Lagrange stability of semilinear differential-algebraic equations and application to nonlinear electrical circuits

We study a semilinear differential-algebraic equation (DAE) with the focus on the Lagrange stability (instability). The conditions for the existence and uniqueness of global solutions (a solution exists on an infinite interval) of the Cauchy problem, as well as conditions of the boundedness of the global solutions, are obtained. Furthermore, the obtained conditions for the Lagrange stability of the semilinear DAE guarantee that every its solution is global and bounded, and, in contrast to theorems on the Lyapunov stability, allow to prove the existence and uniqueness of global solutions regardless of the presence and the number of equilibrium points. We also obtain the conditions of the existence and uniqueness of solutions with a finite escape time (a solution exists on a finite interval and is unbounded, i.e., is Lagrange unstable) for the Cauchy problem. We do not use constraints of a global Lipschitz condition type, that allows to use the work results efficiently in practical applications. The mathematical model of a radio engineering filter with nonlinear elements is studied as an application. The numerical analysis of the model verifies the results of theoretical investigations

Qualitative analysis of nonregular differential-algebraic equations and the dynamics of gas networks

The conditions for the existence, uniqueness and boundedness of global solutions, as well as ultimate boundedness of solutions, and the conditions for the blow-up of solutions of nonregular semilinear differential-algebraic equations are obtained. An example demonstrating the application of the obtained results is considered. Isothermal models of gas networks are proposed as applications

IBVP for the Maxwell-Bloch equations with an arbitrary inhomogeneous broadening and the periodic boundary function

The initial-boundary value problem (IBVP) for the Maxwell-Bloch equations with an arbitrary inhomogeneous broadening and periodic boundary condition is studied. This IBVP describes the propagation of an electromagnetic wave generated by periodic pumping in a resonant medium with distributed two-level atoms. We extended the inverse scattering transform method in the form of the matrix Riemann-Hilbert problem for solving the considered IBVP. First, using the system of Ablowitz-Kaup-Newell-Segur equations equivalent to the system of the Maxwell-Bloch (MB) equations, we construct the associated matrix Riemann-Hilbert (RH) problem. Then we obtain the representation of a solution of the MB equations in terms of a solution of the associated RH problem. Theorems on the existence, uniqueness and smoothness properties of a solution of the constructed RH problem are proved. The significance of this method also lies in the fact that, having studied the asymptotic behavior of the constructed RH problem and equivalent ones, we can obtain formulas for the asymptotics of a solution of the corresponding IBVP for the MB equations