Motivation

Let $\mathcal{G} = (\mathcal{V},\mathcal{E})$ be a graph where $\mathcal{V}$ is the set of nodes and $\mathcal{E} \subset \mathcal{V} \times \mathcal{V}$ is the set of edges. An edge $(v_j,v_k) \in \mathcal{E}$ is read as being from $v_j$ to $v_k$. Let $\mathcal{N}_k \subset \mathcal{V}$ be the set of neighbors of node $v_k \in \mathcal{V}$, that is, $v_j \in \mathcal{N}_k$ if $(v_j,v_k) \in \mathcal{E}$. If the graph is undirected than $(v_i,v_j) \in \mathcal{E}$ implies that $(v_j,v_i) \in \mathcal{E}$.
Let $\psi : \mathcal{E} \mapsto \mathbb{R}$ be the edge weights associated with each edge in $\mathcal{G}$. Let $\mathcal{D} \subset \mathcal{V}$ be the driver nodes, i.e., those nodes which directly receive an external signal. Let $\mathcal{T} \subset \mathcal{V}$ be the target node, i.e., those nodes whose states in which we are interested. We would like to solve the following optimal control problem, $$ \begin{array}{lll} \min & J = \int_0^{t_f} \sum_{v_k\in\mathcal{D}} u_k^2(t) dt\\ \text{s.t.} & \dot{x}_i(t) = -p_i x_i(t) + \sum_{v_j \in \mathcal{N}_i} \psi(v_j,v_i) x_j(t) + \sum_{v_k \in \mathcal{D}} \delta_{i,k} u_k(t), & \forall v_i \in \mathcal{V}\\ & x_i(0) = x_{i,0}, & \forall v_i \in \mathcal{V}\\ & x_i(t_f) = x_{i,f}, & \forall v_i \in \mathcal{T} \end{array} $$ To express the solution, we must define the controllability Gramian which satisfies the differential Lyapunov equation, $$ \dot{W}_{i,j}(t) = -(p_i+p_j)W_{i,j}(t) + \sum_{v_k \in \mathcal{N}_i} \psi(v_k,v_i) W_{k,j}(t) + \sum_{v_k \in \mathcal{N}_j} \psi(v_k,v_j) W_{i,k}(t) + \sum_{v_k \in \mathcal{D}} \delta_{i,k} \delta_{j,k} $$ The output controllability Gramian consists of only those entries of $W_{i,j}(t)$ where $v_i,v_j \in \mathcal{T}$. Let $\bar{W} \in \mathbb{R}^{n_t \times n_t}$ be the symmetric output controllability Gramian. Also, let $\gamma_i(t)$ be the control action applied to node $v_i \in \mathcal{T}$, defined as the difference between the desired final state $x_{i,f}$, $v_i \in \mathcal{T}$, and what the state would be in the absence of any control input, $\hat{x}_i(t)$. When the number of nodes is finite, the solution is well-known from control theory and is typically expressed in terms of the adjancecy matrix. Instead, we write the control energy in terms of the output controllability Gramian, $$ E = \boldsymbol{\gamma}^T W_{\mathcal{T}}^{-1} \boldsymbol{\gamma} $$ The main issue when determining the control energy is that the controllability Gramian cannot be computed in terms of the graph properties of $\mathcal{G}$, but rather must be computed numerically. Instead, if we consider lattice graphs, we can solve for the individual elements of the controllability Gramian in terms of the graph properties. We can then show that the control energy found from the lattice graph can approximate the control energy in complex networks.