Second-order Quasi-linear System Research Articles

Parametric Control to Second-Order Quasi-Linear Systems Based on Dynamic Compensator and Multi-Objective Optimization

This paper presents the parametric design approach to the second-order quasi-linear (SQ) systems by using dynamic compensator and multi-objective optimization. Based on the solutions to a type of the second-order generalized Sylvester equations (SGSEs), the generally parametric form of the dynamic compensator is established, the completely parameterized expressions of left and right eigenvector matrices are obtained, and it also provides two groups of arbitrary parameter matrices. With the parametrized method, the closed-loop system is converted into a linear constant one. Simultaneously, it also investigates a novel technique to multi-objective design and optimization. Multiple performance indexes, such as regional pole assignment, low sensitivity, disturbance attenuation, robustness degree, and low gains, are formulated by arbitrary parameters. Based on the above indexes, robustness criteria and low gain criteria can be expressed by a synthetic objective function which includes each performance index weighted. By using the degrees of freedom (DOFs) in arbitrary parameters, a dynamic compensator can be established by solving a multi-objective optimization problem. Finally, the spacecraft rendezvous problem is proposed to verify the feasibility and effectiveness of the parametrized approach.

Well-posedness of Hall-magnetohydrodynamics system forced by L $$\acute{\mathrm{e}}$$ e ´ vy noise

We establish the existence and uniqueness of a local smooth solution to the Cauchy problem for the Hall-magnetohydrodynamics system that is inviscid, resistive, and forced by multiplicative L $$\acute{\mathrm {e}}$$ vy noise in the three dimensional space. Moreover, when the initial data is sufficiently small, we prove that the solution exists globally in time in probability; that is, the probability of the global existence of a unique smooth solution may be arbitrarily close to one given the initial data of which its expectation in a certain Sobolev norm is sufficiently small. The proofs go through for the two and a half dimensional case as well. To the best of the authors’ knowledge, an analogous result is absent in the deterministic case due to the lack of viscous diffusion, exhibiting the regularizing property of the noise. Our result may also be considered as a physically meaningful special case of the extension of work of Kim (J Differ Equ 250:1650–1684, 2011) and Mohan and Sritharan (Pure Appl Funct Anal 3:137–178, 2018) from the first-order quasilinear to the second-order quasilinear system because the Hall term elevates the Hall-magnetohydrodynamics system to the quasilinear class, in contrast to the Naiver–Stokes equations that has most often been studied and is semilinear.

Localization of Exact Solutions of Second-Order Quasilinear Systems

Localization of Exact Solutions of Second-Order Quasilinear Systems

Simulation of Oscillatory Processes in a Second-Order Quasilinear System with Delay and Random Perturbations

Oscillatory processes are considered that arise in systems with signals delayed under the action of external random perturbations. A simulation algorithm whose results are represented in numerical and graphical forms is proposed and realized as a component of the "GeoPoisk" software package used for interpretation of geological prospecting data.

ASYMPTOTIC STABILITY OF EQUILIBRIUM STATES FOR MULTICOMPONENT REACTIVE FLOWS

We consider the equations governing multicomponent reactive flows derived from the kinetic theory of dilute polyatomic reactive gas mixtures. Using an entropy function, we derive a symmetric conservative form of the system. In the framework of Kawashima and Shizuta's theory, we recast the resulting system into a normal form, that is, in the form of a symmetric hyperbolic–parabolic composite system. We also characterize all normal forms for symmetric systems of conservation laws such that the null space associated with dissipation matrices is invariant. We then investigate an abstract second-order quasilinear system with a source term, around a constant equilibrium state. Assuming the existence of a generalized entropy function, the invariance of the null space naturally associated with dissipation matrices, stability conditions for the source term, and a dissipative structure for the linearized equations, we establish global existence and asymptotic stability around the constant equilibrium state in all space dimensions and we obtain decay estimates. These results are then applied to multicomponent reactive flows using a normal form and the properties of Maxwellian chemical source terms.

Regularity results for quasilinear elliptic systems with nonlinear boundary conditions

It is proved that a solution of the boundary-value problem for a second-order quasilinear system with controlled order of nonlinearity is partially smooth all the way to the boundary of a domain. The boundary condition is imposed by means of a second-order nonlinear operator which can be regarded as a generalization of the “directional derivative” to the case of quasilinear systems. Bibliography: 6 titles.