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Control Strategies for Multi-Species Wasserstein Gradient Flows

Dante Kalise, Lucas M. Moschen, Grigorios A. Pavliotis

arXiv:2609.11422v1math.OCcs.NAmath.APmath.NA

TL;DR

The paper addresses the uncharacterized problem of locally exponentially stabilizing multi-species Wasserstein gradient flows by acting on a single species. It extends a Hessian-based Riccati feedback design to product spaces and characterizes when interactions permit stabilization at a prescribed rate. For two species, the criterion requires the cross-interaction to detect the relevant modes of the unactuated species.

  • Problem

    Local exponential stabilization of multi-species Wasserstein gradient flows under single-species actuation had not been characterized.

    Source: §I · p. 1

  • Method

    The authors extend a feedback design combining the Wasserstein Hessian spectrum and an algebraic Riccati equation to coupled systems on product Wasserstein spaces.

    Source: §I · p. 1

  • Results

    Single-species stabilization at rate δ is possible when the cross-interaction detects every relevant unactuated-species mode; for two species this is expressed as injectivity on eigenspaces.

    Source: B. Multi-species stabilization via… · p. 3, C. Two-species stabilization criterion · p. 4

  • Takeaways & Limitations

    The interactions must transmit control to all relevant modes for one species' actuation to stabilize the full system.

    Source: C. Two-species stabilization criterion · p. 4, §V · p. 6

Abstract

from arXiv · show

We stabilize stationary states of coupled multi-species McKean--Vlasov equations by feedback control, extending a spectral approach based on the Wasserstein Hessian and a Riccati equation from a single density to a product of Wasserstein spaces. We give necessary and sufficient conditions for local exponential stabilization at a prescribed rate when only one species is actuated, together with the minimum number of control inputs required. For two species, the condition reduces to a rank condition on the Fourier coefficients of the cross-interaction kernel over the eigenspaces of the unactuated species' linearized dynamics. A two-community noisy Kuramoto model and a cell-sorting model illustrate the results.

I. INTRODUCTION

The paper addresses feedback control of multi-species mean-field systems, for which local exponential stabilization under single-species actuation had not been characterized. It extends a Wasserstein-Hessian and Riccati-based stabilization approach from a single density to coupled systems.

I. INTRODUCTION

  • The paper situates the problem among multi-species mean-field control, cross-population feedback, and Wasserstein dynamics research.The introduction cites prior work on multi-population control and leader-follower feedback, while noting that single-species stabilization had not been characterized.Source: §I · p. 1
  • The paper characterizes local exponential stabilization of multi-species Wasserstein gradient flows when control acts on only one species.The authors identify this as an uncharacterized problem and extend their earlier single-density approach.Source: §I · p. 1
  • The Hessian-based feedback design extends to coupled systems when interactions satisfy a reciprocity condition.Under reciprocity, the system is a Wasserstein gradient flow on a product of Wasserstein spaces.Source: §I · p. 1
  • The work determines when single-species control can stabilize the full system and the minimum number of control inputs required.The introduction connects selective actuation to suppressing pathological neural synchronization, where stimulation may target one population while another is monitored.Source: §I · p. 1

II. MULTI-SPECIES WASSERSTEIN DYNAMICS

The multi-species dynamics are formulated as a Wasserstein gradient flow under reciprocal interactions, with stationary states providing the base points for a product-space Hessian and its associated operator. The section establishes existence and regularity of stationary states and the analytic properties needed for that construction.

II. MULTI-SPECIES WASSERSTEIN DYNAMICS

  • The reciprocity condition W_ji(z) = W_ij(-z) makes the coupled dynamics a gradient flow of the stated free energy.The first variation includes the entropy contribution and species interaction potentials.Source: §II · pp. 1–2
  • The free energy has a minimizer that is stationary, and every stationary state has positive, bounded C2 probability densities.The stationarity characterization is given componentwise in terms of the confinement and interaction potentials.Source: §II · p. 2
  • Existence of a free-energy minimizer follows from compactness of the product probability space and lower semicontinuity of the free energy.Finite free energy gives absolutely continuous components, and a minimizer is stationary under the Wasserstein gradient flow.Source: §II · p. 2
  • At each stationary state, the bilinear form H_ms represents the Wasserstein Hessian of the non-entropic free energy and is bounded on the product space.Cross-interaction terms involve differences between the species' spatial gradients.Source: §II · p. 2
  • The resulting Hessian form is closed, symmetric, and lower bounded, and its associated self-adjoint operator has compact resolvent.The form domain is [H˙2(T^d)]^N; the entropy contribution is closed and coercive, while the non-entropic Hessian is bounded.Source: §II · p. 2

III. FEEDBACK CONTROL STRATEGIES

The feedback strategy fixes spatial control shapes and uses their input amplitudes to control the system linearized around a stationary state. A change of variables conjugates the linearized dynamics to an evolution governed by the Wasserstein Hessian.

III. FEEDBACK CONTROL STRATEGIES

  • The design seeks local stabilization of a fixed stationary state at a rate greater than a prescribed threshold δ.The section follows the single-density strategy while working with multi-species variables.Source: §III · p. 2
  • The control operator B_ms is built from the chosen spatial shape functions and the time-dependent amplitudes.The amplitudes u belong to R^m.Source: §III · p. 2
  • After conjugation, the linearized dynamics take the form ∂tξ = -A_msξ - B_msu(t).The state perturbation is ν = μ - μ̄, transformed using the inverse stationary-state operator.Source: §III · p. 2

A. Controlling all species

With all species actuated, the Riccati-based design yields local exponential stabilization at rates below the closed-loop spectral threshold. The section constructs the linear-quadratic feedback using the Hessian eigenbasis and states the conditions supporting nonlinear stabilization.

A. Controlling all species

  • The design uses an orthonormal eigenbasis of A_ms and a bounded, nonnegative self-adjoint cost operator diagonal in that basis.The nonlinear argument assumes the product-space analogue of a regularity condition, obtained by adapting maximal-regularity and remainder estimates.Source: A. Controlling all species · pp. 2–3
  • The algebraic Riccati equation has a unique stabilizing nonnegative self-adjoint solution Π_ms, diagonal in the Hessian eigenbasis.The control Π_msξ(t) is optimal for the corresponding linear-quadratic problem.Source: A. Controlling all species · p. 3
  • The nonlinear closed-loop system is locally exponentially stable at μ̄ for every decay rate γ in (0, λ_Π), with λ_Π > δ.The closed-loop operator is A_ms + B_msB_ms*Π_ms.Source: A. Controlling all species · p. 3
  • The Riccati construction relies on injectivity and positivity conditions that yield stabilizability and detectability for the chosen rate.The spectral projection is diagonal in the Hessian eigenbasis, and product-space spectral-gap and nonlinear-stabilization results complete the argument.Source: A. Controlling all species · p. 3

B. Multi-species stabilization via single-species control

The section characterizes stabilization using controls applied to a single species: it gives a spectral condition for whether that species can stabilize the coupled system at rate δ, and specifies how many control inputs suffice.

B. Multi-species stabilization via single-species control

  • If a relevant eigenspace contains a nonzero mode invisible to species r, no bounded control operator acting only through that species can achieve δ-stabilization.The invisible mode lies in ker(Pr), so the Hautus condition fails.Source: B. Multi-species stabilization via… · p. 3
  • If every eigenspace with λ≤δ has trivial intersection with ker(Pr), then m≥mr,δ=maxλ dλ control profiles form an open dense set of δ-stabilizing tuples.Here the maximum is over the eigenspaces included in Λmsδ.Source: B. Multi-species stabilization via… · p. 3
  • Stabilization through species r is possible for every threshold δ<δ*r.The threshold is defined by the first eigenvalue whose eigenspace intersects ker(Pr) nontrivially, with the stated convention for an empty set.Source: B. Multi-species stabilization via… · p. 3
  • For each qualifying tuple, the Riccati equation has a unique stabilizing nonnegative self-adjoint solution.The result applies to tuples in the open dense set Gm r,δ.Source: B. Multi-species stabilization via… · p. 3
  • The corresponding feedback locally exponentially stabilizes the nonlinear system at the prescribed rate δ, under the stated regularity and proposition hypotheses.The passage requires αj∈W2,∞(Td) for the maximal-regularity and feedback-remainder arguments.Source: B. Multi-species stabilization via… · pp. 3–4

C. Two-species stabilization criterion

For two species, stabilization through species 1 is determined by whether cross-interactions detect every relevant eigendirection of the unactuated species; this can be checked through a Fourier-coefficient rank condition.

C. Two-species stabilization criterion

  • The invisible modes Eλ∩ker(P1) correspond linearly to modes in F2,λ∩ker(L12).The isomorphism maps (0,e2) to Īμ2e2.Source: C. Two-species stabilization criterion · p. 4
  • Stabilization through species 1 at rate δ is possible exactly when F2,λ∩ker(L12)={0} for every λ∈σ(−L22)∩(−∞,δ].Equivalently, L12 must be injective on each such eigenspace of the unactuated species.Source: C. Two-species stabilization criterion · p. 4
  • For λ∈σ(−L22), the condition Eλ∩ker(P1)={0} is equivalent to rank V12,λ=d(2)λ.Thus the criterion for λ≤δ can be checked from Fourier coefficients of W12 and eigenfunctions of L22.Source: C. Two-species stabilization criterion · p. 4

IV. EXAMPLES AND NUMERICAL EXPERIMENTS

The numerical experiments use a Fourier–Galerkin discretization and a finite eigenfunction projection, with common control and quadrature settings reported for both experiments.

IV. EXAMPLES AND NUMERICAL EXPERIMENTS

  • The periodic torus eigenfunctions are computed with a Fourier–Galerkin discretization using cosine and sine basis functions.The basis is indexed through K.Source: §IV · p. 4
  • The nonlinear dynamics are projected onto the first 40 computed eigenfunctions, with Galerkin matrices and nonlinear terms evaluated by trapezoidal quadrature on uniform grids.Source: §IV · p. 4
  • Both experiments use δ=5 and Qms=δ2I; the implementation adapts to the product-space setting.Further details and code are reported in the passage.Source: §IV · p. 4

A. The two-community noisy Kuramoto model

In the two-community noisy Kuramoto example, the species-1 stabilization threshold depends on the stationary state, and feedback suppresses synchronization when cross-coupling is present.

A. The two-community noisy Kuramoto model

B. Cell-sorting of two species

In the cell-sorting model, feedback rapidly brings the sorting coefficient close to its target, while uncontrolled convergence can be slow near the bifurcation. The analysis shows that single-species control can achieve any prescribed exponential rate at nonuniform stationary states, whereas the uniform-state rate is limited by a Fourier-mode condition.

B. Cell-sorting of two species

  • For γ > γc sufficiently close to γc, the model has locally asymptotically stable, out-of-phase nonuniform equilibria in the even-density subspace, but their small spectral gap can make uncontrolled convergence slow.The cross-interaction kernel is heat-regularized to meet the stated regularity assumptions.Source: B. Cell-sorting of two species · pp. 5–6
  • At the uniform stationary state, δ*_1 = 16π²; for the Fourier eigenspace indexed by n, V12,4π²n² is injective exactly when n is odd.Injectivity first fails at n = 2.Source: B. Cell-sorting of two species · p. 5
  • For the uniform equilibrium, the species-2 Fourier modes have eigenvalues 4π²n², and the cross-interaction coefficient is nonzero exactly for odd Fourier indices.These mode properties yield the injectivity test used to identify the rate threshold.Source: B. Cell-sorting of two species · p. 5
  • For nonuniform species-2 equilibria, the Fourier-mode argument establishes injectivity of V12,λ for every λ in σ(−L22).The argument separates even and odd modes and uses nonuniformity of the equilibrium.Source: B. Cell-sorting of two species · pp. 5–6
  • For nonuniform stationary states, δ*_1 = +∞, so single-species control can achieve every δ > 0 when m ≥ m1,δ and the profiles lie in the corresponding open dense set.This follows from injectivity of V12,λ for every eigenvalue λ of −L22.Source: B. Cell-sorting of two species · p. 6
  • At t = 300, the uncontrolled sorting coefficient is 0.375 for ε = 0.005 and 0.998 for ε = 0.01, while both control strategies reach S(t) ≥ 0.99 very quickly.After reaching that level, the control becomes negligible.Source: B. Cell-sorting of two species · p. 6

V. CONCLUSION

The paper extends Hessian-based feedback stabilization to multi-species Wasserstein gradient flows and establishes local stabilization at a prescribed exponential rate. It identifies a trade-off in control-profile design and points to further investigation and biologically inspired applications.

V. CONCLUSION

  • The Wasserstein-Hessian feedback construction from [9] extends to multi-species gradient flows, yielding local stabilization of stationary states at a prescribed exponential rate.Source: §V · p. 6
  • Shape controls from an open dense set offer an alternative to computing Hessian eigenfunctions, but require solving a less structured Riccati equation.Source: §V · p. 6
  • Further investigation and application to biologically inspired problems, including external stimulation to suppress or promote synchronization, remain directions for the methodology.Source: §V · p. 6
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