Elimination of the ODE Integration

The objective function in Problem 7.1.2 depends on rig unknown parameters s and on an unknown function x evaluated at rit different points. The ODE (7.1.6) as constraint provides information about the function x at infinitely many time points. Thus, such a constrained optimization problem can be viewed as a problem with an infinite dimensional constraint. In a first step the problem is reduced to a finite dimensional problem by eliminating these constraints. 

If we could express x ti s) as a function of the initial values and parameters s = (so O), then Problem 7.1.2 would reduce to the finite dimensional problem  

Unfortunately, in most cases x cannot be expressed as a function of s directly. In these cases x is obtained by numerically integrating the ODE 

Thus the ODE constraint (7.1.6) can only be eliminated for a given parameter value. Solving the entire problem requires an iterative procedure, which suggests at every iteration step a value s for the next step as a parametric input of the ODE (7.2.2). 

Choose initial estimates for the unknown parameters and initial values. Set k = 0. 

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