Erratum for “Global Identifiability of Differential Models”,Communications on Pure and Applied Mathematics

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Erratum for “Global Identifiability of Differential Models”
Communications on Pure and Applied Mathematics ( IF 3 ) Pub Date : 2023-09-22 , DOI: 10.1002/cpa.22163
Hoon Hong ₁ , Alexey Ovchinnikov ₂ , Gleb Pogudin ₃ , Chee Yap ₄

Affiliation

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 $\hat{θ} $\hat{\theta }$$ is a vector of constants. However, some of the components of $\hat{θ} $\hat{\bm{\theta }}$$ could be the states of the dynamic under consideration, and the lemma was used in such a setup (i.e., with $\hat{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

{\begin{matrix} x_{1}^{'} = f_{1} (x, μ, u), \\ ⋮ \\ x_{n}^{'} = f_{n} (x, μ, u), \end{matrix} $$\begin{equation} {\begin{cases} x_1^{\prime } = f_1(\bm{x}, \bm{\mu }, \bm{u}),\\ \vdots \\ x_n^{\prime } = f_n(\bm{x}, \bm{\mu }, \bm{u}), \end{cases}}\end{equation}$$

(1)

where

x = (x_{1}, …, x_{n}) $\bm{x} = (x_1, \ldots , x_n)$

and

u = (u_{1}, …, u_{m}) $\bm{u} = (u_1, \ldots , u_m)$

are tuples of differential indeterminates,

μ = (μ_{1}, …, μ_{λ}) $\bm{\mu } = (\mu _1, \ldots , \mu _\lambda )$

are scalar parameters, and

f_{1}, …, f_{n} \in C (x, μ, u) $f_1, \ldots , f_n \in \mathbb {C}(\bm{x}, \bm{\mu }, \bm{u})$

. Let

Q (x, μ, u) \in C [x, μ, u] $Q(\bm{x}, \bm{\mu }, \bm{u}) \in \mathbb {C}[\bm{x}, \bm{\mu }, \bm{u}]$

be the LCM of the denominators of

f_{1}, …, f_{n} $f_1, \ldots , f_n$

. Let

P \in C [x, μ] {u} $P \in \mathbb {C}[\bm{x}, \bm{\mu }]\lbrace \bm{u}\rbrace$

be a nonzero differential polynomial. Then there exist nonzero

P_{1} \in C [x, μ] $P_1 \in \mathbb {C}[\bm{x}, \bm{\mu }]$

and

P_{2} \in C {u} $P_2 \in \mathbb {C}\lbrace \bm{u}\rbrace$

such that, for every tuple

\hat{μ} \in C^{λ} $\hat{\bm{\mu }} \in \mathbb {C}^\lambda$

and every power series solution

(\hat{x}, \hat{u}) $(\hat{\bm{x}}, \hat{\bm{u}})$

of (1) with parameters

\hat{μ} $\hat{\bm{\mu }}$

C [[t]] $\mathbb {C}[\![t]\!]$

such that

Q (\hat{x}, \hat{μ}, \hat{u}) |_{t = 0} \neq 0 $$\begin{equation*} Q(\hat{\bm{x}}, \hat{\bm{\mu }}, \hat{\bm{u}})|_{t = 0} \ne 0 \end{equation*}$$

we have

(P_{1} (\hat{x}, \hat{μ}) {|_{t = 0} \neq 0 & P_{2} (\hat{u}) |}_{t = 0} \neq 0) \Rightarrow P (\hat{x}, \hat{μ}, \hat{u}) \neq 0 . $$\begin{equation*} {\left(P_1(\hat{\bm{x}}, \hat{\bm{\mu }})|_{t = 0} \ne 0 \;\&\; P_2(\hat{\bm{u}})|_{t = 0} \ne 0\right)} \Rightarrow P(\hat{\bm{x}}, \hat{\bm{\mu }}, \hat{\bm{u}}) \ne 0. \end{equation*}$$

Proof.Consider the following differential ideal

I : = ⟨ {(Q x_{i}^{'} - Q f_{i})}^{(j)}, P^{(j)} ∣ 1 ⩽ i ⩽ n, j ⩾ 0 ⟩ : Q^{\infty} \subset C [μ] {x, u} . $$\begin{equation*} I := \langle (Qx_i^{\prime } - Qf_i)^{(j)}, P^{(j)} \mid 1 \leqslant i \leqslant n,\; j \geqslant 0 \rangle : Q^\infty \subset \mathbb {C}[\bm{\mu }]\lbrace \bm{x}, \bm{u}\rbrace . \end{equation*}$$

We claim that I contains a nonzero polynomial of the form

P_{1} P_{2} $P_1P_2$

such that

P_{1} \in C [x, μ] $P_1 \in \mathbb {C}[\bm{x}, \bm{\mu }]$

and

P_{2} \in C {u} $P_2 \in \mathbb {C}\lbrace \bm{u}\rbrace$

. First we will show that, if the claim is true, then P₁ and P₂ satisfy the condition of the lemma. Assume the contrary, that there is a power series solution

(\hat{x}, \hat{u}) $(\hat{\bm{x}}, \hat{\bm{u}})$

of (1) with parameters

\hat{μ} $\hat{\bm{\mu }}$

such that the constant term of

Q (\hat{x}, \hat{μ}, \hat{u}) P_{1} (\hat{x}, \hat{μ}) P_{2} (\hat{u}) $Q(\hat{\bm{x}}, \hat{\bm{\mu }}, \hat{\bm{u}}) P_1(\hat{\bm{x}}, \hat{\bm{\mu }}) P_2(\hat{\bm{u}})$

is nonzero but

P (\hat{x}, \hat{μ}, \hat{u}) = 0 $P(\hat{\bm{x}}, \hat{\bm{\mu }}, \hat{\bm{u}}) = 0$

. Since

(\hat{x}, \hat{μ}, \hat{u}) $(\hat{\bm{x}}, \hat{\bm{\mu }}, \hat{\bm{u}})$

is a zero of differential polynomials P and

Q x_{i}^{'} - Q f_{i} $Qx_i^{\prime } - Qf_i$

for every

1 ⩽ i ⩽ n $1 \leqslant i \leqslant n$

, it is a zero of the ideal

⟨ {(Q x_{i}^{'} - Q f_{i})}^{(j)}, P^{(j)} ∣ 1 ⩽ i ⩽ n, j ⩾ 0 ⟩ . $$\begin{equation*} \langle (Qx_i^{\prime } - Qf_i)^{(j)}, P^{(j)} \mid 1 \leqslant i \leqslant n,\; j \geqslant 0 \rangle . \end{equation*}$$

Since

Q (\hat{x}, \hat{μ}, \hat{u}) |_{t = 0} \neq 0 $Q(\hat{\bm{x}}, \hat{\bm{\mu }}, \hat{\bm{u}})|_{t = 0} \ne 0$

, every element in I, which is the saturation of the above ideal at Q, also vanishes at

(\hat{x}, \hat{μ}, \hat{u}) $(\hat{\bm{x}}, \hat{\bm{\mu }}, \hat{\bm{u}})$

. In particular,

P_{1} P_{2} $P_1P_2$

vanishes at

(\hat{x}, \hat{μ}, \hat{u}) $(\hat{\bm{x}}, \hat{\bm{\mu }}, \hat{\bm{u}})$

, so we arrive at the contradiction with

P_{1} (\hat{x}, \hat{μ}) P_{2} (\hat{u}) \neq 0 $P_1(\hat{\bm{x}}, \hat{\bm{\mu }}) P_2(\hat{\bm{u}}) \ne 0$

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]$$ . Let J be the ideal generated by $I \cap C [x, μ] {u} $I \cap \mathbb {C}[\bm{x}, \bm{\mu }]\lbrace \bm{u}\rbrace$$ in R. The definition of I via the saturation at Q implies that

J \cap C [x, μ] {u} = I \cap C [x, μ] {u} . $$\begin{equation*} J \cap \mathbb {C}[\bm{x}, {\bm{\mu }}]\lbrace \bm{u}\rbrace = I \cap \mathbb {C}[\bm{x}, {\bm{\mu }}]\lbrace \bm{u}\rbrace . \end{equation*}$$

Thus, it is sufficient to prove that there is an element of the form

P_{1} P_{2} $P_1 P_2$

with

P_{1} \in C [x, μ] $P_1 \in \mathbb {C}[\bm{x}, \bm{\mu }]$

and

P_{2} \in C {u} $P_2 \in \mathbb {C}\lbrace \bm{u}\rbrace$

in J. We define a derivation

L $\mathcal {L}$

on R (which is basically the Lie derivative) by

L (g) : = \sum_{i = 1}^{n} f_{i} \frac{\partial g}{\partial x_{i}} + \sum_{ℓ = 1}^{m} \sum_{j = 0}^{\infty} u_{ℓ}^{(j + 1)} \frac{\partial g}{\partial u_{ℓ}^{(j)}} for g \in R . $$\begin{equation*} \mathcal {L}(g) := \sum \limits _{i = 1}^n f_i\frac{\partial g}{\partial x_i} + \sum \limits _{\ell = 1}^m \sum \limits _{j = 0}^\infty u_\ell ^{(j + 1)} \frac{\partial g}{\partial u_\ell ^{(j)}}\quad \text{ for } g\in R. \end{equation*}$$

Since

Q x_{1}^{'} - Q f_{1}, …, Q x_{n}^{'} - Q f_{n} \in I $Qx_1^{\prime } - Qf_1, \ldots , Qx_n^{\prime } - Qf_n \in I$

and I is a differential ideal, J is invariant under

L $\mathcal {L}$

Let $\tilde{R} $\widetilde{R}$$ be the localization of R with respect to $C {u} $\mathbb {C}\lbrace \bm{u}\rbrace$$ and $\tilde{J} $\widetilde{J}$$ be the ideal generated by J in this localization. The derivation $L $\mathcal {L}$$ can be naturally extended to $\tilde{R} $\widetilde{R}$$ , and $\tilde{J} $\widetilde{J}$$ is also $L $\mathcal {L}$$ -invariant. It is sufficient to prove that $\tilde{J} \cap C [x, μ] \neq {0} $\widetilde{J}\cap \mathbb {C}[\bm{x}, \bm{\mu }] \ne \lbrace 0\rbrace$$ . Consider a nonzero element of $\tilde{J} \cap C [x, μ] {u} $\widetilde{J} \cap \mathbb {C}[\bm{x}, \bm{\mu }]\lbrace \bm{u}\rbrace$$ with the smallest number of monomials and, among such elements, an element of the smallest total degree. We will call it S. If $S \in C [x, μ] $S\in \mathbb {C}[\bm{x}, \bm{\mu }]$$ , 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 $\tilde{R} $\widetilde{R}$$ is a Noetherian ring, there exists $N > 0 $N > 0$$ such that

L^{N} (S) \in ⟨ S, L (S), …, L^{N - 1} (S) ⟩ . $$\begin{equation*} \mathcal {L}^N(S) \in \langle S, \mathcal {L}(S), \ldots , \mathcal {L}^{N - 1}(S) \rangle . \end{equation*}$$

We have

{ord}_{u_{1}} L^{i} (S) < N + h $\operatorname{ord}_{u_1} \mathcal {L}^i(S) < N + h$

for

i < N $i < N$

and

L^{N} (S) = \frac{\partial S}{\partial u_{1}^{(h)}} u_{1}^{(h + N)} + T, where {ord}_{u_{1}} T < N + h . $$\begin{equation*} \mathcal {L}^N(S) = \frac{\partial S}{\partial u_1^{(h)}} u_1^{(h + N)} + T, \quad \text{ where } \operatorname{ord}_{u_1}T < N + h. \end{equation*}$$

Therefore, we have

\frac{\partial S}{\partial u_{1}^{(h)}} \in ⟨ S, L (S), …, L^{N - 1} (S) ⟩ \subset \tilde{J} . $$\begin{equation*} \frac{\partial S}{\partial u_1^{(h)}} \in \langle S, \mathcal {L}(S), \ldots , \mathcal {L}^{N - 1}(S) \rangle \subset \widetilde{J}. \end{equation*}$$

If S were divisible by

u_{1}^{(h)} $u_1^{(h)}$

, then

S / u_{1}^{(h)} \in \tilde{J} $S / u_1^{(h)} \in \widetilde{J}$

would have the same number of monomials but smaller degree, this contradicts to the choice of S. Therefore,

\frac{\partial S}{\partial u_{1}^{(h)}} $\frac{\partial S}{\partial u_1^{(h)}}$

has fewer monomials than S thus contradicting the choice of S.

□ $\Box$

The following corollary is equivalent to [1, Lemma 3.5, p. 1848] but explicitly highlights that some of the entries of $\hat{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)}) \in C [μ, x] {u} $P(\bm{\mu }, \bm{x}, u, \ldots , u^{(N)})\in \mathbb {C}[\bm{\mu }, \bm{x}] \lbrace u \rbrace$$ be nonzero. Then there exist nonempty Zariski open subsets $Θ \subset C^{s} $\Theta {\subset }\mathbb {C}^{s}$$ and $U \subset C^{\infty} (0) $U\subset \mathbb {C}^{\infty }(0)$$ such that, for every $\hat{θ} = (\hat{μ}, {\hat{x}}^{*}) \in Θ $\hat{\bm{\theta }} = (\hat{\bm{\mu }}, \hat{\bm{x}}^\ast )\in \Theta$$ , $\hat{u} \in U $\hat{u}\in U$$ , and the corresponding $\hat{x} = X (\hat{θ}, \hat{u}) $\hat{\bm{x}} = X(\hat{\bm{\theta }}, \hat{u})$$ , the function $P (\hat{μ}, \hat{x}, \hat{u}, …, {(\hat{u})}^{(N)}) $P(\hat{\bm{\mu }}, \hat{\bm{x}}, \hat{u},\ldots ,(\hat{u})^{(N)})$$ is a nonzero element of $C^{\infty} (0) $\mathbb {C}^{\infty }(0)$$ .

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 })$$ and P₂(u). We define Zariski open sets Θ and U by $P_{1} \neq 0 $P_1 \ne 0$$ and $P_{2} {(u) |}_{t = 0} \neq 0 $P_2(\bm{u})|_{t = 0} \ne 0$$ , respectively. Then the lemma implies that, for $(\hat{μ}, {\hat{x}}^{*}) \in Θ $(\hat{\bm{\mu }}, \hat{\bm{x}}^*) \in \Theta$$ and $\hat{u} \in U $\hat{u} \in U$$ , $P (\hat{μ}, \hat{x}, \hat{u}, …, {(\hat{u})}^{(N)}) $P(\hat{\bm{\mu }}, \hat{\bm{x}}, \hat{u},\ldots ,(\hat{u})^{(N)})$$ will be a nonzero function. $□ $\Box$$

更新日期：2023-09-22

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