“微分模型的全局可辨识性”的勘误

IF 4.3 3区材料科学 Q1 ENGINEERING, ELECTRICAL & ELECTRONIC ACS Applied Electronic Materials Pub Date : 2023-09-22 DOI:10.1002/cpa.22163

Hoon Hong, Alexey Ovchinnikov, Gleb Pogudin, Chee Yap

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引用次数: 0

摘要

我们感谢彼得·汤普森指出了[1，引理3.5，第1848页]中的一个错误。最初的证明只有在θ´$\hat{\ θ}$是一个常数向量的假设下才有效。然而，θ θ $\hat{\bm{\theta}}$的一些分量可以是所考虑的动态状态，引理在后面的[1，命题3.4]中被用于这样的设置(即θ θ $\hat{\bm{\theta}}$涉及状态)。我们给出了一个更明确的版本，并提供了一个正确的证明。我们想要的表述将从下面推导出来:引理1。考虑一个微分方程组x1 ' = f1 (x,μ,u),⋮xn”= fn (x,μ,u), $ $ \开始{方程}{\{病例}开始x_1 ^ {\ '} = f (\ bm {x} \ bm{\μ}\ bm{你}),\ \ \ vdots \ \ x_n ^ {\ '} = fn (\ bm {x} \ bm{\μ}\ bm{你}),结束\{病例}}\{方程}$ $结束(1)x = (x1,…,xn)美元\ bm {x} = (x_1、\ ldots x_n)美元和u = (u1,……,嗯)美元\ bm{你}= (u_1 \ ldots, u_m)美元的元组微分不定,μ=(μ1,…,μλ)美元\ bm{\μ}=(\μ_1 \ ldots \μ_ \λ)美元是标量参数,和f1,…,fn∈C (x,μ,u) $ f \ ldots,f_n \in \mathbb {C}(\bm{x}， \bm{\mu}， \bm{u})$。让问(x,μ,u)∈C (x,μ,u)问美元(\ bm {x} \ bm{\μ}\ bm{你})中\ \ mathbb {C} [\ bm {x} \ bm{\μ}\ bm{你}]美元的LCM f1的分母,…,fn $ f \ ldots fn美元。设P∈C[x，μ]{u}$P \in \mathbb {C}[\bm{x}， \bm{\mu}]\lbrace \bm{u}\rbrace$是一个非零微分多项式。然后存在非零P1∈C (x,μ)P_1美元\ \ mathbb {C} [\ * bm {x} \ bm{\μ}]美元和P2∈C{你}$ P_2 \ \ mathbb {C} \ lbrace \ bm{你}\ rbrace美元,每一个元组μ̂Cλ∈\美元的帽子{\ bm{\μ}}\中\ mathbb {C} ^ \λ美元每个幂级数解(x̂,u) $ ({\ * bm {x}} \帽子,帽子\ {\ bm{你}})美元(1)的参数μ̂美元\帽子{\ bm{\μ}}在C [[t]]美元\ mathbb {C} [\ [t] \ !]美元这样问(x̂,μ̂,u) | t = 0≠0 $ $ \{方程*}开始问(\帽子{\ * bm {x}} \帽子{\ bm{\μ}},\帽子{\ bm{你}})| _ {t = 0} \ 0 ne \{方程*}$ $结束我们P1 (x̂,μ̂)| t = 0≠0,P2 (u) | t = 0≠0⇒P (x̂,μ̂,u)≠0。$ ${方程*}{\ \开始离开(P_1 ({\ * bm {x}} \帽子,帽子\ {\ bm{\μ}})| _ {t = 0} \ 0 ne \; \, \;P_2(\帽子{\ bm{你}})| _ {t = 0} \ 0 ne \右)}\ Rightarrow P(\帽子{\ * bm {x}} \帽子{\ bm{\μ}},帽子\ {\ bm{你}})\ 0。结束\{方程*}$ $的证明。考虑下面的微分理想我:=⟨(Qxi−Qfi) (j)、P (j)∣1⩽我⩽n, j⩾0⟩:问∞⊂C(μ){x, u}。$ $ \{方程*}我开始:= \ langle (Qx_i ^ {\ '} - Qf_i) ^ {(j)}, P ^ {(j)} \ 1 \ leqslant我\ leqslant n,中期\;j \geqslant 0 \rangle: Q^\ inty \子集\mathbb {C}[\bm{\mu}]\lbrace \bm{x}， \bm{u}\rbrace。我们声明I包含一个形式为P1P2$P_1P_2$的非零多项式，使得P1∈C[x，μ]$P_1 \in \mathbb {C}[\bm{x}， \bm{\mu}]$和P2∈C{u}$P_2 \in \mathbb {C}\lbrace \bm{u}\rbrace$。首先，我们将证明，如果断言为真，那么P1和P2满足引理的条件。相反,假设有一个幂级数解(x̂,u) $ ({\ * bm {x}} \帽子,帽子\ {\ bm{你}})美元(1)的参数μ̂美元\帽子{\ bm{\μ}}$的常数项Q (x̂,μ̂,u) P1 (x̂,μ̂)P2 (u)问美元({\ * bm {x}} \帽子,帽子\ {\ bm{\μ}},帽子\ {\ bm{你}})P_1 ({\ * bm {x}} \帽子,帽子\ {\ bm{\μ}})P_2(\帽子{\ bm{你}})美元零但P (x̂,μ̂,u) = 0 $ P ({\ * bm {x}} \帽子,帽子\ {\ bm{\μ}},帽子\ {\ bm{你}})= 0美元。由于(x³，μ³，û)$(\hat{\bm{x}}， \hat{\bm{\mu}}， \hat{\bm{u}})$是微分多项式P和Qxi ' - Qfi$Qx_i^{\素数}- Qf_i$的零，对于每一个1≤i≤n$1 \leqslant i \leqslant n$，它是理想⟨(Qxi ' - Qfi)(j)，P(j)∣1≤i≤n,j≠0⟩的零。$ ${方程*}\ \开始langle (Qx_i ^ {\ '} - Qf_i) ^ {(j)}, P ^ {(j)} \中期1 \ leqslant \ leqslant n \;J \geqslant 0 \rangle。\{方程*}$ $结束以来Q (x̂,μ̂,u) | t = 0≠0美元Q ({\ * bm {x}} \帽子,帽子\ {\ bm{\μ}},帽子\ {\ bm{你}})| _ {t = 0} \ 0美元,在我每一个元素,这是上面的理想的饱和,也消失在(x̂,μ̂,u)美元({\ * bm {x}} \帽子,帽子\ {\ bm{\μ}},帽子\ {\ bm{你}})美元。特别是,P1P2 P_1P_2消失在美元(x̂,μ̂,u)美元(\帽子{\ * bm {x}} \帽子{\ bm{\μ}},\帽子{\ bm{你}})美元,我们到达的矛盾与P1 (x̂,μ̂)P2 (u)≠0美元P_1(\帽子{\ * bm {x}} \帽子{\ bm{\μ}})P_2(\帽子{\ bm{你}})\ 0美元。现在我们来证明这个说法。考虑环R: = C (x,μ){你}[1 / Q] $ R: = \ mathbb {C} [\ * bm {x} \ bm{\μ}]\ lbrace \ bm{你}\ rbrace [1 / Q]美元。设J是r中I∩C[x，μ]{u}$I \cap \mathbb {C}[\bm{x}， \bm{\mu}]\lbrace \bm{u}\rbrace$生成的理想。通过Q处的饱和定义I意味着J∩C[x，μ]{u}=I∩C[x，μ]{u}。$ $ \{方程*}开始J \帽\ mathbb {C} [\ bm {x}, {\ bm{\μ}}]\ lbrace \ bm{你}\ rbrace =我\帽\ mathbb {C} [\ bm {x}, {\ bm{\μ}}]\ lbrace \ bm{你}\ rbrace。因此，足以证明存在一个形式为P1P2$P_1 P_2$的元素，其中P1∈C[x，μ]$P_1 \in \mathbb {C}[\bm{x}， \bm{\mu}]$， P2∈C{u}$P_2 \in \mathbb {C}\lbrace \bm{u}\rbrace$ in j。我们在R上定义一个导数L$\mathcal {L}$(基本上是李氏导数):=∑i=1nfi∂g∂xi+∑j=0∞u R (j+1)∂g∂u R (j)forg∈R。$ ${方程*}\ \开始mathcal {1} (g): = \ \和限制_ {i = 1} ^ n f_i \压裂{\部分g}{\部分x_i} + \ \和限制_{\魔法= 1}^ m \ \和限制_ {j = 0} ^ \ infty u_ \字母L ^ {(j + 1)} \压裂{\部分g}{\部分u_ \魔法^ {(j)}} \四\文本为}{g \ R。由于Qx1 '−Qf1，…，Qxn '−Qfn∈I$Qx_1^{\素数}- Qf_1， \ldots, Qx_n^{\素数}- Qf_n \in I$和I是微分理想，J在L$\mathcal {L}$下是不变的。设R ~ $\ widdetilde {R}$是R对C{u}$\mathbb {C}\lbrace \bm{u}\rbrace$的定位，J ~ $\ widdetilde {J}$是J在这个定位中产生的理想值。推导L$\mathcal {L}$可以自然地扩展到R ~ $\ widdetilde {R}$，并且J ~ $\ widdetilde {J}$也是L$\mathcal {L}$-不变的。足以证明J ~∩C[x，μ]≠{0}$\ widdetilde {J}\cap \mathbb {C}[\bm{x}， \bm{\mu}] \ne \lbrace 0\rbrace$。考虑一个J ~∩C[x，μ]{u}$\ widdetilde {J} \cap \mathbb {C}[\bm{x}， \bm{\mu}]\lbrace \bm{u}\rbrace$的非零元素，它具有最小的单项式个数，并且在这些元素中，有一个总度最小的元素。我们称之为S。如果S∈C[x，μ]$S\in \mathbb {C}[\bm{x}， \bm{\mu}]$，我们就完成了。否则，u中的一个出现在S中，比如u1。设h=ordu1S$h = \operatorname{ord}_{u_1}S$。由于R ~ $\widetilde{R}$是一个诺瑟环，因此存在N>0$N >0$使得LN(S)∈⟨S,L(S)，…，LN−1(S)⟩。$ ${方程*}\ \开始mathcal {L} ^ N (S) \ \ langle年代\ mathcal {1} (S) \ ldots \ mathcal {L} ^ {N - 1} \捕杀。$$我们有ordu1Li(S)<N+h$\operatorname{ord}_{u_1} \mathcal {L}^i(S) <N$i <N$i <N和LN (S) =∂年代美元∂u1 (h) u1 (h + N) + T, whereordu1T< N + h。$ ${方程*}\ \开始mathcal {L} ^ N (S) = \压裂{\部分年代}{\部分u_1 ^ {(h)}} u_1 ^ {(h + N)} + T,{在}\ \四\文本operatorname{奥德}_ {u_1} T & lt;N + h。\{方程*}$ $因此结束,我们有∂年代∂u1 (h)∈⟨年代,L (S),…,LN−1 (S)⟩⊂J∼。$$\begin{equation*} \frac{\partial S}{\partial u_1^{(h)}} \in \langle S， \mathcal {L}(S)， \ldots， \mathcal {L}^{N - 1}(S) \rangle \子集\ widdetilde {J}。$$ $如果S可以被u1(h)$u_1^{(h)}$整除，那么S/u1(h)∈J ~ $S / u_1^{(h)} \in \ widdetilde {J}$将具有相同数量的单项式，但阶数更小，这与S的选择相矛盾。因此，∂S∂u1(h)$\frac{\偏S}{\偏u_1^{(h)}}$的单项式比S少，因此与S的选择相矛盾。□$\Box$P. 1848]但明确地强调了θ n $\hat{\bm{\theta}}$的一些条目可能是初始条件，而不仅仅是系统参数。推论1。(澄清版本的[[1],引理3.5,p . 1848])的符号(1,2.2节),让p(μx, u,…,u (N))∈C[μ,x]{你}$ p (\ bm{\μ}\ bm {x}, u, \ ldots u ^ {(N)})中\ \ mathbb {C} [\ * bm{\μ}\ bm {x}] \ lbrace u \ rbrace美元是零。然后存在非空的Zariski开子集Θ⊂Cs美元\θ{\子集}\ mathbb {C} ^{年代}$和U⊂C∞(0)美元U \ \子集mathbb {C} ^ {\ infty}(0),美元每θ̂=(μ̂,x̂*)∈Θ美元\帽子{\ bm{\θ}}=(\帽子{\ bm{\μ}},帽子\ {\ bm {x}} ^ \ ast) \ \θ,美元U∈\帽子{你}\ U的美元,和相应的x̂= x(θ̂,U)美元\帽子{\ bm {x}} = x(\帽子{\ bm{\θ}},{你}\帽子),美元函数P(μ̂,x̂,U,…,(U) (N)) $ P(\帽子{\ bm{\μ}},{\ * bm {x}} \帽子,\帽子{你}\ ldots,帽子(\{你})^ {(N)})是一个非零元素的C∞(0)美元\ mathbb {C} ^ {\ infty} (0) .Proof美元。我们将引理1应用于模型Σ和命题中的多项式P，得到多项式P1(x，μ)$P_1(\bm{x}， \bm{\mu})$和P2(u)。我们分别用P1≠0$P_1 \ne 0$和P2(U)|t=0≠0$P_2(\bm{U})|_{t =0} \ne 0$来定义Zariski开集Θ和U。引理意味着,(μ̂,x̂∗)∈Θ美元(\帽子{\ bm{\μ}},帽子\ {\ bm {x}} ^ *) \中\θ和u∈\美元的帽子在u ${你}\ P(μ̂,x̂,u,…,(u) (N)) $ P(\帽子{\ bm{\μ}},帽子\ {\ bm {x}},{你}\帽子,\ ldots,帽子(\{你})^ {(N)})将一个非零函数。□\美元美元

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Erratum for “Global Identifiability of Differential Models”

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

The following corollary is equivalent to [1, Lemma 3.5, p. 1848] but explicitly highlights that some of the entries of $\hat{θ}$ may be initial conditions, not only system parameters.

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