A non-classical class of variational problems

Abstract

Abstract:We study a new non-classical class of variational problems that is motivated by some recent research on the non-linear revenue problem in the field of economics. This class of problem can be set up as a maximising problem in the calculus of variations (CoV) or optimal control. However, the state value at the final fixed time, ( ), y T is a priori unknown and the integrand is a function of the unknown ( ).y T This is a non-standard CoV problem. In this paper we apply the new costate boundary conditions ( ) p T in the formulation of the CoV problem. We solve a sample example in this problem class using the numerical shooting method to solve the resulting TPBVP, and incorporate the free ( ) y T as an additional unknown. Essentially the same results are obtained using symbolic algebra software.

Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative

“…Thus, if the C x D β b y term is not present in (9), then for α → 1 we obtain a corresponding result in the classical context of the calculus of variations [29] (see also [35,Corollary 1]). …”

“…The novelty is the dependence of the integrand L on the free end-points y(a), y(b). This class of problems is motivated by applications in the field of economics [29].…”

A non-standard optimal control problem arising in an economics application

“…We iterate the system with the state equation y(t), costate equation p(t), the integral of η(t) and the cost function J (t). As a comparative approach, we used a different nonlinear programming discrete-time technique to solve the same problem [1,7]. We solved the problem using Euler and also Runge-Kutta discretisation, and an optimization algorithm in order to solve the unknown control variables u k at each time t k [1].…”

“…Optimal control channels the paths of the control variables to optimize the cost functional whilst satisfying (in this paper) ordinary differential equations. Economics is a source of interesting applications of the theory of CoV and optimal control [7]. A few classical examples that reflect the use of optimal control are the drug bust strategy, optimal production, optimal control in discrete mechanics, policy arrangement and the royalty payment problem [5,7,9].…”

Quantum Variational Calculus

“…Several quantum variational problems have been recently posed and studied (Aldwoah et al 2012;Almeida and Torres 2009;Almeida and Torres 2011;Bangerezako 2004;Bangerezako 2005; Brito da Cruz et al 2013a;Cresson 2005;Cresson et al 2009;Frederico and Torres 2013;Martins and Torres 2012). The main purpose of this book is to present optimality conditions for generalized quantum variational problems in an unified and a coherent way, and call attention to a promising research area with possible applications in optimal control, physics, and economics (Cruz et al 2010;Malinowska and Martins 2013;Sengupta 1997). The results presented in the book allow to deal with economical problems with a dynamic nature that does not depend on the usual derivative or the forward difference operator, but on the Hahn quantum difference operator D q,x .…”