Fun story: We once had a robot at a very big company you've all heard of that kept getting too close to walls when moving down hallway-like areas. The controls engineer swore he had tuned the controller to death and no further improvements could be had.
A perception engineer took one look at the controller and saw linear error terms for distance left + distance right (distance to walls), changed it to distance left^2 + distance right^2, and the whole thing magically worked beautifully. Exercise for the reader: What position in the hallway minimizes sum of distances squared, vs what position(s) in the hallway minimize sum of distances without square.
That has a unique solution, so it is infinitely better than the linear error, but is not a nice differentiable signal suitable for controls, I'd bet. But that wasn't in the problem statement.
A perception engineer took one look at the controller and saw linear error terms for distance left + distance right (distance to walls), changed it to distance left^2 + distance right^2, and the whole thing magically worked beautifully. Exercise for the reader: What position in the hallway minimizes sum of distances squared, vs what position(s) in the hallway minimize sum of distances without square.
This is essentially the same problem you pose.