# Sparse approximation methods for optimal control and stationary action problems

**Project Leader: **Peter Dower**Collaborators: **Prof. William McEneaney (UCSD)**Primary Contact: **Peter Dower (pdower@unimelb.edu.au)**Keywords: **advanced control architectures; applied control theory; optimisation; systems theory**Disciplines: **Electrical & Electronic Engineering**Domains: ****Research Centre: **Melbourne Systems Laboratory (MSL)

Optimal control and stationary action problems share a common objective in seeking to render an associated cost or integrated Lagrangian (action) functional stationary in a general calculus of variations sense. By applying the tools of optimal control, or their generalisation for stationary action problems, the dynamics of an optimally controlled process or conservative physical system can be encapsulated by the characteristic flow of an attendant Hamilton-Jacobi-Bellman partial differential equation (HJB PDE). Individual open-loop trajectories can subsequently be identified as corresponding solutions of a two-point boundary value problem (TPBVP) constrained by this flow, while a general feedback characterisation of all possible such open-loop trajectories can be described via the solution of the aforementioned HJB PDE. While robustness considerations demand utilisation of this latter feedback characterisation, HJB PDEs are difficult to solve in practice, as numerical methods that approximate their solution routinely suffer from a curse-of-dimensionality that limits their feasibility.

In this research, the algebraic properties of dynamic programming and HJB PDEs will be explored with the objective of developing new theory and methods for attenuating the curse-of-dimensionality via sparse solution representations and approximations. Theory and examples corresponding in particular to idempotent methods, optimisation methods, and relaxation methods, will be developed. Details include the interplay of idempotent linearity, duality, and semigroup properties in idempotent methods for providing a sparse basis approximation for HJB PDE solutions, the utility of a generalised Hopf formula in yielding sparse approximations in optimisation methods, and the role of non-quadratic cost lifting and approximation in relaxation methods.