This paper studies a systemic risk control problem by the central bank, which dynamically plans monetary supply to stabilize the interbank system with borrowing and lending activities. Facing both ...heterogeneity among banks and the common noise, the central bank aims to find an optimal strategy to minimize the average distance between log-monetary reserves of all banks and the benchmark of some target steady levels. A weak formulation is adopted, and an optimal randomized control can be obtained in the system with finite banks by applying Ekeland’s variational principle. As the number of banks grows large, we prove the convergence of optimal strategies using the Gamma-convergence argument, which yields an optimal weak control in the mean field model. It is shown that this mean field optimal control is associated to the solution of a stochastic Fokker–Planck–Kolmogorov (FPK) equation, for which the uniqueness of the solution is established under some mild conditions.
Variational convergence of discrete elasticae Scholtes, Sebastian; Schumacher, Henrik; Wardetzky, Max
IMA journal of numerical analysis,
01/2022, Volume:
42, Issue:
1
Journal Article
Peer reviewed
Open access
Abstract
We discuss a discretization of the Euler–Bernoulli bending energy and of Euler elasticae under clamped boundary conditions by polygonal lines. We show Hausdorff convergence of the set of ...almost minimizers of the discrete bending energy to the set of smooth Euler elasticae under mesh refinement in (i) the $W^{1,\infty }$-topology for piecewise-linear interpolation; and in (ii) the $W^{2,p}$-topology, $p \in 2,\infty $, using a suitable smoothing operator to create $W^{2,p}$-curves from polygons.
A hierarchy of multilayered plate models de Benito Delgado, Miguel; Schmidt, Bernd
ESAIM. Control, optimisation and calculus of variations,
2021, Volume:
27
Journal Article
Peer reviewed
Open access
We derive a hierarchy of plate theories for heterogeneous multilayers from three dimensional nonlinear elasticity by means of Γ-convergence. We allow for layers composed of different materials whose ...constitutive assumptions may vary significantly in the small film direction and which also may have a (small) pre-stress. By computing the Γ-limits in the energy regimes in which the scaling of the pre-stress is non-trivial, we arrive at linearised Kirchhoff, von Kármán, and fully linear plate theories, respectively, which contain an additional spontaneous curvature tensor. The effective (homogenised) elastic constants of the plates will turn out to be given in terms of the moments of the pointwise elastic constants of the materials.
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