313 research outputs found
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
Nonnegative controllability for a class of nonlinear degenerate parabolic equations with application to climate science
Let us consider a nonlinear degenerate reaction-diffusion equation with
application to climate science. After proving that the solution remains
nonnegative at any time, when the initial state is nonnegative, we prove the
approximate controllability between nonnegative states at any time via
multiplicative controls, that is, using as control the reaction coefficient.Comment: arXiv admin note: text overlap with arXiv:1710.0069
On Pareto equilibria for bi-objective diffusive optimal control problems
We investigate Pareto equilibria for bi-objective optimal control problems.
Our framework comprises the situation in which an agent acts with a distributed
control in a portion of a given domain, and aims to achieve two distinct
(possibly conflicting) targets. We analyze systems governed by linear and
semilinear heat equations and also systems with multiplicative controls. We
develop numerical methods relying on a combination of finite elements and
finite differences. We illustrate the computational methods we develop via
numerous experiments
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Challenges in Optimal Control of Nonlinear PDE-Systems
The workshop focussed on various aspects of optimal control problems for systems of nonlinear partial differential equations. In particular, discussions around keynote presentations in the areas of optimal control of nonlinear/non-smooth systems, optimal control of systems involving nonlocal operators, shape and topology optimization, feedback control and stabilization, sparse control, and associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, also aspects of control of fluid structure interaction problems as well as problems arising in the optimal control of quantum systems were considered
Polynomial approximation of high-dimensional Hamilton–Jacobi–Bellman equations and applications to feedback control of semilinear parabolic PDES
© 2018 Society for Industrial and Applied Mathematics. A procedure for the numerical approximation of high-dimensional Hamilton–Jacobi–Bellman (HJB) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed. Its main ingredients are a pseudospectral collocation approximation of the PDE dynamics and an iterative method for the nonlinear HJB equation associated to the feedback synthesis. The latter is known as the successive Galerkin approximation. It can also be interpreted as Newton iteration for the HJB equation. At every step, the associated linear generalized HJB equation is approximated via a separable polynomial approximation ansatz. Stabilizing feedback controls are obtained from solutions to the HJB equations for systems of dimension up to fourteen
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