Erratum for “Global Identifiability of Differential Models”

Abstract

We are grateful to Peter Thompson for pointing out an error in [1, Lemma 3.5, p. 1848]. The original proof worked only under the assumption that θ ̂ $\hat{\theta }$ is a vector of constants. However, some of the components of θ ̂ $\hat{\bm{\theta }}$ could be the states of the dynamic under consideration, and the lemma was used in such a setup (i.e., with θ ̂ $\hat{\bm{\theta }}$ involving states) later in [1, Proposition 3.4]. We give a more explicit version of the statement and provide a correct proof. The desired statement will be deduced from the following: Lemma 1.Consider a system of differential equations Proof.Consider the following differential ideal Now we will prove the claim. Consider the ring R : = C [ x , μ ] { u } [ 1 / Q ] $R := \mathbb {C}[\bm{x}, \bm{\mu }]\lbrace \bm{u}\rbrace [1/Q]$ . Let J be the ideal generated by I ∩ C [ x , μ ] { u } $I \cap \mathbb {C}[\bm{x}, \bm{\mu }]\lbrace \bm{u}\rbrace$ in R. The definition of I via the saturation at Q implies that Let R ∼ $\widetilde{R}$ be the localization of R with respect to C { u } $\mathbb {C}\lbrace \bm{u}\rbrace$ and J ∼ $\widetilde{J}$ be the ideal generated by J in this localization. The derivation L $\mathcal {L}$ can be naturally extended to R ∼ $\widetilde{R}$ , and J ∼ $\widetilde{J}$ is also L $\mathcal {L}$ -invariant. It is sufficient to prove that J ∼ ∩ C [ x , μ ] ≠ { 0 } $\widetilde{J}\cap \mathbb {C}[\bm{x}, \bm{\mu }] \ne \lbrace 0\rbrace$ . Consider a nonzero element of J ∼ ∩ C [ x , μ ] { u } $\widetilde{J} \cap \mathbb {C}[\bm{x}, \bm{\mu }]\lbrace \bm{u}\rbrace$ with the smallest number of monomials and, among such elements, an element of the smallest total degree. We will call it S. If S ∈ C [ x , μ ] $S\in \mathbb {C}[\bm{x}, \bm{\mu }]$ , we are done. Otherwise, one of u appears in S, say u1. Let h = ord u 1 S $h = \operatorname{ord}_{u_1}S$ . Since R ∼ $\widetilde{R}$ is a Noetherian ring, there exists N > 0 $N > 0$ such that The following corollary is equivalent to [1, Lemma 3.5, p. 1848] but explicitly highlights that some of the entries of θ ̂ $\hat{\bm{\theta }}$ may be initial conditions, not only system parameters. Corollary 1. (Clarified version of [[1], Lemma 3.5, p. 1848])In the notation of [1, Section 2.2], let P ( μ , x , u , … , u ( N ) ) ∈ C [ μ , x ] { u } $P(\bm{\mu }, \bm{x}, u, \ldots , u^{(N)})\in \mathbb {C}[\bm{\mu }, \bm{x}] \lbrace u \rbrace$ be nonzero. Then there exist nonempty Zariski open subsets Θ ⊂ C s $\Theta {\subset }\mathbb {C}^{s}$ and U ⊂ C ∞ ( 0 ) $U\subset \mathbb {C}^{\infty }(0)$ such that, for every θ ̂ = ( μ ̂ , x ̂ * ) ∈ Θ $\hat{\bm{\theta }} = (\hat{\bm{\mu }}, \hat{\bm{x}}^\ast )\in \Theta$ , u ̂ ∈ U $\hat{u}\in U$ , and the corresponding x ̂ = X ( θ ̂ , u ̂ ) $\hat{\bm{x}} = X(\hat{\bm{\theta }}, \hat{u})$ , the function P ( μ ̂ , x ̂ , u ̂ , … , ( u ̂ ) ( N ) ) $P(\hat{\bm{\mu }}, \hat{\bm{x}}, \hat{u},\ldots ,(\hat{u})^{(N)})$ is a nonzero element of C ∞ ( 0 ) $\mathbb {C}^{\infty }(0)$ . Proof.We apply Lemma 1 to the model Σ and the polynomial P as in the statement, and obtain polynomials P 1 ( x , μ ) $P_1(\bm{x}, \bm{\mu })$ and P2(u). We define Zariski open sets Θ and U by P 1 ≠ 0 $P_1 \ne 0$ and P 2 ( u ) | t = 0 ≠ 0 $P_2(\bm{u})|_{t = 0} \ne 0$ , respectively. Then the lemma implies that, for ( μ ̂ , x ̂ ∗ ) ∈ Θ $(\hat{\bm{\mu }}, \hat{\bm{x}}^*) \in \Theta$ and u ̂ ∈ U $\hat{u} \in U$ , P ( μ ̂ , x ̂ , u ̂ , … , ( u ̂ ) ( N ) ) $P(\hat{\bm{\mu }}, \hat{\bm{x}}, \hat{u},\ldots ,(\hat{u})^{(N)})$ will be a nonzero function. □ $\Box$

Locations

• Communications on Pure and Applied Mathematics - View - PDF