Consistent systems and pole assignment: I

ISSN 0012-2661, Differential Equations, 2012, Vol. 48, No. 1, pp. 120–135. c© Pleiades Publishing, Ltd., 2012.Original Russian Text c© V.A. Zaitsev, 2012, published in Differentsial’nye Uravneniya, 2012, Vol. 48, No. 1, pp. 117–131.

CONTROL THEORY

Consistent Systems and Pole Assignment: I

V. A. ZaitsevUdmurt State University, Izhevsk, Russia

Received November 1, 2010

Abstract—We study the consistency property of a linear time-invariant control system withincomplete feedback and of a bilinear time-invariant control system in connection with the poleassignment problem. We show that consistency is a sufficient and, in some cases, necessarycondition for the arbitrary pole assignability of systems with coefficients of a special form.

DOI: 10.1134/S001226611110120

1. INTRODUCTION

Consider the linear time-invariant control system

x = Ax + Bu, x ∈ Kn, u ∈ K

m, (1)y = C∗x, y ∈ K

k, (2)

where K = C or K = R. The control in system (1) is constructed in the form

u = Uy,

where U ∈ Mm,k(K) is a constant matrix; here Mm,k(K) is the space of m× k matrices with entriesin the field K [and we write Mn(K) := Mn,n(K)]. We obtain the closed-loop system

x = (A + BUC∗)x. (3)

Consider also the bilinear time-invariant control system

x = (A + u1A1 + · · · + urAr)x, x ∈ Kn, u = (u1, . . . , ur) ∈ K

r. (4)

We identify system (1) with the matrix (A,B) ∈ Mn,n+m(K), system (3) with the matrix Σ =(A,B,C) ∈ Mn,n+m+k(K), and system (4) with the matrix Ω = (A,A1, . . . , Ar) ∈ Mn,n(1+r)(K).System (4) has a more general form than system (3). We denote the characteristic polynomials ofthe matrices of systems (3) and (4) by

�(λ; Σ, U) = det(λI − A − BUC∗), �(λ; Ω, u) = det(λI − A − u1A1 − · · · − urAr),

respectively. The pole assignment problem for systems (3) and (4) is posed as follows. Givena system Σ (or Ω) and a monic polynomial

p(λ) = λn + γ1λn−1 + · · · + γn

with coefficients γi ∈ K, construct a control U ∈ Mm,k(K) in system (3) [respectively, u ∈ Kr

in system (4)] such that the characteristic polynomial �(λ; Σ, U) [respectively, �(λ; Ω, u)] coincideswith p(λ). This problem is also called the eigenvalue assignment problem, or the eigenvalue place-ment problem.

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CONSISTENT SYSTEMS AND POLE ASSIGNMENT: I 121

We identify the set of all monic polynomials p(λ) with the space Kn = {γ = (γ1, . . . , γn)}.

For a given system Σ (respectively, Ω), we introduce the pole assignment map that takes each con-trol U (respectively, u) to the characteristic polynomial of the closed-loop system with this control,

σΣ : Mm,k(K) → Kn, σΣ(U) = �(λ; Σ, U), ωΩ : K

r → Kn, ωΩ(u) = �(λ; Ω, u).

Therefore, if the map σΣ (respectively, ωΩ) is surjective, then the spectrum of eigenvalues of thesystem Σ (respectively, Ω) can be placed arbitrarily. In this case, we say that the system Σ(respectively, Ω) is arbitrarily pole assignable.

Numerous papers deal with the eigenvalue assignment problem. A survey of known results canbe found in [1–5]. Let us mention some of them. The first results pertain to systems of the form (3)with complete feedback, i.e., with C = I.

Assertion 1. The following conditions are equivalent.1. The system (A,B) is completely controllable.2. rank[B,AB, . . . , An−1B] = n.3. The map σΣ, where Σ = (A,B, I), is surjective.

The equivalence 2 ⇐⇒ 3 was proved in [6] for K = C and in [7] for K = R; the equivalence1 ⇐⇒ 2 is the well-known Kalman controllability criterion.

Note that the set of completely controllable systems (A,B) is generic in the space Mn,n+m(K)of matrices, which is identified with K

n(n+m), in the following sense: a subset S ⊂ Kl is called

a generic set if its complement Kl\S is contained in the zero set of some nontrivial polynomial

in x1, . . . , xl.Note that the eigenvalue assignment problem for system (3) with matrix C = I (or with

rankC = n) is essentially linear and can be solved by methods of linear algebra. If rankB < nand rankC < n, then this problem is nonlinear, and its investigation is more complicated. Firstresults on an “almost arbitrary” assignment of max{m,k} eigenvalues (without loss of generality,we assume that m = rankB and k = rankC) were obtained in this case under the assumption of thecomplete controllability and complete observability of system (1), (2) in [8–11]. (These conditionsare necessary for σΣ to be surjective.) For the case in which K = R, it was shown in [12, 13] thatif system (1), (2) is completely controllable and completely observable and the McMillan degree ndoes not exceed m + k − 1, then the image of the map σΣ is everywhere dense.

Various methods of algebraic geometry were used to obtain further results. By comparing thedimensions of the domain and range of the map σΣ, one can establish that the condition mk ≥ n isnecessary for the (almost) surjectivity of the map σΣ. By using the dominant morphism theorem,Hermann and Martin [14] showed that this condition is also sufficient for the case in which K = C :if n ≤ mk, then the map σΣ is almost surjective for a generic set of matrices Σ.

Next, for K = C, it was shown in [15] that if mk ≥ n, then the map σΣ is completely surjective(not only almost surjective) for a generic set of systems Σ ∈ Mn,n+m+k(K) with McMillan degree n.For K = R, it follows from the results in [15] that if mk = n and

d(m,k) =1!2! . . . (k − 1)!(mk)!

m!(m + 1)! . . . (m + k − 1)!

is odd, then the map σΣ is surjective for a generic set of systems Σ with McMillan degree n. IfK = R, m = k = 2, and n = 4, then the eigenvalue assignment problem is not solvable in thegeneric case [16] (see also [17, 18] for other even values of m and k).

Further, a common (for even and odd n) sufficient condition for the arbitrary eigenvalue assign-ability was obtained in [19] for the case in which K = R : if mk > n, then the map σΣ is surjectivefor a generic set of systems Σ with McMillan degree n. A simpler proof of this result was suggestedin [20] (see also the bibliography in [4]).

The eigenvalue assignment problem for the bilinear system (4) is related to the additive inverseeigenvalue problem (see the survey [21]). This problem can be stated as follows. Let L ⊂ Mn(K)

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122 ZAITSEV

be some variety. Given a polynomial p(λ), the problem is to construct an L ∈ L such thatdet(λI − A − L) = p(λ). It is known [22] that, for K = C, the map

ϕA : L → Kn, ϕA(L) = det(λI − A − L), L ∈ L,

is surjective for arbitrary A provided that L coincides with the set Dn(C) ⊂ Mn(C) of diagonalmatrices. It was shown in [23] for K = C that if L is a Lie algebra, then the map ϕA is surjective foran arbitrary matrix A if and only if rankL = n and some element in L has n distinct eigenvalues.It was proved in [24] that, for an affine variety L in Mn(C), the map ϕA is almost surjective fora generic set of matrices A if and only if dimL ≥ n and

L ⊂ sln := {L ∈ Mn(C) : SpL = 0}.

This implies some corollaries for system (4). Some results combining several approaches to theeigenvalue assignment problem can be found in [25].

Thus, the eigenvalue assignment problem for systems (3) and (4) has been quite comprehensivelystudied in the generic case. Nevertheless, as was emphasized in [4], the sufficient conditions [15, 19]obtained for system (3) are mostly of theoretical character. In general, even if the existence ofa compensator U providing the desired eigenvalue assignment is known, there is no usable numericalalgorithm for finding a solution U . This is caused by the intrinsic nonlinearity of this problem forsystem (3) with incomplete feedback [and, so much the more, for system (4)], unlike system (3)with complete feedback (C = I), which has a linear nature and for which such algorithms areknown. In the present paper (which consists of two parts), we continue the research in [26–28] andobtain coefficient conditions for the solvability of the eigenvalue assignment problem in the general(not only generic) case for systems of a special form. Here we prove theorems similar to Assertion 1for systems (3) and (4). These assertions hold both for K = C and K = R. The obtained necessaryand sufficient conditions can be used to construct the above-mentioned numerical algorithms.

2. NOTATION AND DEFINITIONS

Let e1, . . . , en be the canonical basis in the space Kn; i.e., e1 = col(1, 0, . . . , 0), . . . , en =

col(0, . . . , 0, 1); let Mn,m := Mn,m(K) and Mn := Mn,n; let I = [e1, . . . , en] ∈ Mn be the identitymatrix; let T stand for the transposition of a vector or a matrix; let ∗ be the Hermitian conjugation;i.e., H∗ = H T; let J be the unit first superdiagonal matrix; i.e., J :=

∑n−1

i=1 eie∗i+1 ∈ Mn; let SpA

be the trace of a square matrix A; and let vec : Mn,m → Knm be the mapping that “expands”

each matrix H = {hij}, i = 1, . . . , n, j = 1, . . . ,m, over rows into the column vector vec H =col(h11, . . . , h1m, . . . , hn1, . . . , hnm) ∈ K

nm. One can readily see that, for arbitrary L ∈ Mm,n, A ∈Mn,k, and N ∈ Mk,l, the relation D = LAN is equivalent to the relation vec D = (L ⊗ NT) vec A.Here ⊗ is the tensor (Kronecker) product of matrices [29, p. 235]. Note also that the relationSp(A∗B) = (vec A)∗(vec B) holds for matrices A,B ∈ Mn,m.

Consider the linear time-varying control system

x = A(t)x + B(t)u, y = C∗(t)x, t ∈ R, (x, u, y) ∈ Kn × K

m × Kk, (5)

where A(·), B(·), and C(·) are bounded piecewise continuous matrix functions on R. By X(t, s) wedenote the Cauchy matrix of the corresponding homogeneous system x = A(t)x. Let the controlin system (5) be constructed in the form u = U(t)y, where U : R → Mm,k is a bounded piecewisecontinuous function. Then system (5) becomes the closed-loop system

x = (A(t) + B(t)U(t)C∗(t))x. (6)

Consider also the bilinear time-varying control system

x = (A(t) + u1(t)A1(t) + u2(t)A2(t) + · · · + ur(t)Ar(t))x, t ∈ R, x ∈ Kn, (7)

where the matrix functions A(·) and Al(·) and the scalar functions ul(·), l = 1, . . . , r, are piecewisecontinuous and bounded on R. System (5) [respectively, system (7)] is said to be consistent on



the interval [t0, t1] if, for each matrix G ∈ Mn, there exists a piecewise continuous bounded controlU : [t0, t1] → Mm,k(K) (respectively, u = (u1, . . . , ur) : [t0, t1] → K

r) that brings the solution ofthe matrix system

Z = A(t)Z + B(t)U(t)C∗(t)X(t, t0)

[respectively, Z = A(t)Z +(u1(t)A1(t)+ · · ·+ ur(t)Ar(t))X(t, t0)] from the point Z(t0) = 0 into thepoint Z(t1) = G. The definition of consistency of system (5) was introduced in [30] as a general-ization of the notion of complete controllability to systems with incomplete feedback: if C(t) ≡ I,then these properties are equivalent [30]. This property was used in [30–35] to obtain results on thelocal controllability and the stability of Lyapunov exponents of the closed-loop system (6). In [36],this property was generalized to bilinear systems (7), and the results on the local controllability ofLyapunov exponents were transferred to system (7).

The consistency property was introduced in [30, 36] for K = R, and the asterisk was understoodas transposition in these papers. We extend this definition to the case of K = C, and here theasterisk stands for Hermitian conjugation.

In the present paper, we analyze the consistency property of time-invariant systems and establisha relationship of the consistency property with the pole assignment problem for time-invariantsystems of the form (6), (7). Note that if system (7) has the form (6), i.e., if r = mk andAl(t) = bi(t)c∗j (t), l = 1, . . . , r, i = 1, . . . ,m, j = 1, . . . , k, where the bi(t) are the columns of thematrix B(t) and the c∗j(t) are the rows of the matrix C∗(t), then the definition of consistency and allassertions on the consistency property for the system Ω(t) coincide with those for the system Σ(t).[Here we identify both system (5) and system (6) with the matrix Σ(t).]

3. MAIN RESULTS

Suppose that the coefficients of the systems Σ and Ω have the following form: A is in Hessenbergform; i.e.,

A = {aij}ni,j=1, ai,i+1 = 0, i = 1, . . . , n − 1, aij = 0, j > i + 1; (8)

B = {bij}, C = {cis}, i = 1, . . . , n, j = 1, . . . ,m, s = 1, . . . , k,

bij = 0, i = 1, . . . , p − 1, j = 1, . . . ,m,

cis = 0, i = p + 1, . . . , n, s = 1, . . . , k, p ∈ {1, . . . , n};(9)

Al = {alij}n

i,j=1, alij = 0, i = 1, . . . , p − 1, j = 1, . . . , n,

alij = 0, i = 1, . . . , n, j = p + 1, . . . , n, l = 1, . . . , r, p ∈ {1, . . . , n}.

(10)

Theorem 1. Let the coefficients of the system Σ have the form (8), (9). Then the implications1 =⇒ 2 ⇐⇒ 3 hold for the following conditions.

1. Σ is consistent.2. The matrices C∗B, C∗AB, . . . , C∗An−1B are linearly independent.3. Σ is arbitrarily pole assignable.

Theorem 2. Let the coefficients of the system Ω have the form (8), (10). Then the implications1 =⇒ 2 ⇐⇒ 3 hold for the following conditions.

1. Ω is consistent.2. The rank of the matrix Q = {Sp(AjA

i−1)}n,ri,j=1 is equal to n.

3. Ω is arbitrarily pole assignable.

Remark 1. Theorems 1 and 2 are analogs of Assertion 1 for the systems Σ and Ω. The prop-erty of complete controllability of the system (A,B) is replaced by the consistency property of thesystem Σ (respectively, the system Ω); the rank condition 2 in Assertion 1 is replaced by the corre-sponding rank conditions 2 for the systems Σ and Ω. Theorem 1 was announced in [37], where the


124 ZAITSEV

implication 2 =⇒ 1 was claimed as well. The last claim is not true in the general case. The questionof when it is true is considered below.

The implications 2 ⇐⇒ 3 in Theorems 1 and 2 were proved in [26, 28]. The proofs are giventhere for the case in which K = R. They remain valid for K = C. Let us prove the implication1 =⇒ 2 in Theorems 1 and 2. To this end, we start from analyzing the consistency property.

4. CONSISTENT SYSTEMS

First, let us present some properties of consistent systems.

Proposition 1 [30]. If system (5) is consistent on [t0, t1], then it is completely controllable andcompletely observable on [t0, t1].

Remark 2. The converse is not true in general. A time-varying counterexample for n = 1 isgiven in [30]. A time-invariant counterexample is given by system (5) with n = 2, m = k = 1,A = J ∈ M2, B = e2 ∈ K

2, and C = e1 ∈ K2. The inconsistency of this system follows from

Proposition 5 (see below). (However, the converse is true in the time-invariant case for n = 1.)For systems (5) and (7), we construct the so-called “large system” [34, 36]

z = F (t)z + G(t)v, t ∈ R, z ∈ Kn2

,

F (t) = A(t) ⊗ I − I ⊗ AT(t) ∈ Mn2 , I ∈ Mn,(11)

G(t) = B(t) ⊗ C(t) ∈ Mn2,mk, v ∈ Kmk, (12)

G(t) = [vec A1(t), . . . , vec Ar(t)] ∈ Mn2,r, v ∈ Kr. (13)

Here system (11), (12) corresponds to system (5), and system (11), (13) corresponds to system (7).

Proposition 2. System (5) [respectively , system (7)] is consistent on [t0, t1] if and only if thelarge system (11), (12) [respectively , (11), (13)] is completely controllable on [t0, t1].

In particular, it follows from Proposition 2 that if system (5) [or (7)] is time-invariant, thenthe consistency of this system on some interval implies its consistency on an arbitrary interval.Therefore, for time-invariant systems, it is unnecessary to indicate the interval on which the systemis consistent.

Remark 3. Proposition 2 for system (5) with K = R was proved in [34], where the matrix G(t)occurring in formula (12) was given in the form B(t)⊗C(t). To generalize Proposition 2 to the caseof K = C, one should take the matrix G(t) in the form B(t) ⊗ C(t), because, in the definition ofconsistency of system (5) for K = C, the expression B(t)U(t)C∗(t) contains the Hermitian adjointC∗(t) of the matrix C(t). Then the proof in [34] of Proposition 2 for system (5) remains valid forK = C. Proposition 2 for system (7) with K = R was proved in [36], and the proof remains validfor K = C. Note that the consistency of the system Σ(t) = (A(t), B(t), C(t)) does not imply theconsistency of the system Σ1(t) = (A(t), B(t), C(t)) (see Example 1 below).

Proposition 3. System (5) is not consistent on [t0, t1] if and only if there exists a nonzeromatrix H ∈ Mn such that

C∗(t)X(t, t0)HX(t0, t)B(t) ≡ 0, t ∈ [t0, t1]. (14)

Proof. Set B(t) = X(t0, t)B(t) ∈ Mn,m and C(t) = X∗(t, t0)C(t) ∈ Mn,k, t ∈ [t0, t1]. Let usconstruct the controllability matrix of the large system (11), (12) on the interval [t0, t1],

Γ =

t1∫

t0

Z(t0, s)G(s)G∗(s)Z∗(t0, s) ds ∈ Mn2 .



Here Z(t, s) is the Cauchy matrix of the system z = F (t)z. The matrix Γ coincides with the con-sistency matrix [30, 34] of system (5) on the interval [t0, t1]. The positive definiteness of the matrixΓ is equivalent to the consistency of system (5) on [t0, t1]. By the definition of the matrix F (t)in (11), the Cauchy matrix has the form Z(t, s) = X(t, s) ⊗ XT(s, t). Therefore,

Z(t0, s)G(s) = (X(t0, s) ⊗ XT(s, t0))(B(s) ⊗ C(s))

= (X(t0, s)B(s)) ⊗ (XT(s, t0)C(s)) = B(s) ⊗ C(s).

System (5) is not consistent on [t0, t1] if and only if system (11), (12) is not completely controllableon [t0, t1], i.e., if and only if there exists a nonzero vector h ∈ K

n2such that h∗Z(t0, s)G(s) = 0,

s ∈ [t0, t1]. Take the Hermitian conjugate of this relation; then we obtain (B(s) ⊗ C(s))∗h = 0,s ∈ [t0, t1]. Let H∗ ∈ Mn be the preimage of the vector h under the mapping vec; i.e., vec H∗ = h.Then H = 0 and 0 = (B∗(s)⊗ CT(s))(vec H∗) = vec(B∗(s)H∗C(s)), s ∈ [t0, t1], which is equivalentto 0 = C∗(s)HB(s) = C∗(s)X(s, t0)HX(t0, s)B(s), s ∈ [t0, t1].

Proposition 3 and the equivalence of relation (14) to the relation

B∗(t)X∗(t0, t)H∗X∗(t, t0)C(t) ≡ 0, t ∈ [t0, t1],

imply the following assertion.

Proposition 4. System (5) is consistent on [t0, t1] if and only if the dual system

x = −A∗(t)x + C(t)u, y = B∗(t)x, t ∈ R, (x, u, y) ∈ Kn × K

k × Km,

is consistent on [t0, t1].

Proposition 5. If C∗(t)B(t) ≡ 0, t ∈ [t0, t1], then system (5) is not consistent on [t0, t1].

This follows from Proposition 3 if we set H = I.

Proposition 6. If either system (5) is completely controllable on [t0, t1] and detC(t)C∗(t) ≥α > 0, t ∈ [t0, t1], or system (5) is completely observable on [t0, t1] and det B(t)B∗(t) ≥ α > 0,t ∈ [t0, t1], then system (5) is consistent on [t0, t1].

Proof. Let us prove the first part of the proposition. (The second part will follow from theduality in Proposition 4.) Set C2(t) = C∗(t)C1(t), where C1(t) = (C(t)C∗(t))−1 ∈ Mn, t ∈ [t0, t1].Then C(t)C2(t) = I. Assume that system (5) is not consistent on [t0, t1]. Then there existsa nonzero matrix H ∈ Mn such that C∗(t)X(t, t0)HX(t0, t)B(t) ≡ 0, t ∈ [t0, t1]. Therefore,

HX(t0, t)B(t) = X(t0, t)C∗2 (t)C∗(t)X(t, t0)HX(t0, t)B(t) ≡ 0, t ∈ [t0, t1].

Since H is nonzero, it follows that there exists a nonzero row ξ ∈ Kn∗ of matrix H such that

ξX(t0, t)B(t) ≡ 0, t ∈ [t0, t1]. Consequently, the rows of the matrix X(t0, t)B(t) are linearlydependent on [t0, t1]. It follows that system (5) is not completely controllable.

For system (7), an assertion similar to Proposition 3 can be stated as follows.

Proposition 7. System (7) is not consistent on [t0, t1] if and only if there exists a nonzeromatrix H ∈ Mn such that the identity

Sp(HX(t0, t)Al(t)X(t, t0)) ≡ 0, t ∈ [t0, t1], (15)

holds for all l = 1, . . . , r.


126 ZAITSEV

Proof. The following assertions are equivalent: system (7) is not consistent on [t0, t1]; thereexists a nonzero vector h ∈ R

n2such that

h∗Z(t0, s)[vec A1(s), . . . , vec Ar(s)] = 0, s ∈ [t0, t1];

h∗(X(t0, s) ⊗ XT(s, t0)) vec Al(s) = 0, s ∈ [t0, t1], l = 1, . . . , r;h∗ vec(X(t0, s)Al(s)X(s, t0)) = 0, s ∈ [t0, t1], l = 1, . . . , r.

The last equation is equivalent to identity (15) for the matrix H = (vec−1 h)∗ ∈ Mn.If we set H = I, then from Proposition 7, we obtain the following assertion.

Proposition 8. If SpAl(t) ≡ 0, t ∈ [t0, t1], for all l = 1, . . . , r, then system (7) is not consistenton [t0, t1].

5. CONSISTENCY OF TIME-INVARIANT SYSTEMS

Let us establish some necessary and sufficient conditions for the consistency of time-invariantsystems. Consider time-invariant systems Σ and Ω. Without loss of generality, we assume thatm = rankB and k = rankC in the system Σ and that all matrices Al, l = 1, . . . , r, in the system Ωare linearly independent. By [P,Q] := PQ− QP we denote the commutator of matrices P and Q,and by 〈a1, . . . , an〉 we denote the linear span of elements a1, . . . , an of a linear space.

Assertion 2. The following conditions are equivalent.(a) The system Σ = (A,B,C) is not consistent.(b) There exists a nonzero matrix H ∈ Mn such that

C∗eA(t−t0)He−A(t−t0)B ≡ 0, t ∈ [t0, t1], (16)

for any interval [t0, t1] ⊂ R.(c) There exists a nonzero matrix H ∈ Mn such that the relations C∗NνB = 0, where N0 = H

and Nν = [A,Nν−1], ν = 1, . . . , n2 − 1, hold for all ν = 0, . . . , n2 − 1.

Proof. The equivalence (a) ⇐⇒ (b) follows from Proposition 3 applied to a time-invariantsystem on the interval [t0, t1]. Let us prove the implication (b) =⇒ (c). By setting t = t0 inidentity (16), we obtain C∗HB = 0. By differentiating identity (16) n2 − 1 times at the pointt = t0, we obtain C∗NνB = 0, ν = 1, . . . , n2 − 1, i.e., assertion (c).

Let us prove the implication (c) =⇒ (b). Assume that there exists a nonzero matrix H ∈ Mn

such that C∗NνB = 0, ν = 0, . . . , n2 − 1. If the matrices N0, . . . , Nn2−1 are linearly independent,then Nν ∈ 〈N0, . . . , Nn2−1〉 = Mn for each ν ≥ n2. If there exists an s ∈ {1, . . . , n2−1} such that thematrices N0, . . . , Ns−1 are linearly independent and Ns ∈ N := 〈N0, . . . , Ns−1〉, then Nν ∈ N for allν ≥ s. Let us prove this by induction. We have Ns ∈ N. Let Nq ∈ N, q ≥ s; i.e., Nq =

∑s−1

i=0 ciNi.Then

Nq+1 = [A,Nq] =

[

A,

s−1∑

i=0

ciNi

]

=s−1∑

i=0

ci[A,Ni] =s−1∑

i=0

ciNi+1 =s∑

i=1

ci−1Ni

=s−1∑

i=1

ci−1Ni + cs−1Ns ∈ N,

because Ns ∈ N and Ni ∈ N for all i = 1, . . . , s − 1. It follows that C∗NνB = 0 for all ν = 0, 1, . . .and

0 ≡ C∗HB + C∗(AH − HA)B(t − t0) + C∗(A2H − 2AHA + HA2)B · 12!

(t − t0)2 + · · ·

= C∗(

I + A(t − t0) +12!

A2(t − t0)2 + · · ·)

H

(

I − A(t − t0) +12!

A2(t − t0)2 − · · ·)

B

= C∗eA(t−t0)He−A(t−t0)B.

The proof of the assertion is complete.



A similar assertion holds for the system Ω. On the basis of the system Ω, we construct thematrices Ll

0 = Al, Llν = [A,Ll

ν−1], l = 1, . . . , r, ν = 1, . . . , n2 − 1.

Assertion 3. The following conditions are equivalent.(a) The system Ω = (A,A1, . . . , Ar) is not consistent.(b) There exists a nonzero matrix H ∈ Mn such that

Sp(He−A(t−t0)AleA(t−t0)) ≡ 0, t ∈ [t0, t1], l = 1, . . . , r,

for any interval [t0, t1] ⊂ R.(c) There exists a nonzero matrix H ∈ Mn such that the relations Sp(HLl

ν) = 0 hold for alll = 1, . . . , r and ν = 0, . . . , n2 − 1.

The equivalence (a) ⇐⇒ (b) follows from Proposition 7. The equivalence (a) ⇐⇒ (c) isequivalent to the following assertion proved in [36] : the system Ω is consistent if and only if〈Ll

ν , l = 1, . . . , r, ν = 0, . . . , n2 − 1〉 = Mn.Note that condition (c) in Assertion 3 is equivalent to the following condition: there exists

a nonzero matrix H ∈ Mn such that the relations Sp(AlNν) = 0, where N0 = H and Nν = [A,Nν−1],hold for all l = 1, . . . , r and ν = 0, . . . , n2 − 1. Indeed, by the definition of the matrices Ll

ν , we have

Llν =

ν∑

i=0

(−1)i(ν

i

)Aν−iAlA

i;

here the(

νi

)are binomial coefficients. Therefore,

Sp(HLlν) = Sp

(ν∑

i=0

(−1)i(ν

i

)HAν−iAlA

i

)

= Sp

(ν∑

i=0

(−1)i(ν

i

)AlA

iHAν−i

)

= |i �→ ν − i| = Sp

(

Al

ν∑

i=0

(−1)ν−i(ν

i

)Aν−iHAi

)

= (−1)ν Sp(AlNν).

Example 1. Consider the time-invariant system Σ : n = 4, m = k = 2, A = [e2, e3, e4, ae3],B = [e1, e2], C = [e2 + ae4, e3], ej ∈ K

4, and a ∈ C\R is an arbitrary number. Let us showthat the system Σ = (A,B,C) is consistent and the system Σ1 = (A,B,C) is not. For thesystem Σ, we construct the large system (11), (12), where F ∈ M16 and G ∈ M16,4. By evaluatingd = det[G,FG,F 2G,F 3G], we obtain d = 144( Im a)4 = 0. Therefore, rank[G,FG, . . . , F 15G] = 16;consequently, the large system is completely controllable; therefore, the system Σ is consistent. Letus show that the system Σ1 is not consistent. We construct the matrix H = [e4 − ae2, 0, 0, 0] ∈ M4.Then we obtain C∗HB = CTHB = 0 ∈ M2. Next, we have AH − HA = 0; i.e., Nν = 0, ν ∈ N.Consequently, condition (c) of Assertion 2 holds; therefore, the system Σ1 is not consistent.

Corollary 1. The following assertions are equivalent.(a) The system Σ = (A,B,C) is consistent.(b) Identity (16) is possible only for H = 0 ∈ Mn.(c) The relations C∗NνB = 0, ν = 0, . . . , n2−1, where N0 = H and Nν = [A,Nν−1], are possible

only if H = 0.

Remark 4. The relations C∗NνB = 0 are equivalent to the relations C∗NνB = 0, where N0 = H

and Nν = [Nν−1, A] = [−A, Nν−1]. Therefore, the consistency of the system Σ = (A,B,C) is equiv-alent to the consistency of the system Σ1 = (−A,B,C). Likewise, the system Ω = (A,A1, . . . , Ar)is consistent if and only if so is the system Ω1 = (−A,A1, . . . , Ar).


128 ZAITSEV

Corollary 2. If the system Σ = (A,B,C) is consistent , then the system Σ1 = (A,BT,CR) isconsistent as well for arbitrary nonsingular matrices T ∈ Mm and R ∈ Mk.

Proof. If the system Σ1 is not consistent, then, by Assertion 2, there exists a nonzero matrixH ∈ Mn such that

R∗C∗eA(t−t0)He−A(t−t0)BT ≡ 0, t ∈ [t0, t1].

By multiplying this relation by the matrix (R∗)−1 on the left and by the matrix T−1 on the right,we obtain relation (16); consequently, the system Σ = (A,B,C) is not consistent.

Corollary 3. If the system Σ = (A,B,C) [respectively , the system Ω = (A,A1, . . . , Ar)] isconsistent , then for each nonsingular matrix S ∈ Mn, the system Σ := SΣS−1 := (A, B, C), whereA = SAS−1, B = SB, and C∗ = C∗S−1 [respectively , the system Ω := SΩS−1 := (A, A1, . . . , Ar),where A = SAS−1 and Al = SAlS

−1, l = 1, . . . , r] is consistent as well.

This corollary follows from Assertions 2 and 3, the corresponding relations

C∗eA(t−t0)He−A(t−t0)B = C∗eA(t−t0)He−A(t−t0)B, t ∈ [t0, t1],

Sp(He−A(t−t0)AleA(t−t0)) = Sp(He−A(t−t0)Ale

A(t−t0)), t ∈ [t0, t1], l = 1, . . . , r,

for the systems Σ and Ω, where H = SHS−1, and the fact that the matrices H and H aresimultaneously zero or not.

Remark 5. Corollaries 2 and 3 are analogs of the following assertions for completely controllablesystems: (1) if the pair (A,B) is completely controllable, then the pair (A,BT ) is completelycontrollable for any nonsingular matrix T ∈ Mm; (2) if the pair (A,B) is completely controllable,then the pair (SAS−1, SB) is completely controllable for any nonsingular matrix S ∈ Mn.

Propositions 1 and 6 imply the following assertion.

Corollary 4. For rankC = n, the system Σ = (A,B,C) is consistent if and only if system (1)is completely controllable. For rankB = n, the system Σ = (A,B,C) is consistent if and only ifsystem (1), (2) is completely observable.

Therefore, if one of the matrices B and C has rank n, then the property of consistency isobvious. In the general case, to verify the consistency of the system Σ (respectively, the system Ω),one should construct the large system z = Fz + Gv and verify the condition

rank[G,FG, . . . , F n2−1G] = n2.

This condition is necessary and sufficient for consistency, but it is difficult to use this conditionbecause of high dimensions. In what follows, we obtain some consistency conditions in terms ofsmall dimensions of the order of n.

Assertion 4. Let i1(λ), . . . , is(λ) be nontrivial invariant polynomials1 of a matrix A of degreesn1 ≥ · · · ≥ ns > 0, respectively (n1 + · · · + ns = n). If the system Σ (respectively , the system Ω) isconsistent , then mk ≥ n0 := n1 + 3n2 + · · · + (2s − 1)ns (respectively , r ≥ n0).

Proof. By [38, p. 192], the set of matrices commuting with the matrix A is a linear subspaceM ⊂ Mn of dimension n0. Take a basis P1, . . . , Pn0 in M. Set H := α1P1 + · · ·+ αn0Pn0 , where theαi ∈ R are found from the condition

n0∑

i=1

αiC∗PiB = 0. (17)

1 See [38, p. 134].



Relation (17) is a system of mk equations with n0 unknown coefficients αi. If mk < n0, thensystem (17) always has a nontrivial solution α = (α1, . . . , αn0). In this case, we have H = 0 andC∗HB = 0, and since H commutes with A, it follows that Nν = 0 for all ν ∈ N. Consequently,C∗NνB = 0, ν ∈ N. By Assertion 2, the system Σ is not consistent.

For the system Ω, instead of conditions (17) for the coefficients αi we take the conditions∑n0

i=1 αi Sp(AlPi) = 0, l = 1, . . . , r. This system has a nontrivial solution α = (α1, . . . , αn0) ifr < n0. Set H =

∑n0

i=1 αiPi. Then H = 0, H commutes with A, Nν = 0 for all ν ∈ N, Sp(AlH) = 0and Sp(AlNν) = 0, l = 1, . . . , r, ν ∈ N. By Assertion 3, the system Ω is not consistent.

Assertion 4 and the inequality n0 ≥ n imply the following fact.

Corollary 5. If mk < n (respectively , r < n), then the system Σ (respectively , the system Ω)is not consistent.

Therefore, the condition mk ≥ n (respectively, r ≥ n) is necessary for the consistency of a time-invariant system Σ (respectively, Ω), unlike time-varying systems, which can be consistent for anyn and for m = k = 1 (respectively, r = 1).

We say that A is a cyclic matrix [5, p. 69] if the geometric multiplicity of each eigenvalue isequal to unity. This is equivalent to the coincidence of the characteristic polynomial χ(A;λ) of thematrix A with the minimal polynomial ψ(A;λ) of A and to the linear independence of the matricesI,A, . . . , An−1. We have the following necessary consistency condition for systems Σ and Ω withsuch a matrix A. For the system Ω = (A,A1, . . . , Ar), let us construct the matrix Q = {Qij},Qij = Sp(AjA

i−1), i = 1, . . . , n, j = 1, . . . , r.

Assertion 5. Let A be a cyclic matrix. Then the following assertions are true.1. If Σ is consistent , then the matrices

C∗B, C∗AB, . . . , C∗An−1B (18)

are linearly independent.2. If Ω = (A,A1, . . . , Ar) is consistent , then rankQ = n.

Proof. Suppose that the matrices (18) are linearly dependent. Then there exists a vectorc = (c1, . . . , cn) = 0 such that

c1C∗B + c2C

∗AB + · · · + cnC∗An−1B = 0. (19)

Take the matrix H := c1I + c2A + · · · + cnAn−1. Then H = 0, because I,A, . . . , An−1 are linearlyindependent matrices, and C∗HB = 0. Further, the matrix H commutes with A; therefore, Nν = 0,ν ∈ N, and consequently, C∗NνB = 0 for any ν = 0, 1, . . . By Assertion 2, the system Σ is notconsistent.

Let us prove the assertion for the system Ω. Suppose that rankQ < n. Then there existsa vector c = (c1, . . . , cn) = 0 such that

c1 Sp(Al) + c2 Sp(AlA) + · · · + cn Sp(AlAn−1) = 0, l = 1, . . . , r.

Set H := c1I +c2A+ · · ·+cnAn−1. Then H = 0 and Sp(AlH) = 0, l = 1, . . . , r. Next, the matrix Hcommutes with A; therefore, Nν = 0 and Sp(AlNν) = 0, l = 1, . . . , r, ν = 0, 1, . . . By Assertion 3,the system Ω is not consistent.

Remark 6. In the general case, we have the following assertion. Let deg ψ(A;λ) = l. If thesystem Σ (respectively, the system Ω) is consistent, then the matrices C∗B,C∗AB, . . . , C∗Al−1B arelinearly independent (respectively, rankQ = l). This can be proved in the same way as Assertion 5.

Remark 7. The converse of Assertion 5 for the system Σ is not true (for n > 2) in general(it is true for the system Σ if n = 2); i.e., the fact that the matrices (18) are linearly independentdoes not necessarily imply the consistency of the system Σ, although it naturally follows that A is


130 ZAITSEV

a cyclic matrix. Indeed, if the matrix A is not cyclic, then the matrices I,A, . . . , An−1 are linearlydependent; i.e., there exists a vector c = (c1, . . . , cn) = 0 such that c1I+c2A+· · ·+cnAn−1 = 0 ∈ Mn.By multiplying this relation by the matrix C∗ on the left and by B on the right, we obtain (19),which contradicts the linear independence of the matrices (18). Now let us present an example ofa system for which the matrices (18) are linearly independent but the system Σ is not consistent.

Example 2. Let n = 3, m = k = 2, and

A =

∥∥∥∥∥∥∥

0 1 0

0 0 1

6 −11 6

∥∥∥∥∥∥∥

, B =

∥∥∥∥∥∥∥

1 0

0 1

1 − r r

∥∥∥∥∥∥∥

, C =

∥∥∥∥∥∥∥

0 1

−3 0

1 0

∥∥∥∥∥∥∥

.

Here r ∈ K is an arbitrary number such that r = 4 and r = 3. We construct the matrices (18),

C∗B =

∥∥∥∥∥

1 − r −3 + r

1 0

∥∥∥∥∥

, C∗AB =

∥∥∥∥∥

9 − 3r −11 + 3r

0 1

∥∥∥∥∥

,

C∗A2B =

∥∥∥∥∥

25 − 7r −27 + 7r

1 − r r

∥∥∥∥∥

,

(20)

and the matrix V = [vec(C∗B), vec(C∗AB), vec(C∗A2B)] ∈ M4,3. Let us find third-order minorsof the matrix V . The greatest common divisor of these minors is (r − 3)(r − 4). Consequently,rankV = 3, and therefore, the matrices (20) are linearly dependent. Let us show that the systemin question is not consistent. Consider the matrix

H =

∥∥∥∥∥∥∥

1 − r r −1

−7 + r 12 − 2r −5 + r

−23 + 5r 36 − 8r −13 + 3r

∥∥∥∥∥∥∥

.

Then C∗HB = 0 ∈ M2. Next, for the matrix N1 = AH − HA, we have the relation N1 = −H;consequently, C∗N1B = 0 ∈ M2. Likewise, for all matrices Nν = [A,Nν−1], ν ∈ N, we obtainNν = (−1)νH and hence C∗NνB = 0 ∈ M2. By Assertion 2, the original system is not consistent.

Remark 8. Note that, however, if rankC = n (i.e., consistency coincides with completecontrollability), then the converse of Assertion 5 holds for the system Σ; namely, the matricesB,AB, . . . , An−1B are linearly independent if and only if A is a cyclic matrix and the system(A,B) is completely controllable.

Let us obtain one more necessary condition for the consistency of the system Σ, which is equiv-alent to the linear independence of the matrices (18). Consider an arbitrary interval [t0, t1] ⊂ R

and the matrix

Φ(t) = {ϕij(t)} := C∗eA(t−t0)B, i = 1, . . . , k, j = 1, . . . ,m, t ∈ [t0, t1].

Assertion 6. Let l ∈ {1, . . . , n}. Then the following conditions are equivalent.(a) Of the matrices C∗B,C∗AB, . . . , C∗An−1B, there are l linearly independent.(b) Of the functions ϕij(t), i = 1, . . . , k, j = 1, . . . ,m, there are l functions linearly independent

on the interval [t0, t1].

Proof. Let us verify the implication (a) =⇒ (b). Let condition (a) be satisfied. Considerthe vector function φ(t) = col(φ1(t), . . . , φkm(t)) := vec Φ(t) ∈ K

km. We find the derivatives ofthis vector function of order ≤ n − 1, expand the resulting vectors in rows, and form the matrixΨ(t) ∈ Mn,km from them,

Ψ(t) =

∥∥∥∥∥∥∥∥∥∥

φT(t)

φT(t)

. . .

(φ(n−1))T(t)

∥∥∥∥∥∥∥∥∥∥

=

∥∥∥∥∥∥∥∥∥∥

(vec(C∗eA(t−t0)B))T

(vec(C∗AeA(t−t0)B))T

. . .

(vec(C∗An−1eA(t−t0)B))T

∥∥∥∥∥∥∥∥∥∥

.



Suppose the contrary: every l components of the vector function φ(t) are linearly dependenton [t0, t1]. Take l arbitrary columns of the matrix Ψ(t) with indices i1 < · · · < il ∈ {1, . . . , km}and form the matrix Ψ[i1, . . . , il](t) ∈ Mn,l of these columns. By assumption, the functions φij

(t),j = 1, . . . , l, are linearly dependent on [t0, t1]; consequently, there exists a vector

c = col(c1, . . . , cl) = 0

such that∑l

j=1 cjφij(t) ≡ 0, t ∈ [t0, t1]. By differentiating this relation n − 1 times, we ob-

tain∑l

j=1 cjφ(ν−1)ij

(t) ≡ 0, t ∈ [t0, t1], ν = 1, . . . , n. Consequently, Ψ[i1, . . . , il](t)c ≡ 0 ∈ Kn,

t ∈ [t0, t1]. In particular, Ψ[i1, . . . , il](t0)c = 0; i.e., l columns of the matrix Ψ[i1, . . . , il](t0) arelinearly dependent. Therefore, l arbitrary columns of the matrix Ψ(t0) are linearly dependent; i.e.,

rankΨ(t0) = rank

∥∥∥∥∥∥∥∥∥∥

(vec(C∗B))T

(vec(C∗AB))T

. . .

(vec(C∗An−1B))T

∥∥∥∥∥∥∥∥∥∥

< l,

which contradicts condition (a).Let us prove the implication (b) =⇒ (a). Let condition (b) hold, where l ∈ {1, . . . , n}. We claim

that condition (a) is satisfied as well. Suppose that condition (a) fails. It follows that the dimensions of the linear span L := 〈C∗B, . . . , C∗An−1B〉 is less than l. Note that C∗AνB ∈ L for ν ≥ n bythe Cayley–Hamilton theorem. Note also that s ≥ 1, because if s = 0, then C∗Aν−1B = 0, ν ∈ N;this is equivalent to the identity Φ(t) ≡ 0, which contradicts condition (b). Take a basis D1, . . . ,Ds

in L. Then every matrix C∗Aν−1B, ν = 1, . . . , n, depends linearly on the matrices Di, i = 1, . . . , s.The matrix exp(A(t − t0)) can be represented in the form

exp(A(t − t0)) = α1(t)I + · · · + αn(t)An−1,

where the αi(t), i = 1, . . . , n, are some functions, and

Φ(t) = α1(t)C∗B + · · · + αn(t)C∗An−1B = α1(t)D1 + · · · + αs(t)Ds,

where αi(t), i = 1, . . . , s, are some linear combinations of the functions αi(t), i = 1, . . . , n. Conse-quently,

φ(t) = vec Φ(t) = α1(t) vec(D1) + · · · + αs(t) vec(Ds).

It follows that every function ϕij(t), i = 1, . . . , k, j = 1, . . . ,m, is a linear combination of thefunctions αi(t), i = 1, . . . , s; therefore, l > s arbitrary functions ϕij(t) are linearly dependenton [t0, t1]. The proof of the assertion is complete.

Remark 9. Condition (a) in Assertion 6 does not imply that the first l of the matrices C∗Aν−1B,ν = 1, . . . , n, are linearly independent. By way of example, consider the matrices A = J ∈ M6,B = [e4, e6] ∈ M6,2, and C = [e1, e3, e4] ∈ M6,3, l = 5. Here the matrices C∗Aν−1B, ν = 1, 2, 3, 4, 6,are linearly independent, and C∗A4B = 0.

Corollary 6. The matrices C∗B,C∗AB, . . . , C∗An−1B are linearly independent if and only if ,of the functions ϕij(t), i = 1, . . . , k, j = 1, . . . ,m, there are n functions linearly independent on theinterval [t0, t1].

Corollary 7. Let degψ(A;λ) = l. If the system Σ = (A,B,C) is consistent , then, for anyinterval [t0, t1] ⊂ R, of the entries of the matrix Φ(t) = C∗eA(t−t0)B, there are l functions linearlyindependent on [t0, t1].

Corollary 8. Let A be a cyclic matrix. If the system Σ = (A,B,C) is consistent , then, for anyinterval [t0, t1] ⊂ R, of the entries of the matrix Φ(t) = C∗eA(t−t0)B, there are n functions linearlyindependent on the interval [t0, t1].


132 ZAITSEV

Let us derive one more consistency criterion. Consider the system Σ (respectively, the system Ω)with matrix A = J .

Assertion 7. The system Σ with matrix A = J is consistent if and only if the matrices

C∗B, C∗JB, . . . , C∗Jn−1B (21)

are linearly independent.The system Ω with matrix A = J is consistent if and only if rankQ = n.

Proof. The necessity follows from Assertion 5, because J is a cyclic matrix. Let us prove thesufficiency. Let the matrices (21) be linearly independent. We claim that the system Σ is consistent.Suppose the contrary: the system Σ is not consistent. By Assertion 2, there exists a nonzero matrixH ∈ Mn such that the relations C∗NνB = 0 hold for all ν = 0, . . . , n2 − 1, where N0 = H andNν = [J,Nν−1]. Consider the matrix

N2n−1 =2n−1∑

i=0

(−1)i

(2n − 1

i

)

J2n−1−iHJ i.

It consists of 2n terms. The first n terms are zero for i = 0, . . . , n − 1, because J2n−1−i = 0.The second n terms are zero for i = n, . . . , 2n − 1 as well, because J i = 0. Therefore, N2n−1 = 0.Since N2n−1 = JN2n−2 − N2n−2J , it follows that the matrix N2n−2 commutes with J . The set ofmatrices commuting with J is the linear space 〈I, J, J2, . . . , Jn−1〉 of dimension n; consequently,N2n−2 =

∑n

i=1 ciJi−1. Since C∗N2n−2B = 0, we have

n∑

i=1

ciC∗J i−1B = 0.

However, the matrices (21) are linearly independent. This implies that ci = 0, i = 1, . . . , n,and hence N2n−2 = 0. By proceeding in a similar way, one can show that Nν = 0 for allν = 2n − 1, 2n − 2, . . . , 0. Consequently, H = N0 = 0. We have arrived at a contradiction.Therefore, the system Σ is consistent.

Let us prove the sufficiency for the system Ω. Let rankQ = n. It follows that, for the rowvector ξ = (d1, . . . , dn) ∈ K

n∗, the relation ξQ = 0 [which is equivalent to the system of equationsSp(Al(d1I + d2J + · · · + dnJn−1)) = 0, l = 1, . . . , r] implies that di = 0, i = 1, . . . , n. Supposethe contrary: the system Ω is not consistent. Then, by Assertion 3, there exists a nonzero matrixH ∈ Mn such that Sp(AlNν) = 0, l = 1, . . . , r, ν = 0, . . . , n2−1, where N0 = H and Nν = [J,Nν−1].The matrix N2n−1 is the zero matrix; consequently, the matrix N2n−2 commutes with J . Therefore,

N2n−2 =n∑

i=1

diJi−1.

We have Sp(AlN2n−2) = 0. Since rankQ = n, it follows that di = 0, i = 1, . . . , n; therefore,N2n−2 = 0. In a similar way, one can show that Nν = 0 for all ν = 2n − 1, . . . , 0. Consequently,H = N0 = 0. We have obtained a contradiction. Therefore, the system Ω is consistent. The proofof the assertion is complete.

6. CONCLUSIONS

The implication 1 =⇒ 2 in Theorems 1 and 2 follows from Assertion 5, because the matrixA occurring in the Hessenberg form (8) is cyclic. The proof of Theorems 1 and 2 is complete.The validity of the implications 2 =⇒ 1 in Theorems 1 and 2 is discussed in the following assertions.

Theorem 3. Let the coefficients of the system Σ have the form (8), (9). Then the implication2 =⇒ 1 in Theorem 1 holds if at least one of the following conditions is satisfied : (a) rankB = n;(b) rankC = n; (c) A = J ; (d) rankB + rankC ≥ n + 1; (e) n < 6.



Proof. The implication 2 =⇒ 1 follows from Corollary 4 and Remark 8 under condition (b), fromthe duality principle under condition (a), and from Assertion 7 under condition (c). The implication2 =⇒ 1 under condition (e) was proved in [39] by the exhausting of all possible values of n, m,k, and p. It was shown in [40] that the implication 2 =⇒ 1 is not necessarily true for n ≥ 6.For example, consider the matrices

A = J + enψ ∈ Mn, B = [en−2, en] ∈ Mn,2, C = [e1, e3, e4, . . . , en−2] ∈ Mn,n−3,

where J ∈ Mn, ψ = (0, . . . , 0,−d2, 2d) ∈ Kn∗, ei ∈ K

n, and d ∈ K is an arbitrary number such thatd = 0. It was shown that, for such a system, the matrices (18) are linearly independent for n ≥ 6;however, the system is not consistent by Assertion 2 : for the matrix H ∈ Mn one can take the blockmatrix H = ‖Hij‖, i, j = 1, 2, such that H11 = 0 ∈ M2,n−2, H21 = 0 ∈ Mn−2, H22 = 0 ∈ Mn−2,2,and H12 = [e1 − de2, e2] ∈ M2, e1, e2 ∈ K

2.The implication 2 =⇒ 1 under condition (d) is a consequence of the following assertion: if the

coefficients of the system Σ have the form (8), (9) and rankB + rankC ≥ n + 1, then the systemΣ is consistent (and hence the eigenvalue spectrum of the system Σ is arbitrarily assignable). Thisassertion was proved in [40]. The proof given there is very complicated. A simpler proof ofthis assertion will be given in the second part of the present paper. The proof of the theorem iscomplete.

Theorem 4. Let the coefficients of the system Ω have the form (8), (10). Then the implication2 =⇒ 1 in Theorem 2 holds if at least one of the following conditions is satisfied : (a) A = J ;(b) n < 3.

Proof. The implication 2 =⇒ 1 under condition (a) follows from Assertion 7. Let us provethis implication for n = 2 (the case n = 1 is obvious). By virtue of the assumption rankQ = n,we necessarily have r ≥ 2, and of the matrices Al, l = 1, . . . , r, there are n linearly independentmatrices. Let p = 2. Then, by (10), the matrices Al have the form e2c

∗l , e2 ∈ K

2, cl ∈ K2,

and of the vectors cl, l = 1, . . . , r, there are n linearly independent vectors. Then the systemΩ = (A,A1, . . . , Ar) has the form of the system Σ = (A,B,C), where B = e2 ∈ K

2 and C =[c1, . . . , cr] ∈ M2,r, and assumptions 2 and 1 of Theorem 2 coincide with assumptions 2 and 1 ofTheorem 1, respectively. Since rankC = n, it follows from Theorem 3 that the implication 2 =⇒ 1holds for the system Σ and hence for the system Ω. The case in which p = 1 can be consideredin a similar way. In this case, the system Ω has the form of the system Σ = (A,B,C), whereC = e1 ∈ K

2, B = [b1, . . . , br] ∈ M2,r, and rankB = n. The proof of the theorem is complete.For n ≥ 3, the implication 2 =⇒ 1 is not true for the system Ω in general. For example, let us

take the matrices

A = J + dene∗n ∈ Mn, r = n, Ai = en−1e∗n−i, i = 1, . . . , r − 1, Ar = ene∗n−1,

where J ∈ Mn, ei ∈ Kn, and d ∈ K is an arbitrary number such that d = 0. By constructing the

matrix Q for such a system, we obtain Q = [e1, e2, . . . , en−1, h] ∈ Mn, where

h = col(0, 1, d, d2 , . . . , dn−2) ∈ Kn.

Therefore, rankQ = n. However, the system is not consistent by Assertion 3, where the matrixH = dJn−1 is taken for the matrix H.

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Consistent systems and pole assignment: I