Optimal boundary control of coupled parabolic PDE–ODE systems using infinite-dimensional representation

• Description and formulation of the coupled parabolic PDE–ODE system of interest.
• The eigenvalue problem is solved. Dynamical properties of the model are analysed.
• Optimal control problem is studied using the corresponding Riccati equation.
• The theoretical results are applied to the case study of a cracking reactor.
• Numerical simulations are performed to evaluate the closed loop performance of the designed controller.

The optimal boundary control problem is studied for coupled parabolic PDE–ODE systems. The linear quadratic method is used and exploits an infinite-dimensional state-space representation of the coupled PDE–ODE system. Linearization of the nonlinear system is established around a steady-state profile. Using appropriate state transformations, the linearized system has been formulated as a well-posed infinite-dimensional system with bounded input and output operators. It has been shown that the resulting system is a Riesz spectral system. The linear quadratic control problem has been solved using the corresponding Riccati equation and the solution of the corresponding eigenvalue problem. The results were applied to the case study of a catalytic cracking reactor with catalyst deactivation. Numerical simulations are performed to illustrate the performance of the proposed controller.