Abstract
In solving optimal control problems, there arise situations in which Pontryagin's “Maximum Principle” may provide no information for determining the optimal controls. Such situations are called “Singular” and the corresponding solutions “Singular controls”. The main objective of the present paper is to study the necessary and sufficient conditions for the existence (or non-existence) of singular controls to that class of control problems which, when suitable function spaces are introduced, can be regarded as special versions of the follwing Banach space minimization problem. Let X, Y and Z be real Banach spaces. Let T: X→Y and S: X→Z be bounded linear transformations. Let M be a closed convex body in Z.
[Problem F2] Given ξ∈Y and η∈Z, find an element u∈X statisfying (1) ||u||≤ρ and (2) η-Su∈M, while minimizing ||ξ-Tu||.
Throughout the paper, it is assumed that η∈Z is a regular element, i.e., there exists at least one control u satisfying ||u||≤ρ such that η-Su∈int (M) (the interior of M). Existence theorem, uniqueness theorem, and necessary and sufficient conditions for both optimality and nonexistence of singular controls are demonstrated in a concise manner by using techniques of functional analysis.
Much stress is placed upon application of the theory to final value problems, time optimal problems and minimum fuel problems. Main results obtained to these problems can generally be phrased as follows. In order for the j-th component (u0)j(t) of the optimal solution u0(t) to be a Bang-Bang control (whence free from being singular), it is necessary and sufficient that the value of a criterion functional under consideration be made strictly smaller if the j-th component uj(t) of the control input u(t) is released from the amplitude constraint, while the other components remaining constrained.
