Infinite-Dimensional Hautus Test

• Infinite-Dimensional Hautus Test is a generalization of classical control tests, defining controllability and observability for systems governed by PDEs and infinite-dimensional operators.
• It replaces finite-dimensional rank conditions with algebraic, geometric, and functional analytic criteria to assess system behavior in complex infinite-dimensional settings.
• The framework extends to time-dependent, stochastic, and port-Hamiltonian systems, providing robust and practical insights for modern control system design and analysis.

The Infinite-Dimensional Hautus Test generalizes a fundamental principle in finite-dimensional control and systems theory to systems described by partial differential equations (PDEs), time-varying infinite-dimensional operators, stochastic singular systems, and frameworks with complex input/output or functional constraints. Its modern formulations capture the essence of controllability and observability for linear distributed systems, often using concepts from module theory, functional analysis, and algebraic geometry, and are central to both theoretical analysis and practical system design in infinite-dimensional settings.

1. Historical Foundations and the Classical Hautus Test

The classical Hautus test originates in linear finite-dimensional control theory. For a state-space system described by

$\frac{dx}{dt} = Ax + Bu,$

controllability is equivalently characterized by either the Kalman rank condition or the Hautus matrix rank condition: $$\forall \lambda \in \mathbb{C}, \quad \operatorname{rank}\begin{bmatrix} \lambda I - A & B \end{bmatrix} = n,$$ where $A$ is an $n \times n$ matrix and $B$ aligns with the input dimension. This test ensures that no nontrivial invariant subspace exists on which the system is uncontrollable. For practical verification, it often suffices to check the condition only at eigenvalues of $A$ (Nguyen, 5 May 2025).

The Hautus test is intimately linked to the system's ability to control or observe its modes, and dual statements provide necessary and sufficient conditions for observability. These rank-based conditions form the foundation for a wide array of extensions to more general, potentially infinite-dimensional systems.

2. Generalization to Infinite-Dimensional Systems and Behavioral Approach

In infinite-dimensional systems—such as those governed by PDEs—the notion of rank is not directly meaningful. Instead, one considers modules over rings of differential operators and evaluates controllability via algebraic and geometric conditions. The behavioral approach models the dynamics as a behavior $\mathcal{B}(P)$,

$$\mathcal{B}(P) = \{ f \in \mathcal{F}^k : P(\partial)f = 0 \},$$

where $P(\partial)$ is an $l \times k$ matrix with entries in $A = \mathbb{C}[\partial_1, ..., \partial_n]$, acting on a suitable function space $\mathcal{F}$ (Shankar, 2013).

The Infinite-Dimensional Hautus Test in this context is based on evaluating the matrix $P(X)$, obtained by substituting $\partial_j \mapsto X_j$ for $X \in \mathbb{C}^n$. Controllability is then characterized by the (full row) rank of $P(X)$ outside a "small" algebraic variety,

$$\mathcal{B}(P) \text{ is controllable } \iff \operatorname{rank} P(X) = l \text{ for all } X \ \text{outside a set of dimension} \leq n-2.$$

This equivalently translates, via the l-th determinantal ideal $I_e$, to the geometric property $\dim(V(I_e)) \leq n-2$ (Shankar, 2013).

For strictly underdetermined systems ($l < k$), generic controllability is ensured if the cancellation ideal $I_e$ is nontrivial and its zero set is sufficiently small. Strictly overdetermined behaviors ($l > k$) are generically uncontrollable unless they are trivial.

3. Extensions: Time-Dependence, Observability, and Averaged Tests

When system operators are time-dependent (non-autonomous evolution equations), the classical resolvent-based test does not suffice. Instead, the Hautus test is replaced by inequalities involving the evolution family $U(t, s)$ and time averages. One pivotal formulation for observability becomes: $$\|x\|^2 \leq m^2 \int_0^T \|C(s)x\|^2 ds + M^2 \int_0^T \|(i\lambda + A(s))x\|^2 ds,$$ holding for all $x$ in the domain and all $\lambda \in \mathbb{R}$. For skew-adjoint (unitary) operators $A(t)$ with bounded variation, this "averaged" Hautus condition is equivalent to exact observability, and extensions exist for more general classes of evolution operators under controlled growth conditions (Haak et al., 2018). This framework is essential in PDE models with time-varying coefficients, including Schrödinger and damped wave equations.

In some formulations, the Hautus-type inequalities are directly linked to spectral properties and can be stated as spectral inequalities for the system's eigenfunctions—ensuring that none are "undetectable" over the control region (Martin et al., 2020).

4. Functional, Output, and Stochastic Generalizations

Controllability and observability in infinite dimensions are not limited to the full state or output. The Popov-Belevitch-Hautus (PBH) test has been generalized to functional observability and output controllability:

• Functional observability: Can a target functional $z=Fx$ be reconstructed from observations? The test is

$$\operatorname{rank}\left([\lambda I_n - A \quad C \quad F]\right) = \operatorname{rank}\left([\lambda I_n - A \quad C]\right),\quad \forall \lambda \in \mathbb{C},$$

holding generically if the "lead columns" of $F$ are linearly independent in Jordan coordinates (Montanari et al., 5 Feb 2024).

• Output controllability: The ability to drive $z = F x$ to arbitrary targets is contingent on

$$\operatorname{rank}(F[\lambda I_n - A \quad B]) = \operatorname{rank}(F), \quad \forall \lambda\in\mathbb{C},$$

with intersection conditions required when $(A, B)$ is not fully controllable.

In the context of stochastic singular systems, the PBH criterion is adapted to ensure well-posedness and the finiteness of the stochastic linear-quadratic cost by requiring a corresponding rank condition on the transformed system matrices governing the "dynamic" state components. This ensures that a reduction to a regular stochastic LQ problem is possible and that solutions admit robust state-feedback representations, even for potentially degenerate mass matrices (Li et al., 3 Sep 2024).

5. Observability, IQCs, and Port-Hamiltonian Systems

In input-output frameworks for infinite-dimensional systems, such as those described by partial differential or delay equations, Integral Quadratic Constraints (IQCs) provide a powerful method for assessing robust stability. In these frameworks, the infinite-dimensional Hautus principle is interpreted as a set of operator inequalities—generalizations of the Kalman-Yakubovich-Popov (KYP) lemma to Partial Integral Equations (PIEs). The system is stable if, for all relevant signals, certain quadratic/operator inequalities (certificates of energy dissipation) are satisfied by causally bounded operators and their interactions with uncertainty multipliers (Talitckii et al., 2023). These conditions extend matrix-analytic tests to function space operators and are practically verifiable using convex optimization in the space of positive partial-integral operators.

Observability in infinite-dimensional port-Hamiltonian systems, especially those with internal energy dissipation, is robustly characterized via the Hautus resolvent condition: $$\|(sI - A)x\|^2 + |{\rm Re}\ s| \|C x\|^2 \geq m |{\rm Re}\ s|^2 \|x\|^2,$$ which is stronger than approximate observability but in general weaker than exact observability (Jacob et al., 2021). This distinction is critical for physical systems where only indirect energy measurements are feasible over distributed domains.

6. Applications, Genericity, and Geometric Criteria

Infinite-dimensional Hautus conditions underpin numerous results and methodologies:

• Genericity results: In the behavioral module-theoretic approach, controllability is generic for strictly underdetermined behaviors (system of PDEs with more unknowns than equations), whereas overdetermined systems are generically uncontrollable (Shankar, 2013).
• Geometric control: For fractional Schrödinger equations, spectral inequalities derived from the infinite-dimensional Hautus test lead to geometric control conditions, such as the "thickness" of the control set, necessary and sufficient for exact controllability (Martin et al., 2020).
• RKHS inference and testing: In estimation and hypothesis testing with infinite-dimensional nuisance parameters (such as in RKHS settings), a Hautus-type principle is used to orthogonalize test statistics against infinite-dimensional null spaces, leading to robust and size-correct inference (Sancetta, 2016).

7. Challenges and Open Directions

Transitioning from finite to infinite dimensions presents distinct technical challenges:

• Loss of matrix rank/determinant structure: Operator range and density properties must replace rank, with careful attention to domains and admissibility in unbounded settings.
• Spectral complications: The spectrum of infinite-dimensional operators can be continuous or lack a complete eigenbasis, complicating both algebraic and analytic statements of controllability.
• Genericity and perturbation robustness: Algebraic-geometric criteria must be shown robust across structured system classes, often leveraging concepts from algebraic geometry and module theory.
• Non-autonomous and time-varying systems: Analysis of time-dependent operators (non-autonomous systems) relies on advanced functional analytic tools and energy estimates, as direct analogues of the autonomous resolvent-based Hautus test may fail (Hoang, 2017, Haak et al., 2018).

Despite these challenges, the infinite-dimensional Hautus test and its myriad adaptations offer a unifying conceptual and practical framework for analyzing and synthesizing controllability, observability, and stability in a broad class of distributed, stochastic, and functional settings. These tests remain central to the ongoing development of rigorous theory and algorithms in modern control of complex and infinite-dimensional systems.