MORPHOLOGICAL CONTROL PROBLEMS WITH STATE CONSTRAINTS

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 R-N. Their family K(R-N), however, does not have any obvious linear structure, but in combination with the popular Pompeiu-Hausdorff distance d 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(R-N), d). Now various control problems, such as open-loop, relaxed, and closed-loop control problems, are formulated for compact sets depending on time, 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. Finally, this framework is applied to image segmentation and provides a region growing method without regularity restrictions.