STABILITY AND STABILIZATION OF MULTIDIMENSIONAL INPUT ...

SIAM J. CONTROL OPTIM. Vol. 45, No. 4, pp. 1467–1507

c 2006 Society for Industrial and Applied Mathematics 

STABILITY AND STABILIZATION OF MULTIDIMENSIONAL INPUT/OUTPUT SYSTEMS∗ ULRICH OBERST† Abstract. In this paper we discuss stability and stabilization of continuous and discrete multidimensional input/output (IO) behaviors (of dimension r) which are described by linear systems of complex partial differential (resp., difference) equations with constant coefficients, where the signals are taken from various function spaces, in particular from those of polynomial-exponential functions. Stability is defined with respect to a disjoint decomposition of the r-dimensional complex space into a stable and an unstable region, with the standard stable region in the one-dimensional continuous case being the set of complex numbers with negative real part. A rational function is called stable if it has no poles in the unstable region. An IO behavior is called stable if the characteristic variety of its autonomous part has no points in the unstable region. This is equivalent to the stability of its transfer matrix and an additional condition. The system is called stabilizable if there is a compensator IO system such that the output feedback system is well-posed and stable. We characterize stability and stabilizability and construct all stabilizing compensators of a stabilizable IO system (parametrization). The theorems and proofs are new but essentially inspired and influenced by and related to the stabilization theorems concerning multidimensional IO maps as developed, for instance, by Bose, Guiver, Shankar, Sule, Xu, Lin, Ying, Zerz, and Quadrat and, of course, the seminal papers of Vidyasagar, Youla, and others in the one-dimensional case. In contrast to the existing literature, the theorems and proofs of this paper do not need or employ the so-called fractional representation approach, i.e., various matrix fraction descriptions of the transfer matrix, thus avoiding the often lengthy matrix computations and seeming to be of interest even for one-dimensional systems (at least to the author). An important mathematical tool, new in systems theory, is Gabriel’s localization theory which, only in the case of ideal-convex (Shankar, Sule) unstable regions, coincides with the usual one. Algorithmic tests for stability, stabilizability, and ideal-convexity, and the algorithmic construction of stabilizing compensators, are addressed but still encounter many difficulties; see in particular the open problems listed by Xu et al. Key words. stability, stabilization, multidimensional system, behavior, stable transfer matrix AMS subject classifications. 93D15, 93D25, 93C20, 93C35 DOI. 10.1137/050639004

1. Introduction. Stabilization theory is a part of control theory and usually involves the following ingredients [7, p. 60]. 1. Stability: Select the class of admissible systems and define and characterize the stable systems in this class. 2. Stabilizability: Determine which admissible systems can be stabilized by output feedback. 3. Stabilization: Construct a stabilizing compensator for a given stabilizable system. 4. Parametrization: Classify or construct all stabilizing compensators for a given stabilizable system. In this paper we discuss these problems for continuous and discrete multidimensional input/output (IO) behaviors, which are described by linear systems of complex partial differential equations on Rr (resp., difference equations on Nr ) with constant ∗ Received by the editors August 26, 2005; accepted for publication (in revised form) May 12, 2006; published electronically October 16, 2006. http://www.siam.org/journals/sicon/45-4/63900.html † Institut f¨ ur Mathematik, Universit¨ at Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria ([email protected]).

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coefficients, where the signals are taken from various function spaces—in particular from those of polynomial-exponential functions. Polderman and Willems in [15, sec. 10.8] and Rocha [19] suggest renouncing the IO structure and output feedback in favor of more general behavior interconnections. The first stabilization results for multidimensional systems in this generality are due to Shankar [21], [22]. In contrast to our approach, most multidimensional stabilization papers use a commutative integral domain S of “SISO-stable plants” and describe the admissible systems, often called plants, by a transfer operator or IO map, which is a matrix with coefficients in the quotient field of S; cf., for instance, [26], [6], [23], [25], [31], [32], [7], [8], [16], [17], [18]. This approach to stabilization theory is originally due to Desoer, Kucera, Vidyasagar, Youla, and their coworkers. These systems are called structurally [6] or internally [17] stable if their transfer matrix has entries in S. In their recent paper [29], Wood, Sule, and Rogers treat stability and causality, but not stabilization of continuous multidimensional IO systems. An IO behavior B gives rise to its autonomous part B 0 , its transfer matrix H, and the largest controllable subbehavior Bcont which, in turn, has the autonomous 0 part Bcont . The entries of the transfer matrix are complex rational functions in r indeterminates sρ , i.e., contained in the quotient field C(s) of the polynomial algebra A := C[s] = C[s1 , . . . , sr ]. In general, the transfer matrix does not act on arbitrary inputs as an operator or IO map, and it is an important task to identify those inputs on which it does. Associated with these behaviors is the complex variety sing(B) 0 of rank singularities and the characteristic varieties char(B 0 ) ⊇ char(Bcont ) of the autonomous subbehaviors, the latter coinciding with the variety of poles of H. In the one-dimensional theory, the elements of char(B 0 ) are called the poles, modes, characteristic values, or natural frequencies of the system. Stability and stabilization of an IO system are defined with respect to a disjoint decomposition Cr = Λ1  Λ2 of the complex space into a stable region Λ1 and an unstable region Λ2 , with the standard continuous (resp., discrete) cases being r

(1)

Λ2 = C+ , C+ := {z ∈ C; (z) ≥ 0} or Λ2 = C+ × iRr−1 ([29]), resp., Λ2 = {z ∈ C; |z| ≥ 1}r or Λ2 = {z ∈ C; |z| ≥ 1} × (S 1 )r−1 .

A rational function is called stable if it has no poles in the unstable region. The ring of all stable rational functions or SISO-stable plants is the quotient ring AT ⊆ C(s) with T := {t ∈ A; ∀λ ∈ Λ2 : t(λ) = 0} [25]. The discrete IO maps in the literature are usually assumed to be causal or proper and are considered rational functions in the indeterminates s−1 ρ ; then the set Λ2 from (1) is replaced with the closed unit polydisc [6], [7]. Properness is not assumed in the present paper because it is rather restrictive for partial differential equations, but properness and the ensuing BIBO stability will be discussed in the paper [20]. An IO system B is called stable if the characteristic variety of its autonomous part is contained in the stable region or, equivalently, if all polynomial-exponential trajectories in B 0 are stable, i.e., involve exponents in the stable region only. In [29] this is called a characteristic variety (CV) condition and used to define stable autonomous systems. Stability of B in this sense is equivalent to the stability of the transfer matrix and an additional condition (see Theorem/Definition 5.1 and Remark 5.2). In particular, a one-dimensional IO system is stable if and only if its autonomous part is asymptotically stable or, equivalently, if its transfer matrix is stable and its singular variety is contained in the stable region. One of the reviewers points out that these two equivalent descriptions of the stability of one-dimensional IO systems should be

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called time domain (resp., frequency domain) stability. Multidimensional stable IO systems are externally or IO stable in the sense that the transfer matrix acts as an operator on interesting classes of inputs and generates outputs of the same type as shown in Theorem/Definition 5.3 and Theorem 5.4; the latter applies results on partial differential equations from [4] as do Theorems 6.4 and 7.2 in [29] which, however, hold in special cases only. Theorem 7.6.2 of [15, p. 265], for instance, treats the connection between asymptotic stability and bounded input/bounded output (BIBO) stability of one-dimensional IO systems as done already by Kalman in his fundamental work. The paper [29] contains the interesting idea that stable systems should generate stable outputs from stable inputs and initial conditions, with a necessary requirement that the initial value problem be defined and uniquely solvable. For discrete multidimensional systems this is the case for which we give a partial answer and pose Open Problem 5.13, whose study is also worthwhile for continuous systems and certain function spaces. An IO system B is called stabilizable if there is an IO system B  such that the feedback system (in (20), (21)) of B and B  is well-posed [26] and stable; then B  is called a stabilizing compensator. In Theorems 4.4 and 5.8 we characterize stabilizability of B and construct one stabilizing compensator, whereas Theorems 2.14 and 4.6 describe the parametrization or construction of all stabilizing compensators of a stabilizable IO behavior. The famous prototype of such a parametrization is that of Kucera, Youla, Bongiorno, and Jabr and is detailed by Vidyasagar in [26, Chap. 5]. The theorems on the stabilization of general multidimensional IO systems and their proofs are new but essentially influenced and inspired by and related to the results on the stabilization of IO maps in the references given above. The proofs employ localization, after the work of Gabriel, as a new mathematical tool in systems theory which is described in Stenstr¨ om’s book [24] and in section 3 of this paper. At no time do the results and proofs need or employ the so-called fractional representation approach (i.e., matrix fraction descriptions of the transfer matrix of various kinds and the, sometimes long [26], [17], [18], ensuing matrix computations); thus they seem simpler and of interest even in the one-dimensional case (at least to the author). The localization technique also avoids the difficulties in [29] with the lack of ideal-convexity [23] of the unstable regions Λ2 . Such a region is called ideal-convex if (2)

V (a) ∩ Λ2 = ∅ ⇒ a ∩ T = ∅ ∀ ideals a ⊂ C[s], where V (a) := {λ ∈ Cr ; ∀t ∈ a : t(λ) = 0}

denotes the algebraic variety of a. Ideal-convexity is characterized by the coincidence of Gabriel localization with the standard localization functor M → MT on A-modules M (see Theorem/Definition 5.6). Algorithmic problems are addressed in Remark/Open Problem 5.10. The algorithmic test of stability, stabilizability, and ideal-convexity and the algorithmic construction of one or all stabilizing compensators still encounter many difficulties; see in particular the open problems in [7] and [30] which, however, address these difficulties only for the closed unit polydisc of arbitrary dimension as region of instability. Solutions for the closed unit polydisc are known in interesting special cases [6], [7], [8]. Sections 2, 3, and 4 of this paper consider abstract IO systems whose signal spaces are injective cogenerators over a factorial Noetherian integral domain as in [11, Chap. 7] and use, in particular, Matlis’ theory of injective modules over Noetherian

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rings as in [12]. This generality makes the proofs more transparent and may possibly be used for other types of systems too. As preparation for the stable and stabilizable systems in section 4, section 2 discusses trivial and trivializable systems and the construction of all trivializing compensators. An IO behavior is called trivial if its autonomous part is zero or, equivalently by Theorem/Definition 2.5, if it is controllable and its transfer matrix is polynomial. In section 5 the results of sections 2, 3, and 4 are specialized to multidimensional systems proper as described above. Due to the large number of papers on stabilization theory, we list only those references which are actually used. But the author is fully conscious of many other important contributions and contributors, such as, for instance, Bisiacco, Fornasini, Marchesini, and Valcher of the Padovian school. 2. Triviality and trivialization by feedback. Let A denote a commutative Noetherian integral domain with its quotient field K = quot(A), and let F denote an injective cogenerator which is used as a signal space with its scalar multiplication ◦. In this section we consider F-systems or F-behaviors as introduced and studied in [11], in particular [11, Chap. 7, p. 139]. Refer to the first pages of [28] or to [27] for a newer, more elegant introduction to multidimensional behavioral systems theory. It was shown that many cases of interest for systems theory can be developed in this abstract setting. For instance, the continuous case of systems governed by linear systems of partial differential equations with constant coefficients uses the data

(3)

A := C[s] = C[s1 , . . . , sr ], F  y = y(z), sρ ◦ y = ∂y/∂zρ , z := (z1 , . . . , zr ) ∈ Rr , λ = (λ1 , . . . , λr ) ∈ Cr , λ • z := λ1 z1 + · · · + λr zr , F := C ∞ (Rr , C) or F := D (Rr ) or F := C ∞ (Rr , C)lf := {y ∈ C ∞ (Rr , C); [C[s] ◦ y : C] < ∞} = D (Rr )lf = ⊕λ∈Cr C[z] exp(λ • z),

whereas multidimensional discrete or r-dimensional systems theory applies the polynomial algebra A as in (3) and the signal space F := CN = C[[z]] = C[[z1 , . . . , zr ]]  y = (yμ )μ∈Nr = r

(4)



yμ z μ

μ∈Nr

with (sν ◦ y)μ := yμ+ν , μ, ν ∈ Nr . The corresponding objects over the real field R instead of the complex field C are likewise admissible. Recall that an A-module F is an injective cogenerator if the contravariant duality functor, (5)

D := HomA (−, F) : (ModA )op → ModA , M → HomA (M, F),

preserves and reflects exact sequences, where ModA is the category of A-modules. In [11] we used a large injective cogenerator F, but this is unnecessary, as was observed in [12]. The column vectors in F l are suggestively called trajectories also in the abstract module situation. A matrix R ∈ Ak×l gives rise to the row submodule (6)

U := A1×k R =

k  i=1

ARi− ⊂ F := A1×l ,

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the factor module M := A1×l /U =

l 

j

Aδj , δj := (0, . . . , 0, 1, 0, . . . , 0) ∈ A1×l ,

j=1

and the behavior B := U ⊥ := {w = (wj )j=1,...,l ∈ F l ; ∀f = (f1 , . . . , fl ) ∈ U : f ◦ w = f1 ◦ w1 + · · · + fl ◦ wl = 0} = {w = (wj )j=1,...,l ∈ F l ; R ◦ w = 0} = HomA (M, F), w = (δj → wj ). ident.

Since A is Noetherian, every submodule U ⊂ A1×l arises in this fashion. Like U ⊥ we define, for every submodule B of F l , the orthogonal submodule (7)

B ⊥ := {f = (f1 , . . . , fl ) ∈ A1×l ; ∀w ∈ B : f ◦ w = 0} ⊂ A1×l

of all linear equations which are satisfied by all trajectories in B. If B := U ⊥ is a behavior, the relation B ⊥⊥ = U ⊥⊥⊥ = U ⊥ = B holds since (−)⊥ is a Galois correspondence, whereas the identity U = B ⊥ = U ⊥⊥ is a consequence of the cogenerator property of F [11, Cor. 2.47]. The matrix R with M and B as in (6) also gives rise to its transfer space (= signal flow space in [11, Thm./Def. 2.91]) (8)

 = 0} = HomA (M, K) = HomK (K ⊗A M, K) ⊂ K l B := {w  ∈ K l ; Rw ident.

ident.

and, more precisely, to the contravariant exact functor HomA (−, K) on finitely generated A-modules or to the covariant exact functor B → B on behaviors. Standard  In particular, it linear algebra over the field K can be applied to the K-space B. determines the number (9)

rank(B) := rank(M ) := [K ⊗A M : K] = [B : K] = l − rank(R),

where [F : A] denotes the dimension of a free A-module F .  Let w = uy ∈ F p+m , l = p + m, be a decomposition of the trajectories w into two components y and u, possibly after a permutation of the components wj of w, k×(p+m) be the corresponding decomposition of the matrix and let R = (P, −Q) ∈ A y  R such that B := {w = u ∈ F p+m ; P ◦ y = Q ◦ u}. The matrix P gives rise to the module M 0 := A1×p /A1×k P and the behavior B 0 := {y ∈ F p ; P ◦ y = 0}. Result 2.1 (IO structure and transfer matrix; see [11, Thms. 2.69, 2.94]). The following assertions are equivalent: 1. rank(P ) = rank(R) = p. 2. The projection      y y (10) proj := (0 idm )◦ : B = w = ∈ K p+m ; P y = Q u → K m, → u , u  u  is a K-isomorphism, i.e., for each u  ∈ K m the equation P y = Q u is uniquely solvable for y ∈ K p . 3. The A-module sequence (11)

0 −→ A1×m η

inj:=◦(0 idm )

−→ →

M (0, η), (ξ, η)

proj:=◦(id0p )

−→ →

M0 ξ

−→ 0,

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is exact, and the module M 0 is a torsion module or rank(M 0 ) = 0. 4. The dual sequence of behaviors (12)

0

−→ B0 y

(id0p )◦ −→ →

y  B  y  0 , u

proj:=(0 idm )◦

−→ →

Fm u

−→ 0,

is exact, and the behavior B 0 is autonomous or rank(B 0 ) = 0. If these equivalent conditions are satisfied, the behavior is called an IO behavior  with the IO structure uy , the input u, and the output y. The rank of B is m. There is a unique matrix H ∈ K p×m , the transfer matrix of B, such  that P H = Q. The Hu  inverse of the isomorphism (10) has the graph form u  →  u  , and for every input   u ∈ F m there is a trajectory uy ∈ B. If B is an IO behavior as in the preceding result, the A-sequence ◦(idHm ) ◦(P,−Q) A1×k −→ A1×(p+m) −→ K 1×m   is a complex, i.e., (P, −Q) idHm = P H − Q = 0, and induces an A-epimorphism

(13)

(14)

   (idHm ) H M = A1×(p+m) /A1×k (P, −Q) −→ M := im ◦ = A1×p H + A1×m , idm (ξ, η) → ξH + η.

The A-submodule M of K 1×m is a lattice. Lattices play a decisive part in Quadrat’s treatment of stabilization [17, Eq. (38) and Thm. 3], [18] (see also Quadrat’s earlier papers quoted there). Theorem and Definition 2.2 (controllable behaviors and realization). The module M is torsion free if and only if the sequence (13) is exact or (15)

B ⊥ = A1×k (P, −Q) = {(ξ, η) ∈ A1×(p+m) ; ξH + η = 0}.

If this is the case, the behavior B is called controllable and is indeed the unique controllable IO behavior with transfer matrix H or, in other words, the unique controllable realization of H, and moreover, (16)

A1×k P = {ξ ∈ A1×p ; ξH ∈ A1×m }.

Remark 2.3. As shown by Willems and Rocha for discrete two-dimensional systems and by Pillai and Shankar [14] for continuous multidimensional ones, the term controllable is justified by the concatenability of trajectories in controllable systems. A module is torsion free if and only if it can be embedded into a free module and hence, by duality, the system B is controllable if and only if there is a system epimorphism φ : F m → B, which Pommaret (resp., Willems) calls a parametrization (resp., an image representation) of B. The first reviewer suggests calling a controllable system B torsion free or, more generally, to systematically use the attribute of the system module M also for the system itself. We will stick to the term controllable since it is used by most researchers. If d is a common denominator of the entries of H, i.e., if 0 = d ∈ A and dH ∈ Ap×m , the multiplication with d is an isomorphism on K, and therefore (15) can also be expressed as 

◦ dH 1×k 1×(p+m) (d idm ) 1×m −→ A . (P, −Q) = ker A A

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Proof. The theorem is a reformulation of [11, Thm. 7.24]. We give a slightly simpler and more direct proof. If (13) is exact, then (14) is an isomorphism, and hence M is torsion free. If, conversely, M is torsion free, we obtain the monomorphisms (17) M = A1×(p+m) /A1×k R

→ K ⊗A M = K 1×(p+m) /K 1×k R →

(ξ, η)

1 ⊗ (ξ, η) = (ξ, η)

◦(idHm )

∼ =

K 1×m ,

→

ξH + η,

which imply A1×k R = {(ξ, η) ∈ A1×(p+m) ; ξH + η = 0} and the exactness of (13). The first map in (17) is a monomorphism since its kernel is the torsion submodule of M and thus zero. The second isomorphism follows from (P, −Q) = P (idp , −H) and rank(P ) = p; hence K 1×k (P, −Q) = K 1×p (idp , −H), and the exactness of 0 −→ K 1×p

◦(idp ,−H)

−→

◦(idHm ) K 1×m −→ 0 K 1×(p+m) −→

or K 1×(p+m) /K 1×p (idp , −H)

◦(idHm )

∼ =

K 1×m .

Concerning equality (16), the identity P H = Q ∈ Ak×m implies A1×k P ⊆ {ξ ∈ A1×p ; ξH ∈ A1×m }. If, conversely, η := −ξH ∈ A1×m and thus ξH + η = 0, the monomorphism (17) implies (ξ, −η) = ζ(P, −Q) for some ζ ∈ A1×k and hence ξ = ζP ∈ A1×k P . Definition 2.4. A behavior B is called free (resp., projective) if its module M has this property, and is then controllable. Free behaviors were characterized in [11, Thm. 7.53] and were called strongly controllable by Rocha in her thesis. With these preparations we can now draw the following simple conclusion which, however, is basic for the present paper. Theorem and Definition 2.5 (trivial IO behaviors). For an IO behavior B as in Result 2.1, the following conditions are equivalent: 1. The autonomous behavior B 0 is zero or M 0 = 0 or A1×k P = A1×p ; i.e., the rows of P generate the full free module A1×p . 2. The behavior B is controllable and H ∈ Ap×m . If these equivalent conditions are satisfied, the IO behavior B is called trivial and the linear maps   y m ∼ (18) ◦(0 idm ) : A1×m ∼ (0, η), and proj := (0 id )◦ : B F , → u, M, η →  = = m u are isomorphisms; in particular M and B are free. In accordance with the literature, we reserve the term stable to a more general situation in sections 4 and 5. Proof. ⇒: The exact sequences (11) and (12) and M 0 = 0 imply the isomorphisms (18) and hence the freeness and controllability of B. The equality A1×p = A1×k P = {ξ ∈ A1×p ; ξH ∈ A1×m } implies A1×p H ⊆ A1×m ; hence H ∈ Ap×m .

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r u1 r? + 6

r u2 B1

-ry1

r? +

B2



-ry2 ?

Fig. 2.1.

⇐: The controllability of B implies equality (16) and then, with H ∈ Ap×m , also A = A1×k P . In what follows we consider a feedback system given by the following block diagram [11, Chap. 8] as in Figure 2.1. The two IO subbehaviors Bi , i = 1, 2, of F p+m are given as   y1 B1 := ∈ F p+m ; P1 ◦ y1 = Q1 ◦ u1 , P1 H1 = Q1 , P1 ∈ Ak1 ×p , u1   (19) u2 B2 := ∈ F p+m ; P2 ◦ y2 = Q2 ◦ u2 , P2 H2 = Q2 , P2 ∈ Ak2 ×m . y2 1×p

Let l := p + m. The feedback behavior B := feedback(B1 , B2 ) is the behavior       u2 y1 y ,u = ∈ F p+m satisfy (21) (20) B := ∈ F 2l ; y = y2 u1 u with (21)

P1 ◦ y1 = Q1 ◦ (u1 + y2 ), P2 ◦ y2 = Q2 ◦ (u2 + y1 ). Lemma 2.6. The map

      y2 y y1 , → u 1 + y2 u 2 + y1 u       is a behavior isomorphism with the inverse map uy11 , uy22 → yv , with v1 := u1 − y2 and v2 := u2 − y1 . In particular, B is controllable (resp., projective) if and only if both Bi have these properties. If B is trivial, then both Bi are projective. Proof. It is obvious that the indicated map is indeed the inverse map. Recall that a trivial behavior is free and that projective modules are precisely the direct summands of free ones. From Result 2.1 we know that the transfer space of B1 is   y1 p+m B1 = ∈K ; y1 = H1 u 1 u 1 B = feedback(B1 , B2 ) → B1 × B2 ,

and likewise for B2 . Therefore, the equations of the transfer space of B according to (8) are (22)

u1 + y2 ), y2 = H2 ( u2 + y1 ) y1 = H1 ( ⇔ (idp −H1 H2 )y1 = H1 u 1 + H1 H2 u 2 , y2 = H2 ( u2 + y1 ).

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Theorem 2.7.  The feedback behavior B = feedback(B1 , B2 ) is an IO behavior with input u = uu21 ∈ F p+m if and only if one or, equivalently, all of the matrices   idp −H1 idp −H1 H2 , idm −H2 H1 , −H2 idm are invertible, i.e., contained in Gl• (K). If this is the case, the autonomous part B 0 and the transfer matrix of B are given as      y1 −Q1 P1 ∈ F p+m ; ◦ y = 0 = B1 ∩ B2 , B0 = y = −Q2 P2 y2  −1   idp −H1 0 H1 H := . −H2 idm H2 0 Moreover, B1⊥ ⊕ B2⊥ = (B 0 )⊥ . Vidyasagar then calls the feedback system well-posed [26, p. 100]. Rocha calls B 0 a regular interconnection of B1 and B2 [19, Def. 1]. Direct sum decompositions and regular interconnections of systems have also been treated in [1] and [33] and in papers on the stabilization of IO maps; see, for instance, [17], [18].  1 p+m Proof. It is obvious that (22) are uniquely solvable for yy for given 2 ∈ K u 2  p+m if and only if idp −H1 H2 ∈ Glp (K). The assertion then follows from u 1 ∈ K Result 2.1. The simultaneous invertibility of these matrices is standard and follows trivially by elementary row and column operations. According to  Result 2.1, the equations of B 0 follow from (21) of B by setting the input u = uu21 to zero. The expression for H is implied by the equations      P1 P1 0 idp −H1 −Q1 H= H 0 P2 −Q2 P2 −H2 idm      0 Q1 0 H1 P1 0 = = 0 P2 Q2 0 H2 0  P1 0 by cancelling the matrix 0 P2 of rank l = p + m on the left. The sum (B 0 )⊥ = B1⊥ + B2⊥ follows from B 0 = B1 ∩ B2 by duality. Moreover, rank((B 0 )⊥ ) = p + m = rank(B1⊥ ) + rank(B2⊥ ) implies B1⊥ ∩ B2⊥ = 0; hence (B 0 )⊥ = B1⊥ ⊕ B2⊥ . Remark 2.8. If the feedback behavior of the preceding theorem is well-posed, the matrices   0 H1 G := (23) and (id −G)−1 (id −G) = id in K (p+m)×(p+m) H2 0 imply H = (id −G)−1 G and (id −G)−1 = id +H, and hence that (id −G)−1 and H share many properties, in particular that H belongs to A(p+m)×(p+m) if and only if (id −G)−1 does [26, Chap. 5, Lem. 9].

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Corollary and Definition 2.9. If the feedback behavior of the preceding theorem is well-posed, then B is trivial, i.e., B 0 = 0, if and only if B1⊥ ⊕ B2⊥ = A1×l or, by duality, B1 ⊕ B2 = F p+m . If this holds, the modules Uj = Bj⊥ , j = 1, 2, are complementary direct summands of A1×l and the modules Mj = A1×l /Uj and the behaviors Bj are projective. We say that the IO behavior B2 trivializes (instead of stabilizes) B1 . In the following we assume that the IO behavior B1 is given and construct all trivializing IO behaviors B2 . The set of all these B2 is parametrized by an open subset (in the Zariski topology) of a finitely generated polynomial module. This parametrization generalizes the important Youla–Kuˇcera parametrization of the onedimensional stabilization theory. According to the preceding theorem, we make the necessary assumption that the behavior B1 is projective. We use the exact sequence (24)

F1 := A1×k1

d0 :=◦R1

can

−→ F0 := A1×l −→ M1 := A1×l /U1 → 0

with l := p + m, R1 = (P1 , −Q1 ), U1 := im(d0 ) = A1×k1 R1 .

The following remark establishes the well-known one-to-one correspondence between idempotent endomorphisms e = e2 of a module and direct sum decompositions. If e = e2 ∈ HomA (F0 , F0 ) is an idempotent or a projection, then F0 = im(e) ⊕ ker(e)  x = e(x) + (x − e(x)), x = e(x) ⇔ x ∈ im(e), ker(e) = im(id −e). Conversely, any direct sum decomposition F0 = V1 ⊕ V2  x = x1 + x2 =: e(x) + (x − e(x)) gives rise to the projection e = e2 ∈ HomA (F0 , F0 ). The map {e ∈ HomA (F0 , F0 ); e = e2 } → {F0 = V1 ⊕ V2 }, e → F0 = im(e) ⊕ ker(e) is bijective. By restricting the preceding bijection to decompositions F0 = U1 ⊕ U2 with the fixed U1 from above we obtain the following. Corollary 2.10. For the data from (24) the following map is bijective: {e ∈ HomA (F0 , F0 ); e = e2 , im(e) = U1 } → {F0 = U1 ⊕ U2 }, e → F0 = U1 ⊕ ker(e), ker(e) = im(idl −e). The preceding decompositions can also be described by means of homomorphisms g ∈ HomA (F0 , F1 ) ∼ = Al×k1 , as was already shown more generally in [9, Lem. 6.2, p. 88]. This is important for constructive purposes in particular. Lemma 2.11. The following conditions are equivalent for a submodule U1 of F0 and its factor module M1 = F0 /U1 = cok(d0 ): 1. The module M1 is projective. 2. U1 is a direct summand of F0 . Then U1 is also projective and there is an idempotent e = e2 ∈ HomA (F0 , F0 ) such that U1 = im(e).

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3. There is a homomorphism g ∈ HomA (F0 , F1 ) such that d0 gd0 = d0 . Then e := d0 g is an idempotent with U1 = im(d0 ) = im(e). Proof. The equivalence of 1 and 2 is standard since the projectivity of M1 is equivalent with the splitting of the exact sequence (24). 2 ⇒ 3: Let e be an idempotent with image im(e) = U1 . Then the map e : F0 → U1 = im(e) is surjective. Since U1 is projective and the map d0 : F1 → U1 is surjective, there is a linear section e

s

s : U1 → F1 such that idU1 = d0 s. Define g := se : F0 −→ U1 → F1 . Then e(x) = d0 s(e(x)) ∈ U1 for x ∈ F0 ; hence e = d0 se = d0 g and d0 gd0 = d0 sed0 = ed0 = d0 since e(d0 (x)) = d0 (x) for d0 (x) ∈ U1 = im(e). 3 ⇒ 2: Assume d0 gd0 = d0 and define e := d0 g. Then e ∈ HomA (F0 , F0 ) is an idempotent since e2 = (d0 gd0 )g = d0 g = e and, moreover, im(e) = im(d0 g) ⊆ im(d0 ) = U1 . From d0 = d0 gd0 = ed0 we infer U1 = im(d0 ) ⊆ im(e) in the same fashion; hence U1 = im(e). Theorem 2.12. Assume that M1 = A1×l /U1 is projective and that g1 ∈ HomA (F0 , F1 ) and e1 := d0 g1 with im(e1 ) = U1 are constructed according to the preceding lemma. Then the map ϕ : {h ∈ HomA (F0 , F1 ); d0 hd0 = 0}/{h ∈ HomA (F0 , F1 ); d0 h = 0} → {e ∈ HomA (F0 , F0 ); e = e2 , U1 = im(e)}, h → ϕ(h) := e1 + d0 h, is bijective. It furnishes a parametrization of the set of direct complements U2 = ker(e) with F0 = U1 ⊕ U2 by the finitely generated A-module on the left. If A is a field, then the right side is an affine open subset of the projective Grassmann variety of all mdimensional subspaces of A1×(p+m) . Proof. 1. The equation d0 gd0 = d0 is an inhomogeneous linear equation for g in the A-module HomA (F0 , F1 ) and g1 is one solution. Hence {g ∈ HomA (F0 , F1 ); d0 gd0 = d0 } = g1 + {h ∈ HomA (F0 , F1 ); d0 hd0 = 0}. 2. The map is well defined: If d0 hd0 = 0, then g2 := g1 + h satisfies d0 g2 d0 = d0 , and hence e2 := d0 g2 = d0 g1 + d0 h = e1 + d0 h is an idempotent with im(e2 ) = U1 according to the preceding lemma. If h2 and h3 are homogeneous solutions such that h2 = h3 or d0 (h2 − h3 ) = 0, then e1 + d0 h2 = e1 + d0 h3 . 3. It is obvious that the map ϕ is injective, and it is surjective by the preceding lemma. We reformulate the preceding theorem in matrix terms. We identify Am×n = HomA (A1×m , A1×n ), X = ◦X = (ξ → ξX), and emphasize that (◦X)(◦Y ) = ◦(Y X). Theorem 2.13. Let B1 ⊆ F p+m be an IO behavior with the transfer matrix H1 and the data U1 = B1⊥ = A1×k1 R1 , M1 = A1×l /B1⊥ , R1 = (P1 , −Q1 ) ∈ Ak1 ×(p+m) , rank(P1 ) = p, P1 H1 = Q1 , H1 ∈ K p×m .

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1. The module M1 or the behavior B1 is projective if and only if there is a matrix (25) G1 ∈ Al×k1 with R1 G1 R1 = R1 or, equivalently, with (idp , −H1 )G1 P1 = idp . Then E1 := G1 R1 = E12 ∈ Al×l is an idempotent matrix with B1⊥ = A1×l E1 . This algorithm is related to the algorithms of [33]; its idea goes back at least to MacLane [9] as explained above. 2. Assume that B1 is projective with the data from 1. Then the map (26)

ϕ : {X ∈ Al×k1 ; R1 XR1 = 0}/{X ∈ Al×k1 ; XR1 = 0} → {E ∈ Al×l ; E = E 2 , B1⊥ = A1×l E}, X → ϕ(X) := E1 + XR1 ,

is bijective. Moreover, (27) R1 XR1 = 0 if and only if (idp , −H1 )XP1 = 0 if and only if (t idp , −tH1 )XP1 = 0 and XR1 = 0 if and only if XP1 = 0, where 0 = t ∈ A is a common denominator of the entries of H1 , i.e., tH1 ∈ Ap×m . The direct complement behavior B2 of B1 in F p+m , which is constructed by means of the idempotent E = E1 + XR1 , is (28)

B2 := {w ∈ F p+m ; (idl −E1 − XR1 ) ◦ w = 0} = U2⊥ , B1 ⊕ B2 = F p+m , where U2 := ker(◦(E1 + XR1 )) = A1×l (idl −E1 − XR1 ).

3. Since B1⊥ = A1×l E1 , the bijection (26) holds if R1 is replaced with E1 . 4. If U1 is even free, and if the rows of R1 are a basis of U1 , i.e., if R1 = (P1 , −Q1 ) ∈ Ap×(p+m) and det(P1 ) = 0, then the situation of (26) simplifies to

(29)

G1 ∈ Al×p , R1 G1 = idp , E1 = G1 R1 , and ϕ : {X ∈ Al×k ; R1 X = 0} ∼ = {E ∈ Al×l ; E = E 2 , B1⊥ = A1×l E}, X → E1 + XR1 . Proof. The equivalence of (25) and (27) follows from R1 = (P1 , −Q1 ) = P1 (idp , −H1 ) and rank(P1 ) = rank(idp , −H1 ) = p,

which imply that P1 (resp., (idp , −H1 )) can be cancelled as a left (resp., right) factor. Of course, each direct complement behavior    u2 p+m ∈F ; (idl −E) ◦ w2 = 0 with B2 = w2 = y2 rank(idl −E) = l − rank(E) = l − p = m of the preceding theorem admits an IO structure, but not necessarily that with u2 as inputs, as needed for the feedback construction. To enforce this additional property we proceed as follows. Let X1 , . . . , Xn be a system of A-generators of the submodule (30)

{X ∈ Al×k1 ; R1 XR1 = 0( ⇔ R1 XP1 = 0)}.

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The same matrices generate {X ∈ K l×k1 ; (idp , −H1 )XP1 = 0} as a K-vector space. In the relevant examples the Xi can be computed by means of Gr¨ obner bases. The n linear map θ = (θi )i=1,...,n → i=1 θi Xi induces the A-isomorphism

(31)

ϕ1 : A1×n /V ∼ = {X ∈ Al×k1 ; R1 XP1 = 0}/{X ∈ Al×k1 ; XP1 = 0},   n n   1×n where ϕ1 (θ) := θi Xi and V := θ ∈ A ; θ i X i P1 = 0 . i=1

i=1

Again, in the relevant examples the submodule V and its factor module A1×n /V can be computed by means of Gr¨ obner bases. In addition we consider the polynomial map l

d : K l×m → K (m) , Y → d(Y ) = (dα (Y ))α ,   where the dα (Y ) are the ml m × m minors of the matrix Y . The function d is polynomial in the entries of Y and d(Y ) = 0 if and only if rank(Y ) = m. With the data from above, we obtain the induced map    l 0 . {E ∈ Al×l ; E = E 2 , B1⊥ = A1×l E} → A(m) , E → d (idl −E) idm (32)

If

  0 idl −E =: (−Q2 , P2 ) ∈ Al×(p+m) , hence (idl −E) = P2 , idm

the behavior B2 :=



 w2 =

u2 y2



∈ F p+m ; (idl −E) ◦ w2 = 0 or P2 ◦ y2 = Q2 ◦ u2

is an IO behavior with input u2 if and only if rank(P2 ) = m. Composing the maps just constructed, we obtain the map    l 0 , Φ : K 1×n → K (m) , θ → d (idl −E) idm (33) n  with E := E1 + XR1 , X := θi Xi , i=1

which is polynomial in the components of θ; i.e., its components are contained in K[θ]. Summing up we obtain the final parametrization theorem. 2.14 (parametrization of trivializing behaviors). Let B1 = {w1 =  y1 Theorem p+m ∈ F } be a projective behavior as in Theorem 2.13, and assume that A is u1 infinite. Then, with the data introduced above, the map    u2 ∈ F p+m ; (idl −E) ◦ w2 = 0 , θ → B2 := w2 = y2 n  where E := E1 + XR1 , X := θi Xi , i=1

is a bijection from the nonempty set     0 1×n = 0 θ∈A /V ; Φ(θ) = d (idl −E) idm

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onto the set of all IO behaviors B2 ⊆ F p+m , which trivialize B1 . This signifies that the feedback system B := feedback(B1 , B2 ) is well-posed and trivial. In particular, such behaviors exist. Proof. It remains only to show that such behaviors exist. The matrix P1 of rank p has a left inverse S1 ∈ K p×k1 with S1 P1 = idp .

(34) Define

     idp idp idp −H1 l×l (idp , −H1 ) = S1 ∈ K l×k1 ; ∈K and G := E := 0 0 0 0       H1 0 0 H1 hence idl −E = , (idl −E) = , 0 idm idm idm    0 rank (idl −E) = m, (idp , −H1 )GP1 = idp , and E := GR1 . idm 

Thus X := G − G1 ∈ K l×k1 satisfies   0 = m; (idp , −H1 )XP1 = 0, E = E1 + XR1 and rank (idl −E) idm 

in particular, X is of the form X=

n 

θi Xi with θ ∈ K 1×n and Φ(θ) = 0.

i=1

Since the components of Φ are polynomials in K[θ] and since A is an infinite subset of K, there is also a parameter θ ∈ A1×n with nonzero Φ(θ ), which induces an IO behavior B2 such that the feedback system feedback(B1 , B2 ) is well-posed and trivial. Remark 2.15. We remark that {θ ∈ K 1×n ; Φ(θ) = 0} is a nonempty open subset of K 1×n with respect to the Zariski topology and is therefore dense. This signifies that, generically, the behaviors B2 with F p+m = B1 ⊕ B2 are IO behaviors with the desired input u2 ∈ F p . 3. Localization. The assumptions of the preceding section remain in force. Moreover, we assume that the ring A is factorial. We use Matlis’ structure theory of injective modules over Noetherian rings and refer to [10, pp. 145–150] and [12, sec. 2], where this theory was used in the system-theoretic context. Moreover, we use Gabriel’s theory of localization as detailed in the book [24, Chap. IX]. Let Spec(A) (resp., Max(A)) denote the set of prime (resp., maximal) ideals of A. A prime ideal p is associated with an A-module M if and only if there is an x ∈ M such that p = ann(x) = {a ∈ A; ax = 0} or, in other words, that A/p is a submodule of M up to isomorphism. Let Ass(M ) ⊂ Spec(A) denote the set of prime ideals associated with M . A module M is p-coprimary if Ass(M ) consists exactly of one prime ideal p; then   injective a∈ /p (35) a : M → M, x → ax is ⇔ . locally nilpotent a∈p

STABILITY AND STABILIZATION

1481

Local or almost nilpotency of a on M means that for all x ∈ M there is an index m such that am x = 0. The injective module F is a cogenerator if and only if it contains all simple modules A/m, m ∈ Max(A), up to isomorphism, i.e., if Max(A) ⊂ Ass(F). Each module M has an injective envelope E(M ) ⊃ M which is unique up to noncanonical isomorphism. This signifies that E(M ) is injective and that V ∩ M = 0 for each nonzero submodule V of E(M ). Since A is Noetherian, a direct sum or coproduct of modules is injective if and only if each direct summand has this property. Each injective module E admits a direct sum decomposition into directly indecomposable injective modules, and this decomposition is unique up to an automorphism of E. An indecomposable injective module E is coprimary, and if p is its unique associated prime ideal, then E = E(A/p). The map Spec(A) → {E; E indecomposable injective}/isomorphism, p → E(A/p), Ass(E(A/p)) = {p}

(36)

is a bijection of the prime spectrum Spec(A) onto the set of indecomposable injectives up to isomorphism. Since E(A/p) is injective and indecomposable, the injective map s : E(A/p) → E(A/p), x → sx, s ∈ A \ p, is even bijective, and therefore E(A/p) is an Ap -module, where  a ; a ∈ A, s ∈ A \ p ⊂ K = quot(A) Ap := s is the local ring of the prime ideal p with its unique maximal ideal pp = Ap p. For M ∈ ModA , the adjointness isomorphism HomA (M, E(A/p)) ∼ = HomAp (Mp , E(A/p)) = HomA (Mp , E(A/p)),  x ; x ∈ M, s ∈ A \ p ∈ ModAp Mp := Ap ⊗A M = s

(37)

holds. Together with N = Np for N ∈ ModAp this shows that the duality functor HomAp (−, E(A/p)) on ModAp is exact, and hence E(A/p) is an injective Ap -module. Since it contains the unique simple Ap -module (A/p)p = Ap /pp , the module E(A/p) is the unique minimal injective cogenerator of the category ModAp to which the considerations of section 2 are applicable. In particular, Mp = 0 ⇔ HomA (M, E(A/p)) = 0.

(38) Let (39)

F := ⊕i∈I Fi , Ass(Fi ) = {pi }, P := Ass(F) = {pi ; i ∈ I}

denote a direct sum decomposition of the injective cogenerator F into indecomposable injectives Fi and let (40)

P := Ass(F) = P1  P2 , P1 , P2 = ∅,

be a disjoint decomposition of Ass(F). All subsequent objects depend on the choice of F and of the decompositions (39) and (40).

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In particular, the latter imply direct sum decompositions (41)

F = ⊕p∈P F(p) = F1 ⊕ F2 with F(p) := ⊕i∈I, pi =p Fi ∼ = E(A/p)(μ(p)) , μ(p) := |{i ∈ I; pi = p}|, Fj := ⊕p∈Pj F(p), j = 1, 2,

where E(A/p)(μ(p)) denotes the direct sum of μ(p) copies of E(A/p). All modules in these direct sum decompositions are injective. Like E(A/p), the module F(p) ∼ = E(A/p)(μ(p)) is an injective cogenerator of the category of Ap -modules. The decompositions (41) induce corresponding decompositions of F behaviors. Let (42)

R ∈ Ak×l , U := A1×k R, M := A1×l /U , B := U ⊥ = HomA (M, F) = {w ∈ F l ; R ◦ w = 0}. ident.

Since M is finitely generated, the functor HomA (M, −) preserves direct sums, and therefore (41) induces direct decompositions B = ⊕i∈I Bi ⊂ F l = ⊕i∈I Fil , Bi := B ∩ Fil = HomA (M, Fi ) = {w ∈ Fil ; R ◦ w = 0}, ident.

(43)

ident.

B = ⊕p∈P B(p) = B1 ⊕ B2 , B(p) := B ∩ F(p)l = HomA (M, F(p)) = {w ∈ F(p)l ; R ◦ w = 0} ∼ = HomA (Mp , E(A/p))(μ(p)) , p

Bj := B ∩

Fjl

= HomA (M, Fj ) = {w ∈ Fjl ; R ◦ w = 0}, j = 1, 2.

We use the following suggestive system-theoretic terminology. The elements of F (resp., F1 ) are called signals (resp., stable signals). If y = y1 + y2 ∈ F = F1 ⊕ F2 is a signal, then y1 is called its stable part and y2 its steady state. Example 3.1. We indicate the data for the important complex continuous case. The algebra A is the complex polynomial algebra C[s] = C[s1 , . . . , sr ]. The map Cr → Max(C[s]), r  λ = (λ1 , . . . , λr ) → mλ := C[s](sρ − λρ ) = {t ∈ C[s]; t(λ) = 0} ρ=1

is bijective. As an injective cogenerator we take the module (44)

F := D (Rr )lf := {y ∈ D (Rr ); [C[s] ◦ y : C] < ∞} = ⊕λ∈Cr C[z] exp(λ • z), z = (z1 , . . . , zr ) ∈ Rr , λ • z := λ1 z1 + · · · + λr zr

of locally finite distributions or polynomial-exponential functions [12, Thm. 6.6]. Then (45)

Ass(F) = Max(C[s]) = {mλ ; λ ∈ Cr }, F(λ) := F(mλ ) = C[z] exp(λ • z) ∼ = E(C[s]/mλ );

i.e., F is the unique minimal injective cogenerator of the category of C[s]-modules. We choose a disjoint decomposition (46)

Cr = Λ1  Λ2 and Ass(F) = Max(C[s]) = P1  P2 , Pj := {mλ ; λ ∈ Λj }.

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Standard choices for continuous stability theory are r

(47)

Λ2 := C+ , where C+ := {w ∈ C; (w) > 0}, C+ := {w ∈ C; (w) ≥ 0}, or [29, Def. 3.1] Λ2 := C+ × iRr−1 .

A discrete analogue is Λ2 := {w ∈ C; |w| ≥ 1}r ; see the end of section 5. The modules Fj , j = 1, 2, are Fj = ⊕λ∈Λj C[z] exp(λ • z). For r = 1 and time z = z1 the two sets Λ2 from (47) coincide and yield F = ⊕λ∈C C[z] exp(λz) = F1 ⊕ F2 = ⊕ (λ) 1 the system B 0 ⊂ F1p contains, in general, many polynomial-exponential functions which are not stable in a naive sense. This is unavoidable due to the interplay of conditions 2 and 4(i) of the theorem. Indeed, with increasing Λ1 the sets of stable polynomialexponential functions, polynomials, and rational functions also grow; hence the generally desired existence of sufficiently many stable rational functions implies the same for the stable polynomial-exponential functions. For instance, for Λ1 = ∅ a stable system is trivial, whereas for Λ1 = Cr each system is stable. 8. Consider Λ2 := {(λ1 , λ2 ) ∈ C2 ; (λ1 ), (λ2 ) ≥ 0} and the simple system

(s1 + 1) ◦ y = u with the characteristic variety char(B 0 ) = {−1} × C and B 0 = ⊕λ2 ∈C C[z2 ] exp(−z1 ) exp(λ2 z2 ). This is obviously Λ2 -stable and B 0 contains the functions f (z) = exp(−z1 )g(z2 ) with g(z2 ) := exp(λ2 z2 ), (λ2 ) > 0, which grow exponentially with increasing z2 and are not stable in a naive sense. However, they share this property with their initial part f (0, z2 ) = g(z2 ) and such initial conditions are not permitted according to [29, pp. 1499–1500] and the idea that a stable system should generate stable outputs for stable inputs and initial conditions. This requires that the initial value problem can be formulated and uniquely solved. Open Problem 5.13 below addresses this problem for discrete systems. We are going to show next that a stable system is also IO stable in the sense that inputs of various types generate outputs of the same type. Since the used signals are not necessarily bounded, we do not use the acronym BIBO. The modules F(λ) = C[z] exp(λ • z) = E(A/mλ ) (resp., F2 ) are A(λ)-modules (resp., AT -modules). The module F(λ) has the canonical increasing filtration of

STABILITY AND STABILIZATION

1495

finite-dimensional A(λ)-submodules [12, Eq. 1.17, Thm. 1.25, Thm. 6.6] F(λ) = ∪∞ k=0 F(λ)k with (79)

◦ y = 0} = C[z]≤k exp(λ • z), where F(λ)k := {y ∈ D (Rr ); mk+1 λ C[z]≤k := {f ∈ C[z]; deg(f ) ≤ k}

and deg denotes the total degree of a polynomial in C[z1 , . . . , zr ]. Theorem and Definition 5.3 (IO stability). 1. Let t ∈ A \ mλ or t(λ) = 0 and y ∈ F(λ)k . Then t is invertible in the C finitedimensional local ring A/mk+1 or At + mk+1 = A. Via Gr¨ obner bases one constructs λ λ t1 ∈ A such that k+1 t1 t ≡ 1(mk+1 λ ) or t1 t = 1 ∈ A/mλ .

Then the differential operator t◦ : F(λ)k → F(λ)k is bijective and its inverse is the differential operator t1 ◦ : F(λ)k → F(λ)k . Therefore the scalar multiplication with a t ∈ A(λ) = C[s]mλ on F(λ)k is given by the differential operator a ◦ : F(λ)k → F(λ)k , y → (at1 ) ◦ y. t 2. If H ∈ Ap×m is a stable rational matrix and u ∈ F2m = ⊕λ∈Λ2 C[z]m exp(λ • z) T is an input of the form  fλ (z) exp(λ • z) with fλ ∈ C[z]m u= ≤k ∀λ ∈ Λ2 λ∈Λ2

and fλ = 0 for almost all λ, then the corresponding output y := H ◦ u has the same form  gλ (z) exp(λ • z) with gλ ∈ C[z]p≤k ∀λ ∈ Λ2 y= λ∈Λ2

and can be computed with the algorithm from part 1. This property of the operator H◦ : F2m → F2p is suggestively called its IO stability with respect to the decomposition (70). 3. Item 2 is applicable to any stable system B according to Theorem/Definition    m 5.1 with its stable transfer matrix H; in particular B2 := B ∩ F2l = H◦u u ; u ∈ F2 is its steady state part. Proof. Item 1 follows from k+1 tt1 + a = 1, a ∈ mk+1 ◦ y = 0 ⇒ t1 ◦ t ◦ y = y. λ , mλ

Items 2 and 3 are direct consequences of 1 and Theorem/Definition 4.2. The next IO stability theorems are just reformulations of the results [4, Thm. on p. 16 and Thm. 2 on p. 24]. These results also play an essential part in the stability paper [29, subsections 5, 6, 7]. As in [4, pp. 10–12] let S (⊃ C0∞ (Rr )) denote the space of rapidly decreasing C ∞ functions and S  (⊂ D (Rr )) its topological dual space of temperate distributions. Further we consider the space O (⊃ S) of slowly increasing C ∞ functions f for which there is an index m > 0 and C > 0 such that |f (μ) (z)| ≤ C(1 + |z|)m ∀ z ∈ Rr , μ ∈ Nr ,

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ULRICH OBERST

and finally the space O (⊂ S  ) of rapidly decreasing distributions. In the paper [29] the first variable t := z1 is distinguished as time. In this case the spaces F3,+

Rr+ := {z ∈ Rr ; t = z1 ≥ 0} and := {y ∈ F3 ; supp(y) ⊂ Rr+ }, F3 = S, S  , O, O ,

play an important part in [4] and [29]. In the next theorem we use the data from (70)–(79). Theorem 5.4 (IO stability). 1. Let F3 be one of the spaces S, S  , O, O . If Λ2 ⊃ iRr , then F3 is an AT module. In particular, if H ∈ Ap×m is a rational matrix and u ∈ F3m is an input, T p then y := H ◦ u ∈ F3 is an output of the same type. If B is stable, then according to Theorem/Definition 5.1 this holds. 2. If the variable t := z1 is distinguished and if Λ2 ⊃ C+ × iRr−1 , C+ := {λ ∈ C; (λ) ≥ 0} as in [29], then the statement of item 1 applies to the spaces S+ and (O )+ . If, in addition, B 0 is time-autonomous in the sense of [29, Def. 2.1, Thm. 2.2], then   H ◦u p F3m ∼ , F3 = S+ , (O )+ . (80) = B ∩ ((D )+ × F3m ), u → u 3. If in the situation of item 2 Λ2 ⊃ C+ × iRr−1 , C+ := {λ ∈ C; (λ) > 0}, then the statement of item 1 applies to the space (S  )+ . Proof. 1. Since iRr ⊂ Λ2 , each t ∈ T has no zero in iRr . According to [4, Thm. on p. 16] this is necessary and sufficient for t◦ : F3 → F3 to be bijective, and this in turn implies that F3 is an AT -module with the scalar multiplication a ◦ u =: y with t ◦ y = a ◦ u. t 2, 3. These assertions follow from [4, Thm. 2, p. 22] in the same fashion. It remains to show the surjectivity of (80). Assume that   y1 m  p ∈ B, i.e., P ◦ y1 = Q ◦ u. Then u ∈ F3 , y := H ◦ u, y1 ∈ (D )+ , and u y1 − y ∈ (D )p+ and P ◦ (y1 − y) = 0. The time-autonomy and [29, Thm. 2.2] imply y1 = y = H ◦ u and hence the assertion. Remarks 5.5. 1. The preceding theorem suggests to require iRr ⊂ Λ2 for the decompositions (70) in context with stability questions of partial differential equations. The standard set Λ2 := {λ ∈ Cr ; ∀ρ = 1, . . . , r : (λρ ) ≥ 0} = C+ satisfies the assumptions of items 1–3.

r

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STABILITY AND STABILIZATION

  ∈ B ∩F3p+m implies in particular that the input u can be 2. The condition H◦u u m freely chosen in F3 and therefore condition 1 of Theorem 7.2 of [29] which, however, was shown only for p = m = 1 or r ≤ 2. The isomorphism (80) is the IO stability of B according to [29, Def. 7.1]. Theorem/Definition 5.1 and Theorem 5.4 show that stability in the sense of the present paper for Λ2 = C+ × iRr−1 implies the various stability notions of that paper. The next theorem characterizes those sets Λ2 ⊂ Cr for which the localization functor M → MT from section 3 is perfect, i.e., isomorphic to M → MT . It turns out that this coincides with ideal-convexity of Λ2 in the sense of [23, Def. on p. 25] or [29, Def. 5.4]. Recall from (62) that MT = 0 implies MT = (MT )T = 0, in particular {a; a ∩ T = ∅} ⊆ T. Theorem and Definition 5.6 (perfect localization and ideal-convexity). For the decomposition (70) and A := C[s] the following assertions are equivalent: 1. An A-module is T -torsion if and only if it is T-torsion, i.e., MT = 0 ⇔ MT = 0 ⇔ ∀λ ∈ Λ2 : Mmλ = 0. (50)

2. T = {a; a ideal of A, T ∩ a = 0}. 3. For each M ∈ ModA the canonical map (61) MT = AT ⊗A M → MT is an isomorphism. 4. ModAT = ModA,T . 5. The set Λ2 is ideal-convex according to [29, Def. 5.4]; i.e., for each ideal a of A the implication V (a) ∩ Λ2 = ∅ ⇒ a ∩ T = ∅ or ∃t ∈ a with V (t) ∩ Λ2 = ∅ holds. 6. The ideals (mλ )T ⊂ AT , λ ∈ Λ2 , are the only maximal ideals of AT . 7. The module F2 from (70) is a cogenerator in ModAT . Recall from Lemma 3.3 that F2 is an injective AT -module and an injective cogenerator in ModA,T . Stenstr¨ om [24, Chap. XI, Prop. 3.4] talks about perfect localization in this context. Proof. The equivalence of items 1–4 follows from [24, Chap. XI, Prop. 3.4]. 2 ⇔ 5 follows from the equivalences V (a) ∩ Λ2 = ∅ ⇔ ∀λ ∈ Λ2 : a ⊂ mλ or (A/a)mλ = 0 ⇔ a ∈ T. (50)

The equivalence of items 5 and 6 is a consequence of [23, Prop. 3.1.19], but we give the short proof for completeness. 5 ⇒ 6: The maximal ideals of AT are exactly the ideals n = mT , where m is an ideal of A maximal with respect to m ∩ T = ∅. Such an m is always prime, but not necessarily a maximal ideal of A [10, Thm. 4.1]. Recall that mλ ∩ T = ∅ for λ ∈ Λ2 . Consider such an n = mT . If V (m) ∩ Λ2 = ∅, condition 5 implies m ∩ T = ∅, a contradiction. Hence V (m) ∩ Λ2 = ∅ and therefore there is a λ ∈ Λ2 such that m ⊆ mλ and mλ ∩ T = ∅; hence m = mλ by the maximality of m. 6 ⇒ 5: Assume indirectly that a ∩ T = ∅ and let m be maximal with a ⊂ m and m ∩ T = ∅. Then mT is a maximal ideal of AT and therefore by 6 of the form mT = (mλ )T , λ ∈ Λ2 ⇒ a ⊂ m = mλ ⇒ λ ∈ V (a) ∩ Λ2 . 4 ⇒ 7: The module F2 = ⊕λ∈Λ2 E(A/mλ ) is an injective cogenerator in ModA,T and hence by 4 in ModAT .

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7 ⇒ 6: Let n = mT be a maximal ideal of AT as before. The primeness of m implies A/m ⊂ (A/m)T = AT /mT = AT /n. Since F2 is an injective AT -cogenerator it contains the simple AT -module AT /n and thus A/m; hence m ∈ AssA (F2 ) = {mλ ; λ ∈ Λ2 } and m = mλ for some λ ∈ Λ2 . Examples 5.7. 1. If Λ2 ⊂ Cr is ideal-convex, then (81)

Λ2 := {λ ∈ Cr ; ∀t ∈ T : t(λ) = 0} or Λ1 = Cr \ Λ2 = ∪t∈T V (t),

i.e., Λ1 is a union of hypersurfaces. Equation (81) is false for Λ2 := {λ ∈ C2 ; (λ1 ) ≥ 0 or (λ2 ) ≥ 0}, Λ1 = {λ ∈ Cr ; (λ1 ) < 0, (λ2 ) < 0}, T = C \ {0}, ∪t∈T V (t) = ∅. 2. In dimension 1 each Λ2 is ideal-convex. 3. If r = 2, (81) characterizes ideal-convexity. In particular, the noncompact regions C2+ and C+ × iR from Theorem 5.4 are ideal-convex, whereas for r > 2 this is an open question [29, Thm. 5.5]. 4. Each polynomially convex compact subset Λ2 of Cr is ideal-convex [23, Prop. 3.1.20, Rem. on p. 28]. Polynomial convexity signifies that  r Λ2 = λ ∈ C ; ∀ f ∈ C[s] : |f (λ)| ≤ sup |f (z)| . z∈Λ2 r

In particular, the closed unit polydisc U , U := {z ∈ C; |z| ≤ 1}, is idealconvex [23, Prop. 3.1.20, Rem. on p. 28]. 5. Each compact cuboid Λ2 of the form (82)

Λ2 := {λ ∈ Cr ; ∀ρ = 1, . . . , r : aρ ≤ (λρ ) ≤ bρ , cρ ≤ (λρ ) ≤ dρ }, where aρ ≤ bρ , cρ ≤ dρ in R, ρ = 1, . . . , r,

is ideal-convex. The proof is similar to that of the preceding item in [23] and uses the cohomological theorems A and B concerning coherent modules on Stein spaces for these cuboids and the approximation of holomorphic functions by polynomials on such Λ2 . The proof is omitted. The proofs of the last two examples use the compactness of K in different places essentially, in particular for the approximation of analytic functions on Λ2 by polynomials. It seems difficult to derive constructive algorithms from these proofs; see Remark/Open Problem 5.10 below. The theorems A and B, however, are valid in much more generality and can possibly be used to prove ideal-convexity also for certain noncompact subsets Λ2 of Cr . Theorem 5.8 (stabilizability and ideal-convexity). 1. If B is stabilizable with respect to (70), then sing(B) ∩ Λ2 = ∅ or rank(R(λ)) = rank(R) ∀λ ∈ Λ2 .

STABILITY AND STABILIZATION

1499

2. If Λ2 is ideal-convex, the following assertions are equivalent: (a) B is stabilizable. (b) MT is a projective AT -module. 1×(p+m)  H  1×p 1×m (c) i. The AT -lattice M := AT ⊂ C(s)1×m idm = AT H + AT is projective. ii. MT is torsion free; i.e., B2 := B ∩ F2p+m is a controllable AT F2 behavior. (d) sing(B) ∩ Λ2 = ∅ or rank(R(λ)) = rank(R) ∀λ ∈ Λ2 . Proof. 1. According to Theorem 4.4, UT is a direct summand of A1×l T . Hence for 1×l all λ2 ∈ Λ2 also Umλ = (UT )mλ,T is a direct summand of (AT )1×l mλ,T = Amλ . Therefore Mmλ = A1×l mλ /Umλ is projective and thus free, and λ2 is not contained in sing(B) according (75). 2. (a) ⇔ (b): Due to the assumed ideal-convexity, we have UT = UT , MT = MT = A1×l T /UT . Hence B is stabilizable ⇔ UT is a direct summand ⇔ MT is projective. Thm. 4.4

(b) ⇔ (c): The assumed torsion freeness of MT implies the exact sequence (13) ◦(P,−Q)

H 1×(p+m) ◦(idm )

−→ C(s)1×m , hence   H 1×(p+m) 1×(p+m) ∼ MT = AT = M. /A1×k (P, −Q) A = T T idm A1×k T

−→

AT

Hence MT is AT -projective if and only if the lattice M has this property. (a) ⇒ (d) follows from 1. (d) ⇒ (b): Condition (d) implies that for all λ ∈ Λ2 Mmλ = (MT )mλ,T is projective. Since, by Theorem/Definition 5.6(6), the mλ,T , λ ∈ Λ2 , are all maximal ideals of AT , the projectivity of MT follows from [10, Thm. 7.12]. Remarks 5.9. 1. For general, not ideal-convex unstable regions, stabilizability is characterized in Theorem/Definition 4.3 and Theorem 4.4. 2. For r = 1 and Λ2 := {λ ∈ C; (λ) ≥ 0} condition 2(d) coincides with condition 4(ii) from Theorem/Definition 5.1; see item 4 of Remark 5.2. The equivalence (a) ⇔ (d) is Theorem 5.2.30 of [15], where the stabilizability of B is, however, defined by asymptotic controllability to zero of its trajectories. 3. The condition 2(c)i of the preceding theorem characterizes internally stabilizable systems according to Quadrat [17, Thm. 3, Eq. (38)] with AT as a ring of SISO-stable plants. Remarks and open problems 5.10 (algorithmic questions). The assumptions are those from (64) and (70)–(75). It turns out that basic algorithmic problems cannot currently be solved. Refer to [7] and to [30] where the history, the state of the art, and open problems concerning algorithms in multidimensional stability theory are described. 1. According to Theorem/Definition 5.1, the stability of B is checked via char(B 0 ) ⊂ Λ1 or char(B 0 ) ∩ Λ2 = ∅, where char(B 0 ) = V (a) and a := ann(M 0 ) = {f ∈ C[s]; f M 0 = 0},

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and a can be computed from the matrix P ∈ C[s]k×p via Gr¨ obner bases. Recall from Theorem/Definition 5.6 that V (a) ∩ Λ2 = ∅ ⇔ a ∩ T = ∅ ∀ ideals a if and only if Λ2 is ideal-convex. If this is not the case, it is the analogue of the open problem 3 from [7, p. 73] for Λ2 instead of the closed unit polydisc to decide whether V (a) ∩ Λ2 = ∅. No algorithm is presently known. If Λ2 is ideal-convex, then to decide a ∩ T = ∅ and to actually construct a t ∈ a ∩ T is the analogue of the open problem 4 from [7, p. 73] or problem 1 from [30]. For compact, polynomially convex sets Λ2 , such as the closed unit polydisc, this requires a constructive version of the proof of Example 5.7(4), and even this seems hard to obtain. If [M 0 : C] < ∞ ⇔ a is Krull–zero-dimensional ⇔ V (a) = char(B 0 ) is finite, then the finite variety V (a) can be computed via Gr¨ obner bases, and V (a) ∩ Λ2 = ∅ can be decided (compare [7, p. 73, Rem.]). 2. Theorem 4.4 is used to test stabilizability and to construct a stabilizing compensator. Presently the computation of a matrix Rst ∈ Akst ×l such that Ust = {x ∈ A1×l ; ∀λ ∈ Λ2 ∃sλ ∈ A with sλ (λ) = 0 and sλ x ∈ U = A1×k R} st Rst = A1×kst Rst and hence UT = A1×k T

is unsolved. If, however, Λ2 is ideal-convex and MT = MT for all A-modules M , then UT = UT = A1×k T R and the construction of Rst is superfluous. See Theorem 5.14 for a partial result on the construction of Rst . If, in general, a st matrix Rst ∈ Akst ×l with UT = A1×k Rst is known, one computes the module T  l×kst × A of the polynomial linear system L of all solutions (G1 , t1 ) ∈ A (83)

Rst G1 Rst − t1 Rst = 0 ⇔ if t1 = 0 : Rst

G1 Rst = Rst t1

via Gr¨ obner bases and obtains a system of generators of the ideal (84)

a := {t1 ∈ A; ∃(G1 , t1 ) ∈ L}.

According to Theorem 4.4(3), the system is stabilizable if and only if a∩T = ∅. This is again the open problem quoted in item 1. If t1 ∈ a ∩ T and (G1 , t1 ) ∈ L, then G1 :=

G1 st ∈ Al×k T t1

is a matrix according to Theorem 4.4(3), and 1×l E1 := G1 Rst = E12 ∈ Al×l T , UT = AT E1 .

Generically, but  always, this matrix already satisfies the rank condition  not rank((idl −E1 ) id0m ) = m and can be used to construct a stabilizing compensator of B according to Theorem 4.4.

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3. All stabilizing compensators or idempotent matrices    0 l×l 1×l 2 E = E ∈ AT with UT = AT E and rank (idl −E) =m idm are constructed according to Theorems 2.14 and 4.6 in the form E = E1 + XRst , X =

n 

Θi Xi , Θi ∈ AT ,

i=1

where the Xi generate the solution module of the polynomial linear system l×kst Rst X  Rst = 0, X  ∈ A . Generically, these E satisfy the rank condi tion rank((idl −E) id0m ) = m and can be used to construct all stabilizing compensators of B. In the remainder of this section we indicate the necessary modifications of the preceding theory for the discrete case of complex partial difference equations and the r locally finite elements of the A-module CN [12, Thm. 1.25, Cor. 1.26] with the left shift action (sμ ◦ y)(ν) := y(μ + ν), y ∈ CN , μ, ν ∈ Nr . r

For α ∈ C, k, i ∈ N, λ ∈ Cr , μ, ν ∈ Nr define eα,k ∈ CN , eλ,μ ∈ CN by   r i i−k  if α = 0 k α eα,k (i) := , eλ,μ (ν) := eλρ ,μρ (νρ ). Then δi,k if α = 0 ρ=1  eλ,μ−ν if μ ∈ ν + Nr ν (s − λ) ◦ eλ,μ = and 0 otherwise r

(85)

N F := CN lf := {y ∈ C ; [C[s] ◦ y : C] < ∞} = ⊕λ∈Cr , μ∈Nr Ceλ,μ . r

r

Again F is the minimal injective cogenerator in ModA ; indeed F(λ) := ⊕μ∈Nr Ceλ,μ = E(A/mλ ) ∀λ ∈ Cr .

(86)

The Borel isomorphism CN = C[[z]] ∼ = C[[z]], y = r

 μ∈Nr

y(μ)z μ →

 μ∈Nr

y(μ)

zμ μ!

induces the C[s] isomorphism (87)

zμ exp(λ • z). F(λ) = ⊕μ∈Nr Ceλ,μ ∼ = C[z] exp(λ • z), eλ,μ → μ!

The decomposition Cr = Λ1  Λ2 with its implied data from (70) is again arbitrary, but, of course, interesting choices in the discrete case are different from those in the continuous case. Equations (70) to (77) remain valid, and so do Theorems/Definitions 5.1 and 5.3 if F(λ)k from (79) is replaced with (88) F(λ)k := {y ∈ CN ; mk+1 ◦ y = 0} = ⊕μ∈Nr , |μ|≤k Ceλ,μ , |μ| := μ1 + · · · + μr . λ r

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A standard example for Λ2 in the discrete case, in particular for r = 1, is (89)

r

Λ2 := {w ∈ C; |w| ≥ 1}r , but not U with U := {w ∈ C; |w| ≤ 1}, r

where U is the closed unit polydisc [7, p. 60], and for simplicity we assume this Λ2 in what follows. This seeming contrast to [7] and most other papers in this field comes from the fact that these papers do not consider IO systems, but systems defined by a proper or causal IO map H [6], [32, sec. 3.2]. To explain this connection we identify N = C[[z]]. Let zρ := s−1 ρ , A := C[s] ⊂ C(s) = C(z) = quot(A), C[z] ⊂ C r

(90)

n :=C[z] z1 , . . . , zr := {g ∈ C[z]; g(0) = 0}

be the distinguished maximal ideal of C[z]. Then [6, Def. 3.16], [11, sec. 6, Thm. 60] (91)

C[z]n = {n(z)/d(z); n, d ∈ C[z], d(0) = 0} = quot(A) ∩ C[[z]]

is the ring of proper or causal rational functions where the term “causal” is due to r the following property. The power series ring CN = C[[z]] is a module over itself via convolution, also denoted by ◦, and therefore also a module over the ring of proper rational functions. In particular [11, sec. 6, Cor. 45], if the transfer matrix H of B is r proper, any input u ∈ (CN )m = C[[z]]m gives rise to an output   y y := H ◦ u ∈ C[[z]]p with P ◦ y = P ◦ (H ◦ u) = (P H) ◦ u = Q ◦ u or ∈ B. u As noted before, properness is a rather restrictive property in the multidimensional situation and therefore we do not assume it in the present paper. For instance, the z2 of (s1 + s2 ) ◦ y = u and of other standard transfer functions (s1 + s2 )−1 = zz11+z 2 partial difference or differential equations are not proper. With the data from (89) and the terminology from [6, Def. 3.47], [7, p. 60], we see that  n(z) r S := h = (92) ∈ C(z) = C(s); ∀w ∈ U : d(w) = 0 ⊂ C[z]n d(z) is the ring of structurally stable rational functions, whereas the denominators d(z) are called structurally stable polynomials. Lemma 5.11. S ⊆ AT ∩ C[z]n = AT ∩ C[[z]]. Proof. Let h = n(z) d(z) ∈ S be structurally stable with relatively prime n and d. Choose ν ∈ Nr such sν n, sν d ∈ C[s]; hence h=

sν n(z) p(s) n(z) = ν = with relatively prime p, q ∈ C[s]. d(z) s d(z) q(s)

The factoriality of C[s] implies the existence of f (s) ∈ C[s] such that q(s)f (s) = sν d(z) ∈ C[s], hence q(λ)f (λ) = λν d(w) = 0 ∀ r

−1 λ ∈ Λ2 or |λρ | ≥ 1, ρ = 1, . . . , r, and w := (λ−1 1 , . . . , λr ) ∈ U ,

where h ∈ S implies d(w) = 0. Hence q ∈ T and h ∈ AT as asserted.

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Remarks 5.12. 1. Thus structural stability of h implies properness and stability, i.e., h ∈ AT , in the sense of this paper. Equality holds in the preceding lemma if and only if an irreducible polynomial d(z) ∈ C[z] is already structurally stable if d(0) = 0 r and if it has no zero w in U with wρ = 0 ∀ρ = 1, . . . , r. For r = 1 this is the case; in general it is unknown (to the author). This is another example of the difficulty of characterizing and manipulating the multiplicatively closed sets T of polynomials which have no zero in a given subset Λ2 of Cr . 2. The standard one-dimensional continuous (resp., discrete) stabilization theory for systems with a given proper transfer operator uses the Euclidean ring S = AT ∩ C[[z]] [26, Chap. 5] instead of AT here with Λ2 := {λ ∈ C; (λ) ≥ 0}, resp., Λ2 := {λ ∈ C; |λ| ≥ 1}. A counterpart to Theorem 5.4 is the BIBO stability of structurally stable transfer matrices [6, sec. 3.4], i.e., the result that the space  l

∞

:=

u=





uμ z ∈ C μ

Nr

= C[[z]]; ∃M > 0 ∀μ ∈ N : |uμ | ≤ M r

μ∈Nr

of bounded multisequences is an S-submodule of CN . This is “well-known” and follows from the fact that a structurally stable rational function is a convergent power r series in an open polydisc containing U . For general stable discrete systems, we pose the following open problem. Consider r Λ2 from (89), an arbitrary discrete IO system (64) with signals from C[[z]] = CN . Let Γ ⊂ {1, . . . , p} × Nr denote the associated canonical initial region [13, Eq. (14)] with respect to a chosen term order on {1, . . . , p} × Nr . Then the canonical Cauchy problem [13, Thms. 5 and 8] r

P ◦ y = Q ◦ u, u ∈ C[[z]]m , y = (93) y|Γ=







i≤p

 {yi,μ z μ ; (i, μ) ∈ Γ}

μ

∈ C[[z]]p ,

yi,μ z μ

μ∈Nr

= x ∈ CΓ ⊂ C[[z]]p i≤p

has a unique solution y for given input u and initial data x, and this can be computed using Gr¨ obner bases. The space Cz of locally convergent power series is [13, Eq. (54)]  Cz =

u=

 |uμ | ≤ M Rμ

 =



u= 

∞

1

and l , l :=

uμ z μ ∈ C[[z]]; ∃M > 0 ∃R ∈ Rr>0 ∀μ ∈ Nr :

μ∈Nr



 ∈ Rr : uμ z μ ∈ C[[z]]; ∃R >0

μ∈Nr

u=

 μ∈Nr





μ < ∞ |uμ |R

μ∈Nr

uμ z ∈ C[[z]]; μ

 μ∈Nr



|uμ | < ∞

⊂ Cz

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are C[s]-submodules of Cz . The unique solvability of the Cauchy problem (93) also holds for Cz instead of C[[z]] [13, Cor. 25], especially  ∼ =

B 0 = {y ∈ Cz p ; P ◦ y = 0} 

x ∈ CΓ ; ∀i = 1, . . . , p :

{xi,μ z μ ; (i, μ) ∈ Γ} ∈ Cz

 , y → y | Γ.

μ

In particular, if u and x have components in l∞ or l1 , the unique power series solution y satisfies  (94) |yi,μ |Rμ < ∞ for some R ∈ Rr>0 and M > 0. |yi,μ | ≤ M Rμ , resp., μ∈Nr

If all components Rρ are smaller (resp., greater) or equal to 1, then y ∈ (l∞ )p (resp., y ∈ (l1 )p ). Now assume that the IO system B is Λ2 -stable for Λ2 from (89), i.e., that conditions 2–4 of Theorem/Definition 5.1 are satisfied (the function space differs!!). With the idea from [29, pp. 1499–1500] that stability of a system should imply that stable inputs and initial conditions generate stable outputs, we pose the following. Open problem 5.13. Consider Λ2 := {λ ∈ Cr ; ∀ρ = 1, . . . , r : |λρ | ≥ 1}, i = ∞, 1, and a Λ2 -stable IO system B as above with input u ∈ (li )m , initial data x ∈ (li )p , and unique output y ∈ Cz p . When is y ∈ (li )p , i.e., R = (1, . . . , 1) in (94) or, in other words, when is the stable system BIBO- or l∞ -IO-stable (resp., l1 -IO-stable)? This is always true for one-dimensional discrete IO systems—the properness of the transfer matrix is not required. Compare [15, Thm. 7.6.2] for a continuous analogue under the assumption that the transfer matrix is proper. The question can also be asked for other natural unstable regions and is also interesting and reasonable in the continuous case of partial differential equations for those spaces of analytic functions for which the Cauchy problem is uniquely solvable. In particular this holds for the space of entire functions of exponential type [13, Thm. 26] which, by means of the Borel isomorphism, is isomorphic to Cz with the shift action, and for the space Cz , but now with the action by partial differentiation and under the assumption that the term order is graded [13, Thm. 29]. Consider, however, the trivial (in the sense of Theorem/Definition 2.5) and therefore stable one-dimensional system y = s1 ◦ u with an empty initial condition and polynomial transfer function H = s1 . The analytic function u := exp(iz12 ) is bounded on R, but the output y = u = 2iz1 u is not, so BIBO stability does not hold. Likewise, u(z1 ) := (1 + z12 )−1 exp(iz12 ) is in L1 (R), but its derivative is not. So additional assumptions, for instance properness, have to be made in the analogue of the open problem for the general continuous case. The following theorem without a detailed proof improves Theorem 5.8 and item 2 of Remarks/Open Problems 5.10 and was added during the revision process. The data are those from (64) and (70)–(78). According to Theorem/Definition 2.2 we construct the unique controllable realization Bcont of H via 

◦ H ⊥ 1×(p+m) (idm ) 1×m −→ K , hence Ucont := Bcont := ker A (95)

M/t(M ) ∼ = Mcont := A1×(p+m) /Ucont , t(M ) = Ucont /U ⊂ M = A1×(p+m) /U , a := annA (t(M )).

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The submodule Ucont and the annihilator a of the torsion module t(M ) = Ucont /U are computed via Buchberger’s algorithm. For the Gabriel localization UT , Lemma/ Definition 3.6 implies (96)

1×(p+m)

UT ⊂ UT = AT

1×(p+m)

∩ ∩λ∈Λ2 Umλ ⊂ AT

.

Generators of UT are needed for the application of the algorithm in Remarks/Open Problems 5.10(2), and in general are hard to obtain. The next theorem gives a partial solution. Theorem 5.14. We use data as just introduced. (1) All localized modules Mmλ , λ ∈ Λ2 , are torsion free if and only if V (a)∩Λ2 = ∅. If this is the case, then UT = Ucont,T , and B is stabilizable if and only if Mcont,T is a projective AT -module. (2) The module MT is torsion free if and only if a ∩ T = ∅. If this is the case, then 1×p ⊂ C(s)1×m , UT = Ucont,T = UT and MT ∼ = M := AT H + A1×m T

and B is stabilizable if and only if the lattice M is a projective AT -module. According to Quadrat [17, Cor. 3] this characterizes the stabilizability of the transfer matrix H in the usual sense. 6. Conclusion. In contrast to the literature on the stabilization of discrete multidimensional IO maps, the present paper developed the stabilization theory for continuous or discrete multidimensional IO behaviors. An important technical tool in this context was the generalized localization theory due to Gabriel. Algorithmic problems were addressed, but only partially solved. For a complete solution one needs constructive solutions for the following problems from algebraic and analytic geometry. Let Λ2 be an arbitrary subset of the r-dimensional complex space and consider T := {t ∈ C[s1 , . . . , sr ]; ∀λ ∈ Λ2 : t(λ) = 0} =: {stable polynomials}, nonzero polynomials f and ideals a in C[s]. Open algorithmic problems 6.1 (compare [7], [30]). 1. Decide f ∈ T . 2. Decide V (a) ∩ Λ2 = ∅. 3. Decide a ∩ T = ∅ and construct t ∈ a ∩ T . 4. The implication a ∩ T = ∅ ⇒ V (a) ∩ Λ2 = ∅ is obvious. The reverse implication is true for all ideals a if and only if a is ideal-convex. Decide ideal-convexity constructively and construct t ∈ a ∩ T if V (a) ∩ Λ2 = ∅. 5. Of course, the preceding tasks have to be solved only for suitable Λ2 , for instance for the typical continuous (resp., discrete) cases Λ2 := {z ∈ C; (z) ≥ 0}r , resp., Λ2 = {z ∈ C; |z| ≥ 1}r . As shown in Remarks/Open Problems 5.10 and in Theorem 5.14 a special stabilization problem gives rise only to few ideals a for which the preceding tasks have to be solved. The paper [8] treats these problems for the closed unit polydisc and a special class of ideals. 6. If a is Krull–zero-dimensional or V (a) is finite, then this variety can be computed via Buchberger’s algorithm and the constructive tasks can in general be solved.

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