Complementary Extremum Principles for Isoperimetric Optimisation Problems

An isoperimetric problem of the calculus of variations is reviewed. An integral functional, dependent on a function u (x), its derivative u′ (x) and on the independent variable x, is minimised. The minimisation is subject to the isoperimetric constraint that the length of the integration interval remains constant. The well-known Euler equation of the fundamental problem of the calculus of variations is recovered with an additional relationship connecting the values of the function u and its derivative u′ at the ends of the interval. A new complementary extremum principle is derived, that offers an algorithm for determining lower bounds on the minimum value of the original functional. A special treatment of the degenerate case where there is linear dependence on u is presented. Examples of each case are given and the scope for future work is discussed. Key-Words: Isoperimetric, Calculus of Variations, Complementary Principle, Variable End-points.