Controllability Allocation Scores for Linear Infinite-Dimensional Systems
Daisuke Inoue, Kazuhiro Sato
Rigorous but incremental extension of the authors' finite-dimensional CAS framework to PDEs, with a narrow audience and limited validation.
Abstract
We define finite-time controllability allocation scores (CASs) as optimal solutions to two convex optimization problems that allocate a fixed actuation budget among prescribed input functions of linear infinite-dimensional continuum systems with selected target outputs. The objectives are expressed using controllability Gramians for the selected outputs and correspond to maximizing the volume of output changes reachable with unit control energy or minimizing the mean energy required for unit output changes. To compute these CASs, we discretize the system in space, construct the resulting Gramians, and solve the resulting allocation problems by a domain-aware projected gradient descent method. We study the behavior of CASs when the actuator-to-target responses of the spatially discretized systems converge to those of the continuum system as the mesh is refined. If the continuum system is target controllable, we show that the spatially discretized systems are also target controllable on sufficiently fine meshes and that the optimal values of the approximate problems converge to the optimal value for the continuum system. If the allocation problem for the continuum system has a unique CAS, we show that the CASs obtained from the spatially discretized problems converge to it. We also derive quantitative bounds on the errors between these CASs when the Gramian map for the continuum system distinguishes allocations. A two-dimensional advection-diffusion example illustrates how transport dynamics affect target controllability and CASs. Spatial refinement reveals apparent controllability caused by numerical diffusion and supports the theoretical convergence results.
AI Impact Assessments
(1 model)Core Contribution
The paper extends controllability allocation scores (CASs), previously defined for finite-dimensional networks, to linear infinite-dimensional (PDE) systems with fixed distributed actuator functions and finitely many target outputs. A fixed budget on the simplex is allocated among m candidate inputs. The allocation enters through an affine Gramian G(p,T)=Σ p_i G_i(T), which is r×r and therefore small. The volumetric (log-det) and average-energy (trace-inverse) criteria give convex problems. The main novelty is the analysis of how spatial semidiscretization affects the problem. Assuming L² convergence of actuator-to-target responses, the paper proves:
A 2D advection–diffusion room example shows that optimal actuators shift with circulation speed. It also shows that coarse first-order upwinding produces spurious "apparent controllability" when the continuum system is not target controllable over the horizon.
Methodological Rigor
The theory is clean and largely correct. The proofs use standard tools: Weyl's inequality, Rockafellar–Wets epi-convergence, and strong convexity arguments. The extended-value handling of singular Gramians is careful. The error bound proof (A.5) is sound, including the uniform λ_T lower bound. However, much of the structure is inherited from Sato (2026a,b): convexity, existence, PGD convergence, and strong convexity. The new results are essentially a perturbation analysis of finite-dimensional objects under Gramian perturbations. The central assumption (response L² convergence) is simply assumed. It is verified for the example only by citation arguments, with no rate and no computable threshold h̄. The error bound depends on constants (λ_T, χ_T, Λ_T) that are not computable a priori. The numerics are a single example. The observed rates (h^1.56 for responses, h^1.35 for CAS error) are measured against a finer-mesh reference rather than an exact solution. The computed χ_{T,h} decreases from 9.1e-6 to 3.7e-6 across meshes, and the claim that it "stabilizes" is weakly supported. Time quadrature and optimizer error are excluded from the analysis.
Potential Impact
This is a niche contribution within a research line: the Sato group's controllability scores. It is useful for actuator-resource allocation in distributed-parameter systems such as thermal, transport, and vibration problems. The practical insight is a methodological caution: positive definiteness of discretized Gramians on coarse meshes does not imply continuum controllability. This is a known phenomenon (Zuazua), but it is neatly demonstrated for output controllability with upwind schemes. The finite-dimensional reduction to r×r Gramians makes the method computationally attractive and reusable. No real-world validation is given, and the model is an idealized room with prescribed flow.
Timeliness & Relevance
Actuator design for PDEs is an established topic. Allocation of fixed-shape actuators with target outputs is a modest but real gap. Interest in controllability-based network intervention is moderate and growing. The work does not address a major bottleneck.
Strengths
Limitations
Overall it is a solid, careful but incremental extension with modest expected influence.
Generated Oct 21, 2026
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Rigorous but incremental extension of the authors' finite-dimensional CAS framework to PDEs, with a narrow audience and limited validation.