33 research outputs found
On the controlled eigenvalue problem for stochastically perturbed multi-channel systems
In this brief paper, we consider the problem of minimizing the asymptotic
exit rate of diffusion processes from an open connected bounded set pertaining
to a multi-channel system with small random perturbations. Specifically, we
establish a connection between: (i) the existence of an invariant set for the
unperturbed multi-channel system w.r.t. certain class of state-feedback
controllers; and (ii) the asymptotic behavior of the principal eigenvalues and
the solutions of the Hamilton-Jacobi-Bellman (HJB) equations corresponding to a
family of singularly perturbed elliptic operators. Finally, we provide a
sufficient condition for the existence of a Pareto equilibrium (i.e., a set of
optimal exit rates w.r.t. each of input channels) for the HJB equations --
where the latter correspond to a family of nonlinear controlled eigenvalue
problems.Comment: A short paper with 9 pages (a continuation of our previous paper
arXiv:1408.6260
A brief remark on the topological entropy for linear switched systems
In this brief note, we investigate the topological entropy for linear
switched systems. Specifically, we use the Levi-Malcev decomposition of
Lie-algebra to establish a connection between the basic properties of the
topological entropy and the stability of switched linear systems. For such
systems, we show that the topological entropy for the evolution operator
corresponding to a semi-simple subalgebra is always bounded from above by the
negative of the largest real part of the eigenvalue that corresponds to the
evolution operator of a maximal solvable ideal part.Comment: 8 Page
Large deviation principle for dynamical systems coupled with diffusion-transmutation processes
In this paper, we introduce a mathematical apparatus that is relevant for
understanding a dynamical system with small random perturbations and coupled
with the so-called transmutation process -- where the latter jumps from one
mode to another, and thus modifying the dynamics of the system. In particular,
we study the exit problem, i.e., an asymptotic estimate for the exit
probabilities with which the corresponding processes exit from a given bounded
open domain, and then formally prove a large deviation principle for the exit
position joint with the type occupation times as the random perturbation
vanishes. Moreover, under certain conditions, the exit place and the type of
distribution at the exit time are determined and, as a consequence of this,
such information also give the limit of the Dirichlet problems for the
associated partial differential equation systems with a vanishing small
parameter.Comment: 16 page
On a connection between the reliability of multi-channel systems and the notion of controlled-invariance entropy
The purpose of this note is to establish a connection between the problem of
reliability (when there is an intermittent control-input channel failure that
may occur between actuators, controllers and/or sensors in the system) and the
notion of controlled-invariance entropy of a multi-channel system (with respect
to a subset of control-input channels and/or a class of control functions). We
remark that such a connection could be used for assessing the reliability (or
the vulnerability) of the system, when some of these control-input channels are
compromised with an external "malicious" agent that may try to prevent the
system from achieving more of its goal (such as from attaining invariance of a
given compact state and/or output subspace).Comment: 11 pages a brief pape
On the stochastic decision problems with backward stochastic viability property
In this paper, we consider a stochastic decision problem for a system
governed by a stochastic differential equation, in which an optimal decision is
made in such a way to minimize a vector-valued accumulated cost over a
finite-time horizon that is associated with the solution of a certain
multi-dimensional backward stochastic differential equation (BSDE). Here, we
also assume that the solution for such a multi-dimensional BSDE {\it almost
surely} satisfies a backward stochastic viability property w.r.t. a given
closed convex set. Moreover, under suitable conditions, we establish the
existence of an optimal solution, in the sense of viscosity solutions, to the
associated system of semilinear parabolic PDEs. Finally, we briefly comment on
the implication of our results.Comment: 20 pages (Additional Note: This work is, in some sense, a
continuation of our previous papers arXiv:1610.07201, arXiv:1603.03359 and
arXiv:1611.03405
Optimal residence time control for stochastically perturbed prescription opioid epidemic models
In this paper, we consider an optimal control problem for a prescription
opioid epidemic model that describes the interaction between the regular
prescription or addictive use of opioid drugs, and the process of
rehabilitation and that of relapsing into opioid drug use. In particular, our
interest is in the situation, where the control appearing linearly in the
opioid epidemics is interpreted as the rate at which the susceptible
individuals are effectively removed from the population due to an
opioid-related intervention policy or when the dynamics of the addicted is
strategically influenced due to an accessible addiction treatment facility,
while a small perturbing noise enters through the dynamics of the susceptible
group in the population compartmental model. To this end, we introduce a
mathematical apparatus that minimizes the asymptotic exit-rate with which the
solution for such stochastically perturbed prescription opioid epidemics exits
from a given bounded open domain. Moreover, under certain assumptions, we also
provide an admissible optimal Markov control for the corresponding optimal
control problem that optimally effected removal of the susceptible or recovered
individuals from the population dynamics.Comment: 11 pages - Version 2.0 - December, 2018 (Additional Note: This work
is, in some sense, a continuation of our previous papers arXiv:1805.12534 and
arXiv:1806.09502
On the asymptotic of exit problems for controlled Markov diffusion processes with random jumps and vanishing diffusion terms
In this paper, we study the asymptotic of exit problem for controlled Markov
diffusion processes with random jumps and vanishing diffusion terms, where the
random jumps are introduced in order to modify the evolution of the controlled
diffusions by switching from one mode of dynamics to another. That is,
depending on the state-position and state-transition information, the dynamics
of the controlled diffusions randomly switches between the different drift and
diffusion terms. Here, we specifically investigate the asymptotic exit problem
concerning such controlled Markov diffusion processes in two steps: (i) First,
for each controlled diffusion model, we look for an admissible Markov control
process that minimizes the principal eigenvalue for the corresponding
infinitesimal generator with zero Dirichlet boundary conditions -- where such
an admissible control process also forces the controlled diffusion process to
remain in a given bounded open domain for a longer duration. (ii) Then, using
large deviations theory, we determine the exit place and the type of
distribution at the exit time for the controlled Markov diffusion processes
coupled with random jumps and vanishing diffusion terms. Moreover, the
asymptotic results at the exit time also allow us to determine the limiting
behavior of the Dirichlet problem for the corresponding system of elliptic
partial differential equations containing a small vanishing parameter.Comment: 16 Pages. (Additional Note: This work is, in some sense, a
continuation of our previous papers arXiv:1709.04853
Optimal control of diffusion processes pertaining to an opioid epidemic dynamical model with random perturbations
In this paper, we consider the problem of controlling a diffusion process
pertaining to an opioid epidemic dynamical model with random perturbation so as
to prevent it from leaving a given bounded open domain. Here, we assume that
the random perturbation enters only through the dynamics of the susceptible
group in the compartmental model of the opioid epidemic dynamics and, as a
result of this, the corresponding diffusion is degenerate, for which we further
assume that the associated diffusion operator is hypoelliptic. In particular,
we minimize the asymptotic exit rate of such a controlled-diffusion process
from the given bounded open domain and we derive the Hamilton-Jacobi-Bellman
equation for the corresponding optimal control problem, which is closely
related to a nonlinear eigenvalue problem. Finally, we also prove a
verification theorem that provides a sufficient condition for optimal control.Comment: 13 pages - Version 1.0 - June 25, 2018 (Additional Note: This work
is, in some sense, a continuation of our previous paper arXiv:1805.12534
On noncooperative -player principal eigenvalue games
We consider a noncooperative -player principal eigenvalue game which is
associated with an infinitesimal generator of a stochastically perturbed
multi-channel dynamical system -- where, in the course of such a game, each
player attempts to minimize the asymptotic rate with which the controlled state
trajectory of the system exits from a given bounded open domain. In particular,
we show the existence of a Nash-equilibrium point (i.e., an -tuple of
equilibrium linear feedback operators) that is distinctly related to a unique
maximum closed invariant set of the corresponding deterministic multi-channel
dynamical system, when the latter is composed with this -tuple of
equilibrium linear feedback operators.Comment: 4 Page
On the risk-sensitive escape control for diffusion processes pertaining to an expanding construction of distributed control systems
In this paper, we consider an expanding construction of a distributed control
system, which is obtained by adding a new subsystem one after the other, until
all subsystems, where , are included in the distributed control
system. It is assumed that a small random perturbation enters only into the
first subsystem and is then subsequently transmitted to the other subsystems.
Moreover, for any , the distributed control system,
compatible with the expanding construction, which is obtained from the first
subsystems, satisfies an appropriate H\"{o}rmander condition. As a
result of this, the diffusion process is degenerate, i.e., the backward
operator associated with it is a degenerate parabolic equation. Our main
interest here is to prevent the diffusion process (that corresponds to a
particular subsystem) from leaving a given bounded open domain. In particular,
we consider a risk-sensitive version of the mean escape time criterion with
respect to each of the subsystems. Using a variational representation, we
characterize the risk-sensitive escape control for the diffusion process as the
lower and upper values of an associated stochastic differential game. Finally,
we comment on the implication of our results, where one is also interested in
evaluating the performance of the risk-sensitive escape control, when there is
some modeling error in the distributed control system.Comment: 11 Pages (Additional Note: This work is, in some sense, a
continuation of our previous paper arXiv:1409.2751 [math.DS]