On a topological condition for strongly asymptotically stable differential inclusions

In this paper, we will show that a system on a non-contractible manifold cannot be strongly asymptotically stabilized in Filippov’s sense, even if discontinuous feedback is used. The fact is well-known for C
¹
feedback case, and we extend it to the discontinuous feedback case. To consider the stabilization problem on a non-contractible manifold, the assumption convexity or upper semicontinuity is restrictive. We will propose a new type of differential inclusion without upper semicontinuity by defining a function indicating a rate of leaving from discontinuous set. By adopting the new differential inclusion, stabilization problems on non-contractible manifolds become possible for many cases.