Optimal dynamic inversion-based boundary control design for two-dimensional heat equations

This study develops a new boundary control design for a class of distributed parameter systems. A two-dimensional (2D) heat equation is considered and the controller expression is derived for two different types of boundary conditions. The principles of dynamic inversion and optimisation theory are combined to develop an analytical expression for boundary control. To extend the applicability of the controller design to real-life systems, adaptive control theory is used to ensure stability in the presence of parameter uncertainty in the system. L₂ stability is proved using Lyapunov stability theory, which assures that the control drives the system trajectory to the desired temperature profile with uncertainty in the system parameter. Numerical results are presented for a square shape and also a model problem of a 2D cavity.