Identification and control of unknown differential equations

In this thesis, we address control problems involving partially unknown dynamics. The proposed approach is inspired by two closely related fields: Optimal Control and Reinforcement Learning. Our goal is to develop an efficient algorithm capable of both identifying and controlling an unknown system. We assume that the system’s differential equation is observable, given a suitable control input and initial condition. With an initial estimate of the system parameters, we compute the corresponding control using the Differential Riccati Equation within the framework of the Linear Quadratic Regulator (LQR) for linear systems, or the State-Dependent Riccati Equation (SDRE) for nonlinear systems. We then observe the system’s trajectory and update the parameter estimate using Bayesian Linear Regression. This process iterates until the final time is reached, incorporating a stopping criterion to determine when to update the parameter configuration. We address control for both Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs). In particular, the systems resulting from the discretization of PDEs are high-dimensional, so we also focus on the computational efficiency of the algorithm. To reduce the computational burden, we apply Proper Orthogonal Decomposition (POD), a Model Order Reduction technique, to the system. This technique significantly accelerates the control computation while maintaining the desired accuracy. Numerical examples are provided throughout the thesis to demonstrate the effectiveness and accuracy of the proposed methods.