A new mathematical framework treats global circulation and magnetic flux as explicit states, and asks when actuators can control them. It separates the gauge-fixed local state from a physical harmonic state and gives the latter a finite-dimensional actuator. The framework's central rule is that the domain's topology sets the number of global modes, while actuator rank sets which directions are reachable. The number of global modes is set by the relevant Betti number, a mathematical count of independent topological modes.
In the stationary, or time-independent, version, the analysis gives one optimal control pair and uniquely determined associated states, but only under the theorem's stated assumptions. Those conditions include the relevant Hilbert-complex structure, boundedness and isomorphism requirements, and a convex admissible control set. The result is therefore a conditional mathematical statement about well-posedness.
The work is an arXiv preprint, arXiv:2608.25266v1, dated 26 Aug 2026. Its evidence comes from mathematical analysis and deterministic numerical experiments on specified model domains.
A global state with hard boundaries
For a target period, the model's coordinate for a global circulation, exact reachability means that the target belongs to the affine range generated by the actuator matrix. In plain language, an actuator can steer only the global directions it spans. Full control of the harmonic state requires the actuator rank to cover all of those modes; a rank-deficient actuator leaves the remaining directions outside reach.
Stationary tests put the idea under pressure
The stationary numerical program used four domains, with harmonic dimensions of 0, 1, 2 and 1 at the other tested degree. Controls were piecewise-constant DG0 functions, the prescribed bounds were inactive, and the reduced optimization problems were solved with conjugate gradients. This design placed contractible and multiply connected geometries within the same numerical program.
On the contractible L-shaped benchmark, conjugate gradients reached a relative residual of 10^-10 in six iterations from a zero starting point. The reported objective convergence orders were 1.92, 1.96 and 1.97, and the iteration count stayed at six on every mesh. The study did not have an exact mixed-state solution for direct local-error validation, so the result speaks most directly to the reported objective and solver behavior.
Topology appeared more directly in a solid-torus test. In one configuration, the objective contained no period target, topology reference or compatibility-multiplier term, yet the optimal physical state carried circulation of 0.04886029. The computed multiplier itself was of order 10^-16. The example distinguishes a physical global circulation from the separate multiplier used in the formulation.
On a two-hole, or figure-eight, domain, actuator alignment mattered as much as rank. A symmetric rank-one actuator could not improve an antisymmetric target: continuum topological control was zero and the period misfit was 6.250 x 10^-2. The case illustrates why having an actuator is not the same as having access to every topological mode.
The same geometry produced geometry-specific convergence rates. Reported stationary quantities converged at orders 1.31, 1.18 and 1.33. The stated rate for the geometry was 4/3, which the authors associated with a singular harmonic 1-form. These are numerical rates for the tested domain, not universal predictions.
A spherical-shell stationary test made a similar point for flux. With no flux target in the objective, the optimal physical state still carried cavity flux of 6.434808 x 10^-2. The result shows that a global mode can appear in an optimum even when it is not separately requested.
The same logic in Maxwell dynamics
The framework then moved from static optimization to time-dependent Maxwell equations. It treats magnetic cavity flux as conserved when no distinct global or boundary actuator is supplied. To make that flux a controllable global state, the formulation requires such an actuator.
In the torus Maxwell experiment, electric circulation had two modeled routes. It could enter through a harmonic component of distributed current or through a dedicated scalar topological actuator. Both formulations were accompanied by near-round-off Gauss and gradient diagnostics. No physical actuator calibration was reported, so the comparison concerns modeled control channels.
Dynamic tests on the two-hole domain reproduced the rank constraint. With a full-rank actuator, GE = I2, both harmonic circulation coordinates could be resolved. With the rank-one choice GE = (1, 1)^T, the response was restricted to c1(t) = c2(t), leaving a projected-target residual whenever the desired components were unequal.
The shell diagnostics separated conservation from control. In the no-actuator run, GB = 0, maximum flux change was 0. With the controlled choice GB = 1, terminal flux error was 1.43 x 10^-6, but the Gauss defect was 1.60 x 10^-2. That non-round-off defect limits how strongly the controlled-shell result can be treated as a validation of the full dynamic formulation.
What the calculations leave open
The FEEC consistency analysis links discretization quality to control error through consistency of the reduced gradient, state, adjoint and harmonic-reconstruction maps. That transfer requires uniform boundedness and stability assumptions, so it remains tied to the conditions of the analysis.
The numerical evidence is bounded by the tested configurations. The stationary examples did not have exact mixed-system solutions for direct local-state and adjoint checks, while the broader Maxwell refinement changed spatial mesh and time step together and therefore did not separate spatial from temporal convergence. The controlled shell run also retained a non-round-off Gauss defect. These results concern the specified mathematical models, geometries, meshes and modeled actuators, not physical-device performance.
Paper data and sources
Original title: Optimal Control in Hilbert Complex Spaces with Finite Element Exterior Calculus
Authors: Farid Bozorgnia, Michael Holst, Anshu Kumar et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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