Control to of some distributed systems with missing data

Abstract

Unusually, this thesis is divided into two different parts, the objective of the first part is to control the non-linear ODEs, the newly here is the combining the classic method of optimal control with the new concept of average control which is introduced by Zuazua, the modern notion is the average optimal control, thus, we up-date our cost function to an average cost function. So, thanks to one of the important optimality principles which is the Ponteryaguine Maximum Principle, we prove the uniqueness and the existence of the average optimal control, therefore, we arrive at the average optimal control characterization. To precise our results, we must use the shooting method for finding a simulation of that average optimal control. The second part aims to control linear PDEs, where we combine the same notion of average control with the optimal control, we find a new average cost function. Because our distributed system has missing data, the way to characterize the optimality system changes, and it is divided into steps, first we describe the average no-regret control problem, then, using a quadratic perturbation to obtain average low-regret control, which helps us to find an average low-regret control characterization, finally, we can come back to the average no-regret control characterization. The processed example in the first part is controlling the outbreak of an epidemic, To be precise, we study the control of an outbreak of COVID-19 in the city of Wuhan, China in December 2019. In the second part, we control an abstract hyperbolic-parabolic coupled system depending on an unknown parameter