Controllability Allocation Scores for Linear Infinite-Dimensional Systems

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.