Lax-Oleinik-type formulas and efficient algorithms for certain high-dimensional optimal control problems

Abstract

We provide analytical solutions to certain optimal control problems whose running cost depends on the state variable and with constraints on the control. We also provide Lax-Oleinik-type representation formulas for the corresponding Hamilton-Jacobi partial differential equations with statedependent Hamiltonians. Additionally, we present efficient, grid-free numerical solvers based on our representation formulas, which are shown, in some cases, to scale linearly with the state dimension, and thus, to overcome the curse of dimensionality. Using existing optimization methods and the min-plus technique, we extend our numerical solvers to address more general classes of convex and nonconvex initial costs. As a result, our numerical methods have the potential to serve as a building block for solving broader classes of highdimensional optimal control problems in real-time.Keywords Optimal control • Hamilton-Jacobi partial differential equations • Grid-free numerical methods • High dimensions Research supported by NSF 1820821 and AFOSR MURI FA9550-20-1-0358. Authors' names are given in last/family name alphabetical order.

On representation formulas for optimal control: A Lagrangian perspective

“…More recently, various algorithms using the Hopf-Lax type representation formulas have been proposed and shown to efficiently solve some classes of high-dimensional HJB equations [10][11][12][13][14][15]. Furthermore, there have been a few attempts to identify explicit solutions to a certain class of optimal control problems using representation formulas related to the Hopf-Lax formula [16]. Another notable work is [1], where a generalized Lax formula is obtained via DP to handle both state constraints and nonlinear systems.…”

Hopf-type representation formulas and efficient algorithms for certain high-dimensional optimal control problems

“…For example, optimal control problems with state-dependent running costs are, in general, difficult to solve without approximations. To this end, [14] recently provided a Lax-Oleinik-type formula and corresponding exact numerical solver for certain optimal control problems with running costs quadratic in the state variable and certain constraints on the control variable. However, to our knowledge, there is no numerically-computable representation formula in the literature for optimal control problems with non-quadratic, state-dependent running costs.…”

SympOCnet: Solving optimal control problems with applications to high-dimensional multi-agent path planning problems

“…In literature, there are some algorithms for solving high-dimensional optimal control problems (or the corresponding Hamilton-Jacobi PDEs), which include optimization methods [18,24,21,23,15,14,87,60,55], max-plus methods [1,2,29,35,39,67,66,68,69], tensor decomposition techniques [28,44,86], sparse grids [10,37,53], polynomial approximation [51,52], model order reduction [4,57], optimistic planning [9], dynamic programming and reinforcement learning [13,11,3,8,89], as well as methods based on neural networks [5,6,27,47,42,45,46,58,73,78,81,84,62,20,…”