Bang-bang controls for parabolic and pseudoparabolic problems

We study the control of the family of problems where ?? is a bounded domain in Rn, Q = ?? ?? (0, T) and ?? = ???? ?? (0, T) (1 - ????)y t (??) - ??y(??) = v(t)??(x) in Q y (??) (0) = 0 in ?? y (??) | ?? = 0 minimize J(v) = ????(y (??) (x,T;v) - z(x))dx subject to |v(t)| ?? 1 a.e. in [0, T]. Existence, bang-bang, switching, and uniqueness properties are investigated. Further, conditions are studied which imply the almost everywhere convergence of the bang-bang controls as ?? ?? 0.