John Multiplier Theorem for Inequality Constraints

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. 

We begin with an objective functional J dependent on a control function u t) and subject to the constraint K u) ko- As before, we assume that both 

J and K are Gateaux differentiable since we will need the continuity of the differentials. 

Let J be minimum at u among all u satisfying the inequality constraint. Then K u) could be either ko or less. We will consider these two cases as follows  

Case 1 Here K u) = ko and the inequality constraint is said to be active. The augmented objective functional is M = J + pK where /it is a Lagrange multiplier. The Lagrange Multiplier Theorem yields 

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