  This paper presents a framework to jointly optimize the design and control of
an electric race car equipped with a multiple-gear transmission, specifically
accounting for the discrete gearshift dynamics. We formulate the problem as a
mixed-integer optimal control problem, and deal with its complexity by
combining convex optimization and Pontryagin's Minimum Principle in a
computationally efficient iterative algorithm satisfying necessary conditions
for optimality upon convergence. Finally, we leverage our framework to compute
the achievable lap time of a race car equipped with a fixed-gear transmission,
a continuously variable transmission and a multiple-gear transmission with 2 to
4 speeds, revealing that a multiple-gear transmission can strike the best
trade-off in terms of electric motor control, and transmission weight and
efficiency, ultimately yielding the overall best lap time.
