6,654 research outputs found
Perturbations of time optimal control problems for a class of abstract parabolic systems
In this work we study the asymptotic behavior of the solutions of a class of
abstract parabolic time optimal control problems when the generators converge,
in an appropriate sense, to a given strictly negative operator. Our main
application to PDEs systems concerns the behavior of optimal time and of the
associated optimal controls for parabolic equations with highly oscillating
coefficients, as we encounter in homogenization theory. Our main results assert
that, provided that the target is a closed ball centered at the origin and of
positive radius, the solutions of the time optimal control problems for the
systems with oscillating coefficients converge, in the usual norms, to the
solution of the corresponding problem for the homogenized system. In order to
prove our main theorem, we provide several new results, which could be of a
broader interest, on time and norm optimal control problems
On the minimum exit rate for a diffusion process pertaining to a chain of distributed control systems with random perturbations
In this paper, we consider the problem of minimizing the exit rate with which
a diffusion process pertaining to a chain of distributed control systems, with
random perturbations, exits from a given bounded open domain. In particular, we
consider a chain of distributed control systems that are formed by
subsystems (with ), where the random perturbation enters only in the
first subsystem and is then subsequently transmitted to the other subsystems.
Furthermore, we assume that, for any , the
distributed control systems, which is formed by the first subsystems,
satisfies an appropriate H\"ormander condition. As a result of this, the
diffusion process is degenerate, in the sense that the infinitesimal generator
associated with it is a degenerate parabolic equation. Our interest is to
establish a connection between the minimum exit rate with which the diffusion
process exits from the given domain and the principal eigenvalue for the
infinitesimal generator with zero boundary conditions. Such a connection allows
us to derive a family of Hamilton-Jacobi-Bellman equations for which we provide
a verification theorem that shows the validity of the corresponding optimal
control problems. Finally, we provide an estimate on the attainable exit
probability of the diffusion process with respect to a set of admissible
(optimal) Markov controls for the optimal control problems.Comment: 12 Pages. (Additional Note: This work is, in some sense, a
continuation of our previous paper arXiv:1408.6260.
Optimal feedback control infinite dimensional parabolic evolution systems: Approximation techniques
A general approximation framework is discussed for computation of optimal feedback controls in linear quadratic regular problems for nonautonomous parabolic distributed parameter systems. This is done in the context of a theoretical framework using general evolution systems in infinite dimensional Hilbert spaces. Conditions are discussed for preservation under approximation of stabilizability and detectability hypotheses on the infinite dimensional system. The special case of periodic systems is also treated
Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs
This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic PDE. We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on with , and that the PDE operator may depend non-affinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [CDS1, CDS2], we show that the state and the control are analytic as functions depending on these parameters . We
establish sparsity of generalized polynomial chaos (gpc) expansions of both, state and control, in terms of the stochastic coordinate sequence of the random inputs, and prove convergence rates of best -term truncations of these expansions. Such truncations are the key for subsequent computations since they do {\em not} assume that the stochastic input data has a finite expansion. In the follow-up paper [KS2], we explain two methods how such best -term truncations can practically be computed, by greedy-type algorithms
as in [SG, Gi1], or by multilevel Monte-Carlo methods as in
[KSS]. The sparsity result allows in conjunction with adaptive wavelet Galerkin schemes for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [DK, GK, K], see [KS2]
Stability of the solution set of quasi-variational inequalities and optimal control
For a class of quasi-variational inequalities (QVIs) of obstacle-type the
stability of its solution set and associated optimal control problems are
considered. These optimal control problems are non-standard in the sense that
they involve an objective with set-valued arguments. The approach to study the
solution stability is based on perturbations of minimal and maximal elements of
the solution set of the QVI with respect to {monotone} perturbations of the
forcing term. It is shown that different assumptions are required for studying
decreasing and increasing perturbations and that the optimization problem of
interest is well-posed.Comment: 29 page
Global existence, uniqueness and stability for nonlinear dissipative bulk-interface interaction systems
We show global well-posedness and exponential stability of equilibria for a
general class of nonlinear dissipative bulk-interface systems. They correspond
to thermodynamically consistent gradient structure models of bulk-interface
interaction. The setting includes nonlinear slow and fast diffusion in the bulk
and nonlinear coupled diffusion on the interface. Additional driving mechanisms
can be included and non-smooth geometries and coefficients are admissible, to
some extent. An important application are volume-surface reaction-diffusion
systems with nonlinear coupled diffusion.Comment: 21 page
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
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