Yuncheng You

Dynamics of evolutionary equations

Preface * 1 The Evolution of Evolutionary Systems * 2 Dynamical Systems: Basic Theory * 3 Linear Semigroups * 4 Basic Theory of Evolutionary Equations * 5 Nonlinear Partial Differential Equations * 6 Navier Stokes Dynamics * 7 Basic Principles of Dynamics * 8 Inertial Manifolds and the Reduction Principle * Appendices: Basics of Functional Analysis * Bibliography * Notation Index * Subject Index

Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions

A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form. A systematical analysis is made for solving the resultant linear systems of second-order and third-order partial differential equations, along with solution formulas for their representative systems. The key technique is to apply variation of parameters in solving the involved non-homogeneous partial differential equations. The obtained solution formulas provide us with a comprehensive approach to construct the existing solutions and many new solutions including rational solutions, solitons, positons, negatons, breathers, complexitons and interaction solutions of the Korteweg-de Vries equation.

Inertial manifolds: The non-self-adjoint case

Abstract In contrast with the existing theories of inertial manifolds, which are based on the self-adjoint assumption of the principal differential operator, in this paper we show that for general dissipative evolutionary systems described by semilinear parabolic equations with principal differential operator being sectorial and having compact resolvent, there exists an inertial manifold provided that certain gap conditions hold. We also show that by using an elliptic regularization, this theory can be extended to a class of KdV equations, where the principal differential operator is not sectorial.

Some second-order vibrating systems cannot tolerate small time delays in their damping

We show that certain infinite-dimensional damped second-order systems of linear differential equations become unstable when arbitrarily small time delays occur in the damping.

Global Dynamics of the Oregonator System

In this work, the existence and properties of a global attractor for the solution semiflow of the Oregonator system are proved. The Oregonator system is the mathematical model of the celebrated Belousov–Zhabotinskii reaction. A rescaling and grouping estimation method is developed to show the absorbing property and the asymptotic compactness of the solution trajectories of this three-component reaction–diffusion system with quadratic nonlinearity. It is also proved that the fractal dimension of the global attractor is finite and an exponential attractor exists for the Oregonator semiflow. Copyright

Dynamical boundary control for elastic plates of general shape

The control of transverse vibrations of elastic plates of general shape by feedback boundary control is formulated as an abstract evolution equation. Because the control acts locally on the boundary, which possesses a flanged rim with inertial properties of mass and bending moment, the analysis concerns dynamical controllability and stabilizability of a hybrid system. By the approach of energy decay inequalities and Hormander’s global uniqueness theorem, it is shown that the system is strongly stabilizable by a locally supported damping feedback of boundary velocity and boundary angular velocity, and hence the system is approximately controllable.

Optimal control of two-dimensional linear systems

Based on Roessers model of discrete-time two-dimensional (2-D) linear controlled systems, a closed-loop synthesis is presented for quadratic optimal control via the semicausality approach.

Optimal control of bivariate linear Volterra integral type systems

Quadratic optimal control of bivariate and multivariate linear Volterra integral type systems is investigated using a semicausality approach. Closed-loop syntheses can be realized by linear feedback of the semicausal trajectory with the feedback operator determined by solving a linear integral equation.

Global Attractor of a Coupled Two-Cell Brusselator Model

In this work the existence of a global attractor for the solution semiflow of the coupled two-cell Brusselator model equations is proved. A grouping estimation method and a new decomposition approach are introduced to deal with the challenges in proving the absorbing property and the asymptotic compactness of this type of four-variable reaction-diffusion systems with cubic autocatalytic nonlinearity and with linear coupling. It is also proved that the Hausdorff dimension and the fractal dimension of the global attractor are finite.

Pointwise stabilizability of coupled elastic beam systems

A coupled elastic beam system with a controller at the hinged junction that has point-mass is formulated as an abstract evolution equation in the energy space. By the spectral analysis of the hybrid differential operators with mixed boundary-junction conditions, an alternative principle of stabilizability in terms of the beam lengths ρ1 and ρ2 is proved: (I) If ρ1/ρ2 equals a quotient of two positive roots (different or same) of the transcendental equation tanμ = tanhμ, then the pointwise stabilization of the system is impossible. (II) If ρ1/ρ2 does not equal any quotient described above, then the system is strongly stabilized in the energy space by a pointwise damping feedback f(t) = −w t (ρ1,t) at the junction point x = ρ1. Furthermore in the first case it is proved that a combination of the above pointwise damping feedback and an appropriate quasi-pointwise damping feedback on one beam achieves the strong stabilization.