Noise plays a crucial role in human activity, with distinct implications in the arts and sciences. Artists engage with "deliberate noise," crafting masterpieces in literature, music, and fine art that convey clear ideas, often veiled in this complexity. In contrast, the sciences strive to eliminate noise, seeking to extract laws of nature from chaotic data. This work explores methods to combat noise, offering a mathematical perspective on specific engineering challenges through established and novel control and estimation techniques in noisy environments. The focus is on stochastic optimal control and estimation problems, particularly where noise processes depend on the state and observation systems. Despite numerous findings, the topic remains underexplored and systematically unaddressed. The discussion centers on infinite dimensional systems within a linear quadratic framework, applicable to both continuous and finite time horizons. To ensure accessibility, background material is included. The intended audience encompasses applied mathematicians, theoretically inclined engineers developing new technologies, and students in related fields. Additionally, this work serves as a reference for functional analysis relevant to infinite dimensional linear systems theory.
