Specialized Adaptive Algorithms for Model Predictive Control of PDEs

• Applicants: Professor Dr. Lars Grüne; Professor Dr. Anton Schiela
• Subject Area: Mathematics
• Term: from 2017 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 337928467

• Final Report Year: 2020

Final Report Abstract

Subject of this project was the sensitivity analysis and the specialized adaptive discretization for the Model Predictive Control (MPC) of optimal control problems with partial differential equations. In every iteration of an MPC controller, an optimal control problem on a possibly long time horizon is solved. Only an initial part of the optimal solution is used as a feedback for the system to be controlled. This motivated the use of efficient discretization schemes tailored to this approach, i.e., space and time grids, which are fine at the beginning of the time interval and become coarser towards the end. In the course of the project, a comprehensive sensitivity analysis was performed to estimate the influence of perturbations that occur in the far future on the MPC feedback, i.e., the optimal control on an initial part. Under stabilizability conditions on the involved operators it was shown that the influence of perturbations is of local nature, meaning that discretization errors that occur in the far future only have a negligible effect on the MPC feedback. This property has been proven for various problem classes, covering problems governed by strongly continuous semigroups, by non-autonomous parabolic equations or by semilinear parabolic equations. It has further been shown that, in case of an autonomous problem, the exponential decay of perturbations is strongly connected to the turnpike property—a structural feature of optimal solutions stating that solutions of autonomous optimal control problems on a long time horizon reside close to a steady state for the majority of the time. In that context, novel turnpike results for optimal control problems are given. The theoretical analysis served as a foundation for efficient discretization methods for MPC. Thus, we proposed several a priori space and time discretization schemes. Further, we analyzed goal oriented a posteriori error estimation with a specialized objective for refinement, which only incorporates an initial part of the horizon, as a powerful tool for adaptive MPC. We proved under stabilizability assumptions that the error indicators decay exponentially outside the support of this specialized quantity of interest. Finally, we evaluated the behavior and performance of these specialized discretization algorithms in an MPC context by various numerical examples, including problems governed by linear, semilinear, and quasilinear dynamics with distributed and with boundary control.

Publications

• Sensitivity analysis of optimal control motivated by model predictive control. In Proceedings of the 23rd International Symposium on Mathematical Theory of Networks and Systems — Hong Kong, 2018
  L. Grüne, M. Schaller, and A. Schiela
  
• Sensitivity analysis of optimal control for a class of parabolic PDEs motivated by model predictive control. SIAM Journal on Control and Optimization, 57(4):2753–2774, 2019
  L. Grüne, M. Schaller, and A. Schiela
  
• Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs, 2020
  L. Grüne, M. Schaller, and A. Schiela
  
• Exponential sensitivity and turnpike analysis for linear quadratic optimal control of general evolution equations. Journal of Differential Equations, 268(12):7311–7341, 2020
  Grüne, M. Schaller, and A. Schiela
  
• Efficient MPC for parabolic PDEs with goal oriented error estimation, 2020
  L. Grüne, M. Schaller, and A. Schiela