Feedback Diagonalization and Stabilization for Discrete Linear Ensemble Systems

Abstract: We consider a countably infinite collection of linear, scalar control systems, where each system is represented by the pair $(a_n,b_n)$ with $a_n&gt;0$ for $n\in \mathbb{N}$, and all systems are forced by a common scalar control input. We refer to this collection of systems as a discrete linear ensemble system. The ensemble system is feedback stabilizable if there exists a common feedback control input that asymptotically stabilizes every system simultaneously. Unlike finite-dimensional linear systems, an infinite-dimensional linear system is not guaranteed to be stable if its poles lie in the open left half-plane. Stability is guaranteed, however, if (i) its poles are contained in the closed left half-plane and (ii) its infinitesimal generator is diagonalizable. In this paper, we provide necessary and sufficient conditions for the existence of a static, linear feedback control law that ensures the closed-loop ensemble system satisfies (i) and (ii). In particular, we show that the exponential decay of the sequences $(|b_n|)<em>{n\in \mathbb{N}}$ and $(a_n/|b_n|)</em>{n\in \mathbb{N}}$ is necessary and, under an assumption on the desired poles, sufficient.

• The paper develops a constructive infinite-dimensional Ackermann-type feedback that simultaneously assigns prescribed negative poles and diagonalizes the closed-loop generator through Cauchy-matrix factorizations.
• Uniform exponential decay of the input gains is necessary for feedback diagonalizability, while matching conditions on the system, gain, and pole sequences provide sufficiency with a near-sharp characterization.
• The resulting closed loop is stable across the considered sequence spaces and asymptotically stable except on ℓ∞ and c, while gaps remain between the necessary and sufficient ratio conditions and full stabilizability remains open.

This paper studies feedback stabilization of a countably infinite ensemble of scalar linear systems sharing a single control input, and establishes near-matching necessary and sufficient conditions under which a static linear state feedback simultaneously places the closed-loop poles and diagonalizes the closed-loop generator. The work extends the pole placement and stabilizability results of Chen [(2608.12813)'s cited prior work] to give decay-rate characterizations that close much of the gap between what is required for pole placement alone and what is required for diagonalizability.

Problem setting

The ensemble consists of systems $\dot x_n(t) = a_n x_n(t) + b_n u(t)$ for $n \in \mathbb{N}$, aggregated into the infinite-dimensional system $\dot x = Ax + bu$ on a Banach sequence space $X$, where $A = \mathrm{diag}(a_1, a_2, \ldots)$ and $X$ may be $\ell^p$ ($1 \le p \le \infty$), $c$, or $c_0$. Under static feedback $n \in \mathbb{N}$0 with $n \in \mathbb{N}$1, the closed-loop generator $n \in \mathbb{N}$2 is a rank-one perturbation of $n \in \mathbb{N}$3. The authors assume all $n \in \mathbb{N}$4, so the open-loop system is unstable.

Two structural facts motivate the analysis. First, unlike finite-dimensional systems, spectrum containment in the closed left half-plane does not imply stability: an infinite Jordan block structure with uniformly unbounded block sizes yields instability even when all poles equal $n \in \mathbb{N}$5. Second, even placing each pole with algebraic multiplicity one does not guarantee diagonalizability of $n \in \mathbb{N}$6. Consequently, the paper treats feedback diagonalizability — existence of $n \in \mathbb{N}$7 such that $n \in \mathbb{N}$8 is similar to a diagonal operator — jointly with pole placement. Diagonalizability plus poles in the closed left half-plane immediately yields boundedness (stability), and asymptotic stability holds for all considered spaces except $n \in \mathbb{N}$9 and $\dot x = Ax + bu$0, where the essential spectrum at $\dot x = Ax + bu$1 obstructs asymptotic decay.

Feedback construction

The feedback law is an infinite-dimensional analogue of Ackermann's formula:

$\dot x = Ax + bu$2

where $\dot x = Ax + bu$3 is an infinite product over the ratios $\dot x = Ax + bu$4, and $\dot x = Ax + bu$5 is a prescribed null sequence of negative reals. This construction builds on the Cauchy-matrix inversion formula of Schechter, and on Chen's result that if $\dot x = Ax + bu$6 then the discrete spectrum of $\dot x = Ax + bu$7 is exactly $\dot x = Ax + bu$8, with $\dot x = Ax + bu$9 the sole point of the essential spectrum. The key technical object throughout is the pair of infinite products $X$0 and $X$1, whose uniform boundedness governs both necessity and sufficiency.

Necessary conditions

The first main theorem shows that if some negative real sequence $X$2 admits $X$3 with $X$4 diagonalizable, then:

• $X$5 and $X$6 uniformly exponentially decay with constant $X$7;
• $X$8 uniformly exponentially decays;
• $X$9 uniformly exponentially decays;
• additionally, if $A = \mathrm{diag}(a_1, a_2, \ldots)$0 is uniformly bounded below by some $A = \mathrm{diag}(a_1, a_2, \ldots)$1, then $A = \mathrm{diag}(a_1, a_2, \ldots)$2 uniformly exponentially decays and $A = \mathrm{diag}(a_1, a_2, \ldots)$3 is bounded above and below by positive constants.

The proof strategy is notable. Writing $A = \mathrm{diag}(a_1, a_2, \ldots)$4 with $A = \mathrm{diag}(a_1, a_2, \ldots)$5 bounded inverses, the authors conjugate by the diagonal operator $A = \mathrm{diag}(a_1, a_2, \ldots)$6 to reduce to a normalized operator $A = \mathrm{diag}(a_1, a_2, \ldots)$7 on the weighted space $A = \mathrm{diag}(a_1, a_2, \ldots)$8. The columns of $A = \mathrm{diag}(a_1, a_2, \ldots)$9 and rows of $X$0 are right and left eigenvectors expressible via infinite-dimensional Cauchy-type matrices parameterized by scaling factors $X$1. Truncating to finite $X$2 Cauchy matrices and invoking Schechter's closed-form inverse yields the identity $X$3, which requires establishing convergence of truncated eigenvector rows — strong convergence for all spaces except $X$4, where only weak convergence is obtained. From boundedness of $X$5 and $X$6, the scaling factors are shown to be uniformly bounded above and below, which forces the uniform exponential decay of $X$7 and $X$8 via a contradiction argument on the products $X$9. Decay of $\ell^p$0 and $\ell^p$1 follows from bounded-sum inequalities extracted from boundedness of $\ell^p$2, combined with a standalone combinatorial lemma showing that either of two partial-sum conditions on a positive sequence implies its uniform exponential decay.

Implication: uniform exponential decay of $\ell^p$3 is unconditionally necessary for feedback diagonalizability — a strictly stronger requirement than the polynomial decay rates needed for pole placement alone, resolving the discrepancy noted in prior work in the negative direction.

Sufficient conditions

The converse theorem assumes that $\ell^p$4, $\ell^p$5, and $\ell^p$6 satisfy the exponential decay conditions above, together with a refined ratio condition: there exist sequences $\ell^p$7, $\ell^p$8 with $\ell^p$9 such that $1 \le p \le \infty$0 and $1 \le p \le \infty$1 are positive $1 \le p \le \infty$2 sequences. Roughly, this requires $1 \le p \le \infty$3 to be uniformly bounded above and below and to converge to one at an $1 \le p \le \infty$4 rate. Under these hypotheses, $1 \le p \le \infty$5 for every admissible $1 \le p \le \infty$6, and $1 \le p \le \infty$7 is diagonalizable for $1 \le p \le \infty$8 ($1 \le p \le \infty$9) and $c$0.

The proof is constructive. Explicit operators $c$1 and $c$2 — symmetric under swapping $c$3 and $c$4 — are defined with scaling factors $c$5, $c$6. Boundedness follows from Hölder-type estimates controlled by explicit constants $c$7 and $c$8 built from the decay parameters and the $c$9 norms of $c_0$0 and $c_0$1; the $c_0$2 membership of $c_0$3 and the uniform boundedness of both $c_0$4 products follow from logarithmic sum estimates exploiting the exponential decay of $c_0$5 and $c_0$6. Invertibility is established by truncation: finite Cauchy inverses give exact orthogonality $c_0$7, and strong convergence of the truncations passes the identity to the limit.

Combining with the spectral assignment theorem, the closed-loop system is stable for all considered $c_0$8 and asymptotically stable except when $c_0$9 or $n \in \mathbb{N}$00.

Limitations and open questions

Several qualifications bound the scope of the results. The necessity direction for $n \in \mathbb{N}$01 requires the auxiliary assumption that $n \in \mathbb{N}$02 not decay faster than exponentially; without it, only decay of $n \in \mathbb{N}$03 is established. The sufficiency direction excludes $n \in \mathbb{N}$04 from the diagonalizability claim (handled separately via embedding into $n \in \mathbb{N}$05 for stability), and the ratio condition (C5) is stronger than the static two-sided bound that appears among the necessary conditions — so a gap remains between conditions C4–C5 on the necessity side and C5′ on the sufficiency side. Asymptotic stability is unattainable for $n \in \mathbb{N}$06 and $n \in \mathbb{N}$07 under this framework, and exponential stability is impossible for any $n \in \mathbb{N}$08 given $n \in \mathbb{N}$09. The authors explicitly pose the central open problem: whether uniform exponential decay of $n \in \mathbb{N}$10 and $n \in \mathbb{N}$11 is necessary for feedback stabilizability itself, not merely for pole placement combined with diagonalizability.

Conclusion

The paper delivers a constructive, nearly sharp characterization of when a common static feedback can simultaneously place poles and diagonalize the closed-loop generator of a discrete linear ensemble system. Uniform exponential decay of the input gains emerges as the fundamental obstruction, with matching sufficient conditions up to the strength of the prescribed pole-ratio constraint. The proofs contribute reusable machinery — weighted-space conjugation, Cauchy-matrix truncation arguments, and combinatorial decay lemmas — that should apply to related questions in infinite-dimensional stabilization.