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Controllability Allocation Scores for Linear Infinite-Dimensional Systems

Daisuke Inoue, Kazuhiro Sato · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.28205 · cited by 0 works (OpenAlex) · open-access PDF

German

Wir definieren die Differential-Time-Controlability-Targety Allocation Scores (CASs) als optimale Lösungen für zwei konvexe Optimierungsprobleme, die ein festes Aktuationsbudget unter den vorgeschriebenen Eingabefunktionen von linearen infinite-dimensional-Continuum-Systemen mit ausgewählten Zieleausgängen zuweisen. Die Ziele werden unter Verwendung von Controlability-Gramians für die ausgewählten Ausgänge ausgedrückt und entsprechen der Maximierung des Volumens der Ausgangsänderungen, die mit der Einheitskontrollenergie erreichbar sind, oder der Minimierung der durchschnittlichen Energie, die für die Einheitsausgangänderungen erforderlich ist. Um diese CASs zu berechnen, diskretisieren wir das System im Raum, konstruieren die

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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.

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Topics: Controllability, Discretization, Controllability Gramian, Mathematics, Mathematical optimization, Linear system

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