Controllability Through Configuration

• Controllability through configuration is a strategy that achieves system control by adjusting configuration variables such as geometric arrangements and feature selections instead of direct time trajectory tracking.
• It encompasses methodologies like configuration path control in underactuated mechanics, supervisory synthesis in dynamic feature systems, and structural design for actuator-sensor feedback placement.
• This approach leverages geometric insights from shape space and symmetric spaces to redefine reachability, observability, and safety based on inherent configuration structures.

“Controllability through configuration” denotes a family of ideas in which controllability is achieved, analyzed, or improved by acting on configuration variables, architectural arrangements, geometric decompositions, path parameterizations, boundary-actuation layouts, or runtime control settings rather than only by prescribing conventional input trajectories. Across the literature, the phrase covers several technically distinct mechanisms: tracking a configuration path rather than a time trajectory in underactuated mechanics (Pankov, 2022); enforcing safe behavior by synthesizing supervisors over dynamic feature configurations in product lines (Thuijsman et al., 2022); characterizing locomotion by the decomposition of configuration into shape space and group space (Kadam et al., 2016); selecting actuator, sensor, and feedback configurations in large-scale structured systems (Pequito et al., 2013); and treating runtime policies, fallback architecture, or software reconfiguration state as the locus of control in autonomous vehicles, agentic AI, and component-based software (Shinde et al., 5 Jun 2026, Li et al., 26 May 2026, Bouhadiba et al., 2011). The unifying theme is that control authority is mediated by structure in configuration space, feature space, architecture space, or deployment space, with the resulting notions of reachability, observability, and safety depending strongly on how that structure is chosen.

1. Configuration as a control object

In one major usage, configuration is the state representation through which control is organized. “Configuration Path Control” proposes stabilizing a learned policy in the space of configuration paths rather than time trajectories (Pankov, 2022). Its central idea is to replace direct tracking of a time-parameterized reference trajectory with tracking of a configuration path, namely the set of configurations visited by a trajectory, independent of timing. Two trajectories related by a time reparameterization are treated as equivalent if they traverse the same path in configuration space. The target is therefore less restrictive than exact trajectory tracking: instead of matching a desired state $x_d(t)=[q_d(t);\dot q_d(t)]$ at a specific time, the controlled system is required to converge to the same path in the configuration variables $q$, while the timing is allowed to slide (Pankov, 2022).

This formulation is developed for underactuated nonlinear mechanical systems with dynamics

$D(q)\ddot q + H(q,\dot q) = B_{\tau}\tau,$

state

$x \equiv [q;\dot q],$

and control matrix

$B(q) \equiv D^{-1}(q)B_\tau.$

For $N>M$, only a subset of acceleration directions is available at each configuration. The paper isolates the unattainable directions by partitioning $q=[\chi;\psi]$, with $\dim(\chi)=M$, and defining

$$b(q) = [B_{\psi}(q)B_{\chi}^{-1}(q),-I_{N-M}]^\top .$$

Then the achievable acceleration perturbations satisfy

$$b^\top(q)\,\Delta \ddot q = 0 + \mathcal{O}(\Delta\tau\varepsilon),$$

so the achievable acceleration perturbations lie in the nullspace of $q$0 (Pankov, 2022). The same paper states explicitly that it does not provide a full nonlinear controllability analysis in the Lie-algebraic sense, nor formal accessibility or stabilizability tests. Its contribution is a concrete local actuation geometry in configuration coordinates through $q$1 and $q$2, used to define path reachability (Pankov, 2022).

A related but more geometric use of configuration appears in locomotion on principal bundles. For the Purcell swimmer, the full configuration manifold splits as

$q$3

where $q$4 is the shape manifold and $q$5 is the Lie group of body motions (Kadam et al., 2016). This yields two distinct notions: strong controllability, which concerns reachability in the full configuration space $q$6, and weak controllability, which concerns reachability only in the group component $q$7 (Kadam et al., 2016). The paper’s criterion is expressed through the local connection and the sequence of Lie-algebraic subspaces $q$8, with weak controllability when

$q$9

and strong controllability when

$D(q)\ddot q + H(q,\dot q) = B_{\tau}\tau,$0

For the standard 3-link Purcell swimmer, the paper concludes that

$D(q)\ddot q + H(q,\dot q) = B_{\tau}\tau,$1

for all points in shape space, implying local strong controllability everywhere (Kadam et al., 2016).

A broader geometric generalization is given for driftless systems on symmetric spaces. There, the relevant closure is not the usual Lie algebra generated by the controlled vector fields, but the Lie triple system generated by them. The main theorem states that if

$D(q)\ddot q + H(q,\dot q) = B_{\tau}\tau,$2

then

$D(q)\ddot q + H(q,\dot q) = B_{\tau}\tau,$3

is globally controllable on the symmetric space $D(q)\ddot q + H(q,\dot q) = B_{\tau}\tau,$4 (Tiwari et al., 2021). This suggests a recurring principle: when the configuration manifold has a strong intrinsic decomposition, controllability may be governed by geometric closure operations native to that decomposition rather than by a generic state-space rank test.

2. Control by reconfiguration, feature selection, and structural design

A second major usage treats controllability as a property that can be engineered by choosing or synthesizing a valid configuration. In product lines with dynamic feature configuration, a framework is proposed that combines a feature-model view of valid configurations, a behavioral plant model of the system components, and requirements that can depend on both behavior and current configuration, then applies supervisory controller synthesis to compute a controller that restricts behavior so the combined system remains safe, nonblocking, controllable, and maximally permissive (Thuijsman et al., 2022).

The feature-model layer represents static validity through Boolean constraints such as root, mandatory, optional, alternative, or, requires, and excludes. In CIF, each feature is represented by an automaton with a Boolean variable present, and system validity is encoded by an algebraic Boolean expression such as $q=[\chi;\psi]$3 (Thuijsman et al., 2022). Dynamic configuration is introduced by changing a feature’s present value through events come and go, $q=[\chi;\psi]$4 so feature changes are modeled as events in the discrete-event system rather than as an external static parameter (Thuijsman et al., 2022). The paper states explicitly that come/go can be uncontrollable or controllable. If they are uncontrollable, the supervisor must guarantee correctness despite arbitrary reconfiguration; if they are controllable, the supervisor can decide when reconfiguration may occur, depending on the current state (Thuijsman et al., 2022).

The same theme appears in large-scale structural systems, where “configuration” means choosing the structural locations of inputs, outputs, and feedback links. A unified framework addresses three coupled design problems: minimum input selection to ensure structural controllability, minimum output selection to ensure structural observability, and minimum control configuration or feedback selection so that the closed-loop system has no structurally fixed modes (Pequito et al., 2013). The plant is

$D(q)\ddot q + H(q,\dot q) = B_{\tau}\tau,$5

with structural patterns $D(q)\ddot q + H(q,\dot q) = B_{\tau}\tau,$6, and feedback information pattern $D(q)\ddot q + H(q,\dot q) = B_{\tau}\tau,$7; under static output feedback,

$D(q)\ddot q + H(q,\dot q) = B_{\tau}\tau,$8

the closed-loop state matrix becomes

$D(q)\ddot q + H(q,\dot q) = B_{\tau}\tau,$9

The central graph-theoretic characterization is that structural controllability is completely determined by how inputs are placed relative to right-unmatched vertices of a maximum matching and non-top linked strongly connected components of $x \equiv [q;\dot q],$0 (Pequito et al., 2013). For dedicated inputs, if $x \equiv [q;\dot q],$1 is the number of right-unmatched vertices, $x \equiv [q;\dot q],$2 the number of non-top linked SCCs, and $x \equiv [q;\dot q],$3 the maximum top assignability index, then the minimum number of dedicated inputs is

$x \equiv [q;\dot q],$4

(Pequito et al., 2013).

On cograph networks, configuration likewise appears as control-node selection relative to graph structure. For Laplacian dynamics $x \equiv [q;\dot q],$5, where $x \equiv [q;\dot q],$6 is a connected cograph, the minimum number of control nodes is determined by the sibling partition

$x \equiv [q;\dot q],$7

and the network is controllable if one selects exactly $x \equiv [q;\dot q],$8 nodes from each sibling-partition cell $x \equiv [q;\dot q],$9, equivalently all nodes of each cell except one (Mousavi et al., 2018). The minimum number of control nodes is then

$B(q) \equiv D^{-1}(q)B_\tau.$0

where $B(q) \equiv D^{-1}(q)B_\tau.$1 and $B(q) \equiv D^{-1}(q)B_\tau.$2 is the number of sibling cells (Mousavi et al., 2018). In deterministic scale-free networks and Cayley trees, the same principle is expressed through recursive adjacency-matrix rank structure: topology induces repeated row and column patterns, these create rank deficiency, and exact controllability reduces to the number and placement pattern of driver nodes (Xu et al., 2014).

These results do not treat “configuration” as a metaphor. They treat it as an explicit control variable: a feature valuation, an actuator-sensor-feedback pattern, or a selected subset of nodes.

3. Geometric and shape-theoretic formulations

A third line of work studies controllability through geometric structure intrinsic to shape representations. For landmark manifolds,

$B(q) \equiv D^{-1}(q)B_\tau.$3

the control mechanism is not independent actuation of each landmark, but ambient vector fields on $B(q) \equiv D^{-1}(q)B_\tau.$4 acting simultaneously on all landmarks (Grong et al., 2024). The paper proves that for any $B(q) \equiv D^{-1}(q)B_\tau.$5, there exist two vector fields $B(q) \equiv D^{-1}(q)B_\tau.$6 on $B(q) \equiv D^{-1}(q)B_\tau.$7 such that for any $B(q) \equiv D^{-1}(q)B_\tau.$8 and any two distinct landmark configurations, there exists a finite composition of flows of $B(q) \equiv D^{-1}(q)B_\tau.$9 and $N>M$0 sending one configuration to the other (Grong et al., 2024). The statement is exact, global, and uniform in $N>M$1 once $N>M$2 is fixed. The mechanism is Lie-algebraic: lift ambient vector fields to the landmark manifold, compute iterated Lie brackets, show the lifted Lie algebra spans the full tangent space at every configuration, then invoke Chow–Rashevsky (Grong et al., 2024).

This result is especially sharp in the phrase “through configuration” because the configuration space itself is a shape space. The state is an ordered $N>M$3-tuple of pairwise distinct points, and the same pair of ambient control fields acts componentwise: $N>M$4 The paper’s key observation is that the same two preselected vector fields work for every $N>M$5, with the Vandermonde structure induced by distinct landmark positions supplying the needed spanning argument (Grong et al., 2024).

For symmetric spaces, the same configuration-centric viewpoint is expressed in different algebraic language. The tangent space at the base point is modeled by a Lie triple system rather than a Lie algebra, and admissible higher-order motions are synthesized through double brackets

$N>M$6

The paper’s examples on $N>M$7, exponential submanifolds of $N>M$8, and random matrix ensembles all show the same pattern: a comparatively small set of controlled directions suffices because the geometry of the symmetric space closes under the Lie triple operation to generate the full tangent space (Tiwari et al., 2021).

This suggests a broader interpretation. In these geometric papers, controllability through configuration means that the state manifold is not a generic Euclidean state space but a structured configuration manifold—principal bundle, symmetric space, or landmark manifold—and the controllability criterion is correspondingly adapted to that structure.

4. Boundary configuration and domain geometry in PDE control

For distributed-parameter systems, configuration appears as the geometric arrangement of the domain and the placement of controls on its boundary or subregions. In the Grushin-type parabolic problem,

$N>M$9

posed on

$q=[\chi;\psi]$0

null controllability depends crucially on the degeneracy set $q=[\chi;\psi]$1, the control strip lying entirely on the right side of that degeneracy, and the rectangular product geometry that allows Fourier decomposition in $q=[\chi;\psi]$2 (Beauchard et al., 2011). The main theorem states: if \