Dynamic programming and duality applied to an optimal control problem

The solution to a basic problem in time-invariant l1 optimal control is constructed in feedback form. Using ideas from dynamic programming and duality, we find the rule describing how, for optimal l1 regulation, the state at any time instant is to be mapped to the state at the next time instant. This mapping is a function of the state alone and, much like the optimal gain matrix for linear quadratic control, can be computed before system operation.