locations ${\mathcal{D}}= {\{d_1,\dots,d_{{\ell}}\}}$ such that even with ${N}\ll {M}$ and ${\ell}\ll {M}$, the functional evolution of the spatiotemporal model can be estimated robustly, and driven (controlled) to a reference function $f_{\text{ref}}$. Our approach to solve this problem relies on the design of the measurement operator ${\ensuremath{K}}$ such that the pair $(A,{\ensuremath{K}})$ is observable, and the control operator ${{\ensuremath{K}}_{{D}}}$ such that the pair $(A,{{\ensuremath{K}}_{{D}}})$ is controllable.

Theoretical Results {#sec:theory_results}
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In this section, we prove results concerning the observability of spatiotemporally varying functions modeled by the functional evolution and measurement equations [(\[eq:k\_measure\])]{} and [(\[eq:k\_measure1\])]{} formulated in Section \[sec:formulation\]. In particular, observability of the system states implies that we can recover the current state of the spatiotemporally varying function using a small number of sampling locations ${N}$, which allows us to 1) track the function, and 2) predict its evolution forward in time. It should be noted that the results are also applicable to controllability of the system in [(\[eq:k\_measure\_c1\])]{} since the structure of the control matrix $K_{CD}$ is also that of a Kernel matrix. We first show in Proposition \[prop:1\] that if $A$ has a full-rank Jordan decomposition, the kernel matrix meeting a condition called *shadedness* (to be defined below) is 