The maximum principle

Abstract

The maximum principle is formulated and proved first for control problems with fixed terminal time. As an application, a new derivation of the solution to the linear regulator problem is given. Next the maximum principle for impulse control problems and for time-optimal problems are discussed. In the latter case an essential role is played by the separation theorems for convex sets.

Bibliographical notes

Numerous applications of the maximum principle can be found in W.H. Fleming and R.W. Rishel [37], and in E.B. Lee and L. Markus [63]. The proof of the maximum principle for more general control problems is given, for instance, in W.H. Fleming and R.W. Rishel [37], in R. Vinter [96] and B.S. Mordukhovich [71]. A functional analytic approach is presented in I.V. Girsanov [41]. Nonlinear optimization problems can be discussed in the framework of nonsmooth analysis, see F.H. Clarke [16].

Theorem 12.2 is borrowed from A. Blaquiere [11], and its proof follows the paper by R. Rempala and J. Zabczyk [85]. Applications can be found in A. Blaquiere [11].