economically important crop and orchid pests) are of this kind. These insects lay their eggs just before the generation dies out at the end of the season, with the eggs laying dormant, hatching at the start of the next season, and giving rise to a new, nonoverlapping generation. During each generation the population varies continuously (due to mortality, resource consumption, predation, interaction, etc.), while the population varies in a discrete fashion between the end of one generation and the beginning of the nex

Necessary Conditions for a Class of Optimal Control Problems on Time Scales

Based on the Gateaux differential on time scales, we investigate and establish necessary conditions for Lagrange optimal control problems on time scales. Moreover, we present an economic model to demonstrate the effectiveness of our results

Existence for calculus of variations and optimal control problems on time scales

In this paper we prove the existence for optimal control problems with terminal constraints on time scales. A definition of the solution of semi-linear control systems involving Sobolev space W1;2T is proposed and new existence and uniqueness results of this kind of dynamic systems on time scales are presented under a weaker assumption. According to L2T strong-weak lower semi-continuity of integral functionals, we establish the existence of optimal controls. In particular, the existence for calculus of variations on time scales is derived

Hamilton-Jacobi-Bellman equations on time scales

In this paper, we consider a class of optimal control problems on time scales without state constraints, target conditions or the fixed terminal time. We first present and show a time scale version of the Bellman optimality principle. On this basis, using a chain rule of multivariables on time scales, we will derive Hamilton-Jacobi-Bellman equations on a time scale for these kind of optimal control problems. Finally, the quantum time scale is considered as an example to illustrate our results. © 2009 Elsevier Ltd. All rights reserved