On the notions of dilation, controllability, observability, and minimality in the theory of dissipative scattering linear nd systems

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1 On the notions of dilation, controllability, observability, and minimality in the theory of dissipative scattering linear nd systems Dmitriy S. Kalyuzhniy Department of Higher Mathematics Odessa State Academy of Civil Engineering and Architecture Didrihson str Odessa, Ukraine Dmitriy.Kalyuzhniy@p25.f61.n467.z2.fidonet.org Keywords: minimality. Abstract Dissipative scattering nd systems, dilations, We propose two non-equivalent approaches to definitions of dilation, controllability, observability, and minimality for certain type of discrete time-invariant linear nd systems in Hilbert spaces, and discuss interrelations of these notions in both approaches, and their applications in the theory of dissipative scattering linear nd systems. We answer questions on the existence and essential uniqueness of minimal, and minimal dissipative scattering realizations for certain classes of holomorphic operator-valued functions of n complex variables. 1 Introduction and preliminaries In [1] we have introduced n-dimensional (nd) discrete timeinvariant linear systems of the form x(t) = n (A k x(t e k ) + B k φ (t e k )), α : k=1 φ + (t) = n (1) (C k x(t e k ) + D k φ (t e k )), k=1 (t Z n, t > ) where t := n k=1 t k, e k Z n and (e k ) j = δ kj for all k, j {1,..., n}; X, N, N + are separable Hilbert spaces which are called respectively the state space, the input space, and the output space of α; for all k {1,..., n} A k : X X, B k : N X, C k : X N +, D k : N N + are bounded linear operators; and also the following initial condition is given: x(t) = x (t), (t Z n, t = ) (2) where x ( ) is a prescribed X -valued function. It is easy to show that the well-known to nd-systems theorists Fornasini-Marchesini model [2] is a particular case of system (1) (2). Let us note also that for n = 1 the form of equations (1) differs from the standard one by the unit shift of an output sequence φ + ( ), that implies some non-essential changes in formulations of results in the discrete 1D systems theory. If one denotes the n-tuple of operators T k (k = 1,..., n) by T := (T 1,..., T n ) then for system (1) (2) one may use the short notation α = (n; A, B, C, D; X, N, N + ). If z = (z 1,..., z n ) C n then we set zt := n k=1 z kt k. The operator-valued function θ α (z) = zd + zc(i X za) 1 zb (here I X denotes the identity operator on X ) holomorphic on some neighbourhood of z = in C n is called the transfer function of a system α. Recall (see [1]) that α = (n; A, B, C, D; X, N, N + ) is called a dissipative (resp., conservative) scattering system if for any ζ T n (the n-fold unit torus) ζa ζb ζg := [X N, X N + ] (3) ζc ζd is a contractive (resp., unitary) operator (here [H 1, H 2 ] denotes the Banach space of all bounded linear operators mapping a separable Hilbert space H 1 into a separable Hilbert space H 2 ). The condition of dissipativity (resp., conservativity) has a physical sense of the dissipation (resp., the full balance) of energy in a scattering system. Let us note that analogous norm constraints imposed on a system appear also in other approaches to the nd scattering theory (see, e.g., [3, 4, 5, 6, 7]). It was shown in [1] that the transfer function of an arbitrary dissipative scattering nd system of the form (1) (2) is a holomorphic contractive operator-valued function on the open unit polydisk D n. Recall that the Agler-Schur class S n (N, N + ) (see [8]) consists of all functions θ(z) = θ t z t (here Z n + := {t Z n : t k, k = 1,..., n}, z t := for t Z n +) which are holomorphic on D n, take n k=1 zt k k

2 values in [N, N + ], and such that for any separable Hilbert space H, any n-tuple T = (T 1,..., T n ) of commuting contractions on H, and any positive r < 1 one has where θ(rt) 1, θ(rt) = θ(rt 1,..., rt n ) := θ t r t T t, T t := n k=1 T t k k, and the convergence of this series is understood in the sense of norm in [N H, N + H]. For n = 1 and n = 2 by [9] and [1], respectively, S n (N, N + ) coincides with the class of all holomorphic contractive [N, N + ]-valued functions on D n, and for n 3 due to [11] is a proper subclass of the latter. Let Sn(N, N + ) denote the subclass in S n (N, N + ) consisting of functions vanishing at z =. We have proved in [1] that the class of transfer functions of conservative scattering nd systems with the input space N and the output space N + coincides with Sn(N, N + ). We shall call the realization part of this result the conservative scattering nd system realization theorem. Let α = (n; A, B, C, D; X, N, N + ) and α = (n; Ã, B, C, D; X, N, N + ) be two systems such that for each z C n there are subspaces D z and D,z in X such that X = D z X D,z, (4) zãd z D z, z CD z = {}, (zã) D,z D,z, (z B) D,z = {}, za = P X (zã) X, zb = P X (z B), zc = (z C) X. (here P X is the orthoprojector onto X in X ). Then α is called a dilation of α, and α is called a compression of α. It was shown in [12] that a dissipative scattering system α = (n; A, B, C, D; X, N, N + ) allows a conservative dilation if and only if the corresponding linear operator-valued function za zb L G (z) := zg = (z D n ) zc zd belongs to the class S n(x N, X N + ). In this case α is said to be a n-dissipative scattering system. The class of such systems for the cases n = 1 and n = 2 coincides with the class of all dissipative scattering systems, and for any n 3 is a proper subclass of the latter (see [12, 13]). A system α is called minimal if α is not a dilation of any system other than itself. In [12] there were obtained also the following results. Transfer functions of a system α of the form (1) (2) and of its dilation α coincide. For an arbitrary system α of the form (1) (2) there exists a minimal system α min which is a compression of α. Any n-dissipative scattering system α allows a minimal n-dissipative compression α min. Together with the conservative scattering nd system realization theorem, this implies that an arbitrary function θ(z) Sn(N, N + ) has a minimal n-dissipative realization α min, i.e. θ(z) = θ αmin (z) in D n. In the present paper we continue the study of dilations and minimality for nd systems. In Section 2 we prove the theorem on the existence of a nd system realization for an arbitrary operator-valued function holomorphic on some neighbourhood of z = in C n and vanishing at z =. As a corollary, we obtain the theorem on the existence of a minimal nd system realization for such a function. We also introduce the notion of J-conservative scattering linear nd system, and announce two related results: theorems on the existence of a J-conservative scattering linear nd system which is a dilation of a given (arbitrary) system of the form (1) (2), and on the existence of a J-conservative scattering nd system realization for an arbitrary operator-valued function holomorphic on some neighbourhood of z = in C n. In Section 3 we show that minimal nd system realizations are, in general, essentially non-unique, moreover a minimal conservative scattering nd system realization of an arbitrary function θ(z) Sn(N, N + ) can be essentially non-unique. We define the notions of controllable, and observable nd systems, which are natural generalizations of the corresponding notions for 1D systems. We show however that, in contrast to the 1D case, minimal systems are not necessarily controllable and observable, and a system which is controllable and observable can be not minimal. In order to avoid this strange effect one may define the notions of dilation, controllability, observability, and minimality for systems of the form (1) (2) in another way. In Section 4 we introduce the notions of uniform dilation, uniform compression, quasicontrollability, quasiobservability, and quasiminimality for such systems, and show that they have natural interrelations. Namely, an arbitrary system α of the form (1) (2) is quasiminimal if and only if α is quasicontrollable and quasiobservable. Any system α of the form (1) (2) has two canonical quasiminimal uniform compressions whose state spaces are X qco := P Xqo X qc and X qoc := P Xqc X qo, respectively, where X qc (resp., X qo ) is the quasicontrollable (resp., quasiobservable) subspace in the state space X of α. Any minimal system is quasiminimal. On the other hand, a quasiminimal system is not necessarily minimal. In Section 5 we give some concluding remarks in which we compare two presented concepts of dilation and minimality for nd systems. 2 Certain realization theorems Theorem 1 An arbitrary [N, N + ]-valued function θ(z) holomorphic on some neighbourhood Γ of z = in C n and vanishing at z = can be realized as the transfer function of some system α = (n; A, B, C, D; X, N, N + ), i.e. θ(z) = θ α (z) in some neighbourhood (possibly, smaller than Γ) of z =. Proof. For θ(z) under conditions of this theorem there is a number ρ > such that the closure of polydisk ρd n := {ρz z D n } is contained in Γ : ρd n Γ. Then θ(z) =

3 t Z θ + n t z t (z ρd n ), and this series converges absolutely and uniformly on ρd n in the operator norm (e.g., see [14]). Set ψ(z) := θ(ρz), z D n. Then ψ(z) is holomorphic on D n, ψ(z) = t Z θ n t ρ t z t (z D n ), and this series con- + verges absolutely and uniformly on D n in the operator norm. In particular, N := t Z ρ t θ n t <. Hence, for any + separable Hilbert space H, any n-tuple T = (T 1,..., T n ) of commuting contractions on H, and any positive r < 1 ψ(rt) = ρ t θt r t T t ρ t r t θ t T t ρ t θ t = N. Then ξ(z) := N 1 ψ(z) Sn(N, N + ), and there is a conservative scattering system α = (N; Ȧ, Ḃ, Ċ, Ḋ; X, N, N + ) such that ξ(z) = θ α (z), z D n (see Section 1). Since θ(z) = ψ(ρ 1 z) = Nξ(ρ 1 z), z ρd n, we have θ(z) = Nρ 1 zḋ + Nρ 1 zċ(i X ρ 1 zȧ) 1 ρ 1 zḃ for z ρd n. Thus, setting X := X, A k := ρ 1 A k, B k := ρ 1 Ḃ k, C k := Nρ 1 Ċ k, D k := Nρ 1 Ḋ k for all k {1,..., n}, we obtain the system α := (n; A, B, C, D; X, N, N + ) such that θ(z) = θ α (z) for all z ρd n, and the proof is complete. Remark 2 For n = 1 Theorem 1 was proved by D.Z. Arov in [15], however in another way: he obtained the desired realization of θ(z) through the Cauchy integral representation, by using the likeness of a resolvent (I X za) 1 to the Cauchy kernel C(z, ζ) = (1 zζ) 1 for the 1D case. Since any system α of the form (1) (2) has a minimal compression α min, and θ α (z) = θ αmin (z) (see Section 1), Theorem 1 implies the following. Theorem 3 An arbitrary [N, N + ]-valued function θ(z) holomorphic on some neighbourhood Γ of z = in C n and vanishing at z = can be realized as the transfer function of some minimal system α min = (n; A min, B min, C min, D min ; X min, N, N + ), i.e. θ(z) = θ αmin (z) in some neighbourhood (possibly, smaller than Γ) of z =. Let α = (n; A, B, C, D; X, N, N + ), and the operator J [X ] := [X, X ] be given such that J = J = J 1 (such a J is said to be a canonical symmetry on X ). Then J determines on X the new inner product [x 1, x 2 ] J := Jx 1, x 2 (here, is a Hilbert space inner product on X ) which is, in general, indefinite, and the space X with respect to this new inner product has the structure of a Krein space (for more information on Krein spaces see, e.g., [16]). Definition 4 Let α = (n; A, B, C, D; X, N, N + ) and a canonical symmetry J [X ] be given. Define J 1 := J I N [X N ], J 2 := J I N + [X N + ]. We shall call α a J-conservative scattering linear nd system if for each ζ T n the operator ζg from (3) is (J 1, J 2 )-unitary, i.e. ζ T n (ζg) J 2 (ζg) = J 1, (ζg)j 1 (ζg) = J 2. In the particular case when J = I X, a J-conservative scattering system is a conservative scattering one. Let us note that one may consider a J-conservative scattering system as a conservative scattering one, however with a Kreinian state space, i.e. the energy balance equations (see [1]) have the same form, but with the J-metrics [, ] J in the place of the Hilbert space metrics, for states of a system. Now we formulate two results related to the subject of this paper; for the reason of restrictions on the size of a paper we postpone their proofs till our next paper. Theorem 5 For an arbitrary linear system α = (n; A, B, C, D; X, N, N + ) there exists a J-conservative scattering system α = (n; Ã, B, C, D; X, N, N + ), with a canonical symmetry J [ X ], such that α is a dilation of α. Since transfer functions of a system and of its dilation coincide, from Theorems 1 and 5 we obtain the following result. Theorem 6 An arbitrary [N, N + ]-valued function θ(z) holomorphic on some neighbourhood Γ of z = in C n and vanishing at z = can be realized as the transfer function of some J-conservative scattering system α = (n; A, B, C, D; X, N, N + ), with a canonical symmetry J [X ], i.e. θ(z) = θ α (z) in some neighbourhood (possibly, smaller than Γ) of z =. Remark 7 For n = 1 Theorems 5 and 6 were proved by T.Ya. Azizov (see [16]) with use of Arov s realization theorem mentioned in Remark 2. 3 Controllability, observability, and minimality Definition 8 We shall call two systems α = (n; A, B, C, D; X, N, N + ) and α = (n; A, B, C, D ; X, N, N + ) similar if there exists a bounded and boundedly invertible operator W [X, X ] such that for any k {1,..., n} A k = W A k W 1, B k = W B k, C k = C k W 1, D k = D k. For 1D time-invariant linear systems in finite-dimensional spaces it is known [17] that two minimal systems with equal transfer functions are similar. Now we will show that for n > 1, even two minimal conservative scattering nd systems

4 in finite-dimensional spaces with equal transfer functions are not necessarily similar. Example 9 Consider α = (2; A, B, C, D; X := C, C, C) where for any z = (z 1, ) C 2 za zb zg = := zc zd ( z2 z 1 ), and α = (2; A, B, C, D ; X := C 3, C, C) where for any z = (z 1, ) C 2 zg = := ( za zb ) zc zd z 1 / 2 z 1 / 2 / 2 / 2 / 2 z 1 / 2 / 2 z 1 /. 2 It is easy to see that both α and α are conservative scattering 2D system realizations of the function θ(z) = z 1. The system α is minimal. Indeed, if α := (2; Ȧ, Ḃ, Ċ, Ḋ; X, C, C) is a compression of α which not coincides with α then X = {}, Ȧ = (, ), Ḃ = (, ), Ċ = (, ), and Ḋ = D = (, ), hence θ α(z) = z 1 = θ α (z), that is impossible. The system α is also minimal. Indeed, suppose that α := (2; Ȧ, Ḃ, Ċ, Ḋ ; X, C, C) is a compression of α. Then (see (4)) for any z C 2 we have C 3 = X = D z X D,z so that za D z D z, zc D z = {}, (za ) D,z D,z, and (zb ) D,z = {}. In particular, D z (zc ) C = z 1 C, D,z zb C = z 1 hence (D z D,z ) ((zc ) C zb C), and we have X z C 2 ((zc ) C zb C) = C C {}. C, Let ( =) z C 2 be fixed. If D z {} then D z = {} {} C, and za D z = z 1 C D z, that is impossible. If D,z {} then D,z = {} {} C, and (za ) D,z = z 2 z 1 C D,z, that is also impossible. Therefore, D z = D,z = {}, and X = C 3. Thus, α = α, and α is minimal. Since 1 = dim X = dim X = 3, α and α are two minimal conservative scattering 2D system realizations of θ(z) = z 1, which are not similar. Definition 1 Let α = (n; A, B, C, D; X, N, N + ). Define X c := (za) k zbn, z C n, k Z + X o := (za) k (zc) N +, z C n, k Z + which we shall call a controllable subspace, and an observable subspace in X, respectively. The system α is called controllable (resp., observable) if X c = X (resp., X o = X ). A controllable system α is characterized by the property that for any x X and ɛ > there exist t Z n : t >, and a finite multisequence {φ (τ) τ Z n, τ, τ k t k, k = 1,..., n} such that for x(t), which is determined by means of (1) for initial condition x ( ) in (2), we have x x(t) < ɛ. A system α = (n; A, B, C, D; X, N, N + ) is observable if and only if its conjugate system α := (n; A, C, B, D ; X, N +, N ) is controllable (here A := (A 1,..., A n), etc.). Example 11 The system α in Example 9 is minimal, however not controllable, and not observable. Indeed, for any z C 2 and k 1 we have zb C = z 1, (zc ) C = z 1, (za ) k zb C = {}, (za ) k (zc ) C = {}, hence X c = X o = C C {} = C 3 = X. Example 12 Let α = (2; A, B, C, D; C 2, C, C) where for z = (z 1, ) C 2 za zb zg = zc zd z 1 / 2 := / 2 / 2 z 1 /. 2 It is clear that α is a 2-dissipative controllable and observable realization of θ(z), however α is not minimal: it has a trivial minimal compression α = (2;,,, ; {}, C, C). 4 Uniform dilations, quasicontrollability, quasiobservability, and quasiminimality Definition 13 Let α and α be two systems of the form (1) (2), α be a dilation of α. If, additionally, the spaces D z in (4) coincide for all z C n (equivalently, the spaces D,z in (4) coincide for all z C n ) then we call α a uniform dilation of α, and α a uniform compression of α. Definition 14 We shall call a system α of the form (1) (2) quasiminimal if α is not a uniform dilation of any system other than itself.

5 Obviously, any minimal system is quasiminimal, that implies the following two results. Theorem 15 An arbitrary [N, N + ]-valued function θ(z) holomorphic on some neighbourhood Γ of z = in C n and vanishing at z = can be realized as the transfer function of some quasiminimal system α qmin of the form (1) (2), i.e. θ(z) = θ αqmin (z) in some neighbourhood (possibly, smaller than Γ) of z =. Theorem 16 An arbitrary θ(z) S n(n, N + ) can be realized as the transfer function of some quasiminimal n- dissipative scattering system α qmin of the form (1) (2), i.e. θ(z) = θ αqmin (z) in D n. Definition 17 Let α = (n; A, B, C, D; X, N, N + ). Define X qc := (z (k) A) (z (2) A)z (1) BN, k N, {z (j) } k 1 Cn X qo := (z (k) A) (z (2) A) (z (1) C) N +, k N, {z (j) } 1 k Cn which we shall call a quasicontrollable subspace, and a quasiobservable subspace in X, respectively. We shall call a system α quasicontrollable (resp., quasiobservable) if X qc = X (resp., X qo = X ). Since X c X qc, X o X qo, any controllable (resp., observable) system is quasicontrollable (resp., quasiobservable). Theorem 18 An α = (n; A, B, C, D; X, N, N + ) is quasiminimal if and only if α is quasicontrollable and quasiobservable. Proof. Let us note that X X qo contains all subspaces D in X such that zad D, zcd = {} for all z C n, and X X qo itself is one of them; X X qc contains all subspaces D in X such that (za) D D, (zb) D = {} for all z C n, and X X qc itself is one of them. This implies the statement of this theorem. Theorem 19 Let α = (n; A, B, C, D; X, N, N + ). Set X qco := P Xqo X qc. Then the system α qco = (n; A qco, B qco, C qco, D; X qco, N, N + ), where for all k {1,..., n} (A qco ) k := P Xqco A k X qco, (B qco ) k := P Xqco B k, (C qco ) k := C k X qco, is a quasiminimal uniform compression of α. Proof. The argument in the proof of Proposition 4 in [15] (the 1D case of this theorem) is naturally extended to the general nd case. Analogously, we obtain that the system α qoc = (n; A qoc, B qoc, C qoc, D; X qoc, N, N + ), with the state space X qoc := P Xqc X qo and coefficients (A qoc ) k := P Xqoc A k X qoc, (B qoc ) k := P Xqoc B k, (C qoc ) k := C k X qoc (k {1,..., n}) is also a quasiminimal uniform compression of α. Let us remark that, together with Theorem 1 (resp., the conservative scattering nd system realization theorem for functions from the class S n(n, N + )), Theorem 19 implies Theorem 15 (resp., Theorem 16), and these new proofs of these results are independent of the notion of minimal system. Let us remark also that a quasiminimal system is not necessarily minimal, e.g. the system α in Example 12 is controllable and observable, hence quasicontrollable and quasiobservable, i.e., due to Theorem 18 is quasiminimal, however not minimal. 5 Conclusion In this paper we have obtained the general result on a nd system realization of an arbitrary operator-valued function which is holomorphic on some neighbourhood of z = in C n and vanishes at z = (Theorem 1), that gives several consequences. In particular, we have proved the existence of minimal, and minimal n-dissipative scattering system realizations of multivariable holomorphic operator-valued functions from certain classes. We introduced the natural generalizations of the notions of controllability and observability to the case of nd systems (of the form (1) (2)). However, we have discovered that, in contrast to the 1D case, the property of minimality for nd systems of the considered form doesn t correlate with the controllability and observability property (certain related examples are presented). Then we introduce the notion of uniform dilation for such systems, and related notions of quasicontrollability, quasiobservability, and quasiminimality. These notions have more natural interrelations, i.e. analogous to the 1D case, namely, quasiminimality is equivalent to quasicontrollability and quasiobservability. The existence of quasiminimal, and quasiminimal n- dissipative scattering system realizations for the corresponding classes of multivariable holomorphic operator-valued functions is also established. We have demonstrated that all mentioned realizations are, in general, essentially non-unique, even in a finitedimensional situation. References [1] Kalyuzhniy D.S. Multiparametric dissipative linear stationary dynamical scattering systems: Discrete case. J. Operator Theory, Vol.43, No.2, 2. [2] Fornasini E. and Marchesini G. Doubly-indexed dynamical systems: State-space models and structural properties. Math. Systems Theory, Vol.12, pp.59 72, [3] Cotlar M. and Sadosky C. Integral representations of bounded Hankel forms defined in scattering systems

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