216 research outputs found
Flow Stability of Patchy Vector Fields and Robust Feedback Stabilization
The paper is concerned with patchy vector fields, a class of discontinuous,
piecewise smooth vector fields that were introduced in AB to study feedback
stabilization problems. We prove the stability of the corresponding solution
set w.r.t. a wide class of impulsive perturbations. These results yield the
robusteness of patchy feedback controls in the presence of measurement errors
and external disturbances.Comment: 22 page
The patchy Method for the Infinite Horizon Hamilton-Jacobi-Bellman Equation and its Accuracy
We introduce a modification to the patchy method of Navasca and Krener for
solving the stationary Hamilton Jacobi Bellman equation. The numerical solution
that we generate is a set of polynomials that approximate the optimal cost and
optimal control on a partition of the state space. We derive an error bound for
our numerical method under the assumption that the optimal cost is a smooth
strict Lyupanov function. The error bound is valid when the number of subsets
in the partition is not too large.Comment: 50 pages, 5 figure
Nearly Optimal Patchy Feedbacks for Minimization Problems with Free Terminal Time
The paper is concerned with a general optimization problem for a nonlinear
control system, in the presence of a running cost and a terminal cost, with
free terminal time. We prove the existence of a patchy feedback whose
trajectories are all nearly optimal solutions, with pre-assigned accuracy.Comment: 13 pages, 3 figures. in v2: Fixed few misprint
Quasi-optimal robust stabilization of control systems
In this paper, we investigate the problem of semi-global minimal time robust
stabilization of analytic control systems with controls entering linearly, by
means of a hybrid state feedback law. It is shown that, in the absence of
minimal time singular trajectories, the solutions of the closed-loop system
converge to the origin in quasi minimal time (for a given bound on the
controller) with a robustness property with respect to small measurement noise,
external disturbances and actuator noise
Stochastic population growth in spatially heterogeneous environments: The density-dependent case
This work is devoted to studying the dynamics of a structured population that
is subject to the combined effects of environmental stochasticity, competition
for resources, spatio-temporal heterogeneity and dispersal. The population is
spread throughout patches whose population abundances are modelled as the
solutions of a system of nonlinear stochastic differential equations living on
.
We prove that , the stochastic growth rate of the total population in the
absence of competition, determines the long-term behaviour of the population.
The parameter can be expressed as the Lyapunov exponent of an associated
linearized system of stochastic differential equations. Detailed analysis shows
that if , the population abundances converge polynomially fast to a unique
invariant probability measure on , while when , the
population abundances of the patches converge almost surely to
exponentially fast. This generalizes and extends the results of Evans et al
(2014 J. Math. Biol.) and proves one of their conjectures.
Compared to recent developments, our model incorporates very general
density-dependent growth rates and competition terms. Furthermore, we prove
that persistence is robust to small, possibly density dependent, perturbations
of the growth rates, dispersal matrix and covariance matrix of the
environmental noise. Our work allows the environmental noise driving our system
to be degenerate. This is relevant from a biological point of view since, for
example, the environments of the different patches can be perfectly correlated.
As an example we fully analyze the two-patch case, , and show that the
stochastic growth rate is a decreasing function of the dispersion rate. In
particular, coupling two sink patches can never yield persistence, in contrast
to the results from the non-degenerate setting treated by Evans et al.Comment: 43 pages, 1 figure, edited according to the suggestion of the
referees, to appear in Journal of Mathematical Biolog
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