Optimal control of multiscale systems using reduced-order models

We study optimal control of diffusions with slow and fast variables and address a question raised by practitioners: is it possible to first eliminate the fast variables before solving the optimal control problem and then use the optimal control computed from the reduced-order model to control the original, high-dimensional system? The strategy "first reduce, then optimize"--rather than "first optimize, then reduce"--is motivated by the fact that solving optimal control problems for high-dimensional multiscale systems is numerically challenging and often computationally prohibitive. We state sufficient and necessary conditions, under which the "first reduce, then control" strategy can be employed and discuss when it should be avoided. We further give numerical examples that illustrate the "first reduce, then optmize" approach and discuss possible pitfalls

Network-level dynamics of diffusively coupled cells

We study molecular dynamics within populations of diffusively coupled cells under the assumption of fast diffusive exchange. As a technical tool, we propose conditions on boundedness and ultimate boundedness for systems with a singular perturbation, which extend the classical asymptotic stability results for singularly perturbed systems. Based on these results, we show that with common models of intracellular dynamics, the cell population is coordinated in the sense that all cells converge close to a common equilibrium point. We then study a more specific example of coupled cells which behave as bistable switches, where the intracellular dynamics are such that cells may be in one of two equilibrium points. Here, we find that the whole population is bistable in the sense that it converges to a population state where either all cells are close to the one equilibrium point, or all cells are close to the other equilibrium point. Finally, we discuss applications of these results for the robustness of cellular decision making in coupled populations

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

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

Large Deviations for Multiscale Diffusions via Weak Convergence Methods

We study the large deviations principle for locally periodic stochastic differential equations with small noise and fast oscillating coefficients. There are three possible regimes depending on how fast the intensity of the noise goes to zero relative to the homogenization parameter. We use weak convergence methods which provide convenient representations for the action functional for all three regimes. Along the way we study weak limits of related controlled SDEs with fast oscillating coefficients and derive, in some cases, a control that nearly achieves the large deviations lower bound at the prelimit level. This control is useful for designing efficient importance sampling schemes for multiscale diffusions driven by small noise

Controlling pulse stability in singularly perturbed reaction-diffusion systems

The aim of this paper is to investigate the use of Pyragas control on the stability of stationary, localized coherent structures in a general class of two-component, singularly perturbed, reaction-diffusion systems. We use noninvasive Pyragas-like proportional feedback control to stabilize a singular pulse solution to a two-component, singularly perturbed reaction-diffusion system. We show that in a significant region of parameter space, the control can be adjusted to stabilize an otherwise unstable pulse

Fitted non-polynomial spline method for singularly perturbed differential difference equations with integral boundary condition

The aim of this paper is to present fitted non-polynomial spline method for singularly perturbed differential-difference equations with integral boundary condition. The stability and uniform convergence of the proposed method are proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter, ε and mesh size, h. The numerical results are tabulated in terms of maximum absolute errors and rate of convergence and it is observed that the present method is more accurate and uniformly convergent for h ≥ ε where the classical numerical methods fails to give good result and it also improves the results of the methods existing in the literature

Lyapunov stability of a singularly perturbed system of two conservation laws

International audienceThis paper is concerned with a class of singularly perturbed systems of two conservation laws. A small perturbation parameter is introduced in the dynamics and the boundary conditions. By setting the perturbation parameter to zero, the singularly perturbed system of conservation laws can be treated as two subsystems of one conservation law: the reduced system and the boundary-layer system. The asymptotic stability of the complete system is investigated via Lyapunov techniques. A Lyapunov function for the singularly perturbed system is obtained as a weighted sum of two Lyapunov functions of the subsystems

An exponentially fitted finite difference scheme for a class of singularly perturbed delay differential equations with large delays

AbstractThis paper deals with singularly perturbed boundary value problem for a linear second order delay differential equation. It is known that the classical numerical methods are not satisfactory when applied to solve singularly perturbed problems in delay differential equations. In this paper we present an exponentially fitted finite difference scheme to overcome the drawbacks of the corresponding classical counter parts. The stability of the scheme is investigated. The proposed scheme is analyzed for convergence. Several linear singularly perturbed delay differential equations have been solved and the numerical results are presented to support the theory