ISSN:
1573-2878
Keywords:
Existence theorems
;
necessary conditions
;
control theory
;
linear systems
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract 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 (i) J(x,y)⩾-0 for (x,y)teX×Y, (ii) J(θ,θ)=0, (iii) J(x,y)→+∞, as (∥x∥2+∥y∥2)1/2→+∞. Givenζ∈Y andη∈S[core 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00933206
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