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

Understand

Two of the main challenges in optimal control are solving problems with state-dependent running costs and developing efficient numerical solvers that are computationally tractable in high dimension.

• In this paper, 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 state-dependent Hamiltonians.
• Additionally, we present an efficient, grid-free numerical solver based on our representation formulas, which is shown to scale linearly with the state dimension, and thus, to overcome the curse of dimensionality.