## Abstract

A constrained non-homogeneous linear eigenvalue problem is introduced. The application given to the problem is of finding the frequency and amplitudes of exciting forces that impose constraints on the configurations of vibratory modes. The scope of the problem is wider. It is shown that the problem may be transformed to a singular unsymmetric generalized eigenvalue problem. Depending on the given data the problem may have finite, infinite or empty spectrum. The solvability of the problem is analysed. Examples demonstrate the application and the various results.

## 1. Introduction

Consider the following problem:

## Problem 1.1.

*Given: ***A**∈Re^{n×n}; **B**, **C**∈Re^{n×m}; **d**∈Re^{m}, *m*<*n*

*Find: ***x**, λ, **μ**=(*μ*_{1}*μ*_{2} *… μ*_{m}*)*^{T}*; μ*_{m}*=1 satisfying*
1.1
*subject to*
1.2

Our interest is limited to the case where **A** is symmetric positive definite, and **B** and **C** are of full column rank.

The application demonstrated in the paper is of shaping vibratory modes of dynamic systems by external harmonic forces. Problem 1.1 determines the frequency and the amplitudes of the external forces that impose the required constraints. It includes the problem of nodal vibration placement as a special case (e.g. Prells *et al*. 2003).

In §2 a simple algorithm of solution is given. The problem is transformed to a singular generalized eigenvalue problem. The spectrum of this problem may be finite, empty or infinite, as demonstrated in Golub & van Loan (1983, p. 252). Therefore, solvability of the problem is not granted. Applications of problem 1.1 are explored in §§3 and 4. A more complex algorithm is developed in §5. General principles of solvability from physical point of view are discussed in §6. The solvability of the problem when *m*=1 is analysed in §7. It is shown that under certain conditions, important in engineering application, the given constraint induces additional internal constraints. Preliminary results concerning the solvability of problem 1.1 with multiple constraints are given in §8. Concluding remarks are made in §9.

## 2. Simple algorithm

A simple method for solving problem 1.1 is now given. Denote
2.1
where
2.2
Then problem 1.1 is transformed to
2.3
We may thus find the eigenvalues and eigenvectors of
2.4
and obtain a solution to problem 1.1 via eigenvector normalization
2.5
An eigenpair {*λ* **x**} is a *proper* eigenpair of problem 1.1 if and only if *z*_{n+m}≠0.

Owing to the constraint in equation (2.3) there is generally no trivial solution to problem 1.1 and the eigenvectors may not be generally scaled.

Equation (2.4) is an *unsymmetric generalized eigenvalue problem* with singular **M**. Depending on the given data, **K** may become singular as well. In this case the solvability of the problem is convoluted. For problems of large dimension with singular pencil the method for extracting the eigenvalues given in Van Dooren (1979) is numerically stable.

When a constraint is added to problem 1.1, the degree of the characteristic equation which determines the eigenvalues is either decreased or remains unchanged. At the same time, the dimension of **K** and **M** increases. This disadvantage is removed in §5 where an algorithm that keeps the dimension of the problem constant is advised.

The advantage of the formulation (2.3) is in its simplicity. The given data are displayed in their raw form when equation (2.3) is used to solve the problem. It is thus instructive to use this algorithm in the following sections where the application of the problem is introduced. This form is also useful when exploring analytically the properties of problem 1.1.

## 3. Shaping the steady-state response of vibratory systems

In this section four examples are introduced to demonstrate the application and issues related to the solvability of problem 1.1. Example 3.1 is a problem with no solution. Examples 3.2 and 3.3 demonstrate, respectively, problems with homogeneous and non-homogeneous constraints. The problem in example 3.4 includes two constraints, one homogeneous and one heterogeneous. It is shown that the degree of the characteristic equations in these problems varies from problem to problem. The analysis in §7 provides explanation to this phenomenon.

Consider the mass–spring system shown in figure 1. It consists of three masses and four springs, each of unit value. The system is excited by a harmonic force as shown. The time vector **u**(*t*)=(*u*_{1} *u*_{2} *u*_{3})^{T} indicates the displacement of the three masses from the equilibrium position at time *t*.

The equations of motion for the system are
3.1
or
3.2
with the obvious definition of **A** and **b**. Dots denote differentiations with respect to *t*.

The steady-state1 solution of equation (3.2) takes the form
3.3
where **x** is a constant vector. Substituting equation (3.3) into equation (3.2) gives
3.4

## Example 3.1.

We want to find the frequency of the harmonic force which renders *x*_{1}=0. With this frequency of excitation the steady-state motion of the left mass vanishes.

The mathematical problem is thus: given **A**, **b**, as in equations (3.1) and (3.2), **c**=(1 0 0)^{T} and *d*=0, find λ such that equation (3.4) is satisfied subject to the constraint
3.5
With these data equation (2.4) gives
3.6
The eigenvalue problem (3.6) has only one eigenvalue, λ=3. Its corresponding eigenvector **z**=(0 −1 1 0)^{T} shows that there is no proper **x**. The problem has no solution.

At a glance it appears odd that one constraint reduced the degree of the characteristic polynomial by two. It is shown in §7 that when *d*=**c**^{T}**b**=0, as in this example, the given constraint creates an internal constraint, **c**^{T}**Ax**=0, that may reduce the degree of the characteristic polynomial as well.

## Example 3.2.

We want to find the frequency of the harmonic force that shapes the steady-state mode of motion such that *x*_{1}=*x*_{3}. For this case *d*=0, **c**=(1 0 −1)^{T}, so that
3.7
With these data equation (2.3) gives
3.8
The eigenvalue problem (3.8) has two solutions:
3.9
In this example the constraint reduced the degree of the characteristic polynomial by one.

## Example 3.3.

We want to find the frequency of the harmonic force such that at steady state *x*_{1}=2. Here **c**=(1 0 0)^{T}, *d*=2 so that
3.10
With these data equation (2.3) gives
3.11
The eigenvalue problem (3.11) has three finite eigenvalues. One of the eigenvectors is not proper. The physical solutions are
3.12
In this example the constraint did not affect the degree of the characteristic polynomial.

## Example 3.4.

Consider the mass–spring system shown in figure 2. The system is excited by two harmonic forces of the same frequency, . The amplitude of excitation of the force acting on the third mass is *μ*_{1}, and the first mass is excited by a unit harmonic force, as illustrated. We want to find λ and *μ*_{1} such that at steady state *x*_{2}=0 and *x*_{3}=1. For this case
3.13
With these data equation (2.3) gives
3.14
The eigenvalue problem (3.14) has one solution
3.15
In this example two constraints reduced the degree of the characteristic polynomial by two.

Example 3.4 demonstrates the physical meaning of all parameters in problem 1.1. The matrix **A** defines the system. The constraints are expressed by **C** and **D**. Each constraint is enforced by a force vector *μ*_{k}**b**_{k}, where **b**_{k} is the *k*th column of **B**. The enforcement of the constraints is achieved by finding the correct frequency and the force amplitudes *μ*_{k}, *k*=1,2,…,*m*−1.

The change in the degree of the characteristic polynomial of **K**−*λ***M**, exposed in these examples, is intriguing.

## 4. Stationary values of the Rayleigh quotient with homogeneous constraint

A second application of problem 1.1 is now given. Golub & van Loan (1983, pp. 431–432) studied the problem of finding the stationary values of the Rayleigh quotient:
4.1
subject to the constraint
4.2
They have shown that finding the singular value decomposition of **C**
4.3
and partitioning
4.4
the stationary values of *R*(**x**) are the eigenvalues of
4.5
This problem is a special case of problem 1.1.

The stationary values of *R*(**x**) under the constraint (4.2) are the solution of
4.6
subject to
4.7
The Lagrangian associated with this problem is
4.8
where **μ** is a vector of Lagrange multipliers. Differentiating equation (4.8) with respect to **x** gives
4.9
or
4.10
In equation (4.9) the notation
4.11
is used.

Since the constraints in equation (4.2) are homogeneous, the eigenvectors may be scaled arbitrarily. We may choose a normalization , define *λ*=*R*, and transform problem (4.6) and (4.7) to its equivalent
4.12
which is problem 1.1 with
4.13
It thus follows that the constrained stationary values of *R*(**x**) are the finite eigenvalues of
4.14
where |**Y**| is the determinant of **Y**. This result leads to the following application.

## Example 4.1.

The eigenvalue problem associated with the mass–spring system shown in figure 3*a* is
4.15
We want to find the eigenvalues of the system when it is constrained such that *x*_{1}=*x*_{3}. By equation (4.14) the required eigenvalues are the roots of
4.16
i.e.
4.17
By equation (4.5) these eigenvalues are the roots of
4.18
By physical considerations the constrained system behaves like the system shown in figure 3*b*. The eigenvalues of this system are the roots of
4.19
The roots of equations (4.18) and (4.19) are given in equation (4.17).

## 5. Complex algorithm

In this section an algorithm that keeps the dimension of the problem constant, independent of *m*, is developed. Consider first the case where **d**=**0**,
5.1
Let
5.2
be the singular value decompositions of **B** and **C**. Denote
5.3
Then **C**^{T}**x**=**0** implies that
5.4
With these definitions problem (5.1) is congruently equivalent to
5.5
or
5.6
with the obvious definition of and .

By partitioning
5.7
problem (5.6) may be written in the form
5.8
which is equivalent to
5.9
The solution to problem 1.1 with **d**=**0** may thus be obtained by solving the eigenvalue problem (5.9) and normalizing the eigenvector such that *μ*_{m}=1. With this algorithm an eigenvalue is not proper if the *m*th element of its corresponding eigenvector vanishes.

Suppose now that **d**≠**0**. Then **C**^{T}**x**=**d** implies that
5.10
and the system (5.8) is changed to
5.11
The system (5.11) is identical to
5.12
where
5.13
A solution to problem 1.1, with appropriate eigenvector normalization, is determined by a proper eigenpair of equation (5.12).

## Example 5.1.

We now solve the problem in example 3.4 by using equation (5.12).

The singular value decompositions of **B** and **C** are as follows:
5.14
and **d**=(0 1)^{T}. It thus follows that
5.15
5.16
5.17
Equation (5.12) gives
5.18
By inspection, the unique solution of equation (5.18) is
5.19
which leads to
5.20
the same solution as in equation (3.15).

## 6. Principles of solvability

It is clear from the physics of the problem that the solvability of problem 1.1 is affected by the following factors:

— inconsistency of the constraints

— redundancy in the control vectors

— lack of accessibility of the control vectors, and

— controllability.

Inconsistency of constraints includes, for example, requiring *x*_{1}=0, *x*_{2}=1 and *x*_{1}=*x*_{2}. A prerequisite for solvability is that the system of constraints **C**^{T}**x**=**d** is consistent. If **d**=**0** or if rank(**C**)=*m* then the system of constraints is necessarily consistent.

By redundancy of control vectors we mean that there are more essential constraints than essential control vectors. We note in passing that with the forces
6.1
it is possible to impose two constraints, one homogeneous and the other non-homogeneous. But with the forces
6.2
only one essential constraint could be imposed. Therefore, there is no direct correlation between the rank of **B** and the number of essential control vectors.

To illustrate the issue of accessibility of the control vectors consider the simply connected system shown in figure 4*a*. Suppose that by applying two harmonic forces, *f*_{1} and *f*_{2}, as shown, it is desired to achieve the constraints *x*_{4}=0 and *x*_{6}=1. If the constraint *x*_{4}=0 is achieved then mass *m*_{4} is stationary, as shown in figure 4*b*. Consequently, masses *m*_{5} and *m*_{6} are isolated from the control forces. The second constraint *x*_{6}=1 could not be imposed. If, however, there exists an additional spring bridging across the constraint *x*_{4}=0, as in figure 4*c*, the two constraints may be materialized simultaneously.

In the special case **B**=**C**, **d**=**0**, studied in §4, force accessibility is unrestricted since there is a control force at each required constraint.

By controllability we mean that certain symmetry in the system or in the control forces prevents solvability. For example, the system shown in figure 5 is said to be *per-symmetric*. It is symmetric about its mid-point. The control forces follow the same symmetry. The steady-state motion of this system must be symmetric about the mid-point. Therefore, constraints such as *x*_{1}=0, *x*_{3}=1, for example, cannot be achieved simultaneously.

In the special case where **B**=**C** and **d**=**0**, studied in §4, symmetry in the control forces implies similar symmetry in constraints. As a result the problem is solvable unconditionally.

## 7. Solvability of the problem with one constraint

In this section we address the issue of solvability of problem 1.1 in the case that there is a single constraint. When *m*=1 problem 1.1 reduces to

## Problem 7.1.

*Given:* **A***,***b**≠**0**,**c**≠**0***,d*

*Find:*λ,**x** *satisfying*
7.1
7.2

We will call the constraint (7.2) the *external* constraint. Define
7.3
Then the solution to problem 7.1 is given by
7.4
Let
7.5
be the characteristic polynomial of **K**−*λ***M** where *r* is the degree of *Π*_{r}. The problem is not solvable if

—

*r*=0 and*Π*_{0}≠0, so there are no eigenvalues, or—

*r*≠0, but all eigenvectors are improper.

To characterize the relation between *r* and the given data we write equation (7.5) explicitly as
7.6
and obtain:

## Lemma 7.2.

*If d*≠0 *then there are n eigenvalues to problem* 7.1.

## Proof.

Expanding equation (7.6) by its last column gives
7.7
where *D*_{ij} is the (*i*,*j*)-cofactor of the pencil **K**−*λ***M**, e.g. Korn & Korn (1961, p. 12). Since *D*_{n+1,n+1} is a polynomial in *λ* of degree *n* and all other cofactors in equation (7.7) are polynomials of lesser degrees it follows that when *d*≠0 the degree of *Π*_{r} is *r*=*n*. There are, therefore, *n* eigenvalues to the pencil **K**−*λ***M**. ▪

Note that in lemma 7.2, *n* denotes the total number of proper and improper eigenvalues. Example 3.3 demonstrates this case: *d*≠0, *r*=*n*=3, with two proper and one improper eigenpairs.

If *d*=0 then adding the first *n* terms in equation (7.7) the degree of *Π*_{r} may be reduced by cancellations of higher power terms of *λ*, so that *r* is generally in the range 0≤*r*≤*n*−1.

Suppose that
7.8
and that the given constraint is homogeneous
7.9
Then multiplying equation (7.1) by **c**^{T} gives
7.10
The *external* constraint has generated an *internal* constraint
7.11
Equation (7.1) may now be multiplied by **c**^{T}**A**
7.12
by virtue of equation (7.11). If **c**^{T}**Ab**=0 then a second internal constraint is generated
7.13
To reveille all possible other internal constraints the process described above may be repeated as follows. In the *k*th step equation (7.1) is multiplied by **c**^{T}**A**^{k}. The process is terminated when the condition **c**^{T}**A**^{p}**b**≠0, *p*≤*n*−2, is first met.

Assume that the process was repeated *n*−1 times and that the conditions **c**^{T}**A**^{k}**b**=0, *k*=0,1,…,*n*−2 were all satisfied. Then the external constraint has generated *n*−1 internal constraints. Denoting
7.14
where ** Θ** is the

*observability*matrix, we obtain 7.15

## Lemma 7.3.

*If* *ψ**=***0** *and* rank(*Θ**)=n then problem* 7.1 *is not solvable.*

## Proof.

With the stipulations of lemma 7.3 equation (7.15) has a *unique* solution, **x**=**0**. Equation (7.4) reduces in this case to −*y*_{n+1}(**b**^{T}*d*)^{T}=**0**, which gives *y*_{n+1}=0 since **b**≠**0**. There is no proper eigenvector in this case. ▪

## Remark 7.4.

In the theory of control an *n*+1 *output-controllability* vector is defined as follows (Ogata 1970):
7.16
It describes the ability of a control force to manipulate the output of the system in a desired manner. The system is said to be output-controllable if and only if .

Lemma 7.3 does not require the output-controllability vector to vanish since there is no stipulation on the last element **c**^{T}**A**^{n−1}**b**. Example 7.5 demonstrates that this difference is material.

## Example 7.5.

Consider the system shown in figure 6. Suppose that the objective of the control force is to make *x*_{1}=0. The proper eigenpairs of
7.17
are the solution of this problem. By physical considerations it is clear that the problem is not solvable. If *m*_{1} is stationary then the spring *κ*_{1} is inactive. By Newton’s second law if *κ*_{1} is inactive then so is *κ*_{2}. This implies that *m*_{2} is stationary and that the entire system is at rest. But with a non-vanishing force applied to *m*_{3} this scenario is physically unfeasible.

By inspection we see that the characteristic polynomial of equation (7.17) is 7.18 so the set of eigenvalues of equation (7.17) is empty.

The control matrices for this problem are as follows:
7.19
where ** Γ** is the

*controllability*matrix 7.20 It follows from equation (7.19) that in terms of control theory the system is

*controllable, observable, output-controllable*. Yet problem 7.1 is not solvable. This result is in line with lemma 7.3 since

**is invertible and**

*Θ***=**

*ψ***0**.

## Remark 7.6.

Lemma 7.3 requires that ** Θ** be invertible when

**=**

*ψ***0**to ensure that the problem is not solvable. This means that the internal constraints in

**=**

*Θ*x**0**are linearly independent. The following example demonstrates that when

**=**

*ψ***0**and

**is singular the problem may be solvable.**

*Θ*

## Example 7.7.

Consider the system
7.21
In this system ** ψ**=

**0**and

**is singular. The characteristic polynomial vanishes identically,**

*Θ**Π*

_{0}=0. The set of eigenvalues is infinite with proper eigenpairs 7.22 We now focus our attention on the case where

*d*≠0. We multiply equation (7.1) by

**c**

^{T}

**A**

^{k},

*k*=0,2,…,

*n*−2 and, together with

**c**

^{T}

**x**=

*d*, obtain the following system of

*n*equations with

*n*unknowns: 7.23 where 7.24

## Lemma 7.8.

*If d*≠0 *and* rank(** Θ**)=

*n then problem*7.1

*is solvable*.

## Proof.

We need to establish that: (a) the spectrum of (**K**−λ**M**)**y**=**0** is not empty, and (b) that *y*_{n+1}≠0.

By lemma 7.2 the spectrum of **K**−λ**M** has *n* eigenvalues, which establishes the first part of the proof.

By equation (7.23) ** β**(λ)≠

**0**and therefore equation (7.23) has a non-vanishing solution,

**x**≠

**0**, for each λ. Assume that

*y*

_{n+1}=0. Then the first

*n*equations of equation (7.4) imply that

**x**is an eigenvector of

**A**. The last equation in equation (7.4) implies that

**c**is orthogonal to an eigenvector of

**A**, i.e.

**x**. In such a case the observability matrix

**is singular, a contradiction to the stipulation that rank(**

*Θ***)=**

*Θ**n*. The assumption that

*y*

_{n+1}=0 is false. ▪

In example 3.3, *d*≠0 and ** Θ** is singular. As a result, out of the three eigenpairs only two are proper.

## Remark 7.9.

At glance the assertion in lemma 7.8 is puzzling. According to lemma 7.8 when *d*≠0 there are no conditions on **b** that prevent solvability. How can a control force do its function unconditionally?

To answer this question let us examine the extreme, **b**=**0**, where there is no control force at all. In this case problem 7.1 is reduced to

## Problem 7.10.

*Given:* **A**,**c**,*d*≠0

*Find:* **x**,λ *satisfying*
7.25
*subject to*
7.26

With **b**=**0** problem 7.1 reduces to the standard eigenvalue problem. The eigenvalue problem (7.25) has *n* eigenvalues including multiplicity. Since **A** is symmetric there are also *n* linearly independent eigenvectors that may be scaled arbitrarily. The role of the non-homogeneous constraint (7.26) is merely of fixing the scale factors for the eigenvectors.

With this simplicity one may wonder what the role of the observability matrix is. If the observability matrix is singular then **c** may vanish or be orthogonal to one of the eigenvectors of **A**. This creates a contradiction with equation (7.26).

Equations (7.23) and (7.24) are general. We may use them to further investigate the case where *d*=0. In this case equation (7.23) reduces to
7.27

## Lemma 7.11.

*If d*=0,**c**^{T}**b**≠0,rank(** Θ**)=

*n, and there exists an eigenvalue*λ

*to problem*7.1,

*then the corresponding eigenvector is proper.*

## Proof.

With the stipulations in lemma 7.11 for each eigenvalue λ the right-hand-side vector in equation (7.27) does not vanish. Consequently there is a non-vanishing solution, **x**≠**0**. It was established in the proof of lemma 7.8 that this implies that *y*_{n+1}≠0. ▪

We will now show that when *d*=0 and **c**^{T}**b**≠0 there exists an eigenvalue to problem 7.1.

Let
7.28
be the spectral decomposition of **A**, where *δ*_{k} is an eigenvalue of **A**. Then
7.29
Define
7.30
Then equation (7.29) takes the form
7.31
Evaluating *Π*_{r} by its last row gives
7.32
It thus follows that
7.33
where *Q*_{p}(λ) is some polynomial in λ of degree *p*<*n*−1.

## Lemma 7.12.

*If d*=0,**c**^{T}**b**≠0 *and* rank(*Θ**)=n then there exist proper eigenpairs to problem* 7.1.

## Proof.

By equation (7.30) and the stipulation that **c**^{T}**b**≠0 we have
7.34
It thus follows that the leading coefficient in equation (7.33) does not vanish. Hence the degree of *Π*_{r}(λ) is *r*=*n*−1. By lemma 7.11 the corresponding *n*−1 eigenvectors are proper. ▪

Example 3.2 satisfies all stipulations of lemma 7.12. It has two proper eigenpairs.

The assertion in lemma 7.12 suppresses that of lemma 7.11.

Let
7.35
be the characteristic polynomial of |**A**−*δ***I**|. Then it can be shown2 that
7.36

## Example 7.13.

Direct calculation shows that the characteristic polynomial in example 3.1 is *Π*_{1}=−λ+3.

With *Ξ*_{3}=−*δ*^{3}+7*δ*^{2}−13*δ*+3, **c**^{T}**b**=0, **c**^{T}**Ab**=−1 and **c**^{T}**A**^{2}**b**=−4, equation (7.36) gives *Π*_{1}=−λ+3.

Let *ϑ* be the index defined by
7.37
where *ϑ*=0 if *ψ*_{1}=*d*≠0, and *ϑ*=*n* if *ψ*_{k}=0, *k*=1,2,…,*n*. Then it follows that

## Lemma 7.14.

*The degree of Π*

_{r}

*is r=n−ϑ.*

## Proof.

If *d*≠0 then *ϑ*=0, and by lemma 7.2 *r*=*n*−*ϑ*=*n*.

When *d*=0 we may invoke equation (7.36). Noting that , *j*=0,1,…,*n*−2, it follows from equation (7.36) that the degree of *Π*_{r} is *r*=*n*−*ϑ*. ▪

Moreover,

## Lemma 7.15.

*(a) If* ** ψ**=

**0**

*and*rank(

*Θ**)=n then the spectrum of*

*Π*_{0}

*is empty. (b) If*

*ψ**=*

**0**

*and*rank(

*Θ**)<n then the spectrum of*

*Π*_{0}

*is infinite.*

## Proof.

If ** ψ**=

**0**then by lemma 7.14,

*Π*

_{0}=

*γ*,

*γ*some constant.

If ** Θ** is invertible then there is a

*unique*solution to equation (7.15),

**x**=

**0**. This implies that

*y*

_{n+1}=0, so that equation (7.4) has only a trivial solution

**y**=

**0**. The spectrum of

**K**−λ

**M**is empty,

*Π*

_{0}≠0.

If ** Θ** is singular then equation (7.15) has a family of non-vanishing solutions, i.e.

*Π*

_{0}=0. ▪

In example 7.5, ** ψ**=

**0**and

**is invertible. According to lemma 7.15(**

*Θ**a*) the spectrum is empty,

*Π*

_{0}=1. The problem is not solvable.

In example 7.7, ** ψ**=

**0**and

**is singular. According to lemma 7.15(**

*Θ**b*) the spectrum is infinite,

*Π*

_{0}=0. The problem has a family of solutions (7.22).

## 8. Toward solvability of problem 1.1 with multiple constraints

This section comments on the solvability of problem 1.1 when *m*>1 and presents some preliminary results.

When solving classical control problems of driving the dynamic of a system in a desired manner based on observation of the state, increasing the dimension *m* of the output matrix **C** increases the likelihood that the system becomes observable and that the required control is achievable. If the problem is observable with respect to one vector **c**_{k} then it is observable with respect to the entire matrix **C**.

In problem 1.1 the opposite behaviour holds. Increasing the dimension of **C** in problem 1.1 means that more constraints are added. When one constraint is unenforceable the entire problem is unsolvable. The likelihood that the problem is not solvable increases with *m*. A necessary condition for solvability is that the problem is solvable with respect to each individual constraint, **c**_{k}**x**=*d*_{k}, *k*=1,2,…,*m*. This condition is clearly not a sufficient condition for solvability. Other necessary condition is that the system of constraints **C**^{T}**x**=**d** is consistent. The requirement that **C** is of full column rank guarantees the consistency of the constraints.

We denote, 8.1 partition, 8.2 and obtain.

## Lemma 8.1.

*If for a particular k, k*=1,2,…,*m*,*Θ*_{k} *is invertible and* *ψ*_{kj}=**0** *for all j*=1,2,…,*m, then problem* 1.1 *is unsolvable.*

## Proof.

With the above stipulations, equation (7.15) applied to lemma 8.1 gives
8.3
where **x**_{k} is the solution of
8.4
which may be satisfied with *μ*_{m}=1 if and only if
8.5
But since **B** is of full column rank, **b**_{m} cannot be expressed as a linear combination of the vectors in **B**_{1}. The constraint cannot be imposed by the entire control vectors, **Bμ**. The problem is not solvable. ▪

Of particular interest is the case where 8.6 8.7 In this case, 8.8

We multiply equation (1.1) by , *p*=1,2,…,*n*−1, and obtain the following *n*−1 equations:
8.9
and
8.10
where
8.11
Combining equations (8.6), (8.8) and (8.10), we have
8.12
Denote
8.13

## Lemma 8.2.

*If* **d***=***0***,***C**^{T}**B***=***0***,**Ω*_{k} *is invertible and* *for k*=1,2,…,*m, then problem* 1.1 *is not solvable.*

## 9. Concluding remarks

The pole placement problem in the theory of control deals with manipulation of the system’s eigenvalues by control forces. The problem studied here deals with manipulation of the configuration of vibratory mode shapes by harmonic forces. The problem is intriguing when *d*=**c**^{T}**b**=0.

In nodal control (e.g. Prells *et al*. 2003) it is required that the mode shape of a vibrating system vanishes at a certain degree of freedom. The challenge is when the control applies to one part of the system and the required effect is in another part. This case necessarily leads to the condition **c**^{T}**b**=0, since **c** and **b** disjoint. Hence, apart from mathematical curiosity, the solvability of problem 7.1 when *d*=**c**^{T}**b**=0 is significant in engineering applications.

It has been shown that when *d*=**c**^{T}**b**=0 the given constraint induces at least one additional internal constraint, **c**^{T}**Ax**=0. It could generate more constraints. The number of vanishing elements in the vector ** ψ** determines the numbers of induced constraints. The set of constraints may be linearly dependent, or not. This could be determined by the rank of the matrix composed by the rows of the observability matrix corresponding to the vanishing elements of

**. The problem will have no solution if the external constraint induces**

*ψ**n*−1 linearly independent constrains, i.e.

**=**

*ψ***0**and rank(

**)=**

*Θ**n*.

Hence, the role of the observability matrix in problem 1.1 is opposite to the role of its counterpart in control. In control, singular ** Θ** implies that the system is not observable, and hence the control is deficient. In problem 1.1, singular

**implies that the internal constraints induced by the given constraint are not linearly independent. As a result, the problem may be solved even when the number of internal and external constraints is**

*Θ**n*. Example 7.7 demonstrated this case.

In a certain scenario there are other internal constraints that are independent of **c**. If **A** has a repeated eigenvalue *δ*=*η* of multiplicity *p* then **K**−λ**M** has the same eigenvalue, λ=*η*, repeated with multiplicity of at least *p*−1. If **A**−λ**I** and **K**−λ**M** have common eigenvalue the solvability of the problem is impeded.

If **c** is parallel to one of the eigenvectors of **A** then the observability matrix is singular since **c**^{T}(**A**−*δ*_{k}**I**)=**0** implies **c**^{T}**A**=*δ*_{k}**c**^{T}, i.e. **c**^{T}**A** is linearly dependent on **c**^{T}. Consequently the internal constraints induced by **c**^{T}**x**=0 cancel out, and the problem may be solvable. This property in conjunction with the property related to the case where **A** has a repeated eigenvalue were used in constructing the system with infinite spectrum of example 7.7. In that example **A** has an eigenvalue *η*=4 of multiplicity *p*=2, and an eigenvector **c**=(1 0 −1)^{T}. As a result, the internal constraints induced by **c** were nullified and the solution became unrestricted, apart from λ=4 and λ=1 that belong to the spectrum of **A**.

The Cauchy’s interlace theorem states that if a *symmetric* matrix is modified by a *symmetric rank-one* matrix the eigenvalues of the original matrix and its modification interlace (e.g. Parlett 1980, p. 189). As a result, if the original matrix has an eigenvalue of multiplicity *p*>1 then this eigenvalue is also an eigenvalue of the modified matrix. It was stated above that this property holds for problem 7.1 where the modification is *unsymmetric*, *rank-two* of a singular symmetric *generalized* eigenvalue problem. The proof goes as follows: assume without loss of generality that *δ*_{k}=*η*, *k*=1,2,…,*p*, is an eigenvalue of **A**. Then equation (7.32) can be written in the form
9.1
or equivalently
9.2

It thus follows that *η* is an eigenvalue of **K**−λ**M** with multiplicity of at least *p*−1.

Last but not least, the topic of solvability of problem 1.1 when *m*>1 is wide open. The objective of the paper was not to solve the problem on all of its aspects, but rather to open a thread of research in a problem of clear potential and application.

## Footnotes

↵1 The ‘transient response’ of an undamped system is permanent. Sometimes real systems have small damping that is neglected in the mathematical model, but yet over a long time diminishes the effect of the initial conditions. For this reason the particular solution (3.3) of the non-homogeneous problem (3.2) is referred as the ‘steady-state response’.

↵2 Substitute the Newton’s identities, e.g. (Berlekamp 1984), in (7.36) and obtain its equivalent (7.31).

- Received August 12, 2009.
- Accepted October 9, 2009.

- © 2009 The Royal Society