Control of systems to sets and their interiors
For a differential inclusion \(\dot x\in F(t, x)\) and for a target set \(S\subset \mathbb{R}^n\) the problems of local weak (by some trajectory \(x(t)\) of the inclusion) and strong (by all trajectories \(x(t)\)) attainability of \(S\) are considered. The authors give the sufficient conditions for such attainability in terms of proximal normals and proximal subgradients. The proofs are based upon the time discretization method. The authors show that for the problem under consideration, the known state continuity assumption on the right-hand side \(F(t, x)\) of the differential inclusion can be weakened to upper semicontinuity in \(x\). The problems of weak and strong penetration at a point \(x\) are also studied (the penetration property means that for a fixed point \(x\) on the boundary of \(S\) some trajectory \(x(t)\) (for weak penetration) or every trajectory \(x(t)\) (for strong penetration) with a condition \(x(0)= x\) lies in the interior of \(S\) for small \(t>0\)). For these problems the target set \(S\) is assumed to be epicontinuous near \(x\). Three examples are given to illustrate the role of the hypotheses.
scientific article; zbMATH DE number 851108
scientific article