On averaging for switched linear differential algebraic equations

Distributional averaging of switched DAEs with two modes

“…This assumption was also used (and motivated) in [5] to show convergence of the impulse-free part of the solution, but was recently relaxed [8]. However, it is not clear whether this relaxation is also applicable for the convergence of the impulsive part of the solution.…”

“…3]). The available averaging results for switched DAEs Stephan Trenn is with the Technomathematics group, University of Kaiserslautern, Germany, email: trenn@mathematik.uni-kl.de do not allow for Dirac impulses in the solutions and in [5,Rem. 1] the hope was articulated that the effect of the Dirac impulses for fast switching can be neglected, because the Dirac impulses are induced by the jumps in the solutions and the magnitude of the jumps converges to zero for an increasing switching frequency.…”

“…[2] or [3]) that convergence towards an average model is always guaranteed. However, due to the presence of jumps in the solutions of switched DAEs (1) convergence towards an average system does not always takes place and additional assumptions have to be made [5], [4], [9], [7], [8]. In addition to jumps, the solutions of (1) can also contain Dirac impulse (the observation that inconsistent initial values can induce Dirac impulses was already made in [17], for a recent discussion on the effect of inconsistent initial values see [16,Sec.…”

“…The question is now: Does the impulsive part x 3 [·] converges to zero in the distributional sense (as hoped in[5])?For any ϕ ∈ C ∞ 0 with supp ϕ ⊆ [0, T ], T ∈ R, it holds thatx p,3 [·](ϕ) = − T /p k=1 d 2 px 0 1 δ kp (ϕ) = −d 2…”

Stabilization of switched DAEs via fast switching

“…An alternative approach is to directly consider the averaged model for the switched DAE (without introducing the approximation parameter ε > 0), i.e. utilizing our averaging results [2][3][4][7][8][9] for the problem of stabilization via fast switching. Assume the switching signal σ : [0, ∞) → {1, 2, .…”