John Multiplier Theorem

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 this section, we will derive the John Multiplier Theorem, which is a set of necessary conditions for the minimum of an objective functional constrained by inequahties. 

The John Multiplier Theorem can now be expressed by combining the results for both the cases as follows ... 

Hence from the serial application of the John Multiplier Theorem, [similar to that in Section 4.3.3.1 (p. 100)], the final augmented functional is given by... 

The approach in the preceding section may be followed to arrive at the following the John Multiplier Theorem for several inequality constraints ... 

Prom the Generalized John Multiplier theorem, the necessary conditions for the minimum are SM = 0 and... 

The augmented objective functional for this problem is M, which is given by Equation (6.15) on p. 163. From the John Multiplier Theorem (Section 4.5.1, p. 113), the following equations must be satisfied at the minimum of M, and equivalently of / ... 

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