Nonuniform Exponential Detectability

• Nonuniform Exponential Detectability is a property of linear time-varying systems that achieves exponential decay of estimation errors using time-dependent observer gains.
• It is defined through an error dynamic inequality involving specific constants that ensure forward nonuniform exponential stability.
• The concept leverages a duality between observability and controllability to inform robust observer design in nonautonomous systems.

Nonuniform exponential detectability is a nonautonomous observer-theoretic property for linear systems in which the estimation error can be made exponentially stable with time-dependent, nonuniform weights. In the finite-dimensional framework developed for linear time-varying systems on $\mathbb R$, the property is formulated through the existence of an output injection $L(t)$ such that the error dynamics satisfy a forward estimate of the form $$\|\Phi_{A-LC}(t,s)\|\le M e^{\delta|s|}e^{-\beta(t-s)}$$, with $M\ge 1$, $\beta>0$, and $\delta\ge 0$; the factor $e^{\delta|s|}$ distinguishes the theory from the uniform case by allowing nonuniform dependence on the initial time (Huerta et al., 13 Aug 2025). Subsequent work places this notion within a broader reciprocity between nonuniform observability and detectability, while adjacent literature on nonuniform exponential dichotomies, Lyapunov exponents, and cocycles provides a wider dynamical context for interpreting the error system as a nonuniformly hyperbolic or contracting evolution family (Huerta et al., 15 Dec 2025).

1. Formal definition and system class

The direct detectability theory is formulated for finite-dimensional nonautonomous linear control systems

$$\begin{cases} \dot x(t)=A(t)x(t)+B(t)u(t),\ y(t)=C(t)x(t)+D(t)u(t), \end{cases} \qquad t\in J\subseteq \mathbb R,$$

with $x(t)\in\mathbb R^n$, $u(t)\in\mathbb R^p$, $L(t)$0, and measurable coefficient functions $L(t)$1 that are bounded on finite intervals. For observability and detectability, the analysis reduces to the homogeneous pair

$L(t)$2

whose transition matrix is denoted by $L(t)$3 (Huerta et al., 13 Aug 2025).

The target stability notion is nonuniform exponential stability forward. For a linear system

$L(t)$4

the system is nonuniformly exponentially stable forward if there exist constants

$L(t)$5

such that, for all $L(t)$6 and $L(t)$7,

$L(t)$8

Nonuniform exponential detectability is then defined through the observer error equation

$L(t)$9

The system is nonuniformly exponentially detectable if there exists an output injection gain $$\|\Phi_{A-LC}(t,s)\|\le M e^{\delta|s|}e^{-\beta(t-s)}$$0 such that the error system is nonuniformly exponentially stable forward; equivalently, there must exist $$\|\Phi_{A-LC}(t,s)\|\le M e^{\delta|s|}e^{-\beta(t-s)}$$1 and constants $$\|\Phi_{A-LC}(t,s)\|\le M e^{\delta|s|}e^{-\beta(t-s)}$$2, $$\|\Phi_{A-LC}(t,s)\|\le M e^{\delta|s|}e^{-\beta(t-s)}$$3, $$\|\Phi_{A-LC}(t,s)\|\le M e^{\delta|s|}e^{-\beta(t-s)}$$4 such that

$$\|\Phi_{A-LC}(t,s)\|\le M e^{\delta|s|}e^{-\beta(t-s)}$$5

A notable distinction from the uniform theory is that the gain $$\|\Phi_{A-LC}(t,s)\|\le M e^{\delta|s|}e^{-\beta(t-s)}$$6 is not required to be bounded (Huerta et al., 13 Aug 2025).

This places nonuniform exponential detectability as the direct analogue of uniform exponential detectability, but with uniform exponential stability replaced by nonuniform exponential stability forward. The corresponding observer can be written as

$$\|\Phi_{A-LC}(t,s)\|\le M e^{\delta|s|}e^{-\beta(t-s)}$$7

so that $$\|\Phi_{A-LC}(t,s)\|\le M e^{\delta|s|}e^{-\beta(t-s)}$$8 obeys the injected error dynamics (Huerta et al., 15 Dec 2025).

2. Nonuniform observability framework

The observability side of the theory is organized around nonuniform growth estimates. One basic hypothesis is nonuniform bounded growth,

$$\|\Phi_{A-LC}(t,s)\|\le M e^{\delta|s|}e^{-\beta(t-s)}$$9

and a related, more general condition is the nonuniform Kalman property

$M\ge 1$0

for some $M\ge 1$1 and $M\ge 1$2 (Huerta et al., 13 Aug 2025).

For the pair $M\ge 1$3, the observability Gramians are

$M\ge 1$4

and

$M\ge 1$5

Uniform complete observability requires fixed-window lower and upper coercivity bounds on both $M\ge 1$6 and $M\ge 1$7. The nonuniform extension, nonuniform complete observability, replaces time-independent constants by exponential weights in $M\ge 1$8. Specifically, the system is nonuniformly completely observable on $M\ge 1$9 if there exist fixed numbers

$\beta>0$0

and positive functions

$\beta>0$1

such that, for every $\beta>0$2, there exists $\beta>0$3 with, for every $\beta>0$4,

$\beta>0$5

and

$\beta>0$6

Within this hierarchy, uniform complete observability implies nonuniform complete observability, and nonuniform complete observability implies complete observability; both implications are strict in general (Huerta et al., 13 Aug 2025).

A structural theorem used repeatedly in the theory states that, on $\beta>0$7, any two among the nonuniform Kalman property, the $\beta>0$8-estimate, and the $\beta>0$9-estimate imply the third. This gives a nonuniform replacement for the classical equivalence between bounded growth plus one Gramian bound and full complete observability (Huerta et al., 13 Aug 2025).

3. Duality with controllability and the implication $\delta\ge 0$0

The principal sufficiency theorem states that nonuniform complete observability on $\delta\ge 0$1 implies nonuniform exponential detectability. The proof is not obtained from an observability Riccati equation directly. Instead, it proceeds through duality with nonuniform complete controllability and an imported nonuniform stabilizability theorem (Huerta et al., 13 Aug 2025).

For the pair $\delta\ge 0$2, the adjoint system is

$\delta\ge 0$3

and the dual system is

$\delta\ge 0$4

Their transition matrices satisfy

$\delta\ge 0$5

If $\delta\ge 0$6 denote the controllability Gramians of the adjoint and dual systems, then the exact identities

$\delta\ge 0$7

and

$\delta\ge 0$8

hold. From these identities one obtains the equivalences: $\delta\ge 0$9 is nonuniformly completely observable if and only if the adjoint system is nonuniformly completely controllable, and likewise if and only if the dual system is nonuniformly completely controllable (Huerta et al., 13 Aug 2025).

The detectability implication is then assembled in four steps. First, nonuniform complete observability of $e^{\delta|s|}$0 implies nonuniform complete controllability of the dual system. Second, a controllability-to-stabilizability theorem yields a feedback $e^{\delta|s|}$1 such that the dual closed-loop system

$e^{\delta|s|}$2

is nonuniformly exponentially stable forward. Third, the paper proves equivalence among forward nonuniform exponential stability of the original system, backward nonuniform exponential stability of the adjoint system, and forward nonuniform exponential stability of the dual system. Fourth, one sets

$e^{\delta|s|}$3

which yields the observer gain for the original error equation

$e^{\delta|s|}$4

Hence the error dynamics become nonuniformly exponentially stable forward, which is exactly nonuniform exponential detectability (Huerta et al., 13 Aug 2025).

This construction is explicitly an observer-design statement. The gain is obtained indirectly from state-feedback stabilization of the dual system, not from a direct observability synthesis on the original pair.

4. Converse results, strictness, and representative examples

The implication $e^{\delta|s|}$5 is sufficient only. A central point of the theory is that the converse fails in general. The standard counterexample uses

$e^{\delta|s|}$6

The transition matrix is

$e^{\delta|s|}$7

so

$e^{\delta|s|}$8

which violates the nonuniform Kalman property and therefore excludes nonuniform complete observability. Yet choosing

$e^{\delta|s|}$9

gives

$$\begin{cases} \dot x(t)=A(t)x(t)+B(t)u(t),\ y(t)=C(t)x(t)+D(t)u(t), \end{cases} \qquad t\in J\subseteq \mathbb R,$$0

and the resulting error transition matrix satisfies a nonuniform exponential decay estimate. Thus the system is nonuniformly exponentially detectable but not nonuniformly completely observable (Huerta et al., 13 Aug 2025).

Other examples in the same work clarify the hierarchy of observability notions. A scalar system with output $$\begin{cases} \dot x(t)=A(t)x(t)+B(t)u(t),\ y(t)=C(t)x(t)+D(t)u(t), \end{cases} \qquad t\in J\subseteq \mathbb R,$$1 is shown to satisfy a nonuniform bounded-growth estimate and to be nonuniformly completely observable, but not uniformly completely observable. Another example, obtained by duality with a controllability example, is completely observable but neither uniformly completely observable nor nonuniformly completely observable. These examples establish that nonuniform complete observability is strictly more general than uniform complete observability and strictly stronger than complete observability (Huerta et al., 13 Aug 2025).

Later work restores a converse implication under additional hypotheses. If the plant $$\begin{cases} \dot x(t)=A(t)x(t)+B(t)u(t),\ y(t)=C(t)x(t)+D(t)u(t), \end{cases} \qquad t\in J\subseteq \mathbb R,$$2 has non-uniform bounded growth

$$\begin{cases} \dot x(t)=A(t)x(t)+B(t)u(t),\ y(t)=C(t)x(t)+D(t)u(t), \end{cases} \qquad t\in J\subseteq \mathbb R,$$3

is nonuniformly exponentially stable backward with constants $$\begin{cases} \dot x(t)=A(t)x(t)+B(t)u(t),\ y(t)=C(t)x(t)+D(t)u(t), \end{cases} \qquad t\in J\subseteq \mathbb R,$$4 satisfying $$\begin{cases} \dot x(t)=A(t)x(t)+B(t)u(t),\ y(t)=C(t)x(t)+D(t)u(t), \end{cases} \qquad t\in J\subseteq \mathbb R,$$5, the output satisfies

$$\begin{cases} \dot x(t)=A(t)x(t)+B(t)u(t),\ y(t)=C(t)x(t)+D(t)u(t), \end{cases} \qquad t\in J\subseteq \mathbb R,$$6

and the observer gain satisfies

$$\begin{cases} \dot x(t)=A(t)x(t)+B(t)u(t),\ y(t)=C(t)x(t)+D(t)u(t), \end{cases} \qquad t\in J\subseteq \mathbb R,$$7

then nonuniform exponential detectability implies nonuniform complete observability (Huerta et al., 15 Dec 2025). The same paper also proves that nonuniform complete observability is preserved under output feedback $$\begin{cases} \dot x(t)=A(t)x(t)+B(t)u(t),\ y(t)=C(t)x(t)+D(t)u(t), \end{cases} \qquad t\in J\subseteq \mathbb R,$$8 when the feedback term decays sufficiently fast relative to the plant growth and output growth. In this sense, the classical equivalence between observability and detectability survives in the nonuniform setting only after the addition of explicit backward-stability and decay assumptions (Huerta et al., 15 Dec 2025).

5. Dynamical, cocycle, and spectral interpretations

Although detectability is formulated directly for finite-dimensional time-varying systems, several adjacent theories suggest a broader interpretation. For linear cocycles over maps and flows, Datko–Pazy type summability and integrability conditions imply negativity of all Lyapunov exponents, and negativity of the largest Lyapunov exponent yields nonuniform exponential stability. Applied to an estimation-error cocycle, this suggests a cocycle-theoretic version of nonuniform exponential detectability: if the error cocycle satisfies the relevant Datko–Pazy condition, then all its Lyapunov exponents are negative and the error dynamics are nonuniformly exponentially stable (Dragicevic, 2017).

A complementary spectral picture comes from nonuniform exponential dichotomy theory. For nonautonomous linear systems, the nonuniform exponential dichotomy spectrum is defined through shifted systems

$$\begin{cases} \dot x(t)=A(t)x(t)+B(t)u(t),\ y(t)=C(t)x(t)+D(t)u(t), \end{cases} \qquad t\in J\subseteq \mathbb R,$$9

and consists of those $x(t)\in\mathbb R^n$0 for which the shifted system admits no nonuniform exponential dichotomy. Under nonuniformly bounded growth, this spectrum is a union of at most $x(t)\in\mathbb R^n$1 closed intervals, and one has the inclusion

$x(t)\in\mathbb R^n$2

This suggests a rate-oriented interpretation of detectability: for an observer error system $x(t)\in\mathbb R^n$3, placing relevant shifts in the resolvent of the nonuniform dichotomy spectrum amounts to enforcing nonuniform hyperbolic separation with prescribed decay margins (Zhu, 2019).

This spectral viewpoint does not by itself define detectability, because it has no output map or observer gain. Its significance is structural. It replaces autonomous eigenspace arguments by invariant bundles, spectral intervals, and nonuniform growth rates, which are the natural geometric objects once the error dynamics cease to be uniform or autonomous.

6. Robustness and broader functional-analytic extensions

Robustness results for nonuniform exponential dichotomies furnish a stability theory for detectability once detectability is encoded as a stability or dichotomy property of the observer error system. For arbitrary Banach-space evolution families, possibly with noninvertible operators, one robustness theorem states that if an evolution family admits a nonuniform exponential dichotomy with constants $x(t)\in\mathbb R^n$4, $x(t)\in\mathbb R^n$5, and if the perturbation $x(t)\in\mathbb R^n$6 satisfies

$x(t)\in\mathbb R^n$7

then the perturbed evolution family

$x(t)\in\mathbb R^n$8

also admits a nonuniform exponential dichotomy (Dragicevic, 14 Dec 2025). Because the perturbation condition is integral rather than pointwise, the theorem allows perturbations that are large at isolated times provided their exponentially weighted convolution remains small. In observer language, this is naturally interpreted as robustness with respect to time-varying injection mismatch or modeling error.

For Banach-space evolution processes, nonuniform exponential dichotomies are likewise stable under weighted short-time perturbations, and the associated invariant projections are unique and depend continuously on the perturbation. The stable space can be characterized as

$x(t)\in\mathbb R^n$9

while the unstable space is characterized by backward bounded solutions. This offers a functional-analytic mechanism for tracking how a stable/unstable splitting of error dynamics deforms under perturbation (Caraballo et al., 2020).

These robustness theorems do not mention detectability explicitly. Their relevance is conditional but strong: if nonuniform exponential detectability is realized by rendering the error process nonuniformly exponentially stable or dichotomic, then the property persists under the perturbation regimes covered by the corresponding dichotomy roughness theory. In that sense, nonuniform exponential detectability sits at the intersection of observer synthesis, nonuniform observability, and nonuniform hyperbolic dynamics.