Topology-Enabled Modal Control

• Topology-enabled modal control is a research direction that uses designed topological structures to regulate controllability and modal behavior across various systems.
• Methodologies include constructing non-Hausdorff state-space topologies, leveraging network graphs, and employing local homotopy invariants to guide multi-vehicle trajectory optimization.
• The approach spans diverse domains—from electromagnetic design to homotopy type theory—illustrating both enabling mechanisms and structural barriers in achieving desired system performance.

Searching arXiv for papers relevant to topology-enabled modal control and adjacent uses of topology/modality. Topology-enabled modal control denotes a family of constructions in which topology is used to regulate controllability, mode selection, or modal membership. In the cited literature, this appears in several distinct forms: by building a topology on a state space so that an attainable set becomes dense (Moustapha, 2019); by analyzing how network topology and repeated eigenvalues constrain controllability in linearly coupled subsystems (Xue et al., 2018) and in structured dynamical-node networks (Mousavi et al., 2021); by imposing a differentiable local homotopy invariant to select interaction modes in multi-vehicle trajectory optimization (Ma et al., 7 Mar 2025); by using density-based topology optimization to manipulate characteristic modes of conducting surfaces (Tucek et al., 6 Feb 2025); and, in homotopy type theory, by giving internal presentations of topological modalities that make modal subuniverses computationally accessible (Williams, 31 Jan 2025). This suggests that the expression names an umbrella of related strategies rather than a single standardized theory.

1. Meanings of topology and modality across the literature

In control on Banach or Hilbert spaces, topology enters directly through the definition of approximate controllability: density of the attainable set depends on the chosen topology on the state space. The central move of the topological method for controllability is therefore to change the topology itself so that the reachable set becomes dense by construction (Moustapha, 2019).

In networked linear systems, topology refers to the interconnection graph and its coupling matrix $G$. There the modal object is the family of block dynamics $A+\lambda_i BC$, indexed by eigenvalues $\lambda_i$ of the network matrix, and controllability is governed by the distinction between network-invariant modes and special-repeat modes (Xue et al., 2018). In structured networks with dynamical nodes, the corresponding graph-theoretic object is the $\Delta$-network graph, and mode-specific controllability is characterized through zero forcing on that graph (Mousavi et al., 2021).

In trajectory optimization, topology is neither a state-space topology nor a graph alone. It is a local homotopy class encoded by the sign of a differentiable invariant evaluated at the closest interaction point. That invariant is then inserted into the optimization objective so that the solver can reproduce different interaction patterns from the same initial values and support user-designed interaction patterns (Ma et al., 7 Mar 2025).

In electromagnetic design, topology is the material distribution of a conducting surface in a fixed design domain, and the modal quantities are the characteristic modes of the structure. Density-based topology optimization is used to manipulate resonance, modal $Q$-factor, and simultaneous resonance of multiple modes, while feeding synthesis is deferred to a posteriori design (Tucek et al., 6 Feb 2025).

In homotopy type theory, “modal” has a different technical meaning. A topological modality is a reflective subuniverse given by nullification at a family of propositions, and a presentation of that modality provides computational access through covers, internal sheaf conditions, and local choice (Williams, 31 Jan 2025). A plausible implication is that “modal control” here concerns the control of reasoning inside a modal subuniverse rather than dynamical steering.

2. Engineered topologies and controllability by density construction

The topological method for controllability studies a control system on a Banach space $X$,

$$\frac{dx(t)}{dt}=f(x(t),u(t)), \qquad x(0)=x_0,\qquad t\ge 0,$$

with admissible controls $u\in U^{ad}$. For fixed $t\in[0,T]$, the attainable set is

$$A(t):=\{x^u(t)\;:\; x^u(\cdot)\text{ solves (2) for some }u\in U^{ad}\}.$$

Approximate controllability on $A+\lambda_i BC$0 means that $A+\lambda_i BC$1 is dense in $A+\lambda_i BC$2, exact controllability means $A+\lambda_i BC$3, and the same distinction is used for $A+\lambda_i BC$4 on an unbounded horizon (Moustapha, 2019).

The distinctive step is to construct a new topology from a strict nonempty subset $A+\lambda_i BC$5. Define

$A+\lambda_i BC$6

Then $A+\lambda_i BC$7 is a topology on $A+\lambda_i BC$8, and the closure of $A+\lambda_i BC$9 is $\lambda_i$0. Every nonempty set is “inflated” by adding $\lambda_i$1, so $\lambda_i$2 becomes dense in the resulting topology. The paper emphasizes several consequences: the induced topology on $\lambda_i$3 is larger than the original one, the topology is not Hausdorff, and it is remarked to be not metrizable. Because the topology need not be metric or first-countable, convergence is formulated with nets, using the closure criterion

$\lambda_i$4

The main abstract conclusion is that if $\lambda_i$5, then there exists a topology on $\lambda_i$6 for which $\lambda_i$7 is dense. The method therefore proves approximate controllability not by observability inequalities, PDE estimates, fixed-point theorems, or degree-theoretic arguments, but by constructing a topology whose closure operator makes the reachable set dense by design (Moustapha, 2019).

The paper extends this mechanism to semilinear systems of the form

$\lambda_i$8

where $\lambda_i$9 generates a $\Delta$0-semigroup on a Hilbert space $\Delta$1. It recalls negative results when $\Delta$2 generates a compact semigroup or an analytic semigroup, and then states that even if the semilinear system is not approximately controllable in the usual Borel topology, there exists a topology on $\Delta$3 such that the system is approximately controllable.

The illustrative examples show the same pattern. For the controlled bilinear one-dimensional Schrödinger equation,

$\Delta$4

the controllability question is cast in terms of the reachable set $\Delta$5 as $\Delta$6 varies in $\Delta$7, and the paper states that the system is controllable even in the special case $\Delta$8. For the Korteweg–de Vries equation and the Saint-Venant water-tank system, the result is approximate controllability in the sense that there exists a topology on the corresponding state space such that the attainable set is dense. In each case, the density statement is achieved through topologizing the state space around the reachable set (Moustapha, 2019).

A common misunderstanding is to read these results as classical exact controllability in the original PDE topology. They are instead statements about density in a specially constructed non-Hausdorff, non-metrizable topology.

3. Network topology, repeated modes, and structural controllability of modes

For networks of $\Delta$9 identical subsystems,

$Q$0

the stacked dynamics are

$Q$1

If $Q$2 is diagonalizable with eigenvalues $Q$3, then the eigenvalues of the full network matrix are exactly the union of the eigenvalues of the matrices $Q$4. This modal decomposition leads to a precise taxonomy of repeated eigenvalue phenomena (Xue et al., 2018).

A network-invariant mode is a complex number $Q$5 such that

$Q$6

Such modes include uncontrollable or unobservable eigenvalues of the local subsystem and every decentralized fixed mode of $Q$7. They can also exist even when the subsystem is controllable and observable and when there are no decentralized fixed modes. Their significance is that they produce high multiplicity in the full network: if $Q$8 is diagonalizable, the same eigenvalue appears in every block $Q$9. The paper proves that if a network-invariant mode exists, controllability requires

$X$0

where $X$1 is the number of actuated subsystems and $X$2 is the input dimension per subsystem. A stronger barrier appears for projection-fixed network-invariant modes, for which controllability requires $X$3. The condition is նաև refined to weakly connected partitions: for a subset $X$4, controllability requires

$X$5

By contrast, a special-repeat mode is shared by $X$6 only for a finite set of $X$7-values. Its network-repeat set is

$X$8

and the paper proves that the number of distinct $X$9 values in $$\frac{dx(t)}{dt}=f(x(t),u(t)), \qquad x(0)=x_0,\qquad t\ge 0,$$0 is at most $$\frac{dx(t)}{dt}=f(x(t),u(t)), \qquad x(0)=x_0,\qquad t\ge 0,$$1. The associated generalized left eigenvectors and their projections onto $$\frac{dx(t)}{dt}=f(x(t),u(t)), \qquad x(0)=x_0,\qquad t\ge 0,$$2 are linearly independent. Consequently, if $$\frac{dx(t)}{dt}=f(x(t),u(t)), \qquad x(0)=x_0,\qquad t\ge 0,$$3 is diagonalizable, all repeated eigenvalues are special-repeat modes, and $$\frac{dx(t)}{dt}=f(x(t),u(t)), \qquad x(0)=x_0,\qquad t\ge 0,$$4 is controllable, then the full network is controllable. Repetition alone is therefore not the barrier; the barrier is the existence of network-invariant modes (Xue et al., 2018).

A complementary structural formulation appears in modal strong structural controllability. For a structured block matrix $$\frac{dx(t)}{dt}=f(x(t),u(t)), \qquad x(0)=x_0,\qquad t\ge 0,$$5, a network is $$\frac{dx(t)}{dt}=f(x(t),u(t)), \qquad x(0)=x_0,\qquad t\ge 0,$$6-SSC if, for every $$\frac{dx(t)}{dt}=f(x(t),u(t)), \qquad x(0)=x_0,\qquad t\ge 0,$$7 and every admissible realization $$\frac{dx(t)}{dt}=f(x(t),u(t)), \qquad x(0)=x_0,\qquad t\ge 0,$$8 in the $$\frac{dx(t)}{dt}=f(x(t),u(t)), \qquad x(0)=x_0,\qquad t\ge 0,$$9-specified pattern class, every appearing eigenvalue $u\in U^{ad}$0 is controllable in the PBH sense: $u\in U^{ad}$1 The key combinatorial object is the $u\in U^{ad}$2-network graph $u\in U^{ad}$3, whose self-loops and interconnection edges are marked as solid or dotted according to the extra spectral information and the full-row-rank status of interconnection blocks. A zero-forcing coloring process propagates zeros of left eigenvectors across subsystems. The main theorem states that the network is $u\in U^{ad}$4-SSC if the set of control subsystems $u\in U^{ad}$5 is a zero forcing set of $u\in U^{ad}$6. For networks of one-dimensional subsystems and $u\in U^{ad}$7, this condition is also necessary (Mousavi et al., 2021).

These results distinguish two roles of topology. In the first, the network graph replicates or suppresses modal obstructions through the spectrum of $u\in U^{ad}$8. In the second, graph topology is a certificate for controllability of selected eigenvalues through zero forcing. A common misconception is that repeated eigenvalues always damage network controllability; the cited dichotomy shows that special-repeat modes are harmless under the stated hypotheses, whereas network-invariant modes are the structural obstruction.

4. Local homotopy invariants and controllable interaction modes

In multi-vehicle trajectory optimization, topology is used as an optimization handle for interaction patterns. Each vehicle trajectory is represented as a piecewise 5th-order polynomial,

$u\in U^{ad}$9

and the baseline objective is

$t\in[0,T]$0

with jerk $t\in[0,T]$1, duration weight $t\in[0,T]$2, and penalty term $t\in[0,T]$3 for kinodynamic and collision-avoidance constraints. The difficulty is that the problem is high-dimensional, nonlinear, non-convex, and sensitive to initialization, so standard optimization does not reliably control which interaction mode is selected (Ma et al., 7 Mar 2025).

The proposed remedy is a differentiable local homotopy invariant. For a single trajectory relative to an obstacle, at the key point closest to the obstacle,

$t\in[0,T]$4

For two moving vehicles, the invariant is defined on relative motion: $t\in[0,T]$5 Its sign classifies the local interaction: $t\in[0,T]$6 corresponds to counterclockwise interaction, $t\in[0,T]$7 to clockwise interaction, and $t\in[0,T]$8 to the boundary between classes. Desired pairwise modes are encoded by $t\in[0,T]$9, with the sign constraint

$$A(t):=\{x^u(t)\;:\; x^u(\cdot)\text{ solves (2) for some }u\in U^{ad}\}.$$0

This constraint is integrated through the penalty

$$A(t):=\{x^u(t)\;:\; x^u(\cdot)\text{ solves (2) for some }u\in U^{ad}\}.$$1

yielding the full objective

$$A(t):=\{x^u(t)\;:\; x^u(\cdot)\text{ solves (2) for some }u\in U^{ad}\}.$$2

The closest-point timestamp $$A(t):=\{x^u(t)\;:\; x^u(\cdot)\text{ solves (2) for some }u\in U^{ad}\}.$$3 is itself an argmin, so the method becomes bi-level. Differentiation through this argmin is carried out with a KKT-based formula, and the implementation uses gradient descent for $$A(t):=\{x^u(t)\;:\; x^u(\cdot)\text{ solves (2) for some }u\in U^{ad}\}.$$4, L-BFGS for the outer problem, and a two-stage optimization in which the topology term is optimized before reintroducing collision avoidance.

The practical consequence is explicit control over interactive homotopy classes. The framework can generate multiple interactive trajectories from the same initial values, reproduce different interaction patterns from the same initial condition, and support user-designed interaction patterns. The narrow-corridor example is important because it shows why a global winding angle can be inadequate when vehicles do not cross each other globally; the local invariant still distinguishes the meaningful local interaction (Ma et al., 7 Mar 2025).

A recurrent misconception is that topological control here means a global winding-number constraint. The cited construction is explicitly local: it is anchored at the nearest-point topology and is differentiable precisely to remain compatible with gradient-based trajectory optimization.

5. Density-based topology optimization for characteristic mode manipulation

In characteristic mode analysis, the method of moments impedance matrix is

$$A(t):=\{x^u(t)\;:\; x^u(\cdot)\text{ solves (2) for some }u\in U^{ad}\}.$$5

and characteristic modes satisfy

$$A(t):=\{x^u(t)\;:\; x^u(\cdot)\text{ solves (2) for some }u\in U^{ad}\}.$$6

The characteristic number $$A(t):=\{x^u(t)\;:\; x^u(\cdot)\text{ solves (2) for some }u\in U^{ad}\}.$$7 distinguishes capacitive, inductive, and resonant modes, the characteristic angle is

$$A(t):=\{x^u(t)\;:\; x^u(\cdot)\text{ solves (2) for some }u\in U^{ad}\}.$$8

and modal significance is

$$A(t):=\{x^u(t)\;:\; x^u(\cdot)\text{ solves (2) for some }u\in U^{ad}\}.$$9

Density-based topology optimization uses a continuous design variable $A+\lambda_i BC$00 on the mesh triangles, with $A+\lambda_i BC$01 representing vacuum and $A+\lambda_i BC$02 PEC. Surface resistivity is interpolated as

$A+\lambda_i BC$03

with $A+\lambda_i BC$04 and $A+\lambda_i BC$05. This produces a modified lossy characteristic-mode problem

$A+\lambda_i BC$06

where $A+\lambda_i BC$07 is the modal dissipation factor (Tucek et al., 6 Feb 2025).

The framework’s key conceptual separation is between geometry synthesis and feeding synthesis. Because characteristic modes are excitation-independent, the optimizer manipulates properties of the structure itself without prescribing a feed location during the optimization. Feeding synthesis is carried out only after the optimized geometry has been obtained.

The optimization problem is posed as a nested eigenvalue-constrained program, with density filtering and Heaviside projection,

$A+\lambda_i BC$08

and updated by MMA. Several objectives are treated. Single-mode resonance control minimizes

$A+\lambda_i BC$09

Resonance tuning under an area constraint adds

$A+\lambda_i BC$10

Modal $A+\lambda_i BC$11-factor control minimizes

$A+\lambda_i BC$12

with

$A+\lambda_i BC$13

Multi-mode optimization is formulated in min-max form by minimizing $A+\lambda_i BC$14 subject to

$A+\lambda_i BC$15

Computational tractability is provided by adjoint sensitivity analysis. After imposing normalization and phase constraints on the complex eigenvectors, the paper derives an adjoint system that eliminates direct dependence on $A+\lambda_i BC$16, $A+\lambda_i BC$17, and eigenvector derivatives. The resulting gradient takes the form

$A+\lambda_i BC$18

The workflow is a standard local gradient loop: initialize $A+\lambda_i BC$19, filter and project, solve CMA, evaluate objectives and constraints, compute adjoint sensitivities, update by MMA, and continue with $A+\lambda_i BC$20-continuation.

The examples show single-mode resonance tuning, resonance under area constraint, modal $A+\lambda_i BC$21-factor optimization, and simultaneous control of three characteristic modes. The paper also identifies limitations: self-penalization from the auxiliary loss term, resonance shift after thresholding, singularity issues when characteristic numbers are not distinct, local-minimum dependence of MMA, jagged boundaries after thresholding, and the heuristic status of the chosen material interpolation (Tucek et al., 6 Feb 2025).

6. Internal presentations of topological modalities

A distinct but technically precise use of topology-enabled modal control appears in homotopy type theory. A reflective subuniverse consists of a subuniverse $A+\lambda_i BC$22, a reflector $A+\lambda_i BC$23, and a unit

$A+\lambda_i BC$24

satisfying modal recursion. A modality is a reflective subuniverse closed under $A+\lambda_i BC$25-types, a lex modality preserves pullbacks, and a topological modality is given by nullification at a family of propositions. Any topological modality is lex (Williams, 31 Jan 2025).

A presentation of a lex modality is a collection $A+\lambda_i BC$26 of types with $A+\lambda_i BC$27 and $A+\lambda_i BC$28 closed under $A+\lambda_i BC$29. The presented modality $A+\lambda_i BC$30 is defined by nullifying at the truncated family $A+\lambda_i BC$31, and the modal types are the $A+\lambda_i BC$32-sheaves. This acts as an internalisation of the notion of a Grothendieck topology.

The presentation becomes computationally useful because membership in the modal subuniverse can be tested by an internal sheaf condition. For $A+\lambda_i BC$33, an $A+\lambda_i BC$34-type $A+\lambda_i BC$35 is a $A+\lambda_i BC$36-sheaf iff for any $A+\lambda_i BC$37-cover $A+\lambda_i BC$38, the map

$A+\lambda_i BC$39

is an equivalence. A $A+\lambda_i BC$40-cover is defined fiberwise: $A+\lambda_i BC$41 These covers are closed under composition, stable under pullback, and every equivalence is a cover.

A projective presentation is one in which every type in $A+\lambda_i BC$42 is projective. Under this hypothesis, propositions admit an explicit sheafification formula,

$A+\lambda_i BC$43

and existential statements satisfy an internal Kripke–Joyal-style rule,

$A+\lambda_i BC$44

The paper then derives a $A+\lambda_i BC$45-local partial choice principle for $A+\lambda_i BC$46-surjective maps, and uses it to compare cohomology in the ambient universe and the modal subuniverse. Modal cohomology is defined by

$A+\lambda_i BC$47

and if $A+\lambda_i BC$48 is a projective presentation and $A+\lambda_i BC$49 is an abelian group satisfying descent for $A+\lambda_i BC$50, then

$A+\lambda_i BC$51

for all projective $A+\lambda_i BC$52. The paper further proves subcanonicity criteria and treats the Zariski, étale, and fppf presentations, including stability statements for $A+\lambda_i BC$53 of quasi-coherent modules (Williams, 31 Jan 2025).

Here topology controls a modality by specifying its internal covers and sheaf conditions. A plausible implication is that this is a formal analog of control by local pieces: the presentation determines what counts as local evidence, what can be chosen locally, and how local data reconstructs global truth.

7. Synthesis, distinctions, and recurrent misunderstandings

The cited works share a common pattern: topology is promoted from background structure to an active design variable. In state-space controllability, the chosen topology determines the closure relation that defines approximate controllability. In network systems, graph topology and interconnection structure determine whether modes are invariant, repeated, or controllable. In multi-vehicle planning, a local homotopy invariant selects the interaction basin reached by gradient optimization. In electromagnetic synthesis, topology optimization reshapes the geometry so that characteristic modes exhibit prescribed behavior. In homotopy type theory, a presentation of a topological modality determines the internal covers and sheaf conditions through which the modal subuniverse is computed (Moustapha, 2019, Xue et al., 2018, Ma et al., 7 Mar 2025, Tucek et al., 6 Feb 2025, Williams, 31 Jan 2025).

One recurrent misunderstanding is to treat these results as interchangeable uses of “topology.” They are not. The topology of $A+\lambda_i BC$54 in controllability is a non-Hausdorff topology on a state space; the topology of $A+\lambda_i BC$55 or $A+\lambda_i BC$56 is a network interconnection graph; the topology of $A+\lambda_i BC$57 in trajectory optimization is a local homotopy class; the topology in density-based optimization is the material layout of a conductor; and the topology of a lex modality is an internalized Grothendieck topology (Moustapha, 2019, Mousavi et al., 2021).

A second misconception is to identify “modal control” exclusively with eigenvalue placement. The network papers do concern eigenvalues explicitly, but the topological controllability paper defines the modal question through density of attainable sets, the trajectory-optimization paper treats interaction mode as a homotopy class, and the type-theoretic paper uses “modal” in the sense of reflective subuniverses rather than dynamical modes (Xue et al., 2018, Ma et al., 7 Mar 2025, Williams, 31 Jan 2025).

A third misconception is that topology always relaxes an otherwise hard control problem. The literature is more divided. In the topological controllability construction, the reachable set is made dense by changing the topology. In networked systems, however, topology can create severe barriers: network-invariant modes can force actuation on at least $A+\lambda_i BC$58 subsystems or even on every node in the projection-fixed case. Thus topology can be either an enabling mechanism or a structural obstruction, depending on whether it is being designed, analyzed, or fixed by the physical interconnection (Moustapha, 2019, Xue et al., 2018).

Taken together, these works define topology-enabled modal control as a technically heterogeneous but coherent research direction: topology is used to alter the notion of closeness, certify or obstruct controllability of selected modes, choose among homotopy classes of interaction, synthesize structures with desired characteristic modes, or compute modal subuniverses through internal covers. The unifying principle is not a shared formalism but a shared operational idea: modal behavior is controlled by controlling the topological structure in which that behavior is defined.