Abstract— This paper deals with realization theory of socalled Nash systems, i.e. nonlinear systems the right-hand side of which is defined by Nash functions. A Nash function is a semi-algebraic analytic function. The class of Nash systems is an extension of the class of polynomial and rational systems and it is a subclass of analytic nonlinear systems. Nash systems occur in many applications, including systems biology. We formulate the realization problem for Nash systems and present a partial solution to it. More precisely, we provide necessary and sufficient conditions for realizability of a response map by a Nash system. The concepts of semi-algebraic observability and reachability are formulated and their relationship with minimality is explained. In addition to their importance for systems theory, the obtained results are expected to contribute to system identification and model reduction of Nash systems.

I. I NTRODUCTION Realization theory is one of the central topics of system theory. It serves as a theoretical foundation for model reduction, system identification and filtering/observer design. Its aim is to answer the following questions: (1) Under which conditions is it possible to construct a (preferably minimal) system of a certain class generating the specified input/output behaviour? (2) How to characterize minimal systems of a certain class which generate the specified input/output behaviour? In this paper we investigate realization theory of the class of Nash systems. Nash systems are continuous-time dynamical systems with Nash submanifolds as state-spaces and such that the right-hand sides of the differential equations which determine their dynamics, and the output functions are Nash functions. By a Nash function we mean an analytic function satisfying an algebraic equation. A Nash submanifold of Rn is a smooth manifold which is defined by polynomial equalities and inequalities. A. Motivation Nash systems are interesting both from a theoretical and a practical point of view. Theoretical relevance The class of Nash systems lies between polynomial/rational systems and analytic systems. While more general than the former, it still admits a constructive description by means of finitely many polynomial equalities and inequalities. Hence, it might still be possible to derive computational methods for control and analysis of This work was partially supported by the ITEA project Twins 05004 and NWO project 613.000.442. J. Nˇemcov´a and J.H. van Schuppen are with Centrum Wiskunde & Informatica (CWI), P.O. Box 94079, 1090 GB Amsterdam, The Netherlands

[email protected], [email protected] M. Petreczky is with Maastricht University, Department of Knowledge Engineering, P.O. Box 616, 6200 MD Maastricht, The Netherlands

[email protected]

Nash systems. It is well-known that (piecewise-)polynomial systems can be used to approximate the behaviour of nonlinear systems, see [14]. Since polynomial systems are also Nash, it follows that Nash systems might be used to approximate general nonlinear systems. Nash systems might allow an algebraic system theory which is capable of handling topological aspects such as stability and robustness by algebraic means. This is in contrast to polynomial/rational systems structure theory of which is unable to incorporate those topological notions. More precisely, the language of real algebraic geometry, which is a natural tool for studying Nash systems, appears to be able to express such concepts of topology of real numbers as open neighbourhoods, smoothness and continuity. Furthermore, Nash systems appear to be one of the simplest representatives of the class of semi-algebraic systems, i.e. systems defined by semi-algebraic vector fields (statetransition map) and semi-algebraic readout maps. The latter class is very general, and it contains several subclasses of non-smooth and hybrid systems. Relevance for systems biology Polynomial and rational systems, and thus Nash systems, are used in systems biology to model metabolic, signaling, and genetic networks. In [23], [24] Savageau proposes modeling metabolic and generegulatory networks by dynamical systems which still belong to the class of Nash systems, but are more general than polynomial and rational systems. More specifically, in [23], [24] models are proposed where the right-hand sides of the differential equations describing the kinetics are powerlaw functions, i.e. sums of products of the state variables taken to a rational power, i.e. xq11 xq22 · · · xqnn where qi , i = 1, . . . , n are rational numbers. Such functions are a special case of Nash functions. For further examples and details on application of power-law, and thus Nash, systems in biology see [28], [27], [13]. Note that considering Nash submanifolds as state-spaces allows a natural implementation of possible conservation laws and restrictions on the state variables. B. Contribution of the paper Our approach to realization theory for Nash systems is an extension of the one in [5] for polynomial systems and the one in [20], [19] for rational systems. We introduce the framework of Nash systems and the concepts of semialgebraic reachability and semi-algebraic observability. We link semi-algebraically reachable, semi-algebraically observable, and minimal Nash realizations. We expect that the results on realization theory for Nash systems will be useful for system identification, model reduction, filtering and control

design of Nash systems.

A. Definition of Nash systems A Nash system is a dynamical system with inputs and C. Related work outputs whose dynamics is given by Nash functions. In To the best of our knowledge, both the class of Nash systhis paper we consider Nash systems with the fixed input tems and the presented results on realization theory are new. and output spaces. The input space U is a subset of Rm . However, the idea of studying semi-algebraic systems, i.e. r systems described by polynomial equalities and inequalities, The output space is R . Further, we restrict attention to is quite well-established in the hybrid systems community, piecewise-constant inputs. see [18], [17], [22]. In [21] realization theory of discrete-time Definition III.1 (Piecewise-constant inputs) We denote by semi-algebraic hybrid systems is investigated. Since the class U pc the set of all piecewise-constant functions u : [0, Tu ] → m of systems in [21] is different from the class of Nash systems U ⊆ R , where Tu ∈ [0, +∞) is the maximal finite time considered in this paper, the results of this paper neither instance for which u is defined, imply nor are implied by those of [21]. For semi-algebraic Any u ∈ U pc can be identified with a finite sequence hybrid systems, problems related to realization theory are u = (α1 , t1 ) . . . (αn , tn ) where αi ∈PU , ti P ∈ [0, ∞) investigated in [7], [8], [10], [11], [16]. For polynomial for i = 1, . . . , n. Then, for t ∈ [ ij=0 tj , i+1 j=0 tj ), and rational systems, realization theory, reachability and u(t) = αP i+1 ∈ U for i = 0, 1, . . . , n − 1, t0 = observability are treated in [2], [3], [4], [5], [26], [20], [19]. 0, and u( nj=0 tj ) = αn . Note that the same u can have several different representations as a sequence of D. Overview of the paper In §II the necessary background on real algebraic geometry tuples (α, t) where α ∈ U , t ∈ [0, ∞). Every u = , tn ) ∈ U pc has a time domain [0, Tu ] is presented. The framework for Nash systems is introduced (α1 , t1 ) . . . (αnP n where T = u j=1 tj . If u = (α1 , t1 ) . . . (αn , tn ), v = in §III. In this section we also formulate the realization (β , s ) . . . (β , s ) ∈ U pc then by (u)(v) ∈ U pc we denote 1 1 k k problem for Nash systems. The overview of the main results the input which we get by concatenating v to u, i.e. (u)(v) = of the paper is given in §IV. §V contains the proofs of these (α , t ) . . . (α , t )(β 1 1 n n 1 , s1 ) . . . (βk , sk ). To express that the results and §VI concludes the paper. input u was applied only on time-domain [0, t] ⊆ [0, Tu ] we II. P RELIMINARIES write a subindex [0, t] to u like u[0,t] . The empty input e is In this paper we follow the notation and terminology of such an input that Te = 0. [29], [15], [6] on commutative algebra and real-algebraic The precise definition of a Nash system is as follows. geometry. We refer to [29], [15] for the notion of polynomial, Definition III.2 (Nash systems) A Nash system Σ with an algebra over the field R, integral domain and transcendence input space U and an output space Rr is a quadruple degree of a field. In particular, recall that if A is an integral (X, f, h, x ) where 0 domain then the transcendence degree of A, denoted by (i) the state-space X is a semi-algebraically connected trdeg A, is well-defined and it equals the greatest number Nash submanifold of Rn , of algebraically independent elements of A. (ii) the dynamics of the system is given by x(t) ˙ = A subset S ⊆ Rn is semi-algebraic [6] if it is of the form n f (x(t), u(t)), u ∈ U , where f : X × U → R is such pc mi d ^ _ that for each α ∈ U the ith coordinate fα,i : X → R, S = {(x1 , . . . , xn ) ∈ Rn | (Pi,j (x1 , . . . , xn ) i,j 0)}, i = 1, . . . , n of the vector field fα : X 3 x 7→ i=1 j=1 where for each i = 1, . . . , d and j = 1, . . . , mi the symbol f (x, α) = (fα,1 (x), . . . , fα,n (x)) ∈ Rn is a Nash W i,j ∈ {<, >, ≤, ≥, =} and Pi,j ∈ R[X function, V 1 , . . . , Xn ]. Here stands for the logical or operator and stands for the logical (iii) the output of the system is given as y = h(x(t)), where and operator. We say that a semi-algebraic subset S ⊆ Rn the components of h = (h1 , . . . hr ) are Nash functions is semi-algebraically connected if it cannot be written as an on X, union of two disjoint closed semi-algebraic sets in S. Let (iv) x0 = x(0) ∈ X is the initial state of Σ. A ⊆ Rn , B ⊆ Rm be two semi-algebraic sets. A mapping Example III.3 The following model of glycolysis and lacf : A → B is called semi-algebraic if its graph graph(f ) = tate production is adopted from the supplements of [28]. {(x> , y > )> ∈ Rn+m | x ∈ A, y ∈ B, f (x) = y} is a x˙ 1 = 0.3592x−1.2906 x0.2168 Glc1.1287 − 0.3115x2.17 AT P 0.8152 4 1 1 semi-algebraic set in Rn+m . By a Nash function on a semix˙ 2 = 0.3115x2.17 AT P 0.8152 − 0.4698x1.0297 Pi0.2377 1 2 algebraic subset S of Rn we mean an analytic function from 1.0297 0.2377 3.5453 Pi + 1.1452x4 − 2.167x2.1649 2 3 S to R which is also semi-algebraic. We denote the ring of x˙ 3 = 0.9396x2.1649 −1.2906 0.2168 1.1287 n x ˙ = 2.167x − 0.3592x x Glc 4 3 4 1 Nash functions on S by N (S). A Nash submanifold X of R −1.1452x3.5453 − 0.9375x0.8744 x0.0991 P i−0.0005 4 2 4 is a semi-algebraic set which is also an analytic manifold. −0.2087x0.0002 4 Recall from [6] that the dimension of a semi-algebraically −1.2906 0.2168 x4 Glc1.1287 − 0.0417x0.6202 x0.9264 connected Nash submanifold X coincides with the dimension x˙ 5 = 0.3592x1 5 2 n 0.8744 0.0991 −0.0005 1.5255 of an affine variety given as the Zariski-closure of X in R . +0.9375x2 x4 Pi − 1.3258x5 III. NASH SYSTEMS The goal of this section is to define the notion of Nash systems and related concepts.

x˙ 6 = 0.0417x0.6202 x0.9264 5 2

where x1 , . . . , x6 are the concentrations of the respective metabolites (G6P, FBP, PGA3, PEP, pyruvate, lactate). The

state-space is X = (0, ∞)6 . The initial state is x1 (0) = · · · = x6 (0) = 1. The output function is the outflow of the pathway, i.e. h(x1 , . . . , x6 ) = x6 . The inputs are the concentrations of Glc, AT P and P i of external glucose, ATP, and inorganic phosphate, respectively. Note that this system is neither polynomial nor rational, but still Nash. Indeed, h is linear and hence Nash function of the state. The right-hand sides of the differential equations are linear combinations of terms of the form xq11 · · · xq66 where qi , i = 1, . . . , 6 is a rational number. Since Nash functions are closed under linear combination, multiplication and division, it is left to show that the map n g(x) = x d , where x ∈ (0, +∞) and n, d are positive integers, is Nash. Since g(x) is analytic, it is enough to show that g is semi-algebraic. For all x ∈ (0, +∞), y = g(x) if and only if P (y, x) = 0, where the polynomial P is defined as P (Y, X) = Y d −X n . Thus, g is a semi-algebraic function. To introduce the relevant system theoretic concepts we first define the notion of a state trajectory. Definition III.4 (State trajectory) The state trajectory of a Nash system Σ corresponding to an input u : [0, Tu ] → U from U pc is a piecewise-differentiable continuous function xΣ (.; x0 , u) : [0, Tu ] → X such that xΣ (0; x0 , u) = x0 and d xΣ (t; x0 , u) = f (xΣ (t; x0 , u), u(t)) (1) Pdti Pi+1 for t ∈ ( j=0 tj , j=0 tj ), t0 = 0, i = 0, . . . , n − 1 if u ∈ U pc is such that u = (α1 , t1 ) . . . (αn , tn ) with αi ∈ U , ti ∈ [0, +∞) for i = 1, . . . , n. Note that for any x0 ∈ X, α ∈ U such that the Nash functions fα,i , i = 1, . . . , n are defined at x0 there exists a unique trajectory of Σ corresponding to the constant input u = (α, T ) defined on the maximal interval [0, T ] (T may be infinite). Nevertheless, a unique solution of (1) need not exist for all u ∈ U pc . In order to deal with this phenomenon we introduce the notion of admissible inputs for a Nash system. Definition III.5 (Admissible inputs of Σ) A set U pc (Σ) of admissible inputs for Σ is a subset of U pc such that for all u ∈ U pc (Σ) there exists a unique trajectory xΣ (·; x0 , u) : [0, Tu ] → X of Σ corresponding to the input u. B. Response maps of Nash systems A response map characterizes the external behaviour of a system by evaluating the outputs after applying the inputs to the system only for finite time. For our purposes, the response maps introduced in [20], [19] represent an appropriate formalization of the external behaviour of Nash systems. Below we recall the definition and basic properties of response maps from [20], [19]. Potential response maps need not be defined for all piecewise-constant inputs. To formalize this property we introduce the notion of admissible inputs for response maps. g Definition III.6 (Admissible inputs) A subset U pc of the set U pc is called a set of admissible inputs if: g g (i) ∀u ∈ U pc ∀t ∈ [0, Tu ] : u[0,t] ∈ U pc , g g (ii) ∀u ∈ U pc ∀α ∈ U ∃t > 0 : (u)(α, t) ∈ U pc , g (iii) ∀u = (α1 , t1 ) . . . (αk , tk ) ∈ U ∃δ > 0 ∀ti ∈ [0, ti + pc g δ], i = 1, . . . , k : u = (α1 , t1 ) . . . (αk , tk ) ∈ U pc .

The first two properties of the definition above assure the well-definedness of the derivative at switching times of differentiable functions which are defined on a set of admissible inputs. To be more precise, let us define the class of functions analytic in switching times and their derivatives at switching times. Definition III.7 (Maps analytic in switching times) g We say that a function ϕ : U pc → R is analytic in the g switching times of the inputs from U pc if for every input g u = (α1 , t1 ) . . . (αk , tk ) ∈ U pc the function ϕα1 ,...,αk (t1 , . . . , tk ) = ϕ((α1 , t1 ) . . . (αk , tk )) is analytic. We denote the set of all such real functions ϕ by g A(U pc → R). Definition III.8 (Derivatives) Consider a function ϕ ∈ g A(U pc → R). For any α ∈ U we define the derivative d g (Dα ϕ)(u) = ϕ((u)(α, t))|t=0+ for all u ∈ U pc . dt The condition (iii) of Definition III.6 is important for the proof of the following statement. Proposition III.9 ([20]) With pointwise addition and multig plication the set A(U pc → R) forms a commutative algebra over R and it is an integral domain. The class of response maps we will deal with is defined as follows: g Definition III.10 (Response maps) Let U pc be a set of adg missible inputs. Consider a map p : U pc → Rr with the g g components pi : U pc → R, i = 1, . . . , r, i.e. ∀u ∈ U pc : p(u) = (p1 (u), . . . , pr (u)). The map p is called a response g map if pi ∈ A(U pc → R) for all i = 1, . . . , r. In the following definition we recall the notion of observation algebra of a response map which plays a key-role in developing the theory of Nash realizations. r g Definition III.11 ([20]) Let p : U pc → R be a response map. The observation algebra Aobs (p) of p is the smallest g subalgebra of the algebra A(U pc → R) which contains the components pi , i = 1, . . . , r of p, and which is closed with respect to the derivations Dα , α ∈ U . I.e., Aobs (p) is the smallest algebra such that p1 , . . . , pr ∈ Aobs (p), and if ϕ ∈ Aobs (p) then Dα ϕ ∈ Aobs (p) for all α ∈ U . C. Realization problem for Nash systems To state the realization problem for Nash systems formally we first define the notion of dimension of a Nash system and the notion of (minimal) Nash realizations of response maps. Definition III.12 The dimension of a Nash system Σ = (X, f, h, x0 ) is the dimension of the Nash submanifold X, i.e. dim Σ = dim X, see §II. The dimension of Σ is related to the number of statevariables as follows. If dim Σ = d then there exists a local semi-algebraic coordinate transformation around any x ∈ X into Rd , i.e. locally, Σ can be described by d state variables. Notice that our definition of dimension is similar to the classical definition for nonlinear systems [12], [25]. r g Definition III.13 (Realization) Let p : U pc → R be a response map. A Nash system Σ = (X, f, h, x0 ) is called a Nash realization of p if

g g p(u) = h(xΣ (Tu ; x0 , u)) for all u ∈ U pc , U pc ⊆ U pc (Σ). That is, Σ is a realization of p, if (a) all the inputs for which p is defined are admissible for Σ, i.e. there exists a unique state trajectory corresponding to those inputs, and (b) the value of p for the input u equals the output of Σ for u. Definition III.14 (Minimality) We say that a Nash realization Σ = (X, f, h, x0 ) of a response map p is a minimal 0 Nash realization of p if for any Nash realization Σ of p it 0 holds that dim Σ ≤ dim Σ . Next we state the realization problem for Nash systems. r g Problem III.15 (Realization problem) Let p : U pc → R be a response map. The realization problem for p consists of the following three subproblems: Existence Find necessary and sufficient conditions for the existence of a Nash realization of p. Minimality Find necessary and sufficient conditions for minimality of a Nash realization of p. Determine whether a minimal Nash realization of p exists, and whether it is unique in any sense (for example, up to isomorphism). Realization algorithm Formulate algorithms for computing a Nash realization of p from finite data directly obtainable from p. In addition, formulate algorithms for checking minimality of a Nash realization and for transforming a Nash realization of p to a minimal one. The rest of the paper is devoted to solving the existence and minimality problems of Nash realizations. In this paper we will not deal with realization algorithms. IV. M AIN RESULTS In this section we provide an overview of our results. The proofs are stated in the subsequent section. A. Existence of Nash realizations First let us introduce the notion of Nash extension of a finite subset of maps analytic at switching times. Definition IV.1 (Nash extension) Let X ⊆ Rn be a semialgebraically connected Nash submanifold and let A = g {ϕ1 , . . . , ϕn } be a subset of A(U pc → R). Assume that g for all u ∈ U pc , (ϕ1 (u), . . . , ϕn (u)) ∈ X. The Nash extension AN ash (X) of A with respect to X is the subalgebra g g of A(U pc → R) generated by the maps U pc 3 u 7→ q(ϕ1 (u), . . . , ϕn (u)) ∈ R, where q ∈ N (X). Thus, the Nash extension of A is obtained by substituting the elements of A into Nash functions defined on X. Note that the set of substitutions into linear forms (polynomials) yields the linear space (resp. algebra) generated by A. That is, the Nash extension of A can be thought of as the generalization of the notions of linear space and algebra generated by A. Two theorems below provide sufficient and necessary conditions for the existence of Nash realizations. Theorem IV.2 (Realization theorem) A response map p : r g U pc → R has a Nash realization if and only if there exist a g finite subset A = {ϕ1 , . . . , ϕn } of A(U pc → R) and a semialgebraically connected Nash submanifold X ⊆ Rn such that AN ash (X) has the following properties: (i) pi ∈ AN ash (X) for all i = 1, . . . , r,

(ii) ∀α ∈ U ∀ϕ ∈ A : Dα ϕ ∈ AN ash (X). The conditions of the theorem above are quite difficult to check. However, we can formulate a necessary condition for the existence of Nash realizations which is easier to check. Theorem IV.3 (Necessary condition) If Σ = (X, f, h, x0 ) is a Nash realization of a response map p then trdeg Aobs (p) ≤ dim X and thus trdeg Aobs (p) < +∞. Note that this necessary condition is analogous to the finite Hankel-rank condition for linear systems. In particular, the transcendence degree of Aobs (p) can be seen as generalization of the rank of the Hankel-matrix for linear systems. Remark IV.4 (Sufficient condition) We conjecture that the condition trdeg Aobs (p) < +∞ of Theorem IV.3 is not only necessary but also sufficient. In fact, in [26] it was shown that trdeg Aobs (p) < +∞ is a sufficient condition for the existence of a realization by an input-affine rational system, if p itself has a representation by Fliess-series expansion. Since input-affine rational systems are Nash systems, the results of [26] are a strong indication that the condition of Theorem IV.3 might be a sufficient one too. B. Reachability, observability, and minimality Below we define and link a number of system theoretic properties of Nash systems. Semi-algebraic reachability We define semi-algebraic reachability of Nash realizations by a slight modification of the usual concept of reachability. Namely, instead of requiring that the whole state-space is reachable from the initial state we require that the set of reachable states is a sufficiently large (in some sense) subset of the state-space. r g Definition IV.5 Let p : U pc → R be a response map and let Σ be a Nash realization of p. We denote by R(x0 ) the reachable set of Σ given as g R(x0 ) = {xΣ (Tu ; x0 , u) | u ∈ U pc ⊆ U pc (Σ)}. Hence, R(x0 ) is the set of states of Σ which are reachable by g admissible inputs from U pc . This set may be smaller than the set of all states of Σ reachable by the inputs from U pc (Σ). Definition IV.6 (Semi-algebraic reachability) We say that a Nash realization Σ of a response map p is semialgebraically reachable if no non-zero element of N (X) vanishes on the set of states of Σ reachable from x0 , i.e. ∀g ∈ N (X) : (g = 0 on R(x0 ) ⇒ g = 0) . If a Nash system is reachable, i.e. if all its states can be reached by a suitable input, then it is also semi-algebraically g reachable provided that U pc = U pc . A nonlinear smooth system is called accessible if the set of reachable states contains an open set, see [25]. Accessibility of nonlinear systems admits a characterization in terms of the rank of the Lie-algebra generated by the vector fields of the system. Since the definition of accessibility and the corresponding Lie-rank condition can directly be applied to Nash systems, the following proposition implies that Lie-rank condition yields a sufficient condition for semi-algebraic reachability. Proposition IV.7 If there exists an open set ∅ = 6 S ⊆ X such that S ⊆ R(x0 ) then Σ is semi-algebraically reachable.

Semi-algebraic observability In order to define semialgebraic observability of Nash systems we introduce the notion of observation algebra and its Nash extension. Definition IV.8 (Observation algebra) The observation algebra Aobs (Σ) of a Nash system Σ is the smallest subalgebra of N (X) which contains hi , i = 1, . . . , r and which is closed with respect to Lie-derivatives along the vector fields fα : X 3 x 7→ f (x, α) ∈ Rn , α ∈ U . The definition of the observation algebra is analogous to the one for rational and polynomial systems [19], [20], [26], [5]. ash Definition IV.9 We define the Nash extension AN obs (Σ) of the observation algebra Aobs (Σ) of a Nash system Σ as ash AN obs (Σ) = {g : X → R | ∃k ∈ N ∃ϕ1 , . . . , ϕk ∈ Aobs (Σ) ∃q ∈ N (Rk ) : g = q(ϕ1 , . . . , ϕk )}. Definition IV.10 (Semi-algebraic observability) We say that a Nash system Σ = (X, f, h, x0 ) is semi-algebraically ash observable if AN obs (Σ) = N (X). A system theoretic interpretation of semi-algebraic observability is provided by the following proposition. Proposition IV.11 If Σ is semi-algebraically observable then any two states x1 6= x2 of Σ are distinguishable by an element of Aobs (Σ), i.e. ∃g ∈ Aobs (Σ) : g(x1 ) 6= g(x2 ). Corollary IV.12 Let Σ be semi-algebraically observable. Assume that for all x ∈ X and for all u ∈ U pc the trajectory xΣ (Tu ; x, u) starting at x is well-defined. Then Σ is observable in a sense that it has no indistinguishable states. Formally, if x1 6= x2 ∈ X, then there exists u ∈ U pc such that h(xΣ (Tu ; x1 , u)) 6= h(xΣ (Tu ; x2 , u)). This implies that the differential geometric conditions for observability of nonlinear systems, see [12], yield necessary conditions for semi-algebraic observability of Nash systems. Definition IV.13 (Canonicity) We say that a Nash realization Σ of a response map p is canonical if it is both semialgebraically reachable and semi-algebraically observable. Minimality Recall that minimal Nash realization is such that the dimension of its state-space is the smallest one within the class of all Nash realizations of the same response map. Theorem IV.14 (Minimality) A canonical Nash realization Σ of a response map p is minimal. V. P ROOFS OF MAIN RESULTS In this section we present the proofs of the main results on Nash realizations stated in §IV. A. Properties of dual input-to-state-maps The dual input-to-state map of a Nash system Σ = (X, f, h, x0 ) maps each Nash function g on X to the 0 response map which is generated by the Nash system Σ = (X, f, g, x0 ), i.e. the system where g instead of h is used as the readout map. The formal definition goes as follows. Definition V.1 The dual input-to-state map of a Nash realr g ization Σ of a response map p : U pc → R is the map ∗ g τΣ : N (X) → A(U pc → R) such that ∗ g ∀g ∈ N (X) ∀u ∈ U pc : τΣ (g)(u) = g(xΣ (Tu ; x0 , u)). The dual input-to-state map plays a role similar to the observability Grammian of linear systems. In particular, it

allows us to relate the properties of the ring of Nash functions over the state-space to the properties of the corresponding response maps. Proposition V.2 Let τΣ∗ be as in Definition V.1. Then (i) ∀g ∈ N (X) ∀α ∈ U : Dα τΣ∗ (g) = τΣ∗ (Lfα g), (ii) Aobs (p) = τΣ∗ (Aobs (Σ)), (iii) ∀f ∈ N (Rk ) ∀ϕ1 , . . . , ϕk ∈ N (X) : τΣ∗ (f (ϕ1 , ϕ2 , . . . , ϕk ))(u) = (2) = f (τΣ∗ (ϕ1 )(u), τΣ∗ (ϕ2 )(u), . . . , τΣ∗ (ϕk )(u)). Proof: g (i) We will show that ∀g ∈ N (X) ∀α ∈ U ∀u ∈ U pc : ∗ ∗ Dα τΣ (g)(u) = τΣ (Lfα )(u). Indeed, d Dα τΣ∗ (g)(u) = g(xΣ (Tu + t; x0 , (u)(α, t)))|t=0+ dt = (Lfα g)(xΣ (Tu ; x0 , u)) = τΣ∗ (Lfα g)(u). g (ii) It follows that pi (u) = hi (xΣ (Tu ; x0 , u)) for all u ∈ U pc and i = 1, . . . , r. Therefore, pi = τΣ∗ (hi ) for all i = 1, . . . , r.

(3)

Further, from (i) it follows that for all α1 , . . . αk ∈ U, u ∈ g U pc , and for all i = 1, . . . , r we have Dα1 · · · Dαk pi (u) = Dα1 · · · Dαk τΣ∗ (hi )(u) = = τΣ∗ (Lfα1 · · · Lfαk hi )(u)

(4)

Because Aobs (p) is generated by the elements of the set {pi , Dα1 · · · Dαk pi |i = 1, . . . , r, k ∈ N, α1 , . . . , αk ∈ U }, because Aobs (Σ) is generated by the elements of the set {hi , Lfα1 · · · Lfαk hi |i = 1, . . . , r, k ∈ N, α1 , . . . , αk ∈ U }, and because τΣ∗ is a homomorphism, we derive from (3) and (4) that Aobs (p) = τΣ∗ (Aobs (Σ)). (iii) Since f (ϕ1 (xΣ (Tu ; x0 , u)), . . . , ϕk (xΣ (Tu ; x0 , u))) = τΣ∗ (f (ϕ1 , . . . , ϕk ))(u), (iii) follows from Definition V.1. B. Proofs of the results on existence of Nash realizations Proof: Proof of Theorem IV.2 (⇒) Assume that Σ = (X, f, h, x0 ) with X ⊆ Rn is a Nash realization of p. Consider the projection maps πi : Rn → R which map any vector in Rn to its ith entry, i = 1, . . . , n. Let ϕi = τΣ∗ (πi ), i = 1, . . . , n. We define A = {ϕ1 , . . . , ϕn }. g From Proposition V.2(i),(iii) it follows that for all u ∈ U pc pi (u) = hi (ϕ1 (u), ϕ2 (u), . . . , ϕn (u)) Dα ϕi (u)

=

τΣ∗ (Lfα (πi ))(u) = τΣ∗ (fα,i )(u)

= fα,i (ϕ1 (u), ϕ2 (u), . . . , ϕn (u)). Thus, from Definition IV.1, the conditions (i) and (ii) of the theorem are fulfilled. g (⇐) Let ϕ1 , . . . , ϕn ∈ A(U pc → R) be such that A = {ϕ1 , . . . , ϕn } satisfies the conditions (i) and (ii) of the theorem. Then there exist a Nash submanifold X ⊆ Rn and hi , fα,i ∈ N (X), i = 1, . . . , n, α ∈ U such that g ∀u ∈ U pc : pi (u) = hi (ϕ1 (u), . . . , ϕn (u)) (5) Dα ϕi (u) = fα,i (ϕ1 (u), . . . , ϕn (u)). Consider the system Σ = (X, f, h, x0 ) such that for all x ∈ Rn , α ∈ U it holds that f (x, α) = (fα,1 (x), . . . , fα,n (x)), h(x) = (h1 (x), . . . , hr (x)), and x0 = (ϕ1 (e), . . . , ϕn (e)).

We prove that this system is a Nash realization of p. For any g u∈U pc , the solution of x˙ = f (x, u), x(0) = x0 equals x(t) = (ϕ1 (u|[0,t] ), . . . , ϕn (u|[0,t] )) for t ∈ [0, Tu ]. g Thus, xΣ (Tu ; x0 , u) = (ϕ1 (u), . . . , ϕn (u)) for u ∈ U pc . Then, from (2) and the first equation of (5) it follows that pi (u) = τΣ∗ (hi ), i = 1, . . . , r. That is, Σ is indeed a Nash realization of p. In order to prove Theorem IV.3, we need the following simple corollary of [6, Proposition 8.1.9, Lemma 2.6.3]. Lemma V.3 Let X ⊆ Rn be a semi-algebraically connected Nash submanifold. Then the ring N (X) of Nash functions on X is algebraic over the ring R[X] of polynomials on X. Proof: Proof of Theorem IV.3 Let Σ = (X, f, h, x0 ) r g be a Nash realization of p : U pc → R . Let V be the Zariski closure of X. Then dim V = dim X. The algebra R[X] of all polynomials on X coincides with the coordinate ring R[V ] of the variety V . Since X is semi-algebraically connected Nash submanifold of Rn , by [6, Proposition 8.4.1], V is irreducible and hence trdeg R[V ] = dim V . Further, R[X] ⊆ N (X). From Lemma V.3 it follows that trdeg N (X) = trdeg R[X] = trdeg R[V ] = dim V = dim X. Because Aobs (Σ) ⊆ N (X), we derive by Proposition V.2(ii) that trdeg Aobs (p) ≤ trdeg τΣ∗ (N (X)) ≤ trdeg N (X) = dim X. C. Proof of the main results on minimality Proposition V.4 A Nash realization Σ of a response map p is semi-algebraically reachable if and only if τΣ∗ is injective. ash Lemma V.5 AN obs (Σ) is algebraic over Aobs (Σ) and conash sequently trdeg AN obs (Σ) = trdeg Aobs (Σ). Proposition V.6 If a Nash system Σ is semi-algebraically observable, then trdeg Aobs (Σ) = dim X. Proof: Proof of Theorem IV.14 Let Σ be a canonical Nash realization of a response map p. It follows from Proposition V.4 and semi-algebraic reachability of Σ that τΣ∗ is injective. From Proposition V.2(ii), we get that τΣ∗ : Aobs (Σ) → Aobs (p) is an isomorphism and hence trdeg Aobs (Σ) = trdeg Aobs (p). Further, by Proposition V.6, semi-algebraic observability of Σ implies that dim Σ = trdeg Aobs (p). From 0 Theorem IV.3 we derive that for any Nash realization Σ of 0 p it holds that dim Σ = trdeg Aobs (p) ≤ dim Σ . Hence, Σ is a minimal Nash realization of p. VI. C ONCLUSIONS We have defined the class of Nash systems and we have formulated the realization problem for this class. Our motivation for studying Nash systems is their relevance both for theory and applications. We have presented a partial solution to the realization problem by formulating necessary and sufficient conditions for the existence of a Nash realization and sufficient conditions for the existence of a minimal Nash realization. Further research aims at extending these results to a complete solution of the realization problem. In addition, we would like to investigate the applications of the obtained results to system identification and model reduction of Nash systems.

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