${\ensuremath{K}}_{t_i}$. Note that ${\mathcal{O}}_{{\Upsilon}}\in{\mathbb{R}}^{{N}{L}\times{M}}$. Similarly, we can define the *generalized controllability matrix* as $$\begin{aligned}
{\label{eq:control_mat}}
 {\Psi}_{{\Upsilon}} = 
 \begin{bmatrix}
   {A^{t_1}}^T {{{\ensuremath{K}}_{{D}}}}_{t_1} & {A^{t_2}}^T {{{\ensuremath{K}}_{{D}}}}_{t_2} & \cdots {A^{t_{L}}}^T {{{\ensuremath{K}}_{{D}}}}_{t_{L}}
 \end{bmatrix},\end{aligned}$$ ${\Psi}_{{\Upsilon}}\in{\mathbb{R}}^{{M}\times{L}{\ell}}$ A linear system is said to be observable if ${\mathcal{O}}_{{\Upsilon}}$ has full column rank (i.e. $\mathrm{Rank}~{\mathcal{O}}_{{\Upsilon}}={M}$) and is controllable if ${\Psi}_{{\Upsilon}}$ has full row rank, for ${\Upsilon}= \{0, 1, \dots, {M}-1\}$ [@zhou:bk:96].

Observability guarantees that a feedback-based observer can be designed such that the estimate of $w$ denoted by $\hat {w_k}$ converges exponentially fast to the true state $w_k$. In particular, observability is the necessary condition for the existence of a unique solution to the Riccatti equation required in designing a Kalman filter. Therefore, when $\eta,\zeta$ have a zero mean Gaussian distribution, a Bayes optimal filter can be designed for estimating $w$ if and only if $\mathrm{Rank}~{\mathcal{O}}_{{\Upsilon}}={M}$. Similarly, controllability guarantees that a feedback-based controller can drive the current functional state of the system $f_{k}$ to a reference function $f_{\text{ref}}$, as long as $f_{\text{ref}}\in{\ensuremath{{\mathcal{H}}_{{C}}}}$.

We are now in a position to formally state the spatiotemporal monitoring and control problem considered: Given a spatiotemporally evolving system modeled using [(\[eq:ideal\_lin\_evol\_control\])]{}, choose a set of ${N}$ sensing locations ${\mathcal{X}}= {\{x_1, \dots, x_{{N}}\}}$ and ${\ell}$ actuating 