We study a two-sided space-time $L^1$ optimization problem and show how to reformulate the problem within the framework of optimal control theory for polynomial systems. This yields insight on the structure of the optimal solution. We prove existence and uniqueness of the optimal solution, and we characterize it by means of the Pontryagin maximum principle. The cost function and the control converge when the polynomial degree tends to $+∞$. We illustrate the theory with numerical simulations, which show that our optimal control interpretation leads to efficient algorithms.