A note on the minimum-norm in dual space approach to some classical linear optimal control problems

Some classical optimal control problems of linear systems can be characterized as finding minimum-norm vectors from within linear varieties in appropriate dual spaces. Then, the solution to such an optimal control problem can be derived from the alignment between the optimal vector in the dual space and the optimal vector in the primal space, and the dual maximization problem of the minimum-norm problem. This note presents a detailed introduction to this minimum-norm in dual space approach by examples of minimum-supremum-norm, minimum-energy, and minimum-time optimal control problems of linear systems. Connections and differences between these problems in light of the introduced approach are discussed.