Controllability And Stabilization Of ID Schr(o|¨)dinger Equations

Schrodinger equation is a basic equation in quantum mechanics, who can describe free state, bounded state or localized phenomena about atom-s, molecules and subatomic particles. In this dissertation, we mainly discuss how to control bilinear Schrodinger equation with different boundary value conditions. We present some results about controllability and stabilization of1D infinite dimensional Schrodinger equation.This dissertation is divided by four chapters.In the first chapter, we mainly introduce the definition of the bilinear control system, the basic concept on controllability and stabilization, the ap-pearance and development of bilinear Schrodinger equation, the importance and necessary of studying the control problem about Schrodinger equation with different boundary value conditions.In the second chapter, we discuss the local exact controllability of1D Schrodinger equation with Sturm-Liouville boundary value condition. We con-sider1D bilinear Schrodinger equation: in which ai2+bi2≠0, i=1,2. Such an equation arise in the modelization of a quantum particle in po-tential V(x) with Sturm-Liouville boundary value condtion, coupled to an external electric field w(t). The above system is a bilinear control system, in which the state is y:R+×Râ†'C and the control is the real valued function w:[0, T]â†'R, acting on dipole moment μ:(0, Ï€)â†'R.In [8], K. Beauchard presented the exact controllability of1D infinite dimensional Schrodinger equation under Dirichlet boundary value condition (b1=b2=0) when u(x)=1,μ(x)=x. It has been pointed out by G. Turinici in [90] that if the state function belongs to H2∩H01and control function belongs to L2, then1D infinite dimensional Schrodinger equation is not exactly controllable with Dirichlet boundary condition. K. Beauchard got the local exact controllability of1D infinite dimensional bilinear Schrodinger equation in some higher regularity space H(0)7(I; C):={φ∈H7(I, C); φ(2i)∈H01(I, C),i=0,1,2,3}. The method she used relied on the Nash-Moser implicit function theorem in order to deal with a priori loss of regularity. With the same method, K. Beauchard got the local exact controllability of1D infinite dimensional Schrodinger equation with variable domain in [9]. Almost global results have been proved by K. Beauchard and J. M. Coron in [12]. In [14], K. Beauchard and C. Laurent proposed an important simplification of the above proofs with a more general μ(x), and they got the exact controllability by classical inverse mapping theorem under a hidden regularizing effect. When V(x)≠0, V. Nersesyan got the global approximate controllability of ID infinite dimensional Schrodinger equation in H3[76]. He pointed that such Schrodinger equation is local exact controllable in H(V)3+ε(I,C)[75].In this chapter, we consider1D infinite dimensional bilinear Schrodinger equation with Neumann boundary value condition(a1=a2=0), Dirichlet-Neumann boundary value condition (a1=b2=0or a2=b1=0) and general boundary value condition (a1>0, b1>0, a2>0, b2>0). We find that it is necessary to consider Dirichlet boundary value problem in H(0)3space. How-ever, H(0)2regularity is enough with our boundary value conditions. We get the local exact controllability of1D infinite dimensional Schrodinger equation under Sturm-liouville boundary value condition in which Dirichlet boundary value condition is not included. Our proof relies on the linearization principle, by applying the classical inverse mapping theorem to the end-point map. Con-trollability of the linearized system around the ground state is the consequence of classical results about trigonometric moment problems.In the third chapter, we study approximate stabilization of1D bilinear Schrodinger equation on inhomogeneous media. The content of this chapter has already been published in [94]. We consider1D x-dependent coefficient Schrodinger equation:In [15], K. Beauchard and M. Mirrahimi got the approximate stabilization of system (0.0.4) when u(x)=1, μ(x)=x. As we known, LaSalle invariance principle is a powerful tool to prove the asymptotic stability of equilibrium point in finite dimensional dynamic system. However, in infinite-dimensional case, trying to adapt the convergence analysis, based on the use of the LaSalle invariance principle, the precompactness of the trajectories in L2constitutes a major obstacle, because closed and bounded subset is not always compact. K. Beauchard and M. Mirrahimi avoided the loss of compact by proving the ap-proximate convergence. They applied the implicit feedback control method to get the approximate stabilization of bilinear Schrodinger equation in infinite-dimensional systems, similar to [73], where the author prevented the popula-tion form going through the very high energy levels, while trying to stabilize the system around the ground state.In this chapter, we use the similar method to get the approximate stabi-lization of ID infinite dimensional bilinear Schrodinger equation by construct-ing a suitable Lyapunov function when u(x) is a variable coefficient. In the fourth chapter, we study the approximate stabilization of1D bi-linear Schrodinger equation under periodic boundary value conditions. The periodic phenomenon exists everywhere in nature. But it is a big challenge to study the control of periodic boundary value problem, who has two different eigenvectors for one eigenvalue.We consider1D infinite dimensional Schrodinger equation: with periodic boundary value conditionWe get the approximate stabilization of1D infinite dimensional Schrodinger equation under periodic boundary value condition again by using LaSalle in-variance principle and trigonometric moment problem when μ1and μsatisfy some parity conditions.