Compactness of Fixed Point Maps and the Ball-Marsden-Slemrod Conjecture

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Abstract
Given a parameter dependent fixed point equation [Formula: see text], we derive an abstract compactness principle for the fixed point map [Formula: see text] under the assumptions that (i) the fixed point equation can be solved by the contraction principle and (ii) the map [Formula: see text] is compact for fixed [Formula: see text]. This result is applied to infinite-dimensional, semilinear control systems and their reachable sets. More precisely, we extend a noncontrollability result of Ball, Marsden, and Slemrod [ SIAM J. Control Optim., 20 (1982), pp. 575–597] to semilinear systems. First we consider [Formula: see text]-controls, [Formula: see text]. Subsequently we analyze the case [Formula: see text].

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