Lagrange Multiplier Rule constraints

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 ... 

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. 

Lagrange Multiplier Rule for Several Equality Constraints... 

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. 

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. 

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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