Correction to: Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations

1 Correction to: Math. Program. Ser. B (2018) 168:717–757 https://doi.org/10.1007/s10107-016-1093-4

1.1 Closure of the set of primitives of critical directions

We need to correct Proposition 6 as follows. Define the new set

$$\begin{aligned} \widehat{PC_2}(\hat{u}) := \left\{ \begin{array}{l} (w,h)\in L^2(0,T)\times {\mathcal {R}};\, w \text { is constant over boundary arcs,}\\ w=0\text { over an initial boundary arc,}\\ w= h\text { over a terminal boundary arc} \end{array} \right\} .\nonumber \\ \end{aligned}$$

(E1)

Proposition 6

Let $\hat{u}\in {\mathcal {U}}_{ad}$ satisfy (4.24)–(4.25). Then $PC_2(\hat{u})$, defined before (3.45), satisfies

$$\begin{aligned} PC_2(\hat{u}) = \{ (w,h) \in \widehat{PC_2}(\hat{u});\, w\text { is continuous at bang-bang junctions} \}. \end{aligned}$$

(E2)

Proof

The proof is a simplified version of the one of Proposition 4 in [3]. That result dealt with problems with both upper and lower bounds on the control, as well as state constraints, the latter being absent in the present setting.

Remark 1

When $\hat{u}$ has no bang-bang switch, the cones $PC_2(\hat{u})$ and $\widehat{PC_2}(\hat{u})$ coincide.

1.2 Sufficient optimality conditions

The statements of Theorem 8, 10 and 11 have to be modified in the following way:

Theorem 8

Let $\hat{u}$ be a weak minimum for problem (P), satisfying (4.24)–(4.26). Then the following assertions hold.

• (a) If there exists $\alpha >0$ such that

  $$\begin{aligned} \hat{Q}(\xi [w],w,h) \ge \alpha ( \Vert w\Vert ^2_2 + h^2), \quad \text {for all }(w,h) \in \widehat{PC_2}(\hat{u}), \end{aligned}$$

  (E3)

  then the quadratic growth condition (4.29) is satisfied.

• (b) If the quadratic growth condition (4.29) is satisfied, then (4.28) holds.

Remark 2

(i) The proof of the above theorem is essentially identical to the one in the published version of the article. It is enough to change $PC_2(\hat{u})$ into $\widehat{PC_2}(\hat{u})$, two lines from below, on p. 741, and at the end of step 1.

(ii) When $\hat{u}$ has no bang-bang switch, the cones $PC_2(\hat{u})$ and $\widehat{PC_2}(\hat{u})$ coincide and, therefore, the necessary and sufficient conditions have no gap. Hence, in the absence of bang-bang switchings Proposition 6 and Theorem 8 hold as they are in the published version.

Change item (iii) of Theorem 10 into

• (iii) if additionally (4.24)–(4.26) are satisfied, then the following assertions hold.

  • (a) If there exists $\alpha >0$ such that (E3) holds, then the quadratic growth condition (4.29) is satisfied.

  • (b) If the quadratic growth condition (4.29) is satisfied, then (4.28) holds.

Change item (iii) of Theorem 11 into

• (iii) if additionally (4.24)–(4.26) are satisfied, then the following assertions hold.

  • (a) If there exists $\alpha >0$ such that (E3) holds, then the quadratic growth condition (4.29) is satisfied.

  • (b) If the quadratic growth condition (4.29) is satisfied, then (4.28) holds.

Remark 3

Analogously as stated in Remark 2, when the optimal control has no bang-bang switch, Theorem 10 and 11 hold as they are in the published version.

The authors thank Gerd Wachsmuth, from Brandenburgische Technische Universität Cottbus—Senftenberg (BTU), for mentioning a counter example to the former version of Proposition 6.