Abstract: | Let f: X Y be a nonlinear differentiable map, X,Y are Hilbert spaces, B(a,r) is a ball in X with a center a and radius r. Suppose f
(x) is Lipschitz in B(a,r) with Lipschitz constant L and f
(a) is a surjection: f
(a)X=Y; this implies the existence of >0 such that f
(a)*
y![Verbar](/content/u865k17680996915/xxlarge8214.gif) ![ge](/content/u865k17680996915/xxlarge8805.gif) ![ngr](/content/u865k17680996915/xxlarge957.gif) y , y Y. Then, if r, /(2L), the image F=f(B(a, )) of the ball B(a, ) is convex. This result has numerous applications in optimization and control. First, duality theory holds for nonconvex mathematical programming problems with extra constraint x–a![Verbar](/content/u865k17680996915/xxlarge8214.gif) ![le](/content/u865k17680996915/xxlarge8804.gif) . Special effective algorithms for such optimization problems can be constructed as well. Second, the reachability set for small power control is convex. This leads to various results in optimal control. |