Information-Theoretic Guarantees on Forward Set Invariance: Application to Energy Control in Kuramoto-Sivashinsky Equation

Flow control presents fundamental challenges including operating with limited sensor information and finite actuation capacity of systems with many degrees of freedom that are chaotic. Existing approaches for controlling nonlinear systems that yield theoretical guarantees inevitably fail to operate under the finite sensor and actuation assumptions, leading to the question: what guarantees can we make if we have incomplete state information and actuation capacity? In this work, control is framed in information-theoretic terms by envisioning the tandem sensor-actuator as a device that reduces the unknown information about the state to be controlled. We present information-theoretic bounds and guarantees for achieving 1) forward set invariance (maintaining a quantity of interest in a desired set) and 2) maximal observability, providing a principled framework for control design under sensor and actuation constraints. These information-theoretic bounds are utilized to determine the optimal (1) locations of sensors, (2) locations of actuators, and (3) actuation mechanism that provides forward set invariance guarantees in the control of energy with the Kuramoto-Sivashinsky Equation.