Pontryagins Minimum Principle

One of the most profound results of applied mathematics, Pontryagin s minimum principle provides the necessary conditions for the minimum of an optimal control problem. The elegance of the principle lies in the simplicity of its application to a vast variety of optimal control problems. Boltyanskii et al. (1956) developed the principle originally as a maximum principle requiring the Hamiltonian to be maximized at the minimum. 

In this chapter, we will present the proof of the minimum principle. The minimum principle uses a positive multiplier for the objective functional in the Hamiltonian formulation. With this provision, the minimum principle concludes that the minimum of the problem requires minimization of the Hamiltonian in an optimal control problem whose minimmn needs to be determined. 

Some readers may first want to get the essence of the minimum principle and go cursorily over the derivation in Section 5.4. This section may be skipped during the initial reading. 

Before delving into the proof, let us take the simplest optimal control problem and examine the application of Pontryagin s minimum principle. We will realize that we already have been applying the minimum principle to our optimal control problems. 

A negative multiplier, on the other hand, leads to the maximum principle, which stipulates that the Hamiltonian be maximized in a minimum problem. 

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The above formulation is a well posed problem in optimal control theory and its solution can be obtained by the application of Pontryagin s Minimum Principle (Sage and White (1977)). 

According to Pontryagin s minimum principle, if u is optimal, then the corresponding Hamiltonian H y,X,u) at each time instant is minimum over all admissible choices for u. Thus, if u = m + (5m is any admissible control, then... 

Case 5 The integrand F in Equation (3.4) is a function of m. Now, when u = 0 the partial derivative of does not exist, and we cannot apply the stationarity condition, Hu = 0. However, Pontryagin s minimum principle does not require the partial derivatives and Qu to exist. According to the principle,... 

The above cases show that Pontryagin s minimum principle provides an overarching necessary condition for the minimum. Appreciating this fact, we present a general optimal control problem involving a wide class of controls for which we will derive Pontryagin s minimum principle. 

Pontryagin s minimum principle is a statement of necessary conditions for the control to be optimal. The main conclusion of this principle is that the optimal control minimizes the Hamiltonian at each point in the time interval whether or not the control there is continuous. 

Very versatile in applications, Pontryagin s minimum principle is among the most profound and difficult results to derive. Figure 5.1 presents the outline of the derivation of the principle. 

Figure 5.1 Outline of the proof of Pontryagin s minimum principle...

Optimal control problems involving multiple integrals are constrained by partial differential equations. A general theory similar to the Pontryagin s minimum principle is not available to handle these problems. To find the necessary conditions for the minimum in these problems, we assume that the variations of the involved integrals are weakly continuous and find the equations that eliminate the variation of the augmented objective functional. 

Table 5.2. Summary of Minimum Time Problem Using Pontryagin s Maximum Principle...

It might happen that the first condition is not satisfied, e.g., when the objective functional depends on juj or the set of admissible controls contains discrete elements. To obtain necessary conditions for the minimum in that situation, we can rely on Pontryagin s minimmn principle, which does not depend on the existence of or its continuity (see Section 5.3.1, p. 128). According to the principle, the optimal control u minimizes H, i.e.,... 

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