Authors’ Reply to “Comments on “On optimal control of spatially distributed systems””

We thank the author in [1] for careful reading of our paper and pointing out the errors. It is very unfortunate that both of us and the reviewers missed these errors. In [2], the notion of spatially-decaying (SD) matrices is introduced. These are classes of infinite matrices defined by the off-diagonal decay of their entries. Families of coupling characteristic functions (e.g. exponential and algebraic off-diagonal decay) are defined to quantify off-diagonal decay of matrices. The objective of [2] is to show that the unique solution of Lyapunov and Riccati equations with SD coefficients are SD. These results are reported in Theorems 5 and 6 in [2], respectively. The proofs of these theorems are mainly based on convergence properties of the corresponding differential Lyapunov and Riccati equations and the result of Corollary 1 in [2]. The statement of Corollary 1 in [2] asserts that if a sequence of operators in the Banach algebra of SD matrices is convergent in the sense of `p , then it is also convergent in the Banach algebra of SD matrices. As pointed in [1] Corollary 1 in [2] is not true in general, resulting in the collapse of the proofs of Theorems 5 and 6. Even though some of the proofs as appeared in [2] are wrong, the statements of Theorem 5 in [2] and a modified version of Theorem 6 still hold under some additional assumptions on the coupling characteristic functions and the distance function. We assume that the following assumptions hold: the GRS-condition (2) which essentially requires that the coupling characteristic function decays sub-exponentially, a weak growth condition (5) on the coupling characteristic function, a growth condition (6) on the volume of a ball in the index set, and A being self-adjoint. We provide a proof for Theorem 5 in [2] which does not rely on Corollary 1 in [2]. A modified and more restrictive version of Theorem 6 in [2] remains valid. Essentially, we prove that the solution of the operator Riccati differential equation is in the Banach algebra on finite time intervals. Whether the result is true for infinite time, remains to be seen.