Local Observability of Systems on Time Scales

Abstract

Analytic systems on an arbitrary time-scale are studied. As particular cases they include continuous-time and discrete-time systems. Several local observability properties are considered. They are characterized in a unified way using the language of real analytic geometry, ideals of germs of analytic functions, and their real radicals. It is shown that some properties related to observability are preserved under various discretizations of continuous-time systems.

1. Introduction

Local observability for nonlinear systems is defined in various ways [1–4]. As most of these concepts are introduced in the general system-theoretic setting via indistinguishability relation, they mean the same for continuous- and discrete-time systems. We show here that some of these concepts may be studied in a unified way in the framework of systems on time-scales. A time-scale is a model of time, which can be continuous, discrete, or even mixed. Calculus on time-scale is a unification of ordinary differential calculus and the calculus of finite differences. Delta differential equations may be used to model continuous- and discrete-time systems. In the discrete case, time-scale may be nonhomogeneous. It may be applied to systems that are obtained by nonuniform sampling or nonuniform Euler discretization of continuous-time systems.

We concentrate on strong, weak, and robust local observability of analytic systems. We show that the results obtained in [2, 3, 5] may be extended to systems on arbitrary time-scales. This is due to the fact that all the properties are characterized with the aid of the observation algebra of the system, which may be introduced in a universal way on all time-scales. Since the observation algebra consists of real functions defined on the state space, we can use the common procedures to derive the criteria of local observability. As in [2, 3] we use the language of local analytic geometry and real algebra to characterize weak and robust local observability. Ideals of germs of analytic functions and real radicals of these ideals are used to express the criteria.

As an application of time-scale approach to local observability we consider discretization of continuous-time systems. This means replacing the standard derivative by the delta derivative on an appropriate discrete time-scale. We show that some of the properties related to observability are preserved under this operation. We will allow arbitrary discretizations: the discrete time-scale will not have to be homogeneous. Such nonuniform discretizations behave in a better way in many computations.

In Appendices we provide necessary information on time-scale calculus, local analytic geometry, and real algebra.

2. Preliminaries

Let 𝕋 be a time-scale. We will assume that 𝕋 is forward infinite; that is, for every 𝑡₀∈𝕋 there are infinitely many elements of 𝕋 that are greater than 𝑡₀. This will allow us to compute delta derivatives of arbitrary order at 𝑡₀. Let us consider a control system with output 𝑥Σ∶^Δ(𝑡)(𝑥(𝑡)(𝑡,𝑦(𝑡)(𝑥(𝑡,=𝑓,𝑢))=ℎ)) (1) where 𝑡∈𝕋, 𝑥(𝑡)∈ℝ^𝑛, 𝑦(𝑡)∈ℝ^𝑟, and 𝑢(𝑡)∈Ω—arbitrary set. For 𝜔∈Ω, let 𝑓_𝜔 be defined by 𝑓_𝜔(𝑥)∶=𝑓(𝑥,𝜔). We assume that the maps ℎ and 𝑓_𝜔 for every 𝜔∈Ω are analytic and that controls 𝑢 are piecewise constant functions of time.

If 𝕋=ℝ, then (1) is the standard continuous-time system 𝑥̇(𝑡)(𝑥(𝑡)(𝑡,𝑦(𝑡)(𝑥(𝑡.=𝑓,𝑢))=ℎ)) (2) For 𝕋=ℤ (1) takes the form 𝑥()(𝑡)(𝑥(𝑡)(𝑡,𝑦(𝑡)(𝑥(𝑡.𝑡+1−𝑥=𝑓,𝑢))=ℎ)) (3) This can be rewritten in a more standard shift form as 𝑥()(𝑥(𝑡)(𝑡(𝑡)(𝑥(𝑡)(𝑡,𝑦(𝑡)(𝑥(𝑡.𝑡+1=𝑓,𝑢))+𝑥=∶𝑔,𝑢))=ℎ)) (4) As there is a simple passage from 𝑓 to 𝑔 and vice versa, all statements for (3) may be translated to statements for (4).

Remark 1. The equation 𝑥(𝑡+1)=𝑔(𝑥(𝑡),𝑢(𝑡)) may be studied on an arbitrary set 𝑋 or on an analytic manifold 𝑀, if analyticity of the system is essential. But then we cannot pass to form (3), as to do this we need a linear space structure. Thus, one can argue that (4) is more general than (3). However, we concentrate here on local analytic problems, for which ℝ^𝑛 is general enough.
By 𝛾(𝑡,𝑡₀,𝑥₀,𝑢) we denote the solution of the equation 𝑥^Δ=𝑓(𝑥,𝑢) corresponding to control 𝑢 and the initial condition 𝑥(𝑡₀)=𝑥₀ and evaluated at time 𝑡.
Let 𝑥₁, 𝑥₂∈ℝ^𝑛 and 𝑡₀∈𝕋. Then 𝑥₁ and 𝑥₂ are called indistinguishable at time 𝑡₀ if ℎ𝛾𝑡,𝑡₀,𝑥₁,𝑢=ℎ𝛾𝑡,𝑡₀,𝑥₂,𝑢 (5) for every control 𝑢 defined on [𝑡₀,𝑡₁)_𝕋 for some 𝑡₁∈𝕋, and every 𝑡∈[𝑡₀,𝑡₁]_𝕋 for which both sides of the equation are defined. Otherwise 𝑥₁ and 𝑥₂ are distinguishable at time 𝑡₀.
The states 𝑥₁ and 𝑥₂ are called indistinguishable if they are indistinguishable at time 𝑡₀ for every 𝑡₀∈𝕋. Otherwise 𝑥₁ and 𝑥₂ are distinguishable. Thus, 𝑥₁ and 𝑥₂ are distinguishable if they are distinguishable at some time 𝑡₀.

Remark 2. If the time-scale is not homogeneous, indistinguishability at time 𝑡₀ may depend on 𝑡₀. Though the systems we consider have “constant coefficients,” that is, the map 𝑓 does not depend on time, inhomogeneity of the time-scale results in the behavior found in time-variant systems. This may be observed even for linear systems (see [6]).
Let 𝐶^𝜔(ℝ^𝑛) denote the algebra of all real analytic functions on ℝ^𝑛 and let 𝑔∶ℝ^𝑛→ℝ^𝑛 be analytic. Let us fix 𝑡₀∈ℝ^𝑛. In [7] the following operator Γ^𝑡0_𝑔∶𝐶^𝜔ℝ^𝑛⟶𝐶^𝜔ℝ^𝑛 (6) was introduced as Γ^𝑡0_𝑔𝜑(𝑥)∶=¹₀𝜑^′𝑥+𝑠𝜇𝑡₀𝑔(𝑥)𝑑𝑠⋅𝑔(𝑥), (7) where 𝜑∈𝐶^𝜔(ℝ^𝑛) and 𝜑^′ is the gradient of 𝜑 (a row vector).

Example 3. Let 𝜑=𝑥^𝑖 be the 𝑖th coordinate function on ℝ^𝑛. Then 𝜑^′=𝑒_𝑖—the (row) vector of the standard basis of ℝ^𝑛 with 1 at the 𝑖-th position. For any 𝑡₀∈𝕋 we have Γ^𝑡0_𝑔𝑥^𝑖(𝑥)=𝑒_𝑖𝑔(𝑥)=𝑔_𝑖(𝑥). (8)
Observe that if 𝜇(𝑡₀)>0 then Γ^𝑡0_𝑔𝜑(𝑥)=1𝜇𝑡₀𝜑𝑥+𝜇𝑡₀𝑔(𝑥)−𝜑(𝑥). (9) On the other hand for 𝜇(𝑡₀)=0 we obtain (Γ^𝑡0_𝑔𝜑)(𝑥)=𝜑^′(𝑥)𝑔(𝑥), which is the standard Lie derivative of the function 𝜑 along the vector field 𝑔. In general, when operator Γ^𝑡0_𝑔 does not depend on 𝑡₀, we will denote it by Γ_𝑔. This happens, for instance, if the time-scale is homogeneous.
Let 𝜎^𝑡0_𝑔∶𝐶^𝜔(ℝ^𝑛)→𝐶^𝜔(ℝ^𝑛) be another map related to the function 𝑔 and the time 𝑡₀ defined by 𝜎^𝑡0_𝑔𝜑(𝑥)∶=𝜑𝑥+𝜇𝑡₀𝑔(𝑥). (10) If 𝜇(𝑡₀)=0, then 𝜎^𝑡0_𝑔 is the identity map. In general we have an obvious property

Proposition 4. The map 𝜎^𝑡0_𝑔 is an endomorphism of the algebra 𝐶^𝜔(ℝ^𝑛).

For 𝜇(𝑡₀)>0 the operators Γ^𝑡0_𝑔 and 𝜎^𝑡0_𝑔 are related by the following equality: 𝜎^𝑡0_𝑔𝜑=𝜑+𝜇𝑡₀Γ^𝑡0_𝑔𝜑. (11) We also have the following generalization of the Leibniz rule.

Proposition 5. Γ^𝑡0_𝑔𝜑𝜓=Γ^𝑡0_𝑔𝜑𝜎^𝑡0_𝑔𝜓+𝜑Γ^𝑡0_𝑔𝜓. (12)

This property means that Γ^𝑡0_𝑔 is a skew derivation of the algebra 𝐶^𝜔(ℝ^𝑛) with respect to 𝜎^𝑡0_𝑔 or, in other words, that Γ^𝑡0_𝑔 is a 𝜎^𝑡0_𝑔-derivation of this algebra. For 𝜇(𝑡₀)=0 it is an ordinary derivation.

3. Local Observability

Let 𝐻^𝑡0 be a subset of 𝐶^𝜔(ℝ^𝑛) consisting of functions of the form Γ^𝑡0_𝑔𝑘…Γ^𝑡0_𝑔1ℎ_𝑖, (13) where 𝑖=1,…,𝑝, 𝑘≥0 and 𝑔_𝑖=𝑓_𝜔𝑖 for some 𝜔_𝑖∈Ω and 𝑖=1,…,𝑘. If 𝑘=0 then this function is just ℎ_𝑖. Let 𝐻=_𝑡0∈𝕋𝐻^𝑡0. (14)

Proposition 6. (i) The states 𝑥₁ and 𝑥₂ are indistinguishable at time 𝑡₀ if and only if for every 𝜑∈𝐻^𝑡0, 𝜑(𝑥₁)=𝜑(𝑥₂).
(ii) The states 𝑥₁ and 𝑥₂ are indistinguishable if and only if for every 𝜑∈𝐻, 𝜑(𝑥₁)=𝜑(𝑥₂).

Proof. The statement (ii) was shown in [1] for continuous-time systems (𝕋=ℝ). In this case (i) and (ii) mean the same since indistinguishability at some 𝑡₀ is equivalent to indistinguishability. For an arbitrary time-scale (ii) follows from (i). The statement (i) for an arbitrary time-scale was shown in [8]. Analyticity of the control system and the functions from 𝐻 is essential in the proof.

Remark 7. Proposition 6 allows us to use the same language of analytic functions on ℝ^𝑛 to study different observability properties of analytic systems on arbitrary time-scales as long as these properties are defined via the indistinguishability relation. It also implies that indistinguishability is an equivalence relation. This is not true for smooth systems (see [9]) or for analytic partially defined systems (see [10], where a different definition was developed to preserve this property for partially defined systems).
Let ℋ denote the subalgebra of 𝐶^𝜔(ℝ^𝑛) generated by 𝐻. It will be called the observation algebra of the system Σ. The elements of ℋ are obtained by substituting functions from 𝐻 into polynomials of several variables with real coefficients. In particular, all constant functions belong to ℋ.
From Proposition 6 we get the following.

Proposition 8. The states 𝑥₁ and 𝑥₂ are indistinguishable if and only if for every 𝜑∈ℋ, 𝜑(𝑥₁)=𝜑(𝑥₂).

We say that Σ is observable if any two distinct states are distinguishable.

From the definition and Proposition 8 we obtain the following characterization.

Proposition 9. Σ is observable if and only if for any distinct 𝑥₁, 𝑥₂∈ℝ^𝑛 there is 𝜑∈ℋ such that 𝜑(𝑥₁)≠𝜑(𝑥₂).

The condition stated in Proposition 9 is difficult to check. This is one of the reasons that a weaker concept of local observability seems to be more interesting. There are many different concepts of local observability and one concept has often a few different names. For the first two concepts we follow the terminology used in [4].

We say that Σ is weakly locally observable at 𝑥₀ (WLO(𝑥₀)) if there is a neighborhood 𝑈 of 𝑥₀ such that for every 𝑥∈𝑈, 𝑥 and 𝑥₀ are distinguishable.

Remark 10. Weak local observability at 𝑥₀ is in fact a weak property. It holds, for example, for the system 𝑥^Δ=0,𝑦=𝑥²₁+⋯+𝑥²_𝑛, (15) at 𝑥₀=0∈ℝ^𝑛. Since all solutions of 𝑥^Δ=0 are constant, time does not influence indistinguishability relation. To distinguish points we have to use only the output function. Clearly, it takes different values at 0 and any other point, so we can distinguish 𝑥₀ from any of its neighbors. Observe that local observability fails at any 𝑥₀≠0, if 𝑛≥2.
We say that Σ is strongly locally observable at 𝑥₀ (SLO(𝑥₀)) if there is a neighborhood 𝑈 of 𝑥₀ such that for every distinct 𝑥₁, 𝑥₂∈𝑈, 𝑥₁ and 𝑥₂ are distinguishable.
We say that Σ is robustly locally observable at 𝑥₀ (RLO(𝑥₀)) if there is a neighborhood 𝑈 of 𝑥₀ such that Σ is weakly locally observable at 𝑥 for every 𝑥∈𝑈.
Robust local observability was introduced in [3] for continuous-time systems under the name “stable local observability.” It means that the weak local observability at 𝑥₀ is stable or robust with respect to small perturbations of the initial condition 𝑥₀.
We call Σ weakly locally observable (WLO) (strongly locally observable (SLO), and robustly locally observable (RLO), resp.), when it is weakly locally observable (strongly locally observable and robustly locally observable, resp.) at every 𝑥∈ℝ^𝑛.
Let 𝑑ℋ(𝑥₀) denote the linear space of the differentials of functions from ℋ taken at 𝑥₀. The following theorem is a simple extension of the result from [1].

Theorem 11. (a) If dim𝑑ℋ(𝑥₀)=𝑛, then Σ is SLO(𝑥₀).
(b)  (𝑓𝑜𝑟𝑎𝑙𝑙𝑥₀∈ℝ^𝑛∶dim𝑑ℋ(𝑥₀)=𝑛)⇒(𝑓𝑜𝑟𝑎𝑙𝑙𝑥₀∈ℝ^𝑛∶Σ is SLO (𝑥₀))⇒(∃𝑋—a real analytic set in ℝ^𝑛∶𝑓𝑜𝑟𝑎𝑙𝑙𝑥₀∉𝑋∶dim𝑑ℋ(𝑥₀)=𝑛).

Let us denote the condition dim𝑑ℋ(𝑥₀)=𝑛 by HK(𝑥₀) (Hermann-Krener condition at 𝑥₀). The second part of Theorem 11 says that if we are interested in strong local observability at large, that is, at each point, then condition HK(𝑥₀) is satisfied almost everywhere, so the gap between sufficient condition and necessary condition for strong local observability at large is quite narrow. However, when one is interested in local observability at a particular point of the state space, the Hermann-Krener condition may be far from being necessary (see [2]).

We have a nice gradation of different local observability concepts.

Proposition 12. HK(𝑥₀)⇒Σ is SLO(𝑥₀)⇒Σ is RLO(𝑥₀)⇒Σ is WLO(𝑥₀).

None of the implications in Proposition 12 may be, in general, reversed.

Example 13. (a) Let 𝑥∈ℝ, 𝑥^Δ=0, and 𝑦=𝑥³. Thus, 𝐻={𝑥³}. The system is observable, so it is also strongly locally observable at any 𝑥. But the Hermann-Krener condition fails at 𝑥=0.
(b) Let 𝑥∈ℝ, 𝑥^Δ=0, and 𝑦=𝑥². Then 𝐻={𝑥²}. The system is robustly locally observable at 0, but it is not strongly locally observable at this point.
(c) Let 𝑥∈ℝ², 𝑥^Δ=0, and 𝑦=𝑥²₁+𝑥²₂. The system is weakly locally observable at 0, but it is not robustly locally observable at this point.
But we have an important, though obvious, global equivalence.

Proposition 14. Σ is RLO(𝑥₀) for every 𝑥₀∈ℝ^𝑛 if and only if Σ is WLO(𝑥₀) for every 𝑥₀∈ℝ^𝑛. In other words, Σ is RLO if and only if Σ is WLO.

By 𝒪_𝑥 we will denote the algebra of germs at 𝑥 of analytic functions on ℝ^𝑛 (see Appendix B). Let 𝐽_𝑥 be the ideal of 𝒪_𝑥 generated by germs at 𝑥 of functions from ℋ that vanish at 𝑥.

For 𝑥∈ℝ^𝑛 and for an ideal 𝐽 of 𝒪_𝑥, let 𝑍(𝐽) be the germ at 𝑥 of the zero-set of 𝐽. Since 𝐽 is finitely generated (𝒪_𝑥 is Noetherian), 𝑍(𝐽) is well defined. Let 𝐼(𝑍(𝐽)) be the ideal of 𝒪_𝑥 consisting of all germs of analytic functions that vanish on 𝑍(𝐽).

Lemma 15. The following conditions are equivalent:(a)𝑍(𝐽_𝑥0)≠{𝑥₀};(b)arbitrarily close to 𝑥₀ there is 𝑥 such that 𝜑(𝑥₀)=𝜑(𝑥) for every 𝜑∈ℋ.

Proof. (a) holds if and only if arbitrarily close to 𝑥₀ there is 𝑥 such that all representatives of germs 𝜑∈𝐽_𝑥0 (all defined in some neighborhood of 𝑥₀) are 0 at 𝑥. This is equivalent to the fact that all functions 𝜑∈ℋ take on the same values at 𝑥₀ and this 𝑥, which means precisely (b).

To characterize weak local observability we use the concept of real radical (see Appendix B).

Theorem 16.  (a)Σ is WLO(𝑥₀) if and only if ^ℝ𝐽_𝑥0=𝑚_𝑥0.(b)HK(𝑥₀) if and only if 𝐽_𝑥0=𝑚_𝑥0.

Proof. (a) Σ is not weakly locally observable at 𝑥₀ if and only if arbitrarily close to 𝑥₀ there is 𝑥 such that 𝑥₀ and 𝑥 are indistinguishable. By Lemma 15, the last statement is equivalent to the condition 𝑍(𝐽_𝑥0)≠{𝑥₀}, which in turn means that 𝐼(𝑍(𝐽_𝑥0))≠𝐼({𝑥₀}). But 𝐼({𝑥₀})=𝑚_𝑥0, so from Theorem B.1 the last inequality is equivalent to the condition ^ℝ𝐽_𝑥0≠𝑚_𝑥0. This gives the equivalence of both sides in (a).(b) It is enough to prove the proposition for 𝑥=0 in ℝ^𝑛.
Observe that 𝑑ℋ(𝑥)=𝑑𝐽_𝑥(𝑥) for every 𝑥. Thus, if 𝐽₀=𝑚₀, then 𝑑ℋ(0) contains all the differentials 𝑑𝑥_𝑖 for 𝑖=1,…,𝑛, which are linearly independent.
On the other hand, the condition dim𝑑ℋ(0)=𝑛 implies that in a neighborhood 𝑈 of 0, there are functions 𝜑₁,…,𝜑_𝑛 whose germs belong to 𝐽₀ and the differentials at 0, 𝑑𝜑_𝑖(0) are linearly independent. We may write 𝜑_𝑖=∑_𝑗Φ_𝑖𝑗𝑥_𝑗 for some analytic functions Φ_𝑖𝑗 on 𝑈 (sufficiently small). Then 𝑑𝜑_𝑖(0)=∑_𝑗Φ_𝑖𝑗(0)𝑑𝑥_𝑗. This means that the matrix Φ=(Φ_𝑖𝑗) is invertible at 0 and then in some neighborhood of 0. Let Ψ=Φ⁻¹. Then the germs of elements of  Ψ are in 𝒪_𝑥 and 𝑑𝑥_𝑖=∑_𝑗Ψ_𝑖𝑗𝑑𝜑_𝑗 which means that 𝜑₁,…,𝜑_𝑛 generate 𝑚₀.

Let 𝐺 be a subset of 𝒪_𝑥. By 𝐷(𝐺) we will denote the ideal of 𝒪_𝑥 generated by Jacobians 𝜕(𝜑₁,…,𝜑_𝑛)/𝜕(𝑥₁,…,𝑥_𝑛), where 𝜑_𝑖∈𝐺. Observe that these Jacobians are well defined on germs of functions. If 𝒢 is a family of real analytic functions on an open set 𝑈 in ℝ^𝑛, then similarly we define the Jacobian ideal 𝐷(𝒢) in 𝒪_𝑈. Furthermore, there is a simple relation between Jacobian ideals for functions and germs of functions. If 𝒢 is a family of analytic functions on 𝑈, and 𝑥₀∈𝑈 then we have 𝐷(𝒢)_𝑥0=𝐷𝒢_𝑥0. (16)

Now, for a point 𝑥∈ℝ^𝑛, we define a sequence of ideals in 𝒪_𝑥 related to the system Σ. Let 𝐼_𝑥⁽⁰⁾∶=(0) and 𝐼_𝑥^(𝑘+1)∶=^ℝ𝐷(ℋ_𝑥∪𝐼_𝑥^(𝑘)). It is clear that instead of ℋ_𝑥 in this definition one can take the previously defined ideal 𝐽_𝑥; that is, 𝐼_𝑥^(𝑘+1)=^ℝ𝐷(𝐽_𝑥∪𝐼_𝑥^(𝑘)). This leads to the following generalization of statement (b) of Theorem 16.

Corollary 17. HK(𝑥₀)⇔𝐽_𝑥0=𝑚_𝑥0⇔𝐼_𝑥⁽¹⁾₀=𝒪_𝑥0.

The ideals 𝐼_𝑥^(𝑘) will be the main tools in studying robust local observability. First we prove the basic fact.

Proposition 18. For any 𝑘≥0, 𝐼_𝑥^(𝑘)⊆𝐼_𝑥^(𝑘+1), and there is 𝑠≥0 such that 𝐼_𝑥^(𝑠)=𝐼_𝑥^(𝑠+1).

Proof. To prove the first part we proceed by induction. It is clear that 𝐼_𝑥⁽⁰⁾⊆𝐼_𝑥⁽¹⁾, so assume that 𝐼_𝑥^(𝑘)⊆𝐼_𝑥^(𝑘+1) for some 𝑘≥0. Then also ℋ_𝑥∪𝐼_𝑥^(𝑘)⊆ℋ_𝑥∪𝐼_𝑥^(𝑘+1) and 𝐷(ℋ_𝑥∪𝐼_𝑥^(𝑘))⊆𝐷(ℋ_𝑥∪𝐼_𝑥^(𝑘+1)), so finally ^ℝ𝐷(ℋ_𝑥∪𝐼_𝑥^(𝑘))⊆^ℝ𝐷(ℋ_𝑥∪𝐼_𝑥^(𝑘+1)). This means that 𝐼_𝑥^(𝑘+1)⊆𝐼_𝑥^(𝑘+2). Since the ring 𝒪_𝑥 is Noetherian the sequence of ideals 𝐼_𝑥^(𝑘) must stabilize at some 𝑠.

Now we can characterize robust local observability.

Theorem 19. System Σ is RLO(𝑥₀) if and only if 𝐼_𝑥^(𝑠)₀=𝒪_𝑥0 for some 𝑠>0.

The proof of Theorem 19 will rely on several lemmas. They appeared in a similar form in [3]. However, there were a few flaws in the proofs, which are now corrected.

Let 𝒢 be a family of analytic functions on some open set 𝑈⊂ℝ^𝑛. Denote by 𝑆_𝑥(𝒢) the germ at 𝑥 of the level set of 𝒢 that passes through 𝑥. Thus 𝑆_𝑥(𝒢)=𝑦∈𝑈∶𝜑(𝑥)=𝜑𝑦for𝜑∈𝒢_𝑥. (17) The set-germ 𝑆_𝑥(𝒢) is a germ of analytic set. One of the representatives of 𝑆_𝑥(𝒢) is the analytic set in 𝑈∶{𝑦∈𝑈∶𝜑(𝑦)=𝜑(𝑥)for𝜑∈𝒢}.

Lemma 20. Let 𝑈 be an open subset of ℝ^𝑛 and let 𝑋 be an analytic set in 𝑈. Consider a family 𝒢 of analytic functions on 𝑈. If for every 𝑥∉𝑋∶𝑆_𝑥(𝒢)={𝑥}, then for every 𝑥∈𝑋∶𝑆_𝑥(𝒢)⊂𝑋_𝑥.

Proof. Suppose that there is 𝑥∈𝑋 such that 𝑆_𝑥(𝒢)⊄𝑋_𝑥. This means that for every representatives 𝑆_𝑥(𝒢) and 𝑋_𝑥 we have 𝑆_𝑥(𝒢)⊄𝑋_𝑥. Take 𝑋_𝑥∶=𝑋 and arbitrarily small neighborhood 𝑉 of 𝑥 in 𝑈. Let 𝑆_𝑥(𝒢) be a representative of 𝑆_𝑥(𝒢) in 𝑉. Then there is 𝑦∈𝑆_𝑥(𝒢) such that 𝑦∉𝑋. Take a sequence (𝑦_𝑛) of such points converging to 𝑥. We may assume that all these points belong to some (large enough) representative 𝑌 of 𝑆_𝑥(𝒢) and that 𝑌 is an analytic set. Only finite number of points 𝑦_𝑛 may be isolated points of 𝑌. This means that arbitrarily close to 𝑥 there is a point 𝑦_𝑛 for which 𝑆_𝑦𝑛(𝒢)≠{𝑦_𝑛}. Thus we get a contradiction.

Lemma 21. Let 𝑈 be an open subset of   ℝ^𝑛 and let 𝒢 be a family of analytic functions on 𝑈. For every 𝑥∈𝑈: if 𝑥∉𝑍(𝐷(𝒢)), then 𝑆_𝑥(𝒢)={𝑥}.

Proof. If 𝑥∉𝑍(𝐷(𝒢)), then there are functions 𝜑₁,…,𝜑_𝑛∈𝐺 such that 𝜕𝜑₁,…,𝜑_𝑛𝜕𝑥₁,…,𝑥_𝑛(𝑥)≠0. (18) Thus, the map 𝑥↦(𝜑₁(𝑥),…,𝜑_𝑛(𝑥)) is injective in a neighborhood of 𝑥. This implies that 𝑆_𝑥(𝒢)={𝑥}.

Lemma 22. Assume that 𝑋_𝑥^(𝑘)₀≠𝑋_𝑥^(𝑘+1)₀ for some 𝑘≥0. Then there is a neighborhood 𝑈 of 𝑥₀ in ℝ^𝑛 and a representative 𝑋_𝑥^(𝑘+1)₀ of 𝑋_𝑥^(𝑘+1)₀ in 𝑈 such that for every 𝑥∈𝑈: 𝑥∉𝑋_𝑥^(𝑘+1)₀⟹𝑆_𝑥(ℋ)={𝑥}. (19)

Proof. First observe that 𝑍(𝐷(ℋ_|𝑈)) is a representative of 𝑍(𝐷(ℋ_𝑥0)). We proceed by induction. Let 𝑘=0 and assume that 𝑋_𝑥⁽¹⁾₀≠𝑋_𝑥⁽⁰⁾₀=ℝ^𝑛_𝑥0. Take any neighborhood 𝑈 of 𝑥₀. Since 𝑋_𝑥⁽¹⁾₀=𝑍(𝐷(ℋ_𝑥0)), then 𝑍(𝐷(ℋ_|𝑈))=∶𝑋_𝑥⁽¹⁾₀ is a representative of 𝑋_𝑥⁽¹⁾₀ in 𝑈. If 𝑥∉𝑋_𝑥⁽¹⁾₀, then by Lemma 21, 𝑆_𝑥(ℋ)=𝑆_𝑥(ℋ_|𝑈)={𝑥}.
Now assume that the statement of the lemma holds for 𝑘−1≥0. Hence, there is a neighborhood 𝑈 of 𝑥₀ and a representative 𝑋_𝑥^(𝑘)₀=𝑍(𝜑₁,…,𝜑_𝑠) of 𝑋_𝑥^(𝑘)₀ such that if 𝑥∈𝑈 and 𝑥∉𝑋_𝑥^(𝑘)₀, then 𝑆_𝑥(ℋ)={𝑥}. The functions 𝜑₁,…,𝜑_𝑠 are representatives on 𝑈 of generators of 𝐼_𝑥^(𝑘)₀. The set 𝑋_𝑥^(𝑘+1)₀∶=𝑍(𝐷(ℋ_|𝑈∪{𝜑₁,…,𝜑_𝑠})) is a representative of 𝑋_𝑥^(𝑘+1)₀. Clearly 𝑋_𝑥^(𝑘+1)₀⊆𝑋_𝑥^(𝑘)₀. Take 𝑥∈𝑈 such that 𝑥∉𝑋_𝑥^(𝑘+1)₀. If 𝑥∉𝑋_𝑥^(𝑘)₀, then 𝑆_𝑥(ℋ)={𝑥}. Assume then that 𝑥∈𝑋_𝑥^(𝑘)₀. From Lemma 20 we get 𝑆_𝑥(ℋ)⊆(𝑋_𝑥^(𝑘)₀)_𝑥. Then 𝑆_𝑥(ℋ)=𝑆_𝑥(ℋ)∩(𝑋_𝑥^(𝑘)₀)_𝑥=𝑆_𝑥(ℋ_|𝑈∪{𝜑₁,…,𝜑_𝑠}). Since 𝑥∉𝑍(𝐷(ℋ_|𝑈∪{𝜑₁,…,𝜑_𝑠})), then, by Lemma 21, 𝑆_𝑥(ℋ)=𝑆_𝑥(ℋ_|𝑈∪{𝜑₁,…,𝜑_𝑠})={𝑥}. This finishes the inductive step of the proof.

Lemma 23. Let 𝑥₀∈𝑈⊂ℝ^𝑛, where 𝑈 is open, and let 𝒢 be a family of analytic functions on 𝑈. If 𝐷(𝒢)=0 (zero ideal), then arbitrarily close to 𝑥₀ there is 𝑥∈𝑈 such that 𝑆_𝑥(𝒢)≠{𝑥}.

Proof. Let 𝑠=max_{𝜑𝑥∈𝑈𝑖∈𝒢}𝜕𝜑rank¹,…,𝜑_𝑛𝜕𝑥₁,…,𝑥_𝑛(𝑥). (20) Then 𝑠<𝑛 and arbitrarily close to 𝑥₀ there is 𝑥∈𝑈 and 𝜑₁,…,𝜑_𝑛∈𝒢 such that rank(𝜕(𝜑₁,…,𝜑_𝑛)/𝜕(𝑥₁,…,𝑥_𝑛))(𝑥)=𝑠. This rank is preserved in some neighborhood 𝑉 of 𝑥. Thus, we may assume that gradients of 𝜑₁,…,𝜑_𝑠 are linearly independent at every point of 𝑉 and span 𝑑𝒢(𝑥) for 𝑥∈𝑉. Then by Frobenius Theorem, 𝑉 is a union of integral manifolds of codistribution 𝑑𝒢. The integral manifolds are the level sets of 𝒢 and have dimension greater than or equals to 1. This means that 𝑆_𝑥(𝒢)≠{𝑥}.

Lemma 24. Let 𝑈 be an open subset of ℝ^𝑛 and let 𝜑₁,…,𝜑_𝑘 be analytic functions on 𝑈 whose gradients are linearly independent at each point of 𝑈. Let 𝑌=𝑍(𝜑₁,…,𝜑_𝑘), 𝑥₀∈𝑌, and let 𝒢 be a family of analytic functions on 𝑈.
If 𝑌⊂𝑍(𝐷(𝒢∪{𝜑₁,…,𝜑_𝑘})), then arbitrarily close to 𝑥₀ there is 𝑥∈𝑌 such that 𝑆_𝑥(𝒢)≠{𝑥}.

Proof. Changing the coordinates we can obtain 𝜑_𝑖=𝑥_𝑖, 𝑖=1,…,𝑘. Let 𝜓₁,…,𝜓_𝑛−𝑘∈𝒢. Then for 𝑥∈𝑌 we have 0=det∇𝜑₁(𝑥)⋮∇𝜑_𝑘(𝑥)∇𝜓₁(𝑥)⋮∇𝜓_𝑛−𝑘(𝑥)=det𝐼0𝜕𝜓_𝑖𝜕𝑥_𝑗(𝑥)_{𝑖=1,…,𝑛−𝑘𝑗=1,…,𝑛}=det𝜕𝜓_𝑖(𝑥)𝜕𝑥_{𝑗𝑖=1,…,𝑛−𝑘𝑗=𝑘+1,…,𝑛}. (21) Note that 𝑌 is an analytic manifold and the last term above is actually the Jacobian of a map defined on 𝑌. Hence, after restricting to the manifold 𝑌, we get 𝐷(𝒢_|𝑌)=0. From Lemma 23, arbitrarily close to 𝑥₀ there is 𝑥∈𝑌 such that 𝑆_𝑥(𝒢_|𝑌)≠{𝑥}. But 𝑆_𝑥(𝒢_|𝑌)⊂𝑆_𝑥(𝒢) so also 𝑆_𝑥(𝒢)≠{𝑥}.

Lemma 25. Assume that 𝑋_𝑥^(𝑠)₀=𝑋_𝑥^(𝑠+1)₀ for some 𝑠≥0 and 𝑋_𝑥^(𝑠)₀≠∅. Then in every neighborhood of 𝑥₀ there is 𝑥 such that 𝑆_𝑥(ℋ)≠{𝑥}.

Proof. Let 𝜑₁,…,𝜑_𝑘 be representatives of generators of the ideal 𝐼_𝑥^(𝑠)₀, defined on some common neighborhood 𝑈 of 𝑥₀. Then 𝑋_𝑥^(𝑠)₀∶=𝑍(𝜑₁,…,𝜑_𝑘) is a representative of 𝑋_𝑥^(𝑠)₀ in 𝑈. In every neighborhood of 𝑥₀ one can find a regular point of the analytic set 𝑋_𝑥^(𝑠)₀ (see [11, 12]). Let 𝑥 be such a point and let 𝑉 be a neighborhood of 𝑥 in ℝ^𝑛 such that 𝑌∶=𝑉∩𝑋_𝑥^(𝑠)₀ is an analytic manifold. Then 𝑌=𝑍(𝜑_1|𝑉,…,𝜑_𝑘|𝑉). We may assume that the gradients of 𝜑_1|𝑉,…,𝜑_𝑘|𝑉 are linearly independent on 𝑌. Otherwise, after possible shrinking of 𝑉, we can remove the functions whose gradients are linear combinations of the gradients of other functions. Let 𝑥∈𝑌. Then 𝑥∈𝑋_𝑥^(𝑠+1)₀∶=𝑍(𝐷(ℋ_|𝑈∪{𝜑₁,…,𝜑_𝑘})) so that 𝑌⊂𝑍(𝐷(ℋ_|𝑉∪{𝜑_1|𝑉,…,𝜑_𝑘|𝑉})). The statement of the lemma follows now from Lemma 24.

Proof of Theorem 19. Sufficiency. Assume that 𝐼_𝑥^(𝑠)₀=𝒪_𝑥0 for some 𝑠>0. This means that 𝑋_𝑥^(𝑠)₀=∅. From Lemma 22 it follows that nontrivial set-germs 𝑆_𝑥(ℋ) (i.e., different from {𝑥}) may be found only in 𝑋_𝑥^(𝑠)₀—some representative of 𝑋_𝑥^(𝑠)₀. But 𝑋_𝑥^(𝑠)₀=∅ so in some neighborhood of 𝑥₀ the level set-germs of ℋ must be trivial. This means that Σ is robustly locally observable at 𝑥₀.
Necessity. Assume that 𝑋_𝑥^(𝑠)₀=𝑋_𝑥^(𝑠+1)₀ for some 𝑠≥0. From Lemma 25 it follows that Σ is not robustly locally observable.

The statements of Corollary 17, Proposition 18, and Theorem 19 can be translated into the language of germs of analytic sets. Let 𝑋_𝑥^(𝑘)=𝑍(𝐼_𝑥^(𝑘)). Because the ideals 𝐼_𝑥^(𝑘) are real, we also get 𝐽(𝑋_𝑥^(𝑘))=𝐼_𝑥^(𝑘), so there is one-to-one correspondence between the ideals and the set-germs. We have then the following.

Corollary 26. Let 𝑥₀∈ℝ^𝑛. (a)HK(𝑥₀) if and only if 𝑋_𝑥⁽¹⁾₀=∅.(b)For every 𝑘≥0, 𝑋_𝑥^(𝑘)₀⊇𝑋_𝑥^(𝑘+1)₀, and for some 𝑠≥0, 𝑋_𝑥^(𝑠)₀=𝑋_𝑥^(𝑠+1)₀.(c)Σ is RLO(𝑥₀) if and only if for some 𝑠≥0, 𝑋_𝑥^(𝑠)₀=∅.

Example 27. Consider the following system: 𝑥^Δ₁=𝑥₁𝑥𝑢,_Δ2=𝑥₂,𝑥^Δ₃=0,𝑦=𝑥²₁+𝑥²₂+𝑥²₃, (22) and choose 𝑥₀=0. Then for an arbitrary time-scale we get ℋ=ℝ[𝑥²₁,𝑥²₂,𝑥²₃] and 𝐼_𝑥⁽¹⁾₀=𝐽_𝑥0=(𝑥₁𝑥₂𝑥₃). Thus 𝑋_𝑥⁽¹⁾₀ is the germ of a union of three planes intersecting at 0. In the next step we obtain 𝐼_𝑥⁽²⁾₀=^ℝ(𝑥²₁𝑥²₂,𝑥²₂𝑥²₃,𝑥²₁𝑥²₃)=(𝑥₁𝑥₂,𝑥₂𝑥₃,𝑥₁𝑥₃). A quick calculation shows that 𝑋_𝑥⁽²⁾₀ is the germ of a union of three lines intersecting at 0. Finally, 𝐼_𝑥⁽³⁾₀=𝑚_𝑥0 and then 𝐼_𝑥⁽⁴⁾₀=𝒪_𝑥0, so Σ is robustly locally observable at 𝑥₀.

4. Discretization

As we are going to consider several time-scales, we will denote the graininess function on the time-scale 𝕋 by 𝜇_𝕋. A time-scale 𝕋 is called discrete, if 𝜇_𝕋(𝑡)>0 for all 𝑡∈𝕋.

Let Σ be the continuous-time system 𝑥̇=𝑓(𝑥,𝑢),𝑦=ℎ(𝑥) (23) and let Σ_𝕋 be its discretization 𝑥^Δ=𝑓(𝑥,𝑢),𝑦=ℎ(𝑥) (24) on a discrete time-scale 𝕋. Usually 𝕋 is equal to 𝑐ℤ for some 𝑐>0, but nonhomogeneous time-scales are allowed as well. In the discretized system the ordinary derivative is replaced by the delta derivative on the discrete time-scale.

Thus, (23) is replaced with 𝑥𝑡+𝜇_𝕋(𝑡)−𝑥(𝑡)𝜇_𝕋(𝑡)=𝑓(𝑥(𝑡),𝑢(𝑡)),𝑦(𝑡)=ℎ(𝑥(𝑡)) (25) for 𝑡∈𝕋.

Let ℋ denote the observation algebra of the system Σ and ℋ_𝕋 the observation algebra of the system Σ_𝕋. Observe that each generator of ℋ may be approximated by a corresponding generator of ℋ_𝕋 for 𝜇(𝑡) sufficiently small. This follows from the form of the operators Γ^𝑡0_𝑔 on ℝ and 𝕋, which is used in this procedure.

A natural question is which properties related to observability are preserved under discretization.

Proposition 28. If 𝑥₁ and 𝑥₂ are distinguishable by Σ, then there is 𝜇>0 such that 𝑥₁ and 𝑥₂ are distinguishable by Σ_𝕋 at 𝑡∈𝕋 whenever 𝜇_𝕋(𝑡)<𝜇.

Proof. Suppose that for every 𝜇>0 there are a time-scale 𝕋 and 𝑡∈𝕋 with 0<𝜇_𝕋(𝑡)<𝜇 such that 𝑥₁ and 𝑥₂ are indistinguishable by Σ_𝕋 at 𝑡. This means that that there is a sequence (𝕋_𝑖) of time-scales and a sequence (𝑡_𝑖) of real numbers such that 𝑡_𝑖∈𝕋_𝑖 and 𝜇_𝕋𝑖(𝑡_𝑖)→0 when 𝑖→∞ and for every 𝜑∈ℋ^𝑡𝑖_𝕋𝑖, 𝜑(𝑥₁)=𝜑(𝑥₂). Every function 𝜓∈ℋ may be approximated by functions from ℋ^𝑡𝑖_𝕋𝑖; that is, there is a sequence of functions (𝜙_𝑖) such that 𝜑_𝑖∈ℋ^𝑡𝑖_𝕋𝑖 and 𝜑_𝑖→𝜓 on some compact set containing 𝑥₁ and 𝑥₂. This implies that 𝜓(𝑥₁)=𝜓(𝑥₂), so 𝑥₁ and 𝑥₂ are indistinguishable by Σ.

In particular, distinguishability of 𝑥₁ and 𝑥₂ is preserved for quantum discretization, where 𝕋=𝑞^ℕ, if 𝑞 is sufficiently close to 1.

We will show now that the Hermann-Krener rank condition is preserved under discretization. Let HK(𝑥₀) denote the Hermann-Krener condition at 𝑥₀ for the system Σ and HK_𝕋(𝑥₀) for the system Σ_𝕋.

Proposition 29. If 𝐻𝐾(𝑥₀), then there is 𝜇>0 such that for every time-scale 𝕋 if there is 𝑡∈𝕋 with 0<𝜇_𝕋(𝑡)<𝜇, then 𝐻𝐾_𝕋(𝑥₀).

Proof. Assume that HK(𝑥₀) holds. Then there are 𝜓₁,…,𝜓_𝑛∈ℋ such that 𝑑𝜓₁(𝑥₀),…,𝑑𝜓_𝑛(𝑥₀) are linearly independent. Each 𝜓_𝑖 may be approximated by some 𝜑_𝑖∈ℋ^𝑡_𝕋 for 𝑡∈𝕋 and 𝜇_𝕋(𝑡) sufficiently small. Observe that the functions 𝜑_𝑖 depend actually on the parameter 𝜇_𝕋(𝑡) and this dependence is continuous. For 𝜇_𝕋(𝑡) sufficiently small also 𝑑𝜑₁(𝑥₀),…,𝑑𝜑_𝑛(𝑥₀) will be linearly independent. This means that HK_𝕋(𝑥₀) holds for such 𝕋.

The converse of Proposition 29 does not hold.

Example 30. Let Σ be 𝑥̇₁=−𝑢,𝑥̇₂=3𝑥²₁𝑢,𝑦=𝑥³₁+𝑥₂. (26) The observation algebra ℋ is generated by a single function 𝜓(𝑥₁,𝑥₂)=𝑥³₁+𝑥₂, which means that the Hermann-Krener condition does not hold at any point. The discretized system Σ_𝕋 is given by 𝑥₁𝑡+𝜇_𝕋(𝑡)=𝑥₁(𝑡)−𝜇_𝕋(𝑡)𝑢(𝑡),𝑥₂𝑡+𝜇_𝕋(𝑡)=𝑥₂(𝑡)+3𝜇_𝕋(𝑡)𝑥²₁(𝑡)𝑢(𝑡),𝑦(𝑡)=𝑥₁(𝑡)³+𝑥₂(𝑡). (27) The observation algebra ℋ_𝕋 contains now the functions 𝜓₁(𝑥₁,𝑥₂)=𝑥³₁+𝑥₂ and 𝜓₂(𝑥₁,𝑥₂)=𝑥₁. The Hermann-Krener condition is then satisfied at all points and for all discrete time-scales 𝕋.

Remark 31. Hermann-Krener rank condition is equivalent to the property that the ideal 𝐽_𝑥0 is maximal; that is, it is generated by the coordinate functions. One can show that this property is preserved when the ideal 𝐽_𝑥0 is replaced with the ideals corresponding to systems Σ_𝕋 if 𝕋 contains 𝑡 with 𝜇_𝕋(𝑡) sufficiently small. In characterizations of weak and robust local observability there appear real radicals of ideals. It is not clear whether desired properties of the radicals like maximality (for WLO(𝑥₀)) or nonproperness (for RLO(𝑥₀) are preserved under discretizations. Thus preservation of weak and robust local observability under Euler discretization is still an open problem.

We finish this discussion with a positive example.

Example 32. Let Σ be 𝑥̇₁=𝑥₂𝑢,𝑥̇₂=−𝑥₁𝑢,𝑦=𝑥²₁+𝑥²₂. (28) The observation algebra of Σ is generated by 𝜑(𝑥₁,𝑥₂)=𝑥²₁+𝑥²₂.

The discretization gives Σ_𝕋𝑥^Δ₁=𝑥₂𝑢,𝑥^Δ₂=−𝑥₁𝑢,𝑦=𝑥²₁+𝑥²₂, (29) where 𝑥^Δ(𝑡)=𝑥𝑡+𝜇_𝕋(𝑡)−𝑥(𝑡)𝜇_𝕋(𝑡). (30) The observation algebra of Σ_𝕋 is also generated by 𝜑(𝑥₁,𝑥₂)=𝑥²₁+𝑥²₂.

Thus, Σ and Σ_𝕋 are both weakly locally observable at 𝑥=0. They are not weakly locally observable at any other point.

Example 27 describes a positive behavior of robust local observability under discretization. In fact, the calculations are the same for all time-scales.

5. Conclusions

We have shown that the methods of real analytic geometry and real algebra developed for continuous time systems may be used for systems on arbitrary time-scales, in particular on the scale of integers and on the quantum scales. Different concepts of local observability for systems on arbitrary time-scales have been considered. We have established relations between these concepts and provided characterizations of weak and robust local observability with the aid of certain ideals of the ring of germs of analytic functions and real radicals of those ideals. Equivalent geometric characterizations have been given. Observation algebras from which the ideals are obtained and the ideals themselves depend on the time-scale on which the systems is defined, but once the ideals are computed, the procedures and the criteria of local observability are the same for all time-scales. This allows for unified treatment of observability of systems on arbitrary time-scales.

The language of time-scales allows for a natural description of discretization of continuous-time systems: the ordinary derivative is replaced by delta derivative on a discrete time-scale 𝕋. The paper contains preliminary results on preservation of properties related to observability under discretization. In particular Hermann-Krener rank condition is preserved. Preservation of other properties, in particular weak and robust local observability, is stated as an open problem. To solve the problem one will have to study limit properties of real radicals for rings of germs of analytic functions. This will be a subject of a future research.

Appendices

A. Calculus on Time-scales

A time-scale 𝕋 is an arbitrary nonempty closed subset of the set ℝ of real numbers. In particular ℝ, ℎℤ for ℎ>0 and 𝑞^ℕ∶={𝑞^𝑘,𝑘∈ℕ} for 𝑞>1 are time-scales. We assume that 𝕋 is a topological space with the relative topology induced from ℝ. If 𝑡₀, 𝑡₁∈𝕋, then [𝑡₀,𝑡₁]_𝕋 denotes the intersection of the ordinary closed interval with 𝕋. Similar notation is used for open, half-open or infinite intervals.

For 𝑡∈𝕋 we define the forward jump operator 𝜎∶𝕋→𝕋 by 𝜎(𝑡)∶=inf{𝑠∈𝕋∶𝑠>𝑡} if 𝑡≠sup𝕋 and 𝜎(sup𝕋)=sup𝕋 when sup𝕋 is finite; the backward jump operator 𝜌∶𝕋→𝕋 by 𝜌(𝑡)∶=sup{𝑠∈𝕋∶𝑠<𝑡} if 𝑡≠inf𝕋 and 𝜌(inf𝕋)=inf𝕋 when inf𝕋 is finite; the forward graininess function 𝜇∶𝕋→[0,∞) by 𝜇(𝑡)∶=𝜎(𝑡)−𝑡; the backward graininess function 𝜈∶𝕋→[0,∞) by 𝜈(𝑡)∶=𝑡−𝜌(𝑡).

If 𝜎(𝑡)>𝑡, then 𝑡 is called right-scattered, while if 𝜌(𝑡)<𝑡, it is called left-scattered. If 𝑡<sup𝕋 and 𝜎(𝑡)=𝑡, then 𝑡 is called right-dense. If 𝑡>inf𝕋 and 𝜌(𝑡)=𝑡, then 𝑡 is left-dense.

The time-scale 𝕋 is homogeneous, if 𝜇 and 𝜈 are constant functions. When 𝜇≡0 and 𝜈≡0, then 𝕋=ℝ or 𝕋 is a closed interval (in particular a half-line). When 𝜇 is constant and greater than 0, then 𝕋=𝜇ℤ+𝑐, for some 𝑐∈ℝ.

Let 𝕋^𝜅∶={𝑡∈𝕋∶𝑡isnonmaximalorleft-dense}. Thus 𝕋^𝜅 is obtained from 𝕋 by removing its maximal point if this point exists and is left-scattered.

Let 𝑓∶𝕋→ℝ and 𝑡∈𝕋^𝜅. The delta derivative of 𝑓 at 𝑡, denoted by 𝑓^Δ(𝑡), is the real number with the property that given any 𝜀 there is a neighborhood 𝑈=(𝑡−𝛿,𝑡+𝛿)_𝕋 such that 𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)(𝜎(𝑡)−𝑠)≤𝜀|𝜎(𝑡)−𝑠| (A.1) for all 𝑠∈𝑈. If 𝑓^Δ(𝑡) exists, then we say that 𝑓 is delta differentiable at 𝑡. Moreover, we say that 𝑓 is delta differentiable on 𝕋^𝑘 provided 𝑓^Δ(𝑡) exists for all 𝑡∈𝕋^𝑘.

Example A.1. If 𝕋=ℝ, then 𝑓^Δ(𝑡)=𝑓^′(𝑡). If 𝕋=ℎℤ, then 𝑓^Δ(𝑡)=(𝑓(𝑡+ℎ)−𝑓(𝑡))/ℎ. If 𝕋=𝑞^ℕ, then 𝑓^Δ(𝑡)=(𝑓(𝑞𝑡)−𝑓(𝑡))/((𝑞−1)𝑡).
A function 𝑓∶𝕋→ℝ is called rd-continuous provided it is continuous at right-dense points in 𝕋 and its left-sided limits exist (finite) at left-dense points in 𝕋. If 𝑓 is continuous, then it is rd-continuous.
A function 𝑓∶𝕋→ℝ is called regressive, if 1+𝜇(𝑡)𝑓(𝑡)≠0 for all 𝑡∈𝕋.
A function 𝐹∶𝕋→ℝ is called an antiderivative of 𝑓∶𝕋→ℝ provided 𝐹^Δ(𝑡)=𝑓(𝑡) holds for all 𝑡∈𝕋^𝜅. Let 𝑎, 𝑏∈𝕋. Then the delta integral of 𝑓 on the interval [𝑎,𝑏)_𝕋 is defined by ^𝑏_𝑎𝑓(𝜏)Δ𝜏∶=_{[𝑎,𝑏)𝕋}𝑓(𝜏)Δ𝜏∶=𝐹(𝑏)−𝐹(𝑎). (A.2)

Riemann and Lebesgue delta integrals on time-scales have been also defined (see, e.g., [13]). It can be shown that every rd-continuous function has an antiderivative and its Riemann and Lebesgue integrals agree with the delta integral defined above.

Example A.2. If 𝕋=ℝ, then ∫^𝑏_𝑎𝑓(𝜏)Δ𝜏=∫^𝑏_𝑎𝑓(𝜏)𝑑𝜏, where the integral on the right is the usual Riemann integral. If 𝕋=ℎℤ, ℎ>0, then ∫^𝑏_𝑎𝑓(𝜏)Δ𝜏=∑^{(𝑏/ℎ)−1}_{𝑡=(𝑎/ℎ)}𝑓(𝑡ℎ)ℎ for 𝑎<𝑏.

B. Basic Real Geometry

We assume that the reader is familiar with the concepts of germs of functions and sets, and with fundamentals of the sheaf theory and theory of analytic sets. Necessary definitions can be found, for example, in [14, 15]. If 𝐵 is a “global” object (a set, a function, or a family of functions), 𝐵_𝑥 will always denote its germ at the point 𝑥 (but the precise meaning of the germ will depend on the meaning of the object). If 𝐶 is a germ, 𝐶 will denote one of its representatives. By 𝒪_𝑥 we denote the algebra of germs of real analytic functions at 𝑥, where 𝑥∈ℝ^𝑛 (𝑛 fixed throughout the paper), and by 𝑚_𝑥 the (only) maximal ideal of 𝒪_𝑥, consisting of all germs in 𝒪_𝑥 that vanish at 𝑥. By 𝒪 we denote the sheaf of germs of real analytic functions on ℝ^𝑛.

If 𝑈 is an open subset in ℝ^𝑛, then 𝒪_𝑈 will mean the algebra of real analytic functions on 𝑈. If 𝐴 is a subalgebra of 𝒪_𝑈 and 𝑥₀∈𝑈, then 𝐴_𝑥0 means the set of germs at 𝑥₀ of functions from 𝐴. Of course 𝐴_𝑥0 is again an algebra over ℝ. If 𝐼 is an ideal of 𝒪_𝑈, then 𝐼_𝑥0 means the ideal of 𝒪_𝑥0 generated by the germs at 𝑥₀ of function from 𝐼.

Consider a set-germ 𝐴 in ℝ^𝑛 (at some point 𝑥). Then 𝐽(𝐴) denotes the ideal of 𝒪_𝑥 consisting of germs (at 𝑥) of real analytic functions that vanish on 𝐴. If 𝐼 is an ideal of 𝒪_𝑥, then 𝑍(𝐼) will denote the zero set-germ of 𝐼 (at 𝑥). Let us recall that 𝑍(𝐼) is defined as the intersection of the set-germs 𝑍(𝜑_𝑖), 𝑖=1,…,𝑘, where 𝜑₁,…,𝜑_𝑘 are generators of the ideal 𝐼. Since only finite intersections of set-germs are defined, we must use here the property that 𝒪_𝑥 is Noetherian.

We have a natural duality between ideals and set-germs. If 𝐼₁⊂𝐼₂, then 𝑍(𝐼₂)⊂𝑍(𝐼₁).

Let 𝑃 be any commutative ring with a unit and let 𝐼 be an ideal of 𝑃. Then the real radical of 𝐼, denoted by √^ℝ𝐼, is the set of all 𝑎∈𝑃 for which there is 𝑚∈ℕ, 𝑘∈ℕ∪{0} and 𝑏₁,…,𝑏_𝑘∈𝑃 such that 𝑎^2𝑚+𝑏²₁+⋯+𝑏²_𝑘∈𝐼. (B.1) The real radical is an ideal in 𝑃 and it contains 𝐼. If 𝐼 is a proper ideal of 𝑃, then also √^ℝ𝐼 is proper. An ideal 𝐼 is called real if √^ℝ𝐼=𝐼.

Theorem B.1 (see [16]). Let 𝑥∈ℝ^𝑛. If 𝐼 is an ideal of 𝒪_𝑥, then 𝐽(𝑍(𝐼))=√^ℝ𝐼. (B.2)

Theorem B.1 implies that there is a 1 : 1 correspondence between germs of analytic sets at 𝑥 and real ideals of 𝒪_𝑥.

Acknowledgment

This work was supported by the Bialystok University of Technology Grant S/WI/2/2011.