Monotone viable trajectories for functional differential inclusions

Abstract

This paper is a study on functional differential inclusions with memory which represent the multivalued version of retarded functional differential equations. The main result gives a necessary and sufficient equations. The main result gives a necessary and sufficient condition ensuring the existence of viable trajectories; that means trajectories remaining in a given nonempty closed convex set defined by given constraints the system must satisfy to be viable. Some motivations for this paper can be found in control theory where F( t, φ) = { f( t, φ, u)} uϵU is the set of possible velocities of the system at time t, depending on the past history represented by the function φ and on a control u ranging over a set U of controls. Other motivations can be found in planning procedures in microeconomics and in biological evolutions where problems with memory do effectively appear in a multivalued version. All these models require viability constraints represented by a closed convex set.