Control problems for nonlocal set evolutions
نویسندگان
چکیده
In this paper, we extend fundamental notions of control theory to evolving compact subsets of the Euclidean space – as states without linear structure. Dispensing with any restriction of regularity, shapes can be interpreted as nonempty compact subsets of the Euclidean space RN . Their family K(RN ), however, does not have any obvious linear structure, but in combination with the popular Pompeiu-Hausdorff distance dl, it is a metric space. Here Aubin’s framework of morphological equations is used for extending ordinary differential equations beyond vector spaces, namely to the metric space (K(RN ), dl). Now various control problems are formulated for compact sets depending on time: open-loop, relaxed and closed-loop control problems – each of them with state constraints. Using the close relation to morphological inclusions with state constraints, we specify sufficient conditions for the existence of compact-valued solutions. AMS subject classifications. 37N35, 34A60, 49J24, 93B03, 93B52
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