Pontryagin derivation

We will not present here how to derive the first Pontryagin s equation for the probability Q(t, x0) or P(f,x0). The interested reader can see it in Ref. 19 or in Refs. 15 and 18. We only mention that the first Pontryagin s equation may be obtained either via transformation of the backward Kolmogorov equation (2.7) or by simple decomposition of the probability P(t, xq) into Taylor expansion in the vicinity of xo at different moments t and t + t, some transformations and limiting transition to r — 0 [18]. 

Pontryagin s adjoint variables i/j are clearly the partial derivatives of Bellman s F and the continuity of the adjoint variables (as solutions of adjoint differential equations) implies the smoothness of the surface that was lacking in the first paper. 

The maximization problem is to maximize (7.35) subjected to constraints (7.32-7.34). The derivation of the Pontryagin maximum principle for the system and the optimality conditions are given in Appendix D, for one internal collocation point. The derivation of the optimality conditions for N internal collocation points is given by Elnashaie and El-Rifaie (1978). 

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. 

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. 

In the original derivation of the maximum principle, Pontryagin (1986) chose the final costate along the normal p, i. e., away from the set of final states. 

Moreover, quite often are the cases when the derivatives of functions and functionals in specific points and regions do not really exist (e.g. in diseontinuous or broken-line funetions). To get over these difficulties, non-classic methods of ealeulus of variations are applied. Among these the most effective and naturally the more popular ones are Bellman s method of dynamic programming and Pontryagin s principle of maximum. 

Of these three conditions derived by Pontryagin to treat the problem of state restraints, the most significant is that of Eq. (52), the time dependence of the additional Lagrange multiplier. The use of this condition is illustrated in the following section in the context of the xenon shutdown problem. 

Indirect or variational approaches are based on Pontryagin s maximum principle [8], in which the first-order optimality conditions are derived by applying calculus of variations. For problems without inequality constraints, the optimality conditions can be written as a set of DAEs and solved as a two-point boundary value problem. If there are inequality path constraints, additional optimality conditions are required, and the determination of entry and exit points for active constraints along the integration horizon renders a combinatorial problem, which is generally hard to solve. There are several developments and implementations of indirect methods, including [9] and [10]. 

---