Structural Averaged Controllability for Linear Ensemble Systems: Multi-input Case

Technical Summary: Structural Averaged Controllability for Linear Ensemble Systems: Multi-Input Case

Problem Formulation
This paper addresses the problem of structural averaged controllability for linear ensemble systems with multiple inputs. An ensemble system consists of a continuum of linear time-invariant systems indexed by a parameter $\sigma \in \Sigma$, driven by a common control input $u(t)$. The system dynamics are given by:
$$\frac{\partial}{\partial t}x(t, \sigma) = A(\sigma)x(t, \sigma) + B(\sigma)u(t)$$
where $A(\sigma)$ and $B(\sigma)$ are bounded measurable functions. The control objective is averaged controllability: the ability to steer the average state $\bar{\chi}(T) = \int_\Sigma x(T, \sigma) d\mu$ to any target vector in $\mathbb{R}^n$ for any initial profile.

The specific focus is on the structural aspect: given a sparsity pattern (represented by a directed graph $G$) for the matrices $(A, B)$, does there exist at least one realization of these matrices consistent with the pattern that renders the ensemble averaged controllable? While the single-input case was fully characterized in prior work [CG25] using the concept of a "core" graph and a directed spanning path, the general multi-input case remained an open problem. The authors note that a naive extension of the single-input condition fails because multiple input channels interact with the averaged controllability matrix in complex ways.

Methodology and Graph-Theoretic Framework
The authors employ a graph-theoretic approach, representing the sparsity pattern via a digraph $G = (V, E)$ with $n$ state nodes ($\alpha$-nodes) and $m$ input nodes ($\beta$-nodes). The analysis relies on the strong component decomposition of the graph:

1. Strong Components: The graph is decomposed into strongly connected components.
2. Cyclic and Acyclic Parts: The union of nontrivial strong components (those containing cycles) and their successors forms the cyclic part $G^+_{cyc}$. The remaining nodes form the core $G^*$, which is an acyclic subgraph containing all $\beta$-nodes.
3. Truncated Averaged Controllability Matrix: For the core $G^*$, the authors define a truncated averaged controllability matrix $C_{tr}(A, B)$ involving integrals of $B, AB, \dots, A^{n^*-1}B$, where $n^*$ is the number of $\alpha$-nodes in the core.

The core contribution of the methodology is a constructive algorithm for assigning edge weights (values for the non-zero entries of $A$ and $B$) to satisfy rank conditions.

• Core Assignment (Algorithm 1): The authors develop a novel edge-weight assignment algorithm for the core. Unlike the single-input case, which relies on a simple spanning path, the multi-input case requires assigning weights to ensure the truncated controllability matrix achieves full row rank. This involves partitioning the parameter space $\Sigma$ into disjoint measurable sets of equal measure and assigning specific weight functions (often involving a scalar radial function $f$) to edges.
• Extension to Full Graph (Algorithm 2): Once the core is assigned, the weights are extended to the cyclic parts ($G_{cyc}$) and the appendix ($G_{appx}$). This extension utilizes a scalar function $f(\sigma)$ derived from a radial coordinate on the parameter space. By assigning weights proportional to powers of $f(\sigma)$ (specifically $f(\sigma)^{k\gamma}$ for irrational $\gamma$), the authors ensure that the contributions of different walks to the averaged controllability matrix are linearly independent. This leverages the density of polynomials in $L^2$ spaces to separate the contributions of walks with different lengths and cycle traversals.

Key Contributions and Results
The paper provides a necessary and sufficient condition for structural averaged controllability in the general multi-input case.

Theorem 3.7: A graph $G \in \mathcal{G}_{n,m}$ is structurally averaged controllable if and only if:

1. Accessibility: Every $\alpha$-node is reachable from at least one $\beta$-node (i.e., the graph is accessible).
2. Row-Saturating Matching: The row-column bipartite graph $B(S_{G^*})$ associated with the sparsity pattern of the truncated averaged controllability matrix of the core $G^*$ admits a row-saturating matching.

• Definition of Row-Saturating Matching: In the bipartite graph where rows correspond to state nodes and columns correspond to path lengths from inputs, a matching is row-saturating if every row node is incident to exactly one edge in the matching. This condition generalizes the "directed spanning path" condition found in the single-input case (Corollary 3.8).
• Counter-Example: The authors demonstrate via Example 3.9 that the condition is not simply the existence of $m$ disjoint paths covering all nodes. Instead, the matching condition allows for a more flexible combination of paths from different inputs to achieve full rank.

The proof is constructive. The authors explicitly build a pair $(A, B)$ compliant with the sparsity pattern that satisfies the rank condition.

• Theorem 3.1: Proves that for any target matrix $M$ with the sparsity pattern of the core, there exists a realization $(A, B)$ on the core such that $C_{tr}(A, B) = M$.
• Proposition 3.22: Proves that the constructed pair $(A, B)$ for the full graph yields an averaged controllability matrix of full row rank.

Significance and Claims
The paper claims to resolve the open problem of characterizing structural averaged controllability for multi-input linear ensemble systems. The significance lies in:

1. Completeness: It provides a complete graph-theoretic characterization, extending the single-input results of [CG25] to the multi-input setting.
2. Constructive Proof: The result is not merely existential; the paper provides specific algorithms (Algorithm 1 and Algorithm 2) to assign edge weights to realize the controllability property.
3. Subtlety of Multi-Input: The work highlights that the multi-input case is "substantially more delicate" than the single-input case. The necessity of a row-saturating matching, rather than a simple spanning path, reflects the complex interplay of multiple inputs in the averaged controllability matrix.
4. Algorithmic Approach: The use of a radial function on the parameter space to separate walk contributions offers a robust method for handling the infinite-dimensional nature of ensemble systems within a structural framework.

The authors conclude by noting that while their construction guarantees a full-row-rank matrix, it may not be minimal in terms of the number of block columns required. They suggest future work could explore structural conditions for controlling higher-order statistics (moments) of the ensemble.