Optimization—Theory and Applications pp 233-270 | Cite as

# Proofs of the Necessary Condition for Control Problems and Related Topics

Chapter

## Abstract

Let *A* denote the constraint set, a closed subset of the *tx*-space, with *t* in *R*, and the space variable *x* = (*x*^{1},…, *x*^{ n }) in *R*^{ n }.
Let *U*(*t*),the control set, be a subset of the *u*-space *R*^{ m }, *u*= (*u*^{1},
…*u*^{ m })the control variable. Let *M* = [(*t*,*x*,*u*)|(*t*,*x*)∈*A*, *u* ∈ *U*(*t*)] be a closed subset of *R*^{1+n+m}, and let *f* = (*f*_{1},…,*f*_{n}) be a continuous vector function from *M* into *R*^{n}. Let the boundary set *B* be a closed set of points (*t*_{1},*x*_{1},*t*_{2},*x*_{2}) in *R*^{2n+2}, *x*_{1} = (*x* _{1} ^{1} ,…*x* _{1} ^{n} ), *x*_{2} = (*x* _{2} ^{1} ,…*x* _{2} ^{n} ). Let *g* be a continuous function from *B* into *R*.

## Keywords

Control Problem Closed Subset Convex Cone Lebesgue Point Supporting Hyperplane
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag New York Inc. 1983