Controllability of the nonlinear Schrödinger equation in the vicinity of the ground state

Abstract

SUMMARYLocal exact controllability of the one-dimensional NLS (subject to zero-boundary conditions) with distributed control is shown to hold in a H 1 -neighbourhood of the nonlinear ground state. The Hilbert Uniqueness Method (HUM), due to Lions, is applied to the linear control problem that arises by linearization around the ground state. The application of HUM crucially depends on the spectral properties of the linearized NLS operator which are given in detail. INTRODUCTIONThe control properties of many partial differential equations (PDEs) arising in physics and engineering have been studied extensively. Those investigations include exact and/or optimal controllability for the (linear and nonlinear) heat, wave, beam and plate equations as well as equations of elasticity and Navier-Stokes equations, to mention just some of the most prominent examples. For Schrödinger equations, however, the control theory is markedly less developed. For a brief survey on control results for (linear and nonlinear) Schrödinger equations, see e.g. [1]. The purpose of this paper is to establish local exact controllability for the one-dimensional nonlinear Schrödinger (NLS) equation (subject to zero-boundary conditions) with distributed control. Specifically, we will consider the following control problem: where g (x) denotes the indicator function for some fixed, possibly small, open subinterval ⊂ := (0, 1) representing the spatial region in which the control is applied. Given a fixed control 'horizon' T >0 and initial and target states y 0 and y 1 , the objective is to construct a control function u = u(t, x) that will steer the state y from y 0 to y 1 , i.e. the unique solution y = y(t, x) of (1a)- (1c) is to satisfy (1d). The control problem (1a)-(1d) was posed in [2,3]. A small-data controllability result for periodic boundary conditions is contained in [4].In this paper, we will concentrate on the special case of the focusing cubic NLS equation, i.e. f (s) =−sMore general nonlinearities could be treated, but this is not our main interest here. So we restrict ourselves to the prototypical cubic nonlinearity. Our main result states that the control problem (1a)-(1d) is soluble locally within an H 1 -neighbourhood of the ground-state solution (t, x) = e i t (x), if the control time T >0 is sufficiently large. Here (x) denotes the (nonlinear time-independent) ground state, ‡ i.e. the (real and) positive solution of the boundary value problemwhich is known to exist and to be unique (see Section A.1). Our main result reads. Theorem 1Let ⊂ (0, 1), >0 be given and let denote the ground state. Then, there exist T >0 and >0 such that, for any y 0 ,The theorem will be proved by applying the implicit function theorem (IFT) to the nonlinear map :where t → y(t; y 0 , u) ∈ C([0, T ]; H 1 0 (0, 1)) denotes the unique solution of (1a)-(1c). To be able to apply the IFT, it will be verified that the linearization * u ( , e i T , 0) : Lions [5]. The main difficulty in the application of HUM stems from the lack of self-adjointness of the linear...

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