APPROXIMATING OPTIMAL FEEDBACK CONTROLLERS OF FINITE HORIZON CONTROL PROBLEMS USING HIERARCHICAL TENSOR FORMATS

Controlling systems of ordinary differential equations is ubiquitous in science and en-gineering. For finding an optimal feedback controller, the value function and associated fundamental equations such as the Bellman equation and the Hamilton-Jacobi-Bellman equation are essential. The numerical treatment of these equations poses formidable challenges due to their non-linearity and their (possibly) high dimensionality. In this paper we consider a finite horizon control sys-tem with associated Bellman equation. After a time discretization, we obtain a sequence of short time horizon problems which we call local optimal control problems. For solving the local optimal control problems we apply two different methods; one is the well-known policy iteration, where a fixed-point iteration is required for every time step. The other algorithm borrows ideas from model predictive control by solving the local optimal control problem via open-loop control methods on a short time horizon, allowing us to replace the fixed-point iteration by an adjoint method. For high -dimensional systems we apply low-rank hierarchical tensor product approximation/tree-based tensor formats, in particular tensor trains and multipolynomials, together with high-dimensional quadra-ture, e.g., Monte Carlo. We prove a linear error propagation with respect to the time discretization and give numerical evidence by controlling a diffusion equation with an unstable reaction term and an Allen-Cahn equation.

Automated summary

The paper discusses the common occurrence of controlling systems of ordinary differential equations in science and engineering. It focuses on the local optimal control problems in finite horizon control systems, with two methods being applied to solve these problems - policy iteration and open-loop control methods inspired by model predictive control. The use of low-rank hierarchical tensor product approximation and high-dimensional quadrature is also explored for high-dimensional systems, with linear error propagation demonstrated with numerical evidence on diffusion and Allen-Cahn equations.