Maximum principle for the general optimal control problem with phase and regular mixed constraints

Abstract

We prove the maximum principle for the problem outlined in the title with the aid (or more precisely in the class) of so-called sliding variations. INTRODUCTION The by-now classical maximum principle for the simplest nonlinear optimal control problem was formulated by Pontryagin and proved by Boltyanskii over 30 years ago (see [1, 2, 3] and also [8]). Since then, the maximum principle has been derived for various optimal control problems in an enormous number of studies. This is well-known and we do not give a full list of references here; we only mention [4-7, 9-24, 27-30]. In our view, the deepest and most thorough studies are those of Milyutin and Dubovitskii [9-14]. These authors advance very far in their development of the maximum . principle for problems with ordinary differential equations - first to problems with phase constraints [9] and then to the general problem with phase and arbitrary (in general, nonregular) mixed constraints [10-14]. These authors completely resolve the issue of deriving the necessary first-order conditions for these problems. Despite these achievements, we are still witnessing the publication of studies that derive the maximum principle for various problems which are particular cases of the general Dubovitskii-Milyutin problem. The authors of some of these studies possibly do not realize that they are rederiving well-established results.** This is partly due to the fact that, on the one hand, optimal control results are primarily interesting for applied engineers, while the references [9-14] are far from being accessible to every engineer, because they use a fairly complex mathematical apparatus which is not familiar to engineers; on the other hand, optimal control theory as such so far has failed to attract a sufficiently wide followign of mathematicians. "Optimizers" still feel that if a problem originates in engineering practice, then its solution (and preferably the process of its analysis) should remain in the domain of concepts which are familiar in the engineering environment (compare this situation with the totally different state of things in the theory of partial differential equations or in probability theory). This, in our view, accounts for repeated attempts to simplify the proof of the maximum principle. However, even if some studies achieve a certain simplification of the proof, this is the result of extreme simplification of the problem, and not an achievement of the particular method of analysis. The positive impact of these attempts is that they highlight the complexity of the problem. They have shown that the problem cannot be solved by elementary techniques, without invoking new methods (specifically, methods of functional analysis). We are thus forced to admit that the complexity of the work of Dubovitskii-Milyutin is not contrived: it is inherent to the complexity of the underlying problem. Yet there is a fairly wide class of problems for which the maximum principle can be derived relatively simply. These are the problems with pure phase constraints and so-called "regular" mixed constraints. They cover most of the problems considered in the literature. The maximum principle for these problems was derived by Milyutin and Dubovitskii back in the 1960s, immediately after their derivation of the maximum principle for problems with phase constraints [9]. However, it has never been published because, in the authors' opinion, the addition of regular mixed constraints to phase constraints did not constitute a significant advance.