Uniform Boundedness of Norms of Convex and Nonconvex Processes

Abstract

The lower limit F:ℝ ⁿ ℝ ^m of a sequence of closed convex processes F _ν:ℝ ⁿ ℝ ^m is again a closed convex process. In this note, we prove the following uniform boundedness principle: if F is nonempty-valued everywhere, then there is a positive integer ν₀ such that the tail {F _ν}_ν≥ν0 is “uniformly bounded” in the sense that the norms ‖ F _ν‖ are bounded by a common constant. As shown with an example, the uniform boundedness principle is not true if one drops convexity. By way of illustration, we consider an application to the controllability analysis of differential inclusions.

Keywords:

• Controllability
• Convex processes
• Graph-convergence
• Painlevé–Kuratowski limits
• Positively homogeneous maps

AMS Subject Classification:

• 34A60
• 47H04
• 52A20