  This paper presents a mathematical formulation to perform temporal
parallelisation of continuous-time optimal control problems, which are solved
via the Hamilton--Jacobi--Bellman (HJB) equation. We divide the time interval
of the control problem into sub-intervals, and define a control problem in each
sub-interval, conditioned on the start and end states, leading to conditional
value functions for the sub-intervals. By defining an associative operator as
the minimisation of the sum of conditional value functions, we obtain the
elements and associative operators for a parallel associative scan operation.
This allows for solving the optimal control problem on the whole time interval
in parallel in logarithmic time complexity in the number of sub-intervals. We
derive the HJB-type of backward and forward equations for the conditional value
functions and solve them in closed form for linear quadratic problems. We also
discuss other numerical methods for computing the conditional value functions
and present closed form solutions for selected special cases. The computational
advantages of the proposed parallel methods are demonstrated via simulations
run on a multi-core central processing unit and a graphics processing unit.
