# A Note on a Necessary Spectral Condition for $L^2$ Exponential Stabilization of Difference Delay Equations with Distributed Delays

Sébastien Fueyo

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Sébastien Fueyo. A Note on a Necessary Spectral Condition for $L^2$ Exponential Stabilization of Difference Delay Equations with Distributed Delays. 2026. hal-05513088

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# A Note on a Necessary Spectral Condition for $L^2$ Exponential Stabilization of Difference Delay Equations with Distributed Delays

S. Fueyo

January 2026

## Abstract

This note establishes a necessary frequency-domain spectral rank condition for the exponential stabilization of linear difference delay equations with distributed delays by bounded linear feedback laws. Such systems arise in particular from backstepping reductions of one-dimensional linear hyperbolic PDEs, where stabilization of the PDE is equivalent to stabilization of the associated delay equation. The result is obtained by adapting an operator-theoretic framework, extending ideas introduced in [19]. The condition provides an explicit and directly verifiable obstruction to stabilizability. A conjecture formulating a corresponding necessary and sufficient frequency-domain criterion is also presented.

## 1 Introduction

In this note, we consider the linear controlled difference delay equation with distributed delays

$$x(t) = \sum_{j=1}^{N} A_j x(t - \Lambda_j) + \int_{0}^{\Lambda_N} g(s) x(t - s) ds + Bu(t), \quad t \ge 0, \quad (1)$$

where $A_1, \dots, A_N \in \mathbb{R}^{d \times d}$, $g \in L^\infty([0, \Lambda_N]; \mathbb{R}^{d \times d})$, $B \in \mathbb{R}^{d \times m}$, and the delays $\Lambda_1, \dots, \Lambda_N$ belong to $(0, 1]$.

Systems of the form (1) naturally arise through backstepping transformations of one-dimensional linear hyperbolic partial differential equations; see, for instance, [1]. Through this transformation, stability and stabilization properties of the original PDE are equivalent to those of the associated delay system. This connection has recently renewed interest in such equations, notably for stability analysis [18, 2, 5, 4], stabilization [20, 3], and controllability [8, 9]. Foundational results on delay systems and functional differential equations can be found in [16, 14, 7], while systematic treatments of stability and stabilization for time-delay systems are given in [17, 11].

For retarded and neutral delay systems, stabilization theory is now well developed, both in time and frequency domains [12, 19]. However, to the best of our knowledge, no frequency-domain stabilization criterion, not even a necessary one, is available for systems of the form
(1). Establishing such a criterion is of practical interest, since it provides an explicit ob-
struction to stabilizability that can be checked directly from the system parameters, without
constructing a Lyapunov functional or a feedback law.

Compared with classical retarded or neutral delay systems, (1) combines finitely many
pointwise delays with a distributed delay term and does not involve delayed derivatives.
A classical result [14] shows that neutral systems can be viewed as compact perturbations
of difference delay systems (such as (1) without control). In this sense, difference delay
systems capture the most delicate part of the analysis, while the neutral term mainly con-
tributes additional eigenvalues to the solution operator. For this reason, stabilization results
for neutral-type systems often rely on the assumption that the associated difference delay
system is already exponentially stable; see, e.g., [12, 19]. Standard frequency-domain tech-
niques developed for neutral systems therefore do not apply directly, and a dedicated spectral
analysis is required.

From an abstract viewpoint, (1) admits a controlled abstract differential formulation in-
volving an unbounded operator; see, e.g., [21]. More precisely, it can be rewritten as an
infinite-dimensional state-space system posed on a suitable history space, where the asso-
ciated operator generates a strongly continuous semigroup. Such representations are com-
monly used to establish stabilization results for retarded-type systems [6], but they become
more delicate to handle in the presence of unbounded control operators. In this framework,
exponential stabilization is closely related to the spectral properties of the corresponding gen-
erator. Although operator-theoretic methods have proved effective for several classes of delay
equations, they have not yet been fully exploited to derive frequency-domain stabilization
criteria for difference delay equations with distributed delays (1).

The objective of this note is to make explicit how the operator-theoretic framework devel-
oped in [14, 19] can be adapted to (1), and to derive a simple necessary spectral (frequency-
domain) condition for exponential stabilization. In the PDE context, this yields a directly
verifiable necessary condition for boundary feedback stabilization after backstepping trans-
formation, thereby providing a diagnostic tool prior to controller synthesis. Beyond this nec-
essary condition, the analysis also suggests a natural conjecture toward a complete frequency-
domain characterization of exponential stabilization, that is, a necessary and sufficient crite-
rion formulated in terms of the associated characteristic operator. We state this conjecture
and discuss supporting arguments in the final section.

The note is organized as follows. Section 2 introduces the stabilization notion used throughout the paper and states the main result, namely a necessary frequency-domain stabilization criterion for linear difference delay equations with distributed delays. Section 3 contains the proof. Finally, Section 4 presents a conjecture toward a necessary and sufficient stabilization criterion.

## 2 Preliminaries and the main result

Denote by

$$
\mathcal{X} := L^2([-\Lambda_N, 0]; \mathbb{R}^d)
$$
the state space of (1), endowed with the norm

$$ \|\phi\|_{X} = \left( \int_{-\Lambda_{N}}^{0} \|\phi(\theta)\|^{2} d\theta \right)^{1/2}, \qquad (2) $$

where $\| \cdot \|$ denotes the Euclidean norm in $\mathbb{R}^d$ (or $\mathbb{C}^d$ after complexification). For a locally square integrable function $x : [-\Lambda_N, +\infty) \to \mathbb{R}^d$, we define, for $t \ge 0$, the history segment

$$ x_t(\theta) := x(t + \theta), \quad \theta \in [-\Lambda_N, 0]. $$

We first recall a standard well-posedness result for equation (1) (see, e.g., [9]).

**Proposition 2.1.** Let $T > 0$, $\varphi \in \mathcal{X}$ and $u \in L^2([0, T], \mathbb{R}^m)$. Then there exists a unique solution $x \in L^2([-\Lambda_N, T]; \mathbb{R}^d)$ to system (1) such that $x(\theta) = \varphi(\theta)$ for $\theta \in [-\Lambda_N, 0]$. Moreover, the map $t \mapsto x_t$ is continuous from $[0, T]$ into $\mathcal{X}$.

We now introduce the notion of stabilization considered throughout this paper.

**Definition 2.2** (Exponential stabilization). System (1) is said to be exponentially stabilizable if there exists a bounded linear operator

$$ \mathcal{K} : \mathcal{X} \to \mathbb{R}^m $$

such that the feedback law

$$ u(t) = \mathcal{K}(x_t) $$

makes the closed-loop system exponentially stable at the origin. More precisely, there exist constants $M \ge 1$ and $\omega > 0$ such that the corresponding solution satisfies

$$ \|x_t\|_{X} \le M e^{-\omega t} \|\varphi\|_{X}, \quad \forall t \ge 0, $$

for every initial condition $\varphi \in \mathcal{X}$.

We associate with the uncontrolled system ($u \equiv 0$) the characteristic matrix

$$ \hat{Q}(p) = I_d - \sum_{j=1}^{N} A_j e^{-p \Lambda_j} - \int_{0}^{\Lambda_N} g(s) e^{-ps} ds, \quad p \in \mathbb{C}, \qquad (3) $$

where $I_d$ is the identity matrix on $\mathbb{C}^d$. The zeros of $\det \hat{Q}(\cdot)$ correspond to the characteristic exponents of the system. We are now in a position to state the main result of this note.

**Theorem 2.3.** A necessary condition for the exponential stabilization of system (1) in the sense of Definition 2.2 is that

$$ \mathrm{rank}[\hat{Q}(p), B] = d, \quad \forall p \in \mathbb{C}_+ := \{p \in \mathbb{C} \mid \Re(p) \ge 0\}, \qquad (4) $$

where $\Re(p)$ denotes the real part of $p$.

The interpretation of Theorem 2.3 is intuitive. Indeed, the uncontrolled system (1) (i.e., with $u \equiv 0$) can be formulated as an abstract differential equation generating a strongly continuous semigroup with generator $A$. The spectrum of $A$ consists of isolated eigenvalues, which correspond precisely to the zeros of $\det(\hat{Q}(\cdot))$. By projecting the system onto the finite-dimensional invariant subspaces associated with these eigenvalues, the stabilization problem reduces locally to a finite-dimensional one. In this context, the classical Hautus test [15] applies: the rank condition (4) ensures that each such finite-dimensional subspace is exponentially stabilizable. The details of this construction and the corresponding projections are given in the proof of Theorem 2.3 which is the subject of the next section.
### 3 Proof Theorem 2.3

First, in this section, we rely on the spectral decomposition theory developed for retarded and neutral functional differential equations [13], as well as the stabilization result from [19]. These results extend naturally to the difference delay system (1) in the state space $\mathcal{X}$. Since only minor and straightforward adaptations are needed, we omit the proofs and instead provide precise references.

Let us briefly outline the proof. Theorem 2.3 relies on a spectral decomposition of the semigroup generated by the uncontrolled system (1) (with $u \equiv 0$). Its generator $A$ has isolated eigenvalues given by the zeros of $\det(\hat{Q}(\cdot))$. By analyzing the dual equation and constructing suitable dual bases, one obtains finite-dimensional invariant subspaces that are orthogonal with respect to an appropriate bilinear form. Steps 1 and 2 below make this construction explicit and describe the associated spectral projections. Projecting the system onto these subspaces reduces the exponential stabilization problem locally to a finite-dimensional one, where the classical Hautus criterion [15] can be applied. This is carried out in Steps 3 and 4.

**Step 1.** First, consider System (1) without control, i.e.

$$x(t) = \sum_{j=1}^{N} A_j x(t - \Lambda_j) + \int_{0}^{\Lambda_N} g(s) x(t - s) ds, \quad t \ge 0. \quad (5)$$

For simplicity, we rewrite (5) in the abstract form:

$$x(t) = \int_{-\Lambda_N^{-}}^{0^{-}} d\mu(\theta)x(t + \theta), \quad t \ge 0, \quad (6)$$

where

$$\mu = \sum_{j=1}^{N} A_j \delta_{-\Lambda_j} + g(-\cdot)\mathcal{L},$$

and $\mathcal{L}$ denotes the Lebesgue measure and, for $a \in \mathbb{R}$, $\delta_a$ is the Dirac measure at $a$. The integral in (6) is understood as a Stieltjes–Lebesgue integral over the interval $[-\Lambda_N, 0]$. It is standard (see [14, Chapter 7, Lemma 1.1, Lemma 1.2 and Lemma 2.1]) that for any initial state $\varphi \in \mathcal{X}$, system (6) admits a unique solution denoted by $x_t \in \mathcal{X}$ for all $t \ge 0$, with $x_t(\theta) = x(t + \theta)$ for $\theta \in [-\Lambda_N, 0]$. This allows us to define the strongly continuous semigroup of bounded operators $(T(t))_{t \ge 0}$ by $T(t)\varphi(\cdot) = x_t(\cdot)$ for $t \ge 0$. Its generator is $A$, with domain

$$D(A) = \{ \phi \in W^{1,2}([-\Lambda_N, 0], \mathbb{R}^d) \mid \phi(0) = \int_{-\Lambda_N^{-}}^{0^{-}} d\mu(\theta) \phi(\theta) \},$$

where $W^{1,2}([-\Lambda_N, 0], \mathbb{R}^d)$ denotes the usual Sobolev space of absolutely continuous functions with derivative in $L^2([-\Lambda_N, 0], \mathbb{R}^d)$. The domain $D(A)$ is dense in $\mathcal{X}$, and the spectrum of $A$ consists of isolated eigenvalues with finite multiplicity; that is, the spectrum is purely point spectrum. An element $\lambda \in \mathbb{C}$ belongs to the spectrum $\sigma(A)$ if and only if

$$\det(\hat{Q}(\lambda)) = 0. \quad (7)$$
Let $\lambda \in \sigma(A)$, and denote by $k \ge 1$ the algebraic multiplicity of the eigenvalue $\lambda$. Define

$$W_{\lambda} := \ker(A - \lambda I)^k, \quad \mathcal{J}_{\lambda} := \mathrm{Ran}(A - \lambda I)^k,$$

where $\ker$ and $\mathrm{Ran}$ denote the kernel and range, respectively. Moreover, $W_{\lambda}$ and $\mathcal{J}_{\lambda}$ are finite-dimensional subspaces of $\mathcal{X}$. We have the following decomposition

$$\mathcal{X} = W_{\lambda} \oplus \mathcal{J}_{\lambda},$$

and the operator $A$ can be reduced by this decomposition.

All the arguments in [13, Chapter 7, Lemma 1.1, Lemma 1.2 and Lemma 2.1] are stated for the classical retarded framework but adapt straightforwardly to System (1). We now turn to Step 2, where we construct a basis for $W_{\lambda}$ and $\mathcal{J}_{\lambda}$ that are orthogonal to each other with respect to a suitable bilinear form.

**Step 2.** Following [13, Section 7.3], we start by defining the dual equation to (6), namely the equation defined by

$$y(\tau) = - \int_{-\Lambda_N^{-}}^{0} y(\tau - \theta) d\mu(\theta). \qquad (8)$$

Let $\mathcal{X}^* = L^2([0, \Lambda_N], (\mathbb{R}^d)^*)$ the dual space of $\mathcal{X}$ and $(\mathbb{R}^d)^*$ is the dual space of $\mathbb{R}^d$. Let $y$ be a solution of Equation (8) on an interval $(-\infty, \Lambda_N]$ with respect to the initial data $\varphi \in \mathcal{X}^*$, i.e., $y(\theta) = \varphi(\theta)$ for $\theta \in [0, \Lambda_N]$. For each $\tau \in (-\infty, 0]$, we define the forward restriction $y^{\tau} \in \mathcal{X}^*$ by

$$y^{\tau}(\theta) := y(\tau + \theta), \quad \theta \in [0, \Lambda_N].$$

Define

$$T^*(\tau)\varphi := y^\tau, \quad -\infty < \tau \le 0.$$

Then $(T^*(\tau))_{\tau \le 0}$ satisfies the same semigroup properties as the semigroup $(T(t))_{t \ge 0}$ associated with Equation (6), namely it defines a strongly continuous semigroup on $\mathcal{X}^*$ and the generator, denoted $A^*$ is defined on the domain

$$D(A^*) = \{ \phi \in W^{1,2}([0, \Lambda_N], (\mathbb{R}^d)^*) \mid \phi(0) = - \int_{-\Lambda_N^{-}}^{0} \phi(-\theta) d\mu(\theta) \},$$

Moreover, for all $\varphi \in D(A^*)$ and $\tau \le 0$, it holds that

$$\frac{d}{d\tau} T^*(\tau)\varphi = -A^*T^*(\tau)\varphi = -T^*(\tau)A^*\varphi. \qquad (9)$$

and the spectrum of $A^*$ is composed the element $\lambda \in \mathbb{C}$ such that $\det \hat{Q}(\lambda) = 0$. Furthermore, $A^*$ is the dual operator of $A$ with respect to the duality pairing $(\cdot, \cdot)$ on $\mathcal{X}^* \times \mathcal{X}$, which is defined for $\alpha \in \mathcal{X}^*$ and $\psi \in \mathcal{X}$ by

$$(\alpha, \psi) := - \int_{-\Lambda_N^{-}}^{0} \int_{0}^{\theta} \alpha(\zeta - \theta) d\mu(\theta) \psi(\zeta) d\zeta. \qquad (10)$$
With respect to this pairing, $A^*$ satisfies

$$ (A^*\Psi, \xi) = (\Psi, A\xi), \quad \Psi \in \mathcal{X}^*, \xi \in \mathcal{X}. $$

From [13, Lemma 3.2], it is easy to show that a vector $\psi$ belongs to $\text{Ran}(A - \lambda I)^k$ if and only if

$$ (\alpha, \psi) = 0 \quad \text{for all } \alpha \in \text{ker}(A^* - \lambda I)^k. $$

In other words, $\text{Ran}(A - \lambda I)^k$ coincides with the orthogonal complement of $\text{ker}(A^* - \lambda I)^k$ with respect to the above bilinear form. Thus as a consequence, for a fixed eigenvalue $\lambda \in \sigma(A)$, let

$$ \Phi_{\lambda} = (\phi_1, \dots, \phi_k), \quad \Psi_{\lambda} = (\psi_1, \dots, \psi_k) $$

be bases of the generalized eigenspaces $W_\lambda$ and $J_\lambda$, respectively, and assume that these bases are chosen so that

$$ (\Psi_{\lambda}, \Phi_{\lambda}) = I_{d}. $$

With this choice, any element $\varphi \in \mathcal{X}$ can be explicitly decomposed as

$$ \varphi = \varphi_{W_{\lambda}} + \varphi_{J_{\lambda}}, $$

where $\varphi_{W_\lambda} \in W_\lambda$ and $\varphi_{J_\lambda} \in J_\lambda$. Here,

$$ W_{\lambda} := \{ \varphi \in \mathcal{X} \mid \varphi = \Phi_{\lambda} b \text{ for some } b \in \mathbb{C}^{k} \}, $$

$$ J_{\lambda} := \{ \varphi \in \mathcal{X} \mid (\Psi_{\lambda}, \varphi) = 0 \}. $$

The projection of $\varphi$ onto $W_\lambda$ is given by

$$ \varphi_{W_{\lambda}} = \Phi_{\lambda} b, \quad b = (\Psi_{\lambda}, \varphi), $$

and the complementary component in $J_\lambda$ is

$$ \varphi_{J_{\lambda}} = \varphi - \varphi_{W_{\lambda}}. $$

As shown in [13, Chapter 7, Lemma 3.3], every element of the generalized adjoint eigenspace $\text{ker}(A^* - \lambda I)^k$ admits the representation

$$ \psi(s) = \sum_{j=1}^{k} \beta_j \frac{(-s)^{k-j}}{(k-j)!} e^{-\lambda s}, \quad 0 \le s \le \Lambda_N, $$

where $\beta = (\beta_1, \dots, \beta_k)$ is a block row vector with $\beta_j \in (\mathbb{R}^d)^*$ satisfying the left kernel condition

$$ \beta A_k(\lambda) = 0. $$

To define the matrix $A_k(\lambda)$, introduce

$$ P_{j+1}(\lambda) := \frac{1}{j!} \hat{Q}^{(j)}(\lambda), \quad j = 0, 1, 2, \dots, \qquad (11) $$
where

$$ \hat{Q}^{(j)}(\lambda) = \frac{d^j}{d\lambda^j} \hat{Q}(\lambda). $$

For an integer $k \ge 1$, the associated block matrix $A_k(\lambda) \in \mathbb{C}^{kd \times kd}$ is defined by

$$ A_k(\lambda) = \begin{pmatrix} P_1(\lambda) & P_2(\lambda) & \cdots & P_k(\lambda) \\ 0 & P_1(\lambda) & \cdots & P_{k-1}(\lambda) \\ \vdots & \vdots & \ddots & \vdots \\ 0 & \cdots & 0 & P_1(\lambda) \end{pmatrix}. \qquad (12) $$

Hence $\ker(A^* - \lambda I)^k$ is in one-to-one correspondence with the left nullspace of $A_k$. Let $\{\beta^{(1)}, \dots, \beta^{(k)}\}$ be a basis of the left nullspace of $A_k$, with block decompositions $\beta^{(j)} = (\beta_1^{(j)}, \dots, \beta_k^{(j)})$ for $j \in \{1, \dots, k\}$. In particular, it follows that:

$$ \beta_k^{(j)} \hat{Q}(\lambda) = 0, \quad j = 1, \dots, k. \qquad (13) $$

Define the block column function for $s \in [0, \Lambda_N]$:

$$ v(s) = \begin{pmatrix} \frac{(-s)^{k-1}}{(k-1)!} I_d \\ \frac{(-s)^{k-2}}{(k-2)!} I_d \\ \vdots \\ I_d \end{pmatrix} e^{-\lambda s} \in \mathbb{R}^{dk \times d}. $$

Then a basis of $\ker(A^* - \lambda I)^k$ can be written as

$$ \psi_i(s) = \beta^{(i)} v(s), \quad i = 1, \dots, k. $$

Collecting all basis functions into a matrix (or vector) form:

$$ \Psi_{\lambda}(s) := (\psi_{1}(s) \ \psi_{2}(s) \ \dots \ \psi_{k}(s)) = (\beta^{(1)}v(s) \ \beta^{(2)}v(s) \ \dots \ \beta^{(k)}v(s)) \in \mathbb{R}^{d \times (dk)}. \quad (14) $$

Here, each $\psi_i$ is a row-valued generalized adjoint eigenfunction, and the family $\{\psi_1, \dots, \psi_k\}$ forms a basis of $\ker(A^* - \lambda I)^k$. The matrix-valued function $\Psi_{\lambda}(s)$ is obtained by stacking these adjoint eigenfunctions.

**Step 3.** We use the two previous steps to establish a decomposition in the variation of constant formula for the control problem (1). First, let $\Phi_{\lambda}$ as defined in Step 2. There exists a $k \times k$ matrix $\mathcal{B}_{\lambda}$ such that

$$ A\Phi_{\lambda} = \Phi_{\lambda}\mathcal{B}_{\lambda}. \qquad (15) $$

The matrix $\mathcal{B}_{\lambda}$ represents the restriction of $A$ to the generalized eigenspace $W_{\lambda}$ in the basis $\Phi_{\lambda}$. In particular, its spectrum reduces to $\{\lambda\}$; equivalently, $\mathcal{B}_{\lambda} = \lambda I + N_{\lambda}$ with $N_{\lambda}$ nilpotent. Following [13, Chapter 7, Section 7.6], the projection of $x_t(\cdot)$ of (1) onto $W_{\lambda}$ is a function $\theta \mapsto \Phi_{\lambda}(\theta)y(t)$ for $\theta \in [-\Lambda_N, 0]$, and, $y(\cdot)$ satisfies the equation:
$$y(t) = e^{\mathcal{B}_{\lambda}t}y(0) + \int_{0}^{t} e^{\mathcal{B}_{\lambda}(t-s)}\Psi_{\lambda}(0)Bu(s)ds, \quad (16)$$

where $\Psi$ is given by (14), which is after a differentiation an ordinary differential equation:

$$\dot{y}(t) = \mathcal{B}_{\lambda}y(t) + \Psi_{\lambda}(0)Bu(t). \quad (17)$$

Thus, the control problem (1) reduces to the finite-dimensional control system (17). It remains to show that, under assumption (4), system (17) is exponentially stabilizable.

**Step 4.** We end by proving Theorem 2.3. For any $\lambda \in \sigma(A)$ with $\Re(p) \ge 0$, we will show that the projection of $\mathcal{X}$ on $W_{\lambda}$ is exponentially stabilizable if and only if the rank condition (4) holds. More precisely, the projection of a trajectory $x_t$ on $W_{\lambda}$ is $\Phi_{\lambda}y(t)$, with $y$ defined in Equation (16) and it is an ordinary differential equation which is known from the Hautus test to be exponentially stabilizable if and only if

$$\mathrm{rank}[\mathcal{B}_{\lambda} - \lambda I_k, \Psi_{\lambda}(0)B] = k, \quad (18)$$

since the only eigenvalues of $\mathcal{B}_{\lambda}$ are reduced to $\{\lambda\}$.

First, if (18) does not hold, then there exists a nonzero vector $u \in \mathbb{C}^k$ such that

$$u^*\mathcal{B}_{\lambda} = \lambda u^* \quad \text{and} \quad u^*\Psi_{\lambda}(0)B = 0. \quad (19)$$

The first identity in (19) is equivalent to

$$A^*(u^*\Psi_{\lambda}(\theta)) = \lambda u^*\Psi_{\lambda}(\theta). \quad (20)$$

By the definition of the adjoint generator $A^*$, it follows that for $\theta \in [0, \Lambda_N]$,

$$u^*\Psi_{\lambda}(\theta) = e^{-\lambda\theta} u^*\Psi_{\lambda}(0). \quad (21)$$

Defining $\alpha := u^*\Psi_{\lambda}(0)$, relation (21) shows that $\alpha$ is nonzero; otherwise $u^*\Psi_{\lambda}(\theta)$ would vanish for all $\theta$, which would contradict the linear independence of the family defining $\Psi_{\lambda}$. Thus, using (13), we obtain

$$\alpha \hat{Q}(\lambda) = u^* \Psi_{\lambda}(0) \hat{Q}(\lambda) = \sum_{j=1}^{k} u_j \beta_k^{(j)} \hat{Q}(\lambda) = 0. \quad (22)$$

Thus, combining (22) with (19), we have shown that

$$\alpha \neq 0, \quad \alpha \hat{Q}(\lambda) = 0 \quad \text{and} \quad \alpha B = 0. \quad (23)$$

Thus condition (4) is not satisfied.

Conversely, suppose that condition (4) does not hold. Then there exists $\alpha \in (\mathbb{R}^d)^*$ such that (23) is satisfied. The $\{\beta^{(1)}, \dots, \beta^{(k)}\}$ form a basis of the left nullspace of $A_k$, and denote by $\beta_k^{(i)}$ the last block of $\beta^{(i)}$. By the triangular structure of $A_k$, each $\beta_k^{(i)} \in \ker \hat{Q}(\lambda)$, and
since dim ker $\hat{Q}(\lambda) = k$, the family $\{\beta_k^{(1)}, \dots, \beta_k^{(k)}\}$ forms a basis of ker $\hat{Q}(\lambda)$. Therefore, there exists a unique vector of coefficients $u^* = (u_1, \dots, u_k) \in \mathbb{C}^k$ such that

$$ \alpha = u^* \Psi_{\lambda}(0), \quad \Psi_{\lambda}(0) = (\beta_k^{(1)}, \dots, \beta_k^{(k)}). $$

By construction, the function $\theta \in [0, \Lambda_N] \mapsto u^* \Psi_{\lambda}(\theta)$ satisfies (20), and therefore also (19). Moreover,

$$ u^* \Psi_{\lambda}(0) B = \alpha B = 0, $$

which shows that the Hautus condition for the associated finite-dimensional projected system is violated, i.e., condition (18) fails.

We thus conclude that the projection of $\mathcal{X}$ onto $W_{\lambda}$ is exponentially stabilizable if and only if the rank condition (4) holds. Since the exponential stabilization of the full system on $\mathcal{X}$ requires the exponential stabilization on each subspace $W_{\lambda}$ corresponding to every $\lambda \in \sigma(A)$ with $\Re(\lambda) \ge 0$, it follows that condition (4) is necessary for the exponential stabilization of system (1).

# 4 Discussion and Conjecture

Theorem 2.3 provides a necessary condition for exponential stabilization, but it is not sufficient. Indeed, it is currently unclear how to achieve stabilization on the entire state space $\mathcal{X}$ rather than only on the finite-dimensional invariant subspaces. Building on recent developments on the controllability of delay systems without distributed terms (i.e., with $g \equiv 0$), see [10], and by mimicking the finite-dimensional stabilization criterion, we propose the following conjecture.

**Conjecture 1.** A necessary and sufficient condition for the exponential stabilization of system (1) in the sense of Definition 2.2 is that

$$ \mathrm{rank}[F, B] = d, \qquad (24) $$

for all $F \in \overline{\hat{Q}(\mathbb{C}_+)}$, the closure of the set $\hat{Q}(\mathbb{C}_+)$ in $\mathbb{C}^{d \times d}$, where

$$ \mathbb{C}_+ := \{p \in \mathbb{C} \mid \Re(p) \ge 0\}. $$

In other words, the rank condition must hold for every matrix $F$ in the topological closure of the characteristic matrix $\hat{Q}(\cdot)$ evaluated in the half-plane $\mathbb{C}_+$.

Intuitively, the conjecture asserts that stabilization is possible only if every “limit point” of the characteristic matrix in the unstable half-plane is controllable through the input matrix $B$. This can be seen as a natural extension of the finite-dimensional Hautus test to the infinite-dimensional setting of difference delay equations with distributed terms. Although a complete proof remains open, it is expected to be closely related to a frequency-domain controllability criterion for system (1), which essentially amounts to solving a non-trivial corona problem, as discussed in [8].
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