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A C1 function which is nowhere strongly paraconvex and nowhere semiconcave

Zajicek L.

Control and Cybernetics

2007

Vol. 36, no 3

803-810

X be an infinite-dimensional Banach space which admits an equivalent Fréchet smooth norm (or, more generally, a C1 Lipschitz bump function). Then there exists a C1 Lipschitz function ƒ on X such that, on any open ball, ƒ is neither strongly paraconvex nor semiconcave. In particular, ƒ is approximately convex in the sense of Ngai, Luc and Thera, but it is not strongly paraconvex on any ball.

Optimal synthesis via superdifferentials of value function

Frankowska H.

Control and Cybernetics

2005

Vol. 34, no 3

787-803

We derive a differential inclusion governing the evolution of optimal trajectories to the Mayor problem. The value function is allowed to be discontinuous. This inclusion has convex compact right-hand sides.