Fuzzy differential inclusions

Abstract

This paper is devoted to differential inclusions the right-hand sides of which are fuzzy subsets, whose membership functions are cost functions taking their values in [0, ∞] instead of [0, 1]. By doing so, the concept of uncertainty involved in differential inclusions becomes more precise, by allowing the velocities not only to depend in a multivalued way upon the state of the system, but also in a fuzzy way. The viability theorems are adapted to fuzzy differential inclusions and to sets of state constraints which are either usual or fuzzy. The existence of a largest closed fuzzy viability domain contained in a given closed fuzzy subset is also provided.