Abstract
In the present paper, the minimum effort control problem under phase constraint is considered by functional analysis approach. The abstract version of the problem can be stated as follows; Let X, Y and Z be real Banach spaces. Let S, from X into Y, and T, from X onto Z, be bounded linear transformations. With ξ∈Y and η∈Z, find an element (if one exists) u∈X satisfying η=Tu and ||ξ-Su||≤ε, while minimizing ||u||. Throughout the paper, the pair (ξ, η) is assumed to be regular in the sense that there exists at least one element u∈X which satisfies the constraints η=Tu and ||ξ-Su||<ε. Initially X, Y and Z are taken to be arbitrary Banach spaces and the necessary and sufficient conditions for the existence of the solution to a regular pair are discussed.
It is then required that either X be a reflexive Banach space, or each of the three Banach spaces be the conjugate of another normed space. After having established the existence of the solution, the primal problem is transformed into the dual one in the conjugate space by virtue of the Kahn-Banach produced hyperplane of support to a convex body. As a result, the optimal solution is completely characterized in terms of the hyperplane. Incidentally, the conjugate equation and the conjugate problem are derived. It is shown that this conjugate problem provides a useful computational procedure forr obtaining the explicit solution through an example.
