Title: A non-classical class of variational problems

Authors: Pedro A.F. Cruz, Delfim F.M. Torres, Alan S.I. Zinober

Addresses: Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal. ' Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal. ' Department of Applied Mathematics, The University of Sheffield, Sheffield, S10 2TN, UK

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.

Keywords: nonlinear revenue problems; transversality conditions; shooting method; calculus of variations; optimal control; variational problems; economics.

DOI: 10.1504/IJMMNO.2010.031750

International Journal of Mathematical Modelling and Numerical Optimisation, 2010 Vol.1 No.3, pp.227 - 236

Available online: 22 Feb 2010 *

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