The Basic Variable-Endpoint Control Problem is the same as the Basic Fixed-Endpoint Control Problem except the target set is now of the form , where is a -dimensional surface in for some nonnegative integer . As in Section 220.127.116.11 we define such a surface via equality constraints:
where , are functions from to . We also assume that every is a regular point. As two extreme special cases, for we obtain (which gives a free-time, free-endpoint problem) while for the surface reduces either to a single point (as in the Basic Fixed-Endpoint Control Problem) or to a set consisting of isolated points. The difference between the maximum principle for this problem and the previous one lies only in the boundary conditions for the system of canonical equations.
Maximum Principle for the Basic Variable-Endpoint Control Problem Let be an optimal control and let be the corresponding optimal state trajectory. Then there exist a function and a constant satisfying for all and having the following properties:
The additional necessary condition (4.3) is called the transversality condition (we encountered its loose analog in Example 2.4 and Exercise 2.6 in Section 2.3.5). We know from Section 1.2.2 that the tangent space can be characterized as
The maximum principle was developed by the Pontryagin school in the Soviet Union in the late 1950s. It was presented to the wider research community at the first IFAC World Congress in Moscow in 1960 and in the celebrated book [PBGM62]. It is worth reflecting that the developments we have covered so far in this book--starting from the Euler-Lagrange equation, continuing to the Hamiltonian formulation, and culminating in the maximum principle--span more than 200 years. The progress made during this time period is quite remarkable, yet the origins of the maximum principle are clearly traceable to the early work in calculus of variations.