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Existence of minimizers for optical flow based optimal control problems under mild regularity assumptions

Jarde Philipp, Ulbrich Michael

Control and Cybernetics

2019

Vol. 48, No. 2

261--306

Optimal control problems governed by a transport equation are investigated that are motivated by optical flow problems. The control is given by the velocity field, corresponding to the optical flow, while the state corresponds to the brightness of image points. The problem is studied in the setting of spatially BVregular vector fields under very low regularity requirements. Existing stability results for the control-to-state operator are improved and based on this the existence of minimizers for several classes of optimal control problems is proved under mild assumptions on the admissible sets.