A continuous-time linear tracking problem with an asymmetric quadratic objective functional containing a cost-free interval.

We solve an optimal control problem with an asymmetric quadratic objective functional containing a ‘cost-free’ interval. If the state variable falls within the specified ‘cost-free’ interval, the system incurs no cost at all. The cost of overshooting or undershooting this interval is asymmetric quadratic. The problem is transformed into an equivalent control problem with state and control variable inequality constraints. We develop the reciprocal (dual) of the latter, which turns out to be a linear regulator problem with simple control-variable non-negativity constraints. Using the solution to the regulator problem, one can easily generate the solution to the original problem.