Speaker
Description
Hamilton-Jacobi-Bellman partial differential equations (HJB PDEs) arise in various settings in optimal control and model order reduction, and their solutions are notoriously difficult to acquire. For linear time-invariant systems, the HJB PDEs of interest typically simplify to matrix algebraic equations, such as the algebraic Riccati equation or the matrix Lyapunov equation, for which many mature, reliable, and scalable solvers exist.
For weakly nonlinear systems with polynomial dynamics, a popular approach for locally approximating solutions to the HJB PDEs is the method of Al'brekht, which amounts to computing a Taylor series approximation. The first term in the Taylor expansion is given by the solution to a matrix Riccati (or Lyapunov) equation, and then the higher-order terms in the expansion are given by solutions to very large linear algebraic equations.
Despite being dense, it turns out that these equations exhibit a surprising amount of structure reminiscent of the matrix Lyapunov equation, and they can be considered as tensor Lyapunov equations. In this talk, we expand on related works by considering the generalized tensor Lyapunov equation that arises due to the presence of an invertible but non-identity mass matrix. We describe the changes to the problem associated with the inclusion of the mass matrix, some surprising beauty that emerges in the form of the fractal structure of the Sierpinski triangle, and the resulting obstacles that are introduced that preclude the use of existing solvers, before concluding with a new proposed algorithm for solving these structured systems efficiently. We also highlight the open problems and opportunities for future research on the topic.