results on a real-world large dataset and control synthesis results for a diffusion PDE.

Kernel Controllers {#sec:observers}
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This section outlines our modeling framework and presents theoretical results associated with the number of sampling locations required for monitoring functional evolution.

Problem Formulation {#sec:formulation}
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We focus on predictive inference and control over a time-varying stochastic process, whose mean $f$ is temporally evolving: $$\begin{aligned}
{\label{eq:e:bnp_model1}}
& f_{k+1}& \sim \mathbb{F}(f_k,\eta_k) $$ where $\mathbb{F}$ is a distribution varying with time $t$ and exogenous inputs $\eta$. The theory of reproducing kernel Hilbert spaces (RKHSs) provides powerful tools for generating flexible classes of functions with relative ease, and is thus a natural choice for modeling complex spatial functions [@scholkopf:bk:2002]. Therefore, our focus will be on spatiotemporally evolving kernel-based models, such as Gaussian Processes (GPs). In a kernel-based model, ${k}:{\Omega}\times{\Omega}\to{\mathbb{R}}$ is a positive definite kernel on some compact domain ${\Omega}$ that models the covariance between any two points in the input space. A Mercer kernel [@scholkopf:bk:2002] implies the existence of a smooth map ${\psi}:{\Omega}\to{\mathcal{H}}$, where ${\mathcal{H}}$ is an RKHS with the property $$\begin{aligned}
 {k}(x,y) &= {\langle {\psi}(x),{\psi}(y) \rangle_{{\mathcal{H}}}} = {\langle {\psi}({k}(x,\cdot)),{\psi}({k}(y,\cdot)) \rangle_{{\mathcal{H}}}}. \end{aligned}$$ There is a large body of literature on modeling spatiotemporal evolution in ${\mathcal{H}}$ 