Dynamic programming and duality applied to an optimal control problem

The solution to a basic problem in time-invariant l
₁
optimal control is constructed in feedback form. Using ideas from dynamic programming and duality, we find the rule describing how, for optimal l
₁
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.