Correction to: Continuous Differentiability of the Value Function of Semilinear Parabolic Infinite Time Horizon Optimal Control Problems on L²(Ω ) Under Control Constraints

There is an error in Step 3 of the proof of the Lemma 4.8 of the original paper.In the new version this error is corrected by using the fact that for t large enough the control constraint becomes inactive.Lemma 4.8 Let assumptions (A) hold and let ( ȳ, ū), and p denote a local solution and associated adjoint state to (P) corresponding to an initial datum y 0 ∈ B Y ( δ2 ).Then, possibly after further reduction of V , the mappingProof Step1 : For β ∈ V , with V as in Remark 4.2, let (y (β) , u (β) , p (β) , p 1(β) ) be the solution to T (y, u, p, p 1 ) = β.As a consequence of (A5) it is also a solution to T (y, u, p, p(0)) = β with p (β) ∈ W ∞ .Thus the first two equations of this latter equality can be expressed as (4.35a)