735 research outputs found
Fluid limit theorems for stochastic hybrid systems with application to neuron models
This paper establishes limit theorems for a class of stochastic hybrid
systems (continuous deterministic dynamic coupled with jump Markov processes)
in the fluid limit (small jumps at high frequency), thus extending known
results for jump Markov processes. We prove a functional law of large numbers
with exponential convergence speed, derive a diffusion approximation and
establish a functional central limit theorem. We apply these results to neuron
models with stochastic ion channels, as the number of channels goes to
infinity, estimating the convergence to the deterministic model. In terms of
neural coding, we apply our central limit theorems to estimate numerically
impact of channel noise both on frequency and spike timing coding.Comment: 42 pages, 4 figure
Consistent Approximations for the Optimal Control of Constrained Switched Systems
Though switched dynamical systems have shown great utility in modeling a
variety of physical phenomena, the construction of an optimal control of such
systems has proven difficult since it demands some type of optimal mode
scheduling. In this paper, we devise an algorithm for the computation of an
optimal control of constrained nonlinear switched dynamical systems. The
control parameter for such systems include a continuous-valued input and
discrete-valued input, where the latter corresponds to the mode of the switched
system that is active at a particular instance in time. Our approach, which we
prove converges to local minimizers of the constrained optimal control problem,
first relaxes the discrete-valued input, then performs traditional optimal
control, and then projects the constructed relaxed discrete-valued input back
to a pure discrete-valued input by employing an extension to the classical
Chattering Lemma that we prove. We extend this algorithm by formulating a
computationally implementable algorithm which works by discretizing the time
interval over which the switched dynamical system is defined. Importantly, we
prove that this implementable algorithm constructs a sequence of points by
recursive application that converge to the local minimizers of the original
constrained optimal control problem. Four simulation experiments are included
to validate the theoretical developments
Controlled diffusion processes with markovian switchings for modeling dynamical engineering systems
A modeling approach to treat noisy engineering systems is presented. We
deal with controlled systems that evolve in a continuous-time over finite time intervals,
but also in continuous interaction with environments of intrinsic variability. We face the complexity of these systems by introducing a methodology based on Stochastic
Differential Equations (SDE) models. We focus on specific type of complexity derived
from unpredictable abrupt and/or structural changes. In this paper an approach based on
controlled Stochastic Differential Equations with Markovian Switchings (SDEMS) is
proposed. Technical conditions for the existence and uniqueness of the solution of these models are provided. We treat with nonlinear SDEMS that does not have closed
solutions. Then, a numerical approximation to the exact solution based on the Euler-
Maruyama Method (EM) is proposed. Convergence in strong sense and stability are
provided. Promising applications for selected industrial biochemical systems are
showed
Approximate solutions of stochastic differential delay equations with Markovian switching
Our main aim is to develop the existence theory for the solutions to stochastic differential delay equations with Markovian switching (SDDEwMSs) and to establish the convergence theory for the Euler-Maruyama approximate solutions under the local Lipschitz condition. As an application, our results are used to discuss a stochastic delay population system with Markovian switching
- …