  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. 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. We demonstrate the capabilities of our numerical solvers using
implementations on a central processing unit (CPU) and a field-programmable
gate array (FPGA). In several cases, our FPGA implementation obtains over a 10
times speedup compared to the CPU, which demonstrates the promising performance
boosts FPGAs can achieve. Our numerical results show that our solvers have the
potential to serve as a building block for solving broader classes of
high-dimensional optimal control problems in real-time.
