Controllability Of Degenerate Hyperbolic Equations And Insensitizing Controls For Ginzburg-Landau Equations

This dissertation investigates mainly the controllability of one-dimensional degenerate hyperbolic equations and insensitizing controls for Ginzburg-Landau equations.For degener-ate hyperbolic equations,we study the boundary controllability and interior controllability for this system when control acts on different position,respectively.For some other de-generate hyperbolic equations which is not null controllable,we study this system weaker controllability,including regional null controllability and persistent regional null control-lability.Ginzburg-Landau equations can describe a variety of phenomena from nonlinear waves to the superconductivity and plays an important role in the theory of amplitude equa-tions.We mainly investigate the insensitivity problems for the semilinear Ginzburg-Landau equations.There are four parts in this dissertation.In the Chapter 2 of this dissertation,we are devoted to a study of the existence of insensitizing controls for a class of nonlinear complex Ginzburg-Landau equations with ho-mogeneous Dirichlet boundary conditions and arbitrary located internal controller.When the nonlinearity in the equation satisfies a suitable superlinear growth condition at infinity,we prove the existence of insensitizing controls for the corresponding semilinear Ginzburg-Landau equation.Meanwhile,if the nonlinearity in the equation is only a smooth function without any additional growth condition,a local insensitivity result is obtained.As usual,the problem of insensitizing controls is transformed into a suitable controllability problem for a coupled system governed by a semilinear complex Ginzburg-Landau equation and a linear one through one control.The key is to establish an observability inequality for a coupled linear Ginzburg-Landau system with one observer.In the Chapter 3 of this dissertation,we are concerned with the null controllability problems for one-dimensional linear degenerate hyperbolic equations through a boundary controller.Since the degenerate hyperbolic equations still have time-reversibility,the null controllability of it is equivalent to the exact controllability.First,we discuss the well-posedness of linear degenerate hyperbolic equations.Then the null controllability of some degenerate hyperbolic equations is established,when a control acts on the non-degenerate boundary.Different from the known controllability results in the case that a control acts on the degenerate boundary,any initial value in state space is controllable in this case.Also,an explicit expression for the controllability time is given.Furthermore,a counterexample on the controllability is given for some other degenerate hyperbolic equations.In the Chapter 4 of this dissertation,we are devoted to a study of the controllabil-ity problems for one-dimensional semilinear degenerate hyperbolic equations through a dis-tributed controller.By Hilbert unique method,we need to establish an observability estimate for the linear degenerate hyperbolic equation.First,we prove the unique continuation for the degenerate hyperbolic equation by characteristic line method.Then,combining the multi-plier method and the unique continuation for the degenerate hyperbolic equation,we obtain the observability inequality.The key point is to choose a suitable multiplier.The Chapter 5 is addressed to a study of the persistent regional null controllability problems for one-dimensional linear degenerate hyperbolic equations through a distributed controller.Different from non-degenerate hyperbolic equations,the classical null controlla-bility results do not hold for some degenerate hyperbolic equations.Thus,persistent regional null controllability is introduced,which means finding a control such that the corresponding state of the degenerate hyperbolic equation may vanish in a suitable subset of the space domain in a period of time.In order to solve this problem,we need to establish the regional null controllability for degenerate hyperbolic equations.This problem is also reduced to a suitable observability problem of a linear degenerate hyperbolic equation.The key point is to choose a suitable multiplier in order to establish this observability inequality.