Observing a linear process over analog erasure channels using multiple sensors: Necessary and sufficient conditions for mean-square stability

2007 ◽  
Author(s):  
Vijay Gupta ◽  
Nuno C. Martins ◽  
John S. Baras
Keyword(s):  
Sufficient Conditions ◽  
Linear Process ◽  
Necessary And Sufficient Conditions ◽  
Mean Square ◽  
Multiple Sensors ◽  
Erasure Channels ◽  
Mean Square Stability ◽  
Necessary And Sufficient

scholarly journals Stability of the Positive Point of Equilibrium of Nicholson's Blowflies Equation with Stochastic Perturbations: Numerical Analysis

2007 ◽  
Vol 2007 ◽  
pp. 1-25 ◽  
Author(s):  
Nataliya Bradul ◽  
Leonid Shaikhet
Keyword(s):  
Linear Part ◽  
Sufficient Conditions ◽  
Discrete Analogue ◽  
Positive Equilibrium ◽  
Mean Square ◽  
Stochastic Perturbations ◽  
Mean Square Stability ◽  
Nicholson’S Blowflies Equation ◽  
Unstable Solutions ◽  
Necessary And Sufficient

Known Nicholson's blowflies equation (which is one of the most important models in ecology) with stochastic perturbations is considered. Stability of the positive (nontrivial) point of equilibrium of this equation and also a capability of its discrete analogue to preserve stability properties of the original differential equation are studied. For this purpose, the considered equation is centered around the positive equilibrium and linearized. Asymptotic mean square stability of the linear part of the considered equation is used to verify stability in probability of nonlinear origin equation. From known previous results connected with B. Kolmanovskii and L. Shaikhet, general method of Lyapunov functionals construction, necessary and sufficient condition of stability in the mean square sense in the continuous case and necessary and sufficient conditions for the discrete case are deduced. Stability conditions for the discrete analogue allow to determinate an admissible step of discretization for numerical simulation of solution trajectories. The trajectories of stable and unstable solutions of considered equations are simulated numerically in the deterministic and the stochastic cases for different values of the parameters and of the initial data. Numerous graphical illustrations of stability regions and solution trajectories are plotted.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

Sign in / Sign up

Export Citation Format

Share Document