Abstract
It is well known that singular problems may fail to have optimal solution in the class of “ordinary” (say, square-integrable) controls, even in the cases where the cost is bounded from below. In this paper, we suggest a method for overcoming this difficulty by defining an order r of singularity of the problem and extending both the input-trajectory map and the cost functional to an adequate subspace of the Sobolev space H_r-r. We show that the extended problem has a minimum if and only if the infimum of the original problem is finite. The extended problem can be transformed in a “natural” way into a regular L-Q problem with strictly smaller controllable space and (possibly) smaller control space. We use this transformation to describe the structure of relaxed optimal controls and the corresponding relaxed trajectories. We provide a method for computing optimal relaxed solutions from the solution of an adequate Riccati differential equation. We also show how the relaxed minimizers can be approximated by square-integrable functions.
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Guerra, M. Highly Singular L-Q Problems: Solutions in Distribution Spaces. Journal of Dynamical and Control Systems 6, 265–309 (2000). https://doi.org/10.1023/A:1009534204516
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DOI: https://doi.org/10.1023/A:1009534204516