52 research outputs found
Galerkin approximations for the optimal control of nonlinear delay differential equations
Optimal control problems of nonlinear delay differential equations (DDEs) are
considered for which we propose a general Galerkin approximation scheme built
from Koornwinder polynomials. Error estimates for the resulting
Galerkin-Koornwinder approximations to the optimal control and the value
function, are derived for a broad class of cost functionals and nonlinear DDEs.
The approach is illustrated on a delayed logistic equation set not far away
from its Hopf bifurcation point in the parameter space. In this case, we show
that low-dimensional controls for a standard quadratic cost functional can be
efficiently computed from Galerkin-Koornwinder approximations to reduce at a
nearly optimal cost the oscillation amplitude displayed by the DDE's solution.
Optimal controls computed from the Pontryagin's maximum principle (PMP) and the
Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE
systems, are shown to provide numerical solutions in good agreement. It is
finally argued that the value function computed from the corresponding reduced
HJB equation provides a good approximation of that obtained from the full HJB
equation.Comment: 29 pages. This is a sequel of the arXiv preprint arXiv:1704.0042
Galerkin-Koornwinder approximations of delay differential equations for physicists
Formulas for Galerkin-Koornwinder (GK) approximations of delay differential
equations are summarized. The functional analysis ingredients (semigroups,
operators, etc.) are intentionally omitted to focus instead on the formulas
required to perform GK approximations in practice
Ruelle-Pollicott Resonances of Stochastic Systems in Reduced State Space. Part II: Stochastic Hopf Bifurcation
The spectrum of the generator (Kolmogorov operator) of a diffusion process,
referred to as the Ruelle-Pollicott (RP) spectrum, provides a detailed
characterization of correlation functions and power spectra of stochastic
systems via decomposition formulas in terms of RP resonances. Stochastic
analysis techniques relying on the theory of Markov semigroups for the study of
the RP spectrum and a rigorous reduction method is presented in Part I. This
framework is here applied to study a stochastic Hopf bifurcation in view of
characterizing the statistical properties of nonlinear oscillators perturbed by
noise, depending on their stability. In light of the H\"ormander theorem, it is
first shown that the geometry of the unperturbed limit cycle, in particular its
isochrons, is essential to understand the effect of noise and the phenomenon of
phase diffusion. In addition, it is shown that the spectrum has a spectral gap,
even at the bifurcation point, and that correlations decay exponentially fast.
Explicit small-noise expansions of the RP eigenvalues and eigenfunctions are
then obtained, away from the bifurcation point, based on the knowledge of the
linearized deterministic dynamics and the characteristics of the noise. These
formulas allow one to understand how the interaction of the noise with the
deterministic dynamics affect the decay of correlations. Numerical results
complement the study of the RP spectrum at the bifurcation, revealing useful
scaling laws. The analysis of the Markov semigroup for stochastic bifurcations
is thus promising in providing a complementary approach to the more geometric
random dynamical system approach. This approach is not limited to
low-dimensional systems and the reduction method presented in part I is applied
to a stochastic model relevant to climate dynamics in part III
A Girsanov approach to slow parameterizing manifolds in the presence of noise
We consider a three-dimensional slow-fast system with quadratic nonlinearity
and additive noise. The associated deterministic system of this stochastic
differential equation (SDE) exhibits a periodic orbit and a slow manifold. The
deterministic slow manifold can be viewed as an approximate parameterization of
the fast variable of the SDE in terms of the slow variables. In other words the
fast variable of the slow-fast system is approximately "slaved" to the slow
variables via the slow manifold. We exploit this fact to obtain a two
dimensional reduced model for the original stochastic system, which results in
the Hopf-normal form with additive noise. Both, the original as well as the
reduced system admit ergodic invariant measures describing their respective
long-time behaviour. We will show that for a suitable metric on a subset of the
space of all probability measures on phase space, the discrepancy between the
marginals along the radial component of both invariant measures can be upper
bounded by a constant and a quantity describing the quality of the
parameterization. An important technical tool we use to arrive at this result
is Girsanov's theorem, which allows us to modify the SDEs in question in a way
that preserves transition probabilities. This approach is then also applied to
reduced systems obtained through stochastic parameterizing manifolds, which can
be viewed as generalized notions of deterministic slow manifolds.Comment: 54 pages, 6 figure
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