Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1802.04978

Singularly perturbed forward-backward stochastic differential equations: application to the optimal control of bilinear systems

Omar Kebiri·Lara Neureither·Carsten Hartmann
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study linear-quadratic stochastic optimal control problems with bilinear state dependence for which the underlying stochastic differential equation (SDE) consists of slow and fast degrees of freedom. We show that, in the same way in which the underlying dynamics can be well approximated by a reduced order effective dynamics in the time scale limit (using classical homogenziation results), the associated optimal expected cost converges in the time scale limit to an effective optimal cost. This entails that we can well approximate the stochastic optimal control for the whole system by the reduced order stochastic optimal control, which is clearly easier to solve because of lower dimensionality. The approach uses an equivalent formulation of the Hamilton-Jacobi-Bellman (HJB) equation, in terms of forward-backward SDEs (FBSDEs). We exploit the efficient solvability of FBSDEs via a least squares Monte Carlo algorithm and show its applicability by a suitable numerical example.

Explore related subjects

math.DSmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Omar Kebiri, Lara Neureither, Carsten Hartmann. 2018-03-20. Singularly perturbed forward-backward stochastic differential equations: application to the optimal control of bilinear systems. https://arxiv.org/abs/1802.04978

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Toric Differential Inclusions and a Proof of the Global Attractor Conjecture

The global attractor conjecture says that toric dynamical systems have a globally attracting point (up to linear conservation relations), or equivalently, complex balanced systems have a globally attracting point within each stoichiometric compatibility class. A proof of this conjecture implies that a large class of nonlinear dynamical systems on the positive orthant have very simple and stable dynamics. The conjecture originates from the 1972 breakthrough work by Fritz Horn and Roy Jackson, and was formulated in its current form by Horn in 1974. Toric dynamical systems can be embedded into toric differential inclusions. We show that each bounded positive solution of a toric differential inclusion is contained in an invariant region that prevents it from approaching the boundary of the positive orthant. We use this result to prove the global attractor conjecture. In particular, it follows that all detailed balanced mass action systems and all deficiency zero weakly reversible systems have the global attractor property.

math.DS↗
arXiv2026

Pinched Arnol'd tongues for Families of circle maps

We prove that generically for a family of circle maps \begin{equation*} f_{b, ω} (x) = x + ω+ b\, ϕ(x) \end{equation*} with $ϕ$ a piecewise linear forcing with $k>2$ breakpoints there is no pinching in any of its Arnol'd tongues. This is in contrast to a theorem of Campbell, Galeeva, Tresser, and Uherka who showed that with two break points there are always multiple pinchpoints in its rational tongues. We also prove that the absence of pinching is generic for Lipschitz and $C^r$ ($r>0$) forcing. The family $f_{b, ω}$ is used as a simple model for a periodically forced oscillator. The rational tongue $T_{p/q}$ represents parameter values where the system is mode-locked into a $p/q$-periodic response. The pinching of the tongues to a point represents parameter values where the system's periodic response is unstable to all perturbations in the frequency parameter $ω$. The theorems in this paper show that typically this type of instability does not occur in the families under consideration.

math.DS↗
arXiv2026

Mostly nonuniformly sectional expanding systems

We introduce the notion of \emph{mostly nonuniform sectional expanding} (MNUSE) for singular flows which encompasses the notions of sectional hyperbolicity, asymptotically sectional and multisingular hyperbolicity. We construct examples of a vector field of class $C^r, r \ge 1$, whose flow exhibits a nonuniformly sectional hyperbolic set satisfying MNUSE, which is neither sectional hyperbolic nor asymptotically sectional hyperbolic. We obtain sufficient conditions for the existence of physical/SRB measures for asymptotically sectionally hyperbolic attracting sets with any finite codimension, extending the codimension two case. We provide examples of such attractors, either with non-sectional hyperbolic equilibria, or with sectional hyperbolic equilibria of mixed type, i.e., with a Lorenz-like singularity together with a Rovella-like singularity in a transitive set. These are higher-dimensional versions of contracting Lorenz-like attractors (also known as Rovella-like attractors) to which we apply our criteria to obtain a physical/SRB measure with full ergodic basin. We also adapt the previous examples to obtain higher codimensional (i.e. with central direction of dimension greater than $2$) nonuniformly sectional expanding attractors.

math.DS↗