Quadratic Optimal Control: Unifying PDE Solutions

The representation of the unique solution to quadratic optimal control problems with linear PDE constraints, as the solutions of a single higher order PDE using Riesz operators and pivot Hilbert spaces. This has applications to control problems involving non-negative characteristic forms.

In PDE constrained optimisation problems, there are many different approaches as formulations one can consider. One particular approach is to consider the functions which are in the kernal of the first variations of the associated Lagrangian. For quadratic cost functionals subject to linear PDE constrains this yeilds a system referred to as the Primal, Dual and Control equations. Solving these equations simulanteously yields the minimising configuration. 

However, through the application of Riesz operators we are able to demonstrate the solution to this system of equations satisfies a single PDE whose order is at least equal to the Primal or at most twice that of the primal. With the approach we are able to demonstrate that if there exists a unique solution to the constraint, for a given source term, then there exists a unique solution to the optimal control problem. Additionally, we are able to explicitly derive the Riesz operator in a number of key examples using pivot spaces. This analysis has application to many optimal control problems, such as the treatment planning problem for radiation therapy

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