Bang-bang controls for parabolic and pseudoparabolic problems

We study the control of the family of problems where Ω is a bounded domain in Rn, Q = Ω × (0, T) and Σ = аΩ × (0, T) (1 - εΔ)y t (ε) - Δy(ε) = v(t)ε(x) in Q y (ε) (0) = 0 in Ω y (Ω) | Σ = 0 minimize J(v) = ∫Ω(y (ε) (x,T;v) - z(x))dx subject to |v(t)| ε 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 ε → 0.