Solution of the sweeping problem and application to a friction problem
The largest part of the article contains several results concerning existence and some qualitative properties of solutions \(q(\cdot)\in BV(I;H)\) (with bounded variation) to sweeping problems of the form \[ q(t)\in F(t,q(t)) \;\forall \;t\in I=[0,T], \quad -{{dq}\over{d\mu}}(t)\in N_{F(t,q(t))}q(t) \text{ a.e. }(d\mu), \quad q(0)=q_0\in F(0,q_0), \] under several types of regularity assumptions on the convex-valued multifunction \(F(\cdot,\cdot)\); here, \(H\) is either the real line (\(H=\mathbb R\)) or a finite-dimensional Hilbert space (\(H=\mathbb R^n\)), \(dq\) denotes the Stieltjes differential measure associated to the mapping \(q(\cdot)\in BV(I;H)\), \(d\mu\) is a positive measure on \(I\) for which there exists \(q'(\cdot)\in L^1(I;H)\) such that \(dq=q'd\mu\) and \(N_Aa\) denotes the normal cone in the sense of convex analysis to the convex subset \(A\subset H\) at the point \(a\in A\). Most of the results in the paper seem to be slight extensions of previous results, mainly from \textit{M. Kunze} and \textit{M. D. P. Monteiro Marques} [Topol. Methods Nonlinear Anal. 12, No. 1, 179--191 (1998; Zbl 0923.34018)]; the last part of the paper contains an application to a dry friction problem of Coulomb type.
scientific article; zbMATH DE number 1751363