exact controllability in projections of the bilinear schrodinger equation we consider the bilinear schrodinger equation with discrete spectrum drift we show for n n arbitrary exact controllability in projections on the first n given eigenstates the controllability result relies on a generic controllability hypothesis on some associated finite dimensional approximations the method is based on lie algebraic control techniques applied to the finite dimensional approximations coupled with classical topological arguments issuing from degree theory

a general theorem on local controllability we prove a general sufficient condition for local controllability of a nonlinear system at an equilibrium point earlier results of brunovsky hermes jurdjevic crouch and byrnes sussmann and grossmann are shown to be particular cases of this result also a number of new sufficient conditions are obtained all these results follow from one simple general principle namely that local controllability follows whenever brackets with certain symmetries can be neutralized in a suitable way by writing them as linear combinations of brackets of a lower degree both the class of symmetries and the definition of degree can be chosen to suit the problem

decay of the navier stokes poisson equations we establish the time decay rates of the solution to the cauchy problem for the compressible navier stokes poisson system via a refined pure energy method in particular the optimal decay rates of the higher order spatial derivatives of the solution are obtained the h s 0 s 3 2 negative sobolev norms are shown to be preserved along time evolution and enhance the decay rates as a corollary we also obtain the usual lp l2 1 p 2 type of the optimal decay rates compared to the compressible navier stokes system and the compressible irrotational euler poisson system our results imply that both the dispersion effect of the electric field and the viscous dissipation contribute to enhance the decay rate of the density our proof is based on a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis

spectral controllability for 2d and 3d linear schrodinger equations we consider a quantum particle in an infinite square potential well of rnrn n 2 3n 2 3 subjected to a control which is a uniform in space electric field under the dipolar moment approximation the wave function solves a pde of schrodinger type we study the spectral controllability in finite time of the linearized system around the ground state we characterize one necessary condition for spectral controllability in finite time kal if is the bottom of the well then for every eigenvalue of view the mathml source d the projections of the dipolar moment onto every normalized eigenvector associated to are linearly independent in rnrn in 3d our main result states that spectral controllability in finite time never holds for one directional dipolar moment the proof uses classical results from trigonometric moment theory and properties about the set of zeros of entire functions in 2d we first prove the existence of a minimal time tmin 0tmin 0 for spectral controllability i e if t tmin t tmin one has spectral controllability in time t if condition kal holds true for and if ttmin holds generically with respect to the domain the proof relies on shape differentiation and a careful study of dirichlet to neumann operators associated to certain helmholtz equations we also show that one can recover exact controllability in abstract spaces from this 2d spectral controllability by adapting a classical variational argument from control theory

practical stabilization of a quantum particle in a one dimensional infinite square potential well we consider a nonrelativistic charged particle in a one dimensional infinite square potential well this quantum system is subjected to a control which is a uniform in space time depending electric field it is represented by a complex probability amplitude solution of a schrodinger equation on a one dimensional bounded domain with dirichlet boundary conditions we prove the almost global practical stabilization of the eigenstates by explicit feedback laws

local controllability and non controllability for a 1d wave equation with bilinear control abstract we consider a linear wave equation on the interval 0 1 with bilinear control and neumann boundary conditions we study the controllability of this nonlinear control system locally around a constant reference trajectory we prove that the following results hold generically for every t 2 this system is locally controllable in h 3 h 2 in time t with controls in l 2 0 t r for t 2 this system is locally controllable up to codimension one in h 3 h 2 in time t with controls in l 2 0 t r the reachable set is locally a non flat submanifold of h 3 h 2 with codimension one for every t 2 this system is not locally controllable more precisely the reachable set with controls in l 2 0 t r is contained in a non flat submanifold of h 3 h 2 with infinite codimension the proof of these results relies on the inverse mapping theorem and second order expansions

conditions for the existence of continuous storage functions for nonlinear dissipative systems the problem of existence of continuous storage functions for dissipative nonlinear systems is considered it is shown that if a nonlinear system is dissipative in the state x then under certain assumptions a continuous storage function can be constructedon a set of points accessible from x by concatenation of a finite number of forwardandbackwardmotions of the system most of these assumptions are weaker than certain controllability type properties andcan be checkedusing similar tests 2004 elsevier b v all rights reserved

global solutions of nonlinear magnetohydrodynamics with large initial data a free boundary problem for nonlinear magnetohydrodynamics with general large initial data is investigated the existence uniqueness and regularity of global solutions are established with large initial data in h1 it is shown that neither shock waves nor vacuum and concentration in the solutions are developed in a finite time although there is a complex interaction between the hydrodynamic and magnetodynamic effects an existence theorem of global solutions with large discontinuous initial data is also established

on the controllability of the vlasov poisson system in this paper we study the controllability of the vlasov poisson system in a periodic domain by means of an interior control located in an spatial subdomain the first result proves the local exact zero controllability in the two dimensional torus between two small acceptable distribution functions with an arbitrary control zone a second result establishes the global exact controllability in arbitrary dimension provided the control zone satisfies the condition that it contains a hyperplane of the torus

controllability of a quantum particle in a moving potential well we consider a nonrelativistic charged particle in a 1d moving potential well this quantum system is subject to a control which is the acceleration of the well it is represented by a wave function solution of a schrodinger equation the position of the well together with its velocity we prove the following controllability result for this bilinear control system given 0 close enough to an eigenstate and f close enough to another eigenstate the wave function can be moved exactly from 0 to f in finite time moreover we can control the position and the velocity of the well our proof uses moment theory a nash moser implicit function theorem the return method and expansion to the second order

growth of sobolev norms and controllability of the schrodinger equation in this paper we obtain a stabilization result for both linear and nonlinear schrodinger equations under generic assumptions on the potential then we consider the schrodinger equations with a potential which has a random time dependent amplitude we show that if the distribution of the amplitude is sufficiently non degenerate then any trajectory of the system is almost surely non bounded in sobolev spaces

a partial differential inequality for dissipative nonlinear systems abstract in this paper we show that storage functions for dissipative systems can be characterised by a partial differential inequality in the weak or viscosity sense the storage functions need not be differentiable this characterisation is stable with respect to perturbations we formulate a version of the positive real lemma for passive systems using the weak sense framework

passivity and feedback passification of switched discrete time linear systems abstract this paper focuses on the passivity analysis and feedback passification for switched discrete time linear systems using multiple storage functions under a certain switching law controllers and switching laws are designed based on complete and partial state measurements respectively conditions for strict passivity of a switched discrete time system are obtained without the requirement of strict passivity of subsystems in the case of partial state measurements dynamic output feedback controllers for subsystems are designed and a switching law depending only on the state of the output feedback controllers is constructed to guarantee strict passivity of the closed loop switched system finally a numerical example is provided to demonstrate the feasibility of the theoretical results

input output to state stability this work explores lyapunov characterizations of the input output to state stability ioss property for nonlinear systems the notion of ioss is a natural generalization of the standard zero detectability property used in the linear case the main contribution of this work is to establish a complete equivalence between the ioss property and the existence of a certain type of smooth lyapunov function as corollaries one shows the existence of norm estimators and obtains characterizations of nonlinear detectability in terms of relative stability and of finite energy estimates

passivity and stability of switched systems a multiple storage function method this paper presents a concept of passivity for switched systems using multiple storage functions this passivity property is invariant under compatible feedback interconnection branicky s stability theorem of multiple lyapunov functions is generalized by relaxing the non increasing condition on values of lyapunov like functions using this result we show that a passive switched system is stable in the sense of lyapunov moreover asymptotic stability is reached if all subsystems are asymptotically detectable

dissipativity theory for switched systems a framework of dissipativity theory for switched systems using multiple storage functions and multiple supply rates is set up each subsystem of a switched system is associated with a storage function to describe the energy stored in the subsystem and is associated with a supply rate that represents energy coming from outside the subsystem when the subsystem is active the exchange of energy between the active subsystem and an inactive subsystem is characterized by cross supply rates stability is reached when all supply rates can be made negative as long as the total exchanged energy between the active subsystem and any inactive subsystems is finite in some sense two special forms of dissipativity passivity and l 2 gain are addressed for both cases asymptotic stability is guaranteed under certain negative output feedback plus asymptotic zero state detectability switched passivity conditions and switched l 2 gain inequalities are respectively derived which are generalizations of classical ones feedback invariance of passivity and a small gain theorem are also given

on the regularity of weak solutions to the magnetohydrodynamic equations abstract in this paper we study the regularity of weak solution to the incompressible magnetohydrodynamic equations we obtain some sufficient conditions for regularity of weak solutions to the magnetohydrodynamic equations which is similar to that of incompressible navier stokes equations moreover our results demonstrate that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solution to the magneto hydrodynamic equations

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

controllability of quasi linear hamiltonian nls equations abstract we prove internal controllability in arbitrary time for small data for quasi linear hamiltonian nls equations on the circle we use a procedure of reduction to constant coefficients up to order zero and hum method to prove the controllability of the linearized problem then we apply a nash moser hormander implicit function theorem as a black box

global existence and large time behavior of weak solutions to the compressible magnetohydrodynamic equations with coulomb force abstract in this paper we consider the equations of magnetohydrodynamics with coulomb force which is of hyperbolic parabolic elliptic mixed type by constructing the approximate solutions to the modified system with an artificial pressure term added global existence of finite energy weak solutions is established via the weak convergence method more careful argument has been paid to overcome the new difficulty arising from the poisson term of coulomb force in two dimensions when the adiabatic exponent is close to one we also investigate the large time behavior of such weak solutions after discussing the regularity and uniqueness of solutions to the stationary problem

approximate controllability exact controllability and conical eigenvalue intersections for quantum mechanical systems we study the controllability of a closed control affine quantum system driven by two or more external fields we provide a sufficient condition for controllability in terms of existence of conical intersections between eigenvalues of the hamiltonian in dependence of the controls seen as parameters such spectral condition is structurally stable in the case of three controls or in the case of two controls when the hamiltonian is real the spectral condition appears naturally in the adiabatic control framework and yields approximate controllability in the infinite dimensional case in the finite dimensional case it implies that the system is lie bracket generating when lifted to the group of unitary transformations and in particular that it is exactly controllable hence lie algebraic conditions are deduced from purely spectral properties

exact controllability of a nonlinear korteweg de vries equation on a critical spatial domain we consider the boundary controllability problem for a nonlinear korteweg de vries equation with the dirichlet boundary condition we study this problem for a spatial domain with a critical length for which the linearized control system is not controllable in order to deal with the nonlinearity we use a power series expansion of second order we prove that the nonlinear term gives the local exact controllability around the origin provided that the time of control is large enough

controllability issues for continuous spectrum systems and ensemble controllability of bloch equations we study the controllability of the bloch equation for an ensemble of non interactinghalf spins inastaticmagneticfield withdispersioninthelarmorfrequency this system may be seen as a prototype for infinite dimensional bilinear systems with continuous spectrum whose controllability is not well understood we provide several mathematical answers with discrimination between approximate and exact controlla bility and between finite time or infinite time controllability this system is not exactly controllable in finite time t with bounded controls in l 2 0 t but it is approximately controllable in l in finite time with unbounded controls in l 0 moreover we propose explicit controls realizing the asymptotic exact controllability to a uniform state of spin 1 2o r 1 2

stabilization for an ensemble of half spin systems feedback stabilization of an ensemble of non interacting half spins described by the bloch equations is considered this system may be seen as an interesting example for infinite dimensional systems with continuous spectra we propose an explicit feedback law that stabilizes asymptotically the system around a uniform state of spin 1 2 or 1 2 the proof of the convergence is done locally around the equilibrium in the h 1 topology this local convergence is shown to be a weak asymptotic convergence for the h 1 topology and thus a strong convergence for the c 0 topology the proof relies on an adaptation of the lasalle invariance principle to infinite dimensional systems numerical simulations illustrate the efficiency of these feedback laws even for initial conditions far from the equilibrium

local exact controllability of a one dimensional nonlinear schrodinger equation we consider a one dimensional nonlinear schrodinger equation modeling a bose einstein condensate in an infinite square well potential box this is a nonlinear control system in which the state is the wave function of the bose einstein condensate and the control is the length of the box we prove that local exact controllability around the ground state associated with a fixed length of the box holds generically with respect to the chemical potential mu i e up to an at most countable set of mu values the proof relies on the linearization principle and the inverse mapping theorem as well as ideas from analytic perturbation theory

large time behavior of the solutions to a hydrodynamic model for semiconductors we establish the global existence of smooth solutions to the cauchy problem for the one dimensional isentropic euler poisson or hydrodynamic model for semiconductors for small initial data in particular we show that as t to infty these solutions converge to the stationary solutions of the drift diffusion equations the existence and uniqueness of stationary solutions to the drift diffusion equations are proved without the smallness assumption

optimal bilinear control of nonlinear schrodinger equations with singular potentials abstract we consider an optimal bilinear control problem for the nonlinear schrodinger equations with singular potentials we show the well posedness of the problem and the existence of an optimal control in addition the first order optimality system is rigorously derived our results give an improvement for some recent results

blow up criterion for compressible mhd equations abstract in this paper we study the 3d compressible magnetohydrodynamic equations we obtain a blow up criterion for the local strong solutions just in terms of the gradient of the velocity similar to the beal kato majda criterion see j t beal t kato and a majda 1984 1 for the ideal incompressible flow in addition initial vacuum is allowed in our case

large time behavior of the compressible magnetohydrodynamic equations with coulomb force abstract in this paper we consider the large time behavior of the solutions near a constant equilibrium state to the cauchy problem for the compressible magnetohydrodynamic equations with coulomb force in r 3 under the assumptions that the h 3 norm of the initial data is small but its higher order derivatives could be large we obtain the optimal time decay rates of the solutions and their higher order spatial derivatives by introducing the negative sobolev and besov spaces as an immediate byproduct the l p l 2 1 p 2 type of the decay rates follows without requiring the smallness for l p norm of initial data

optimal bilinear control of gross pitaevskii equations with coulombian potentials in this paper we consider an optimal bilinear control problem for the gross pitaevskii equations with coulombian potentials we show the well posedness of the problem and the existence of an optimal control in addition the first order optimality system is rigorously derived in particular we prove the frechet differentiability of the unconstrained functional we extend the study of hintermuller et al 2013 15 to more general power nonlinearities and unbounded potentials c 2015 elsevier inc all rights reserved

adiabatic control of the schrodinger equation via conical intersections of the eigenvalues in this paper we present a constructive method to control the bilinear schrodinger equation via two controls the method is based on adiabatic techniques and works if the spectrum of the hamiltonian admits eigenvalue intersections and if the latter are conical as it happens generically in this framework we are able to spread on several levels connected by conical intersections a state initially concentrated in a single energy level we provide sharp estimates on the dependence of the error with respect to the controllability time moreover we identify some special curves in the space of controls that improve the precision of the adiabatic approximation when passing through conical intersections with respect to classical adiabatic theory

on the problem of quantum control in infinite dimensions in the framework of bilinear control of the schrodinger equation it has been proved that the reachable set has a dense complement in hence in this setting exact quantum control in infinite dimensions is not possible on the other hand it is known that there is a simple choice of operators which when applied to an arbitrary state generate dense orbits in hilbert space compatibility of these two results is established in this paper and in particular it is proved that the closure of the reachable set of bilinear control is dense in the requirements for controllability in infinite dimensions are also related to the properties of the infinite dimensional unitary group

implicit lyapunov control of finite dimensional schrodinger equations an implicit lyapunov based approach is proposed for generating trajectories of a finite dimensional controlled quantum system the main difficulty comes from the fact that we consider the degenerate case where the linearized control system around the target state is not controllable the controlled lyapunov function is defined by an implicit equation and its existence is shown by a fix point theorem the convergence analysis is done using lasalle invariance principle closed loop simulations illustrate the performance of such feedback laws for the open loop control of a test case considered by chemists

a nash moser hormander implicit function theorem with applications to control and cauchy problems for pdes abstract we prove an abstract nash moser implicit function theorem which when applied to control and cauchy problems for pdes in sobolev class is sharp in terms of the loss of regularity of the solution of the problem with respect to the data the proof is a combination of i the iteration scheme by hormander arma 1976 based on telescoping series and very close to the original one by nash ii a suitable way of splitting series in scales of banach spaces inspired by a simple clever trick used in paradifferential calculus for example by metivier as an example of application we apply our theorem to a control and a cauchy problem for quasi linear perturbations of kdv equations improving the regularity of a previous result with respect to other approaches to control and cauchy problems the application of our theorem requires lighter assumptions to be verified

global variational solutions to the compressible magnetohydrodynamic equations abstract we prove the existence of globally defined variational solutions to the compressible magnetohydrodynamic mhd equations with the coefficients depending on the temperature as a by product we give a simple proof for the nonexistence of nontrivial weak time periodic solutions by the entropy principle of clausius duhem and a new poincare type inequality

a note on global existence of weak solutions to the compressible magnetohydrodynamic equations with coulomb force abstract in this note for the case of 9 33 12 4 3 we prove the existence of global in time finite energy weak solution of the equations of a two dimensional magnetohydrodynamics with coulomb force where denotes the adiabatic exponent the value 9 33 12 is the optimal lower bound of to establish global in time finite energy weak solution under current frame

multi input schrodinger equation controllability tracking and application to the quantum angular momentum abstract we present a sufficient condition for approximate controllability of the bilinear discrete spectrum schrodinger equation in the multi input case the controllability result extends to simultaneous controllability approximate controllability in h s and tracking in modulus the sufficient condition is more general than those present in the literature even in the single input case and allows the spectrum of the uncontrolled operator to be very degenerate e g to have multiple eigenvalues or equal gaps among different pairs of eigenvalues we apply the general result to a rotating polar linear molecule driven by three orthogonal external fields a remarkable property of this model is the presence of infinitely many degeneracies and resonances in the spectrum

exact boundary controllability of the nonlinear schrodinger equation abstract this paper studies the exact boundary controllability of the semi linear schrodinger equation posed on a bounded domain r n with either the dirichlet boundary conditions or the neumann boundary conditions it is shown that if s n 2 or 0 s n 2 with 1 n 2 2 s or s 0 1 with n 2 then the systems are locally exactly controllable in the classical sobolev space h s around any smooth solution of the cubic schrodinger equation

optimal decay rate for the compressible navier stokes poisson system in the critical l p framework abstract in this paper we consider the large time behavior of global solutions to the initial value problem for the compressible navier stokes poisson system in the l p critical framework and in any dimension n 3 we obtain the time decay rates not only for lebesgue spaces but also for a family of besov norms with negative or nonnegative regularity exponents which improves the decay results in high sobolev regularity the proof is mainly based on the littlewood paley theory and refined time weighted inequalities in fourier space

on the existence of a continuous storage function for dissipative systems conditions for nonlocal existence of a continuous storage function for nonlinear dissipative system are presented more precisely it is shown that under the local w uniform reachability assumption at one point x the required supply function is continuous on the set of points reachable from x conditions for the local w uniform reachability based on the local controllability properties of the system are provided

on the controllability of quantum transport in an electronic nanostructure we investigate the controllability of quantum electrons trapped in a two dimensional device typically a metal oxide semiconductor mos field effect transistor the problem is modeled by the schrodinger equation in a bounded domain coupled to the poisson equation for the electrical potential the controller acts on the system through the boundary condition on the potential on a part of the boundary modeling the gate we prove that generically with respect to the shape of the domain and boundary conditions on the gate the device is controllable we also consider control properties of a more realistic nonlinear version of the device taking into account the self consistent electrostatic poisson potential

solving quantum optimal control problems using clebsch variables and lin constraints clebsch variables and lin constraints are applied to the study of a class of optimal control problems for affine controlled quantum systems the optimal control problem will be modelled with controls defined on an auxiliary space where the dynamical group of the system acts freely the reciprocity between both theories the classical theory defined by the objective functional and the quantum system is established by using a suitable version of lagrange s multipliers theorem and a geometrical interpretation of the constraints of the system as defining a subspace of horizontal curves in an associated bundle it is shown how the solutions of the variational problem defined by the objective functional determine solutions of the quantum problem then a new way of obtaining explicit solutions for a family of optimal control problems for affine controlled quantum systems finite or infinite dimensional is obtained one of its main advantages is the the use of clebsch variables allows to compute such solutions from solutions of invariant problems that can often be computed explicitly this procedure can be presented as an algorithm that can be applied to a large class of systems finally some simple examples spin control a simple quantum hamiltonian with an elroy beanie type classical model and a controlled one dimensional quantum harmonic oscillator illustrating the main features of the theory will be discussed

global well posedness of strong solutions with large oscillations and vacuum to the compressible navier stokes poisson equations subject to large and non flat doping profile abstract this paper concerns an initial value problem of compressible navier stokes poisson equations subject to large and non flat doping profile in whole space r 3 the global well posedness of strong solutions with large oscillations and vacuum is established provided the initial data are of small energy and the steady state is strictly away from vacuum a weak strong uniqueness result is also obtained

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

quantum control and representation theory a new notion of controllability for quantum systems that takes advantage of the linear superposition of quantum states is introduced we call such a notion von neumann controllability and it is shown that it is strictly weaker than the usual notion of pure state and operator controllability we provide a simple and effective characterization of it by using tools from the theory of unitary representations of lie groups in this sense we are able to approach the problem of control of quantum states from a new perspective that of the theory of unitary representations of lie groups a few examples of physical interest and the particular instances of compact and nilpotent dynamical lie groups are discussed

global existence and convergence rates of smooth solutions for the compressible magnetohydrodynamic equations abstract in this paper we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations in r 3 we prove the global existence of the smooth solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in h 3 framework moreover if additionally the initial data belong to l p with 1 p 6 5 the optimal convergence rates of the solutions in l q norm with 2 q 6 and its spatial derivatives in l 2 norm are obtained

decay estimates of solutions to the compressible euler maxwell system in r 3 abstract we study the large time behavior of solutions near a constant equilibrium to the compressible euler maxwell system in r 3 we first refine a global existence theorem by assuming that the h 3 norm of the initial data is small but the higher order derivatives can be arbitrarily large if the initial data belongs to h s 0 s 3 2 or b 2 s 0 s 3 2 by a regularity interpolation trick we obtain the various decay rates of the solution and its higher order derivatives as an immediate byproduct the usual l p l 2 1 p 2 type of the decay rates follows without requiring that the l p norm of initial data is small