A POD Approach to Identify and Control PDEs Online Through State Dependent Riccati Equations

Alessandro Alla () and Agnese Pacifico ()
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Alessandro Alla: Sapienza Università di Roma
Agnese Pacifico: Sapienza Università di Roma

Dynamic Games and Applications, 2025, vol. 15, issue 2, No 7, 502 pages

Abstract: Abstract We address the control of partial differential equations (PDEs) with unknown parameters. Our objective is to devise an efficient algorithm capable of both identifying and controlling the unknown system. We assume that the desired PDE is observable provided a control input and an initial condition. The method works as follows, given an estimated parameter configuration, we compute the corresponding control using the state-dependent Riccati equation (SDRE) approach. Subsequently, after computing the control, we observe the trajectory and estimate a new parameter configuration using Bayesian linear regression method. This process iterates until reaching the final time, incorporating a defined stopping criterion for updating the parameter configuration. We also focus on the computational cost of the algorithm, since we deal with high dimensional systems. To enhance the efficiency of the method, indeed, we employ model order reduction through the proper orthogonal decomposition (POD) method. The considered problem’s dimension is notably large, and POD provides impressive speedups. Further, a detailed description on the coupling between POD and SDRE is also provided. Finally, numerical examples will show the accuracy of our method across two test cases: Allen-Cahn and viscous Burgers’ equation.

Keywords: State dependent Riccati equation; Feedback control; System identification; Model order reduction; Proper orthogonal decomposition (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13235-024-00605-3

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