On Stochastic Optimal Control in Ferromagnetism

Dynamic Programming for Finite Ensembles of Nanomagnetic Particles

“…In [5], some of the present authors have studied Problem 1.1 in its weak form and constructed a weak optimal solution Π * := Ω * , P * , F * , {F * t } 0≤t≤T , m * , u * , W * of the underlying problem via Young measure (relaxed control) approach for a compact set U ⊂ (R 3 ) N , a control space, such that 0 ∈ U, which may be generalized to the case U = (R 3 ) N thanks to the coercivity of the cost functional with respect to u; see also [4] for an extension to infinite spin ensembles. To approximate it numerically, implementable strategies may be developed that rest on Pontryagin's maximum principle which characterizes minimizers.…”

“…In [5], a stochastic gradient method is proposed to generate a sequence of functional-decreasing approximate feedback controls, where the update requires to solve a coupled forward-backward SDE system. A relevant part here is to simulate a (time-discrete) backward SDE via the least-squares Monte-Carlo method, which requires significant data storage resources [4,5], and thus limits the complexity of practically approachable Problems 1. 1. In this work, we use an alternative strategy which rests on the dynamic programming principle.…”

“…To approximate ∇ M w through a difference quotient with needed accuracy, we choose a stencil diameter h = O 1/ √ M for a sufficiently large number of Monte-Carlo realizations M ; see Remark 6.1. Importantly, this approach does not require larger data storage resources as [4,5] does, but an ample calculation of iterates from related SDEs. Computational studies for the switching dynamics of single and multiple ferromagnetic particles are reported in Section 6.…”

Stochastic optimal control of a evolutionary $p$-Laplace equation with multiplicative Lévy noise

“…[21]) on stochastic optimal control with SPDEs mainly considers those which has a mild solution, which is not available for problem (1.1). For this reason, we use variational method to construct a minimizer π * of (1.2), see also [7,9]. Being motivated from [7,9,28], our aim is twofold: i) Firstly, we prove existence of a weak solution of the problem (1.1).…”

“…For this reason, we use variational method to construct a minimizer π * of (1.2), see also [7,9]. Being motivated from [7,9,28], our aim is twofold: i) Firstly, we prove existence of a weak solution of the problem (1.1). We construct an approximate solutions ũ∆t := ũ∆t (t); t ∈ [0, T ]} (cf.…”