Two-dimensional First-Order Hyperbolic Partial Differential Equations MPC Design

In this work, the semigroup operator for a two-dimensional transport-reaction model that is described by a first-order hyperbolic system is derived. Two different cases of one spatially varying velocity and two spatially varying velocities are considered. A discrete in-time model setting is formulated without any model reduction or approximation, typically present in controller designs of distributed parameter systems. This setting is obtained by an exact time discretization that makes use of the semigroup operator. Additionally, adjoint operators, which are utilized in the design of the controller, are derived. The system is assumed to have a point observation and a distributed actuation function. A model predictive controller has been designed for this system to account for the constraints present in the input and output. The controller's efficiency in achieving convergence and satisfying both input and output constraints is demonstrated through the use of numerical simulations.