Remark on Numerical Solutions

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. 

Alternatively, increasing penalties may be applied on constraint violations during repeated applications of any computational algorithm used for unconstrained problems. We will use the latter approach in Chapter 7 to solve optimal control problems constrained by (in)equalities. 

When using Lagrange multipliers in the rest of the book, we will skip mentioning the preconditions assuming that they are satisfied. 

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