General mechanical engineering - Dynamics: optimization

The theory of functions (maximum and minimum) provides the basis for what we might call static optimization. Dynamic optimization deals with much more general extremum problems.

Dynamic optimization problems have their historical roots in general mechanics. That's why we're devoting an article to this problem. It is based on the calculus of variations, whose founders were Euler and Lagrange. The Lagrange equations – when the system is Lagrangian – are identical to Euler's formulas. The first problems to be formulated are due to Newton (shape of bodies giving minimum drag) and Bernoulli (brachistochrone problem).

A dynamic optimization problem is based on two fundamental elements:

• a theoretical model representing the nature of the problem in mechanics. This model is provided by the system of differential equations and linkage equations ;

• a quantity whose maximum or minimum value we want to render. This is known as the optimization criterion or performance index.

The long-standing existence of a mathematical model is the fundamental cause of the birth of optimization theory in mechanics. The purpose of this article is twofold: on the one hand, to provide an introduction to optimization problems and, on the other, to provide a directly usable tool. We have left aside the choice of criteria and specific optimization techniques.