Lagrange Multiplier Rule

To proceed, we need to apply the Lagrange Multiplier Rule, the details of which will be provided later in Chapter 4. According to this rule, the above constrained problem is equivalent to the problem of finding the control T t) that maximizes the following augmented functional ... 

The problem at hand has only two constraints, namely. Equations (2.27) and (2.28). With the application of the Lagrange Multiplier Rule, this problem is equivalent to the minimization of the following augmented objective functional ... 

Arising from 6J = 0, both Equations (3.11) and (3.12) are the necessary conditions for the optimum of J. On the basis of the Lagrange Multiplier Rule, these two equations are also the necessary conditions for the constrained optimum of I. Both optima are still subject to the given initial condition, i. e.. Equation (3.6). The equivalence between the two optima will be shown later in Section 4.3.3. 

Using the Lagrange Multiplier Rule, the augmented functional is... 

In this chapter, we introduce the concept of Lagrange multipliers. We show how the Lagrange Multiplier Rule and the John Multiplier Theorem help us handle the equality and inequality constraints in optimal control problems. 

In Section 3.2.1 (p. 59), we had asserted the Lagrange Multiplier Rule that the optimum of the augmented J is equivalent to the constrained optimum of I. This rule is based on the Lagrange Multiplier Theorem, which provides the necessary conditions for the constrained optimum. We will first prove this theorem and then apply it to optimal control problems subject to different types of constraints. 

Observe that Equations (4.2) and (4.3), which arise from 5J y-, 5y) = 0, are also the necessary conditions for the optimmn of J. This fact gives rise to the following Lagrange Multiplier Rule ... 

Lagrange Multiplier Rule for Several Equality Constraints... 

We will apply the Lagrange Multiplier Rule to obtain the set of necessary conditions for the optimum in an optimal control problem constrained by a differential equation. In Section 3.2, we asserted the rule and obtained the following necessary conditions (see p. 60) ... 

We will first show that the differential equation poses a series of equality constraints along the t-direction. Then we will apply the Lagrange Multiplier Rule for the optimum of I subject to those constraints. 

Consider the problem in Example 3.1 (p. 65). According to the Lagrange Multiplier Rule, the minimum of... 

When solving an inequality-constrained optimal control problem numerically, it is impossible to determine which constraints are active. The reason is one cannot obtain a p, exactly equal to zero. This difficulty is surmounted by considering a constraint to be active if the corresponding p < a where a is a small positive number such as 10 or less, depending on the problem. Slack variables may be used to convert inequalities into equalities and utilize the Lagrange Multiplier Rule. 

Based on the Lagrange Multiplier Rule (see Section 4.3.3.2, p. 103), the above problem is equivalent to minimizing the augmented functional... 

Recall from Section 4.3 (p. 88) that it means having continuous partial derivatives of the integrands — a precondition for the Lagrange Multiplier Rule. 

The state equation, G = 0, constitutes a partial differential equation constraint. Applying the Lagrange Multiplier Rule, the equivalent problem is to find the control function D c) that minimizes the augmented objective functional... 

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