85 research outputs found
Stability on a cone in terms of two measures for impulsive differential equations with “supremum”
AbstractThe stability of nonlinear impulsive differential equations with “supremum” is studied. A special type of stability, combining two different measures and a dot product, is defined. The definition is a generalization of several types of stability known in the literature. Razumikhin’s method as well as a comparison method for scalar impulsive ordinary differential equations have been employed
Integral φ
This paper establishes a criterion on integral φ0-stability in terms of two measures for impulsive differential equations with “supremum” by using the cone-valued piecewise continuous Lyapunov functions, Razumikhin method, and comparative method. Meantime, an example is given to illustrate our result
Practical Stability and Vector-Lyapunov Functions for Impulsive Differential Equations with "Supremum"
Stability of nonlinear impulsive differential equations with
"supremum" is studied. A special type of stability, combining two different
measures and a dot product on a cone, is defined. Perturbing cone-valued
piecewise continuous Lyapunov functions have been applied. Method of Razumikhin as well as comparison method for scalar impulsive ordinary differential
equations have been employed
Practical Stability in terms of Two Measures for Impulsive Differential Equations with “Supremum”
The object of investigations is a system of impulsive differential equations with “supremum.” These equations are not widely studied yet, and at the same time they are adequate mathematical model of many real world processes in which the present state depends significantly on its maximal value on a past time interval. Practical stability for a nonlinear system of impulsive differential equations with “supremum” is defined and studied. It is applied Razumikhin method with piecewise continuous scalar Lyapunov functions and comparison results for scalar impulsive differential equations. To unify a variety of stability concepts and to offer a general framework for the investigation of the stability theory, the notion of stability in terms of two measures has been applied to both the given system and the comparison scalar equation. An example illustrates the usefulness of the obtained sufficient conditions
Nonnegative solutions for a system of impulsive BVPs with nonlinear nonlocal BCs
We study the existence of nonnegative solutions for a system of impulsive differential equations subject to nonlinear, nonlocal boundary conditions. The system presents a coupling in the differential equation and in the boundary conditions. The main tool that we use is the theory of fixed point index for compact maps
Viability, Invariance and Reachability for Controlled Piecewise Deterministic Markov Processes Associated to Gene Networks
We aim at characterizing viability, invariance and some reachability
properties of controlled piecewise deterministic Markov processes (PDMPs).
Using analytical methods from the theory of viscosity solutions, we establish
criteria for viability and invariance in terms of the first order normal cone.
We also investigate reachability of arbitrary open sets. The method is based on
viscosity techniques and duality for some associated linearized problem. The
theoretical results are applied to general On/Off systems, Cook's model for
haploinssuficiency, and a stochastic model for bacteriophage lambda.Comment: submitte
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