Application of the Euler-Lagrange Theorem

In this chapter we will look at some applications of the Euler-Lagrange theorem. The theorem transforms the Problem of Bolza into a set of differential equations and attendant boundary conditions. In some cases, simple closed-form solutions are available which completely solve the problem. In other cases, numerical methods are required to solve the “two-point boundary-value problem.” In some instances we find that the Euler-Lagrange theorem does not supply enough conditions to determine the optimal control law. In such cases we appeal to another theorem (the Weierstrass condition or Minimum Principle, discussed in Chap. 5) to solve the problem.