regional optimal control problem of a class of infinite dimensional bi linear systems abstractthe purpose of this paper is to study regional optimal control of an infinite dimensional bi linear system excited by unbounded or bounded controls then we prove the existence of an optimal control that minimises a functional cost and we give the characterisation of such a control also we give the condition that ensures the uniqueness of the optimal control the obtained results lead to an algorithm that we illustrate by simulations for a bi dimensional diffusion system excited by a zone actuator

improved stability results for discrete time switched systems a multiple piecewise convex lyapunov function approach abstract in this paper the stability problem is investigated for a class of discrete time switched systems with unstable subsystems under the mode dependent average dwell time mdadt switching a multiple convex lyapunov function mclf and a multiple piecewise convex lyapunov function mpclf are firstly proposed which are formulated in a convex combination form of positive definite matrices with quasi time dependent coefficients it is pointed out in the paper that the multiple lyapunov function mlf and the multiple discontinuous lyapunov function mdlf can be regarded as special cases of the proposed mclf and mpclf respectively then the exponential stability conditions are derived by the new lyapunov functions both slow switching and fast switching are exerted on stable modes and unstable modes respectively finally two numerical examples are given to demonstrate that larger stability regions and tighter mdadt bounds can be obtained by using our developed techniques compared with some recent results

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

regular propagators of bilinear quantum systems the present analysis deals with the regularity of solutions of bilinear control systems of the type x a u t b x where the state x belongs to some complex infinite dimensional hilbert space the possibly unbounded linear operators a and b are skew adjoint and the control u is a real valued function such systems arise for instance in quantum control with the bilinear schr o dinger equation for the sake of the regularity analysis we consider a more general framework where a and b are generators of contraction semi groups under some hypotheses on the commutator of the operators a and b it is possible to extend the definition of solution for controls in the set of radon measures to obtain precise a priori energy estimates on the solutions leading to a natural extension of the celebrated noncontrollability result of ball marsden and slemrod in 1982 complementary material to this analysis can be found in hal 01537743v1

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

feedback stabilization for a bilinear control system under weak observability inequalities abstract in this paper we discuss the feedback stabilization of bilinear systems under weak observation properties in this case the uniform stability is not guaranteed thus we provide an explicit weak decay rate for all regular initial data applications to schrodinger and wave equations are provided

frechet differentiability for an extensible beam equation and its application to bilinear minimax control problems abstract we consider an extensible beam equation with dirichlet boundary conditions the object is to show the frechet differentiability of nonlinear solution map from a bilinear control input to the solution of the beam equation we used this method to formulate the minimax optimal control problem we also show the existence of optimal pairs and find their necessary optimality conditions

finite controllability of infinite dimensional quantum systems quantum phenomena of interest in connection with applications to computation and communication often involve generating specific transfers between eigenstates and their linear superpositions for some quantum systems such as spin systems the quantum evolution equation the schrodinger equation is finite dimensional and old results on controllability of systems defined on on lie groups and quotient spaces provide most of what is needed insofar as controllability of non dissipative systems is concerned however in an infinite dimensional setting controlling the evolution of quantum systems often presents difficulties both conceptual and technical in this paper we present a systematic approach to a class of such problems for which it is possible to avoid some of the technical issues in particular we analyze controllability for infinite dimensional bilinear systems under assumptions that make controllability possible using trajectories lying in a nested family of pre defined subspaces this result which we call the finite controllability theorem provides a set of sufficient conditions for controllability in an infinite dimensional setting we consider specific physical systems that are of interest for quantum computing and provide insights into the types of quantum operations gates that may be developed

optimal control of pdes in a complex space setting application to the schrodinger equation in this paper we discuss optimality conditions for abstract optimization problems over complex spaces we then apply these results to optimal control problems with a semigroup structure as an application we detail the case when the state equation is the schrodinger one with pointwise constraints on the bilinear control we derive first and second order optimality conditions and address in particular the case that the control enters the state equation and cost function linearly

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

controllability of a 2d quantum particle in a time varying disc with radial data abstract in this article we consider a 2 d quantum particle confined in a disc whose radius can be deformed continuously in time we study the problem of controllability of such a quantum particle via deformations of the initial disc i e when we set the time dependent radius of the disc to be the control variable we prove that the resulting system is locally controllable around some radial trajectories which are linear combinations of the first three radial eigenfunctions of the laplacian in the unit disc with dirichlet boundary conditions we prove this result thanks to the linearisation principle by studying the linearised system which leads to a moment problem that can be solved using some results from nonharmonic fourier series in particular we have to deal with fine properties of bessel functions

null controllability for a semilinear parabolic equation with gradient quadratic growth abstract this article is concerned with the null controllability of a semilinear parabolic equation with the nonlinear term involving the gradient quadratic term the technique in this paper is a combination of cole hopf transformation and some methods from a y khapalov controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term a qualitative approach siam j control optim 41 2003 1886 1900

the second rational homology group of the moduli space of curves with level structures abstract let be a finite index subgroup of the mapping class group of a closed genus g surface that contains the torelli group for instance can be the level l subgroup or the spin mapping class group we show that h 2 q q for g 5 a corollary of this is that the rational picard groups of the associated finite covers of the moduli space of curves are equal to q we also prove analogous results for surface with punctures and boundary components

asymptotic properties of feedback solutions for a class of quantum control problems control of quantum systems described by the schrodinger equation is considered feedback control laws are developed for orbit tracking via controlled hamiltonians and their asymptotic properties are analyzed numerical integration via time splitting is also investigated and used to demonstrate the feasibility of the proposed feedback laws

constructive solution of a bilinear optimal control problem for a schrodinger equation abstract often considered in numerical simulations related to the control of quantum systems the so called monotonic schemes have not been so far much studied from the functional analysis point of view yet these procedures provide an efficient constructive method for solving a certain class of optimal control problems this paper aims both at extending the results already available about these algorithms in the finite dimensional case i e the time discretized case and at completing those of the continuous case this paper starts with some results about the regularity of a functional related to a wide class of models in quantum chemistry these enable us to extend an inequality due to ojasiewicz to the infinite dimensional case finally some inequalities proving the cauchy character of the monotonic sequence are obtained followed by an estimation of the rate of convergence

micro motions of a 2 d swimming model governed by multiplicative controls abstract the goal of this paper is to derive a formula for the micro motions of a simplified model of an object swimming by itself in a 2 d stokes fluid mathematics wise the model equations are described by a nonlinear coupled hybrid system of fluid equations and integro differential equations we view this swimming motion as one governed by multiplicative controls such problems are of interest in biological and engineering applications dealing with propulsion systems in fluids

on conditions for asymptotic stability of dissipative infinite dimensional systems with intermittent damping abstract we study the asymptotic stability of a dissipative evolution in a hilbert space subject to intermittent damping we observe that even if the intermittence satisfies a persistent excitation condition if the hilbert space is infinite dimensional then the system needs not being asymptotically stable not even in the weak sense exponential stability is recovered under a generalized observability inequality allowing for time domains that are not intervals weak asymptotic stability is obtained under a similarly generalized unique continuation principle finally strong asymptotic stability is proved for intermittences that do not necessarily satisfy some persistent excitation condition evaluating their total contribution to the decay of the trajectories of the damped system our results are discussed using the example of the wave equation schrodinger s equation and for strong stability also the special case of finite dimensional systems

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

nonlinear control of a coupled pde ode system modeling a switched power converter with a transmission line we consider an infinite dimensional system modelling a boost converter connected to a load via a transmission line the governing equations form a system coupling the telegraph partial differential equation with the ordinary differential equations model ing the converter the coupling is given by the boundary conditions and the nonlinear controller we introduce we design a nonlinear saturating control law using a lya punov function for the averaged model of the system the main results give the well posedness and stability properties of the obtained closed loop system

simultaneous control of a rod equation and a simple schro dinger equation abstract we show that a corollary of a local approximate controllability result for the bilinear rod equation in 1 is that the controls which steer the modes of the rod equation also move the modes of a controlled schrodinger equation specifying a target point for the schrodinger system restricts but does not determine the outcome of the controlled rod equation the local result is a special case of a general local result for hyperbolic systems which is used in 1 to obtain a global approximate controllability result after modifying the latter we obtain a global result for the two systems

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

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

a stroock varadhan support theorem in non linear filtering theory let t 0 t 1 be the unnormalized filter arising in the filtering theory of correlated diffusions in this article is considered as a stochastic process taking values inc n a description of the support of its law in the frechet spacec 0 1 c n r is given this result is the analogue for stochastic partial differential equations of the celebrated stroock varadhan theorem for diffusion processes the support of the law of the filter is shown to be the closure of the set of trajectories obtained from the zakai equation by replacing the stratonovitch differentialdy by anh1 control herey denotes the observation process

some results on controllability of a nonlinear degenerate parabolic system by bilinear control in this paper we discuss the controllability of a nonlinear degenerate parabolic system with bilinear control based on the shrinking property of the solutions we prove that the system is not globally approximately controllable furthermore we give an approximate null controllability result we also prove that the system is not globally exactly null controllable by a comparison principle

on controllability of delayed boolean control networks this paper is devoted to studying the trajectory and state controllability of boolean control networks bcns with time delay in contrast to bcns without time delay the dynamics of delayed bcns are determined by a sequence of initial states named here trajectories trajectory controllability means that there exists a control signal steering a system from an initial trajectory to a desired trajectory while state controllability means that there exists a control signal steering an initial state to a given state here both trajectory controllability and state controllability will be studied it should be noted that in this paper trajectory controllability does not mean tracking or following a given trajectory in fact it means to control bcns to a destination trajectory of length mu at the k th step using the semi tensor product of matrices the delayed bcns are first converted into an equivalent algebraic description and then some necessary and sufficient conditions are derived for the trajecto

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

notions of controllability for bilinear multilevel quantum systems in this note we define four different notions of controllability of physical interest for multilevel quantum mechanical systems these notions involve the possibility of driving the evolution operator as well as the state of the system we establish the connections among these different notions as well as methods to verify controllability

bilinear optimal control of a kirchhoff plate abstract we consider the problem of optimal control of a kirchhoff plate a bilinear control is used as a force to make the plate close to a desired profile taking into the account a quadratic cost of control we prove the existence of an optimal control and characterize it uniquely through the solution of an optimality system

time optimal control of semilinear parabolic equations via bilinear controls abstract we establish the existence of the time optimal control for semilinear parabolic equations with gradient quadratic growth via bilinear controls it is worth pointing out that there is no restriction on the growth of the nonlinearity f s with respect to the variable s in the equation which is a remarkable difference compared to the semilinear parabolic system with locally distributed controls the technique used in this paper is the combination of the hopf cole transformation the a prior estimates on solutions of parabolic equations and the strategy of the stepwise control

controllability for distributed bilinear systems this paper studies controllability of systems of the form dw dt mathcal a w p t mathcal b w where mathcal a is the infinitesimal generator of a c 0 semigroup of bounded linear operators e mathcal a t on a banach space x mathcal b x to x is a c 1 map and p in l 1 0 t mathbb r is a control the paper i gives conditions for elements of x to be accessible from a given initial state w 0 and ii shows that controllability to a full neighborhood in x of w 0 is impossible for dim x infty examples of hyperbolic partial differential equations are provided

wavefunction controllability for finite dimensional bilinear quantum systems we present controllability results for quantum systems interacting with lasers exact controllability for the wavefunction in these bilinear systems is proved in the finite dimensional case under very natural hypotheses

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

constrained regional control problem of a bilinear plate equation this paper investigates a regional optimal control problem of a plate equation described by a bilinear system evolving in a spacial domain the control is distributed bounded and acts on the vel

derivation of an evolution model for nonlinearly elastic beams by asymptotic expansion methods abstract the asymptotic expansion method has been shown to be a powerful tool in order to mathematically justify well known mechanical models and obtain new ones in this paper we justify the classical semi linear evolution equation for the bending of an elastic beam submitted to axial forces at both ends in addition to a theoretical study of this limit model in the last section we obtain a generalization of this model for more complete situations

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

bilinear minimax control problems for a class of parabolic systems with applications to control of nuclear reactors this paper considers the bilinear minimax control problem of an important class of parabolic systems with robin boundary conditions such systems are linear on state variables when the control and disturbance are fixed and linear on the control or disturbance when the state variables are fixed the objective is to maintain target state variables by taking account the influence of noises in data while a desired power level and adjustment costs are taken into consideration firstly we introduce some classes of bilinear systems and obtain the existence and the uniqueness of the solution as well as stability under mild assumptions afterwards the minimax control problem is formulated we show the existence of an optimal solution and we also find necessary optimality conditions finally to illustrate the abstract results we present two examples of neutron fission systems