Maximum Principle of Optimal Controls

In this chapter, we will present some first-order necessary conditions on time optimal controls for some evolution systems. Such necessary conditions are referred to as Pontryagin’s maximum principle, which provides some information on optimal tetrads. Differing from analysis methods (see L.S. Pontryagin et al. The mathematical theory of optimal processes, ed. by L.W. Neustadt, (Interscience Publishers Wiley, New York, London, 1962) [Translated from the Russian by K. N. Trirogoff], Li and Yong (Optimal control theory for infinite-dimensional systems. Systems & control: foundations & applications. Birkhauser Boston, Boston, MA, 1995), V. Barbu (Optimal control of variational inequalities). Research notes in mathematics, vol 100 Pitman (Advanced Publishing Program), Boston, MA (1984) and V. Barbu, Analysis and control of nonlinear infinite-dimensional systems, In: Mathematics in science and engineering, vol. 190 (Academic, Boston, MA, 1993) ), there are other geometric ways to approach Pontryagin’s maximum principle (see, for instance, Agrachev and Sachkov Control theory from the geometric viewpoint. Encyclopaedia of mathematical sciences, vol. 87. Control theory and optimization, II (Springer, Berlin, 2004) and E. Roxin, A geometric interpretation of Pontryagin’s maximum principle. In: International symposium on nonlinear differential equations and nonlinear mechanics, pp. 303–324. Academic, New York (1963)). Usually, these geometric methods are based on the use of separation theorems and representation theorems. In this chapter, we first introduce geometric methods through studying the problem \((TP)_{min}^{Q_S,Q_E}\) in several different cases. Three different kinds of Pontryagin’s maximum principles for \((TP)_{min}^{Q_S,Q_E}\) are given in order. They are respectively called as: the classical Pontryagin Maximum Principle, the local Pontryagin Maximum Principle, and the weak Pontryagin Maximum Principle. These Pontryagin’s maximum principles are obtained by separating different objects, i.e., separating the target from the reachable set in the state space at the optimal time; separating a reachable set from a controllable set in the state space before the optimal time; separating the target from the reachable set in the reachable space at the optimal time. We then discuss the classical and the local Pontryagin Maximum Principles for \((TP)_{max}^{Q_S,Q_E}\) in the final section, where the methods are similar to those used for \((TP)_{min}^{Q_S,Q_E}\).