Existence theorems and necessary conditions for a general formulation of the minimum effort problem

The basic problem considered may be described briefly as follows. LetX,Y, andZ be normed linear spaces,T:D(T)→Y,S:D(S)→Z linear operators withD(T) \( \subseteq\) X andD(S) \( \subseteq\) X,Ω \( \subseteq\) X a convex set containing the zero elementθ, andJ a real-valued convex function defined onX×Y such that

1. J(x,y)?-0 for (x,y)teX×Y,
2. J(θ,θ)=0,
3. J(x,y)→+∞, as (∥x∥²+∥y∥²)^1/2→+∞.

Givenζ∈Y andη∈Score _T Ω∩;D(S)], find an elementx=x ₀ which minimizesJ(x,ζ?Tx) on the set {x∈Ω∩;D(S)∩;D(T)]:Sx=η}. The abovementioned problem, together with certain special cases, is analyzed using the classical techniques of functional analysis. Existence problems are considered for a certain class of closed linear operators. In particular, existence of an optimal solution is determined by evaluating a generalized Minkowski functional at the point (ζ,η) inY×Z. A necessary condition is presented for special cases, and corresponding characterizations of optimal solutions are made in terms of the adjoint operators. These results are applicable to linear minimum effort problems, constrained variational problems, optimal control of distributive systems, and certain ill-posed variational problems.