Kalman's Controllability Criterion

• Kalman's controllability criterion is an algebraic condition that checks if the controllability matrix spans the entire state space, ensuring full system reachability.
• It extends to various classes including PDEs, delay systems, and second-order models by employing generalized rank and moment conditions.
• The criterion underpins control law synthesis by linking algebraic properties of system matrices to the feasibility of state steering, highlighting both practical design and spectral constraints.

Kalman's controllability criterion provides an algebraic characterization of when a linear (finite- or infinite-dimensional) dynamical system can be driven from any initial state to any final state by a suitable choice of input over a finite time interval. The criterion, first articulated for finite-dimensional systems by R. E. Kalman in 1960, has since been demonstrated to underlie a broad range of system classes, including parabolic and hyperbolic PDEs, delay systems, and discrete or continuous second-order systems. At its core, the Kalman criterion tests whether the range generated by repeated applications of the system matrix and the control matrix covers the entire state space.

1. Classical Kalman Rank Condition: Finite-Dimensional Linear Systems

For the continuous-time linear time-invariant (LTI) system

$$\dot{x}(t) = A x(t) + B u(t), \quad x\in\mathbb{R}^n,\, u\in\mathbb{R}^m,$$

the Kalman controllability matrix is defined as

$$\mathcal{C} = [B\,|\,AB\,|\,A^2B\,|\,\cdots\,|\,A^{n-1}B] \in \mathbb{R}^{n \times nm}.$$

The fundamental criterion is: $$\text{The system is controllable} \iff \operatorname{rank}\mathcal{C} = n.$$ This guarantees that, for any initial and final state, there exists an admissible control steering the system in finite time (Mahmudov, 2019). The Cayley–Hamilton theorem ensures no new directions are obtained beyond $A^{n-1}B$.

2. Extensions to Partial Differential Equations

Parabolic and Degenerate Parabolic Systems

For coupled systems of heat equations or degenerate parabolic PDEs with constant coupling and control matrices, the Kalman criterion generalizes as follows. Approximate or null controllability of the infinite-dimensional system is equivalent to the classical Kalman rank condition applied to the pair $(A,B)$. For the impulse-controlled heat system on a bounded domain $\Omega$,

$$\begin{cases} \partial_t y - A y = 0 & \text{between impulses} \ y(T_k^+) - y(T_k^-) = 1_{\omega} B u_k & \text{at impulse times} \end{cases}$$

one has

$$\text{Approximate controllability} \iff \operatorname{rank}[B, AB, \ldots, A^{n-1}B]=n.$$

The analysis involves representing the reachable state as a linear combination of exponentials and leveraging duality with observability inequalities. For approximate controllability via impulses in coupled heat equations, it suffices to select $n$ impulse times within an interval of width less than $$d_A = \min\{\pi/|\Im \lambda|\,:\,\lambda\in\sigma(A)\}$$, ensuring the associated exponential block matrix $$\mathcal{C} = [B\,|\,AB\,|\,A^2B\,|\,\cdots\,|\,A^{n-1}B] \in \mathbb{R}^{n \times nm}.$$0 has rank $$\mathcal{C} = [B\,|\,AB\,|\,A^2B\,|\,\cdots\,|\,A^{n-1}B] \in \mathbb{R}^{n \times nm}.$$1 (Qin et al., 2017). Null controllability in certain PDE settings may require a spectral family of the classical Kalman rank conditions, i.e., for each spatial mode with eigenvalue $$\mathcal{C} = [B\,|\,AB\,|\,A^2B\,|\,\cdots\,|\,A^{n-1}B] \in \mathbb{R}^{n \times nm}.$$2,

$$\mathcal{C} = [B\,|\,AB\,|\,A^2B\,|\,\cdots\,|\,A^{n-1}B] \in \mathbb{R}^{n \times nm}.$$3

which ensures controllability across all Fourier modes of the infinite-dimensional state (Hassi et al., 2018).

Hyperbolic Systems and Boundary Control

Boundary controllability of systems such as coupled 1-D wave equations with boundary input also relies on a Kalman-type rank condition. If $$\mathcal{C} = [B\,|\,AB\,|\,A^2B\,|\,\cdots\,|\,A^{n-1}B] \in \mathbb{R}^{n \times nm}.$$4 (coupling) and $$\mathcal{C} = [B\,|\,AB\,|\,A^2B\,|\,\cdots\,|\,A^{n-1}B] \in \mathbb{R}^{n \times nm}.$$5 (boundary control input), exact controllability holds if and only if

$$\mathcal{C} = [B\,|\,AB\,|\,A^2B\,|\,\cdots\,|\,A^{n-1}B] \in \mathbb{R}^{n \times nm}.$$6

alongside additional spectral nonresonance and minimum time requirements, due to the clustered spectral structure of the PDE. Modal decomposition reduces the PDE to a countable family of forced scalar ODEs, each steered by the same control, and solvability of the infinite family of moment equations is directly linked to the Kalman condition (Avdonin et al., 2019).

3. Kalman-Type Condition for Systems with Delay and Second-Order Structure

Discrete Linear Delay Systems

For discrete-time systems with a single constant delay,

$$\mathcal{C} = [B\,|\,AB\,|\,A^2B\,|\,\cdots\,|\,A^{n-1}B] \in \mathbb{R}^{n \times nm}.$$7

define recursively the set of matrices $$\mathcal{C} = [B\,|\,AB\,|\,A^2B\,|\,\cdots\,|\,A^{n-1}B] \in \mathbb{R}^{n \times nm}.$$8 and assemble the delayed controllability matrix $$\mathcal{C} = [B\,|\,AB\,|\,A^2B\,|\,\cdots\,|\,A^{n-1}B] \in \mathbb{R}^{n \times nm}.$$9 from blocks $$\text{The system is controllable} \iff \operatorname{rank}\mathcal{C} = n.$$0, $$\text{The system is controllable} \iff \operatorname{rank}\mathcal{C} = n.$$1. The Kalman-type criterion is: $$\text{The system is controllable} \iff \operatorname{rank}\mathcal{C} = n.$$2 where $$\text{The system is controllable} \iff \operatorname{rank}\mathcal{C} = n.$$3 is the input dimension. For $$\text{The system is controllable} \iff \operatorname{rank}\mathcal{C} = n.$$4, this collapses to the standard rank test (Asadzade et al., 18 Aug 2025).

Second-Order Linear Systems

For continuous-time second-order systems (mechanical, mass–damping–stiffness form)

$$\text{The system is controllable} \iff \operatorname{rank}\mathcal{C} = n.$$5

the generalized Kalman test is conducted without introducing the first-order lift. By repeated differentiation and defining the moment matrices $$\text{The system is controllable} \iff \operatorname{rank}\mathcal{C} = n.$$6 via recursion ($$\text{The system is controllable} \iff \operatorname{rank}\mathcal{C} = n.$$7, $$\text{The system is controllable} \iff \operatorname{rank}\mathcal{C} = n.$$8, $$\text{The system is controllable} \iff \operatorname{rank}\mathcal{C} = n.$$9), controllability is equivalent to

$A^{n-1}B$0

A similar structure applies to the discrete-time second-order case, with corresponding reduction in horizon length and moments (Mahmudov, 2019).

4. Role of Duality, Observability, and Gramians

The equivalence between controllability and certain observability inequalities in the dual system is established by standard duality arguments. In the infinite-dimensional (PDE) case, unique continuation properties and Carleman estimates are key analytic tools for establishing equivalence between primal controllability and dual observability properties. Controllability is further characterized by the invertibility (nonsingularity) of associated controllability Gramians: $A^{n-1}B$1 in the classical case, or by its explicit discrete or delayed analogues in difference-delay systems (Asadzade et al., 18 Aug 2025).

5. Illustrative Examples and Modal Decomposition

Examples highlight the precise application of the Kalman rank condition across system classes:

• For two coupled heat equations with $A^{n-1}B$2 a rotation matrix and $A^{n-1}B$3 nonzero, approximate controllability under impulse control fails if control times are separated by precisely the “spectral gap” $A^{n-1}B$4, illustrating the subtlety of the required time/window condition (Qin et al., 2017).
• In degenerate parabolic systems with non-diagonalizable diffusion, the necessity to check rank$A^{n-1}B$5 at each spatial mode is exemplified by a 2x2 Jordan block structure, where the presence of a single “1” in the block provides controllability for all spatial modes (Hassi et al., 2018).
• For boundary control of coupled N-wave equations, the minimal time and nonresonance conditions illustrate that the Kalman condition is necessary but not always sufficient without complementary spectral properties (Avdonin et al., 2019).

6. Explicit Control Synthesis, Implementation, and Limitations

When the Kalman condition is satisfied, explicit formulas for control laws are derivable via inversion of the controllability Gramian (or its discrete/impulsive/delayed/PDE analogs). For delay systems, for example, the unique control achieving a target state is given by

$A^{n-1}B$6

with $A^{n-1}B$7 the “controllable part” of the final state (Asadzade et al., 18 Aug 2025). In second-order systems, invertibility of the moment matrix provides the necessary set of control derivatives. The classical Kalman rank criterion does not guarantee null or exact controllability for all infinite-dimensional or degenerate systems, with spectral and temporal constraints sometimes required. For instance, impulse-controlled parabolic systems may be approximately but not null-controllable.

7. Connections, Context, and Limits

The Kalman controllability criterion is foundational for modern control theory, providing a direct and efficiently testable link between algebraic structure (matrix rank conditions) and system-theoretic properties (controllability, reachability). Its generalizations govern a wide variety of dynamical regimes, subject to proper handling of infinite-dimensionality, spectrum, operator structure, and the temporal support of controls. Extensions to systems with delays, second-order dynamics, impulse or boundary controls, and PDEs reinforce both the necessity and the sharpness of Kalman's rank condition, albeit often in a more nuanced, mode-wise or time-constrained form. Results confirm that whenever the criterion fails, a positive codimension of the state space remains uncontrollable regardless of the sophistication of the control strategy (Qin et al., 2017, Hassi et al., 2018, Avdonin et al., 2019, Mahmudov, 2019, Asadzade et al., 18 Aug 2025).