146 research outputs found
Singularly perturbed piecewise deterministic games
In this paper we consider a class of hybrid stochastic games with the piecewise open-loop information structure. These games are indexed over a parameter which represents the time scale ratio between the stochastic (jump process) and the deterministic (differential state equation) parts of the dynamical system. We study the limit behavior of Nash equilibrium solutions to the hybrid stochastic games when the time scale ratio tends to 0. We also establish that an approximate equilibrium can be obtained for the hybrid stochastic games using a Nash equilibrium solution of a reduced order sequential discrete state stochastic game and a family of local deterministic infinite horizon open-loop differential games defined in the stretched out time scale. A numerical illustration of this approximation scheme is also developed
Control of singularly perturbed hybrid stochastic systems
In this paper, we study a class of optimal stochastic
control problems involving two different time scales. The fast
mode of the system is represented by deterministic state equations
whereas the slow mode of the system corresponds to a jump disturbance
process. Under a fundamental “ergodicity” property for
a class of “infinitesimal control systems” associated with the fast
mode, we show that there exists a limit problem which provides
a good approximation to the optimal control of the perturbed
system. Both the finite- and infinite-discounted horizon cases are
considered. We show how an approximate optimal control law
can be constructed from the solution of the limit control problem.
In the particular case where the infinitesimal control systems
possess the so-called turnpike property, i.e., are characterized by
the existence of global attractors, the limit control problem can be
given an interpretation related to a decomposition approach
On average control generating families for singularly perturbed optimal control problems with long run average optimality criteria
The paper aims at the development of tools for analysis and construction of
near optimal solutions of singularly perturbed (SP) optimal controls problems
with long run average optimality criteria. The idea that we exploit is to first
asymptotically approximate a given problem of optimal control of the SP system
by a certain averaged optimal control problem, then reformulate this averaged
problem as an infinite-dimensional (ID) linear programming (LP) problem, and
then approximate the latter by semi-infinite LP problems. We show that the
optimal solution of these semi-infinite LP problems and their duals (that can
be found with the help of a modification of an available LP software) allow one
to construct near optimal controls of the SP system. We demonstrate the
construction with a numerical example.Comment: 36 pages, 4 figures. arXiv admin note: substantial text overlap with
arXiv:1309.373
Control of singularly perturbed hybrid stochastic systems
In this paper we study a class of optimal stochastic control
problems involving two different time scales. The
fast mode of the system is represented by deterministic
state equations whereas the slow mode of the system
corresponds to a jump disturbance process. Under a
fundamental ”ergodicity” property for a class of ”infinitesimal
control systems” associated with the fast
mode, we show that there exists a limit problem which
provides a good approximation to the optimal control
of the perturbed system. Both the finite and infinite
discounted horizon cases are considered. We show how
an approximate optimal control law can be constructed
from the solution of the limit control problem. In the
particular case where the infinitesimal control systems
possess the so-called turnpike property, i.e. are characterized
by the existence of global attractors, the limit
control problem can be given an interpretation related
to a decomposition approach
Averaging and linear programming in some singularly perturbed problems of optimal control
The paper aims at the development of an apparatus for analysis and
construction of near optimal solutions of singularly perturbed (SP) optimal
controls problems (that is, problems of optimal control of SP systems)
considered on the infinite time horizon.
We mostly focus on problems with time discounting criteria but a possibility
of the extension of results to periodic optimization problems is discussed as
well. Our consideration is based on earlier results on averaging of SP control
systems and on linear programming formulations of optimal control problems. The
idea that we exploit is to first asymptotically approximate a given problem of
optimal control of the SP system by a certain averaged optimal control problem,
then reformulate this averaged problem as an infinite-dimensional (ID) linear
programming (LP) problem, and then approximate the latter by semi-infinite LP
problems. We show that the optimal solution of these semi-infinite LP problems
and their duals (that can be found with the help of a modification of an
available LP software) allow one to construct near optimal controls of the SP
system. We demonstrate the construction with two numerical examples.Comment: 53 pages, 10 figure
Asymptotic Control for a Class of Piecewise Deterministic Markov Processes Associated to Temperate Viruses
We aim at characterizing the asymptotic behavior of value functions in the
control of piece-wise deterministic Markov processes (PDMP) of switch type
under nonexpansive assumptions. For a particular class of processes inspired by
temperate viruses, we show that uniform limits of discounted problems as the
discount decreases to zero and time-averaged problems as the time horizon
increases to infinity exist and coincide. The arguments allow the limit value
to depend on initial configuration of the system and do not require dissipative
properties on the dynamics. The approach strongly relies on viscosity
techniques, linear programming arguments and coupling via random measures
associated to PDMP. As an intermediate step in our approach, we present the
approximation of discounted value functions when using piecewise constant (in
time) open-loop policies.Comment: In this revised version, statements of the main results are gathered
in Section 3. Proofs of the main results (Theorem 4 and Theorem 7) make the
object of separate sections (Section 5, resp. Section 6). The biological
example makes the object of Section 4. Notations are gathered in Subsection
2.1. This is the final version to be published in SICO
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
Singularly Perturbed Markov Chains with Two Small Parameters: A Matched Asymptotic Expansion
AbstractThis work is concerned with asymptotic properties of solutions to forward equations for singularly perturbed Markov chains with two small parameters. It is motivated by the model of a cost-minimizing firm involving production planning and capacity expansion and a two-level hierarchical decomposition. Our effort focuses on obtaining asymptotic expansions of the solutions to the forward equation. Different from previous work on singularly perturbed Markov chains, the inner expansion terms are constructed by solving certain partial differential equations. The methods of undetermined coefficients are used. The error bound is obtained
Final Scientific Report: Control Strategies for Complex Systems for Use in Aerospace Avionics
Coordinated Science Laboratory was formerly known as Control Systems LaboratoryAir Force Office of Scientific Research (AFOSR), U.S. Air Force / AF-AFOSR 73-257
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