Seminars this week

Abstract

The Calculus of Variations was initiated in the 17th Century and forms a basic foundation of modern optimal (maximising or minimising) variational problems, nowadays often called optimal control. An introduction to the Calculus of Variations with some sample examples will be presented. This will include the Euler-Lagrange and Hamiltonian formulation together with the associated final boundary value conditions. A numerical shooting method can be used to solve the resulting Two Point Boundary Value Problem (TPBVP), a set of differential equations. There are many interesting applications including the optimal spending of capital, reservoir control, maintenance and replacement policy of vehicles and machinery, optimal delivery of medicines, drug bust strategies, study for examinations and optimal presentation of a lecture like this one. A new non-classical class of variational problems has been motivated by 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 {em a priori} unknown and the integrand to be maximised is also a function of the unknown $y(T)$. This is a non-standard CoV problem that has not been studied before. New final value costate boundary conditions will be presented for this CoV problem and some results will be shown.