on the linearization via a restricted class of dynamic feedback we deal with the problem of linearization of nonlinear systems using a restricted class of dynamic compensator which is composed of chains of integrators followed by static feedback in the vector field formulation we present an upper bound on the number of integrators to linearize nonlinear systems and show that the bound is sharp when n 2 spl ges m spl ges 2 this results in the checkable necessary and sufficient conditions

quantum gate generation by t sampling stabilisation this paper considers right invariant and controllable driftless quantum systems with state x t evolving on the unitary group u n and m inputs u u1 um the t sampling stabilisation problem is introduced and solved given any initial condition x0 and any goal state find a control law u u x t such that for the closed loop system the purpose is to generate arbitrary quantum gates corresponding to this is achieved by the tracking of t periodic reference trajectories of the quantum system that pass by using the framework of coron s return method the t periodic reference trajectories are generated by applying controls that are a sum of a finite number m of harmonics of sin 2 t t whose amplitudes are parameterised by a vector a the main result establishes that for m big enough x jt exponentially converges towards for almost all fixed a with explicit and completely constructive control laws this paper also establishes a stochastic version of this deterministic control law the key idea

bang bang control design for quantum state transfer based on hyperspherical coordinates and optimal time energy control we present a constructive control scheme for solving quantum state engineering problems based on the parameterization of the state vector in terms of complex hyperspherical coordinates unlike many control schemes based on factorization of unitary operators the scheme gives explicit expressions for all generalized euler angles in terms of the hyperspherical coordinates of the initial and final states the factorization when applicable has added benefits that phase rotations can be combined and performed concurrently the control procedure can be realized using simple bang bang or square wavefunction controls optimal time energy control is considered to find the optimal control amplitude the extension of the scheme to implement arbitrary unitary operators is also discussed

boundary behaviour of dirichlet eigenfunctions of second order elliptic operators we investigate the rate of decay of solutions of a second order elliptic equation tf f in an arbitrary domain r n which satisfy zero boundary conditions and then apply the results to a study of the behaviour of the associated heat semigroup in various weighted l 2 spaces our results concern the behaviour at infinity and at points on the boundary of especially irregular ones like corners and cusps the decay rates we obtain are expressed in terms of weighted l 2 estimates which are governed by the sharpness of the cusps and the properties of the coefficients of also the meaning of the zero boundary condition at a cusp is addressed

a constructive condition for dynamic feedback linearization the paper deals with dynamic feedback linearization of continuous time affine systems a constructive procedure based on prolongations is proposed based on necessary geometric conditions the algorithm computes a set of prolongation indices

left definite sturm liouville problems abstract left definite regular self adjoint sturm liouville problems with either separated or coupled boundary conditions are studied we give an elementary proof of the existence of eigenvalues for these problems for any fixed equation we establish a sequence of inequalities among the eigenvalues for different boundary conditions and estimate the range of each eigenvalue as a function on the space of boundary conditions some of our results here yield an algorithm for numerically computing the eigenvalues of a left definite problem with an arbitrary coupled boundary condition our inequalities imply that the well known asymptotic formula for the eigenvalues in the separated case also holds in the coupled case moreover we study the continuous and differentiable dependence of the eigenvalues of the general left definite problem on all the parameters in its differential equation and boundary condition

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

spectral theory of second order vector difference equations abstract this paper is concerned with spectral problems of second order vector difference equations with two point boundary value conditions where the matrix valued coefficient of the leading term may be singular a concept of self adjointness of the boundary value conditions is introduced the self adjointness of the corresponding difference operator is discussed on a suitable admissible function space and fundamental spectral results are obtained the dual orthogonality of eigenfunctions is shown in a special case rayleigh s principles and the minimax theorems in two linear spaces are given as an application a comparison theorem for eigenvalues of two sturm liouville problems is presented

dynamic observer error linearization in this paper a new framework of observer error linearization problem is proposed the main idea of our approach is twofold the one is to introduce an auxiliary dynamics whose input is the system output and the other is to transform the augmented system into an observable linear system with an injection term which contains the system output as well as the state of the auxiliary dynamics it is a natural extension of the recently developed dynamic observer error linearization where the injection term contains only newly defined output it is also shown that whenever an n dimensional system is immersible into an n d dimensional linear system up to an output injection then it can be also dynamically observer error linearizable in our sense with a d dimensional auxiliary dynamics moreover we show that the converse is not true by providing a counterexample which implies that our approach is applicable to a strictly wider class of systems than that of the system immersion method

dynamic output feedback linearization and global stabilization abstract we present a class of single input single output nonlinear systems which are globally transformable by a dynamic output feedback control and a time varying state space transformation into a linear observable and minimum phase system we then show how those systems can be globally stabilized by a dynamic output feedback nonlinear control and how global output tracking can be achieved as well

a remark on nonlinear accessibility conditions and infinite prolongations the lie rank condition for strong nonlinear accessibility is interpreted via the differential geometry of jets and prolongations of infinite order it yields to a new lie algebraic criterion and to the consideration of first integrals which apply to nonlinear systems in quite general form the latter characterization in particular is valid without distinguishing between the system variables and is independent of time scalings weak accessibility is also considered

discrete linear hamiltonian eigenvalue problems abstract this paper introduces general discrete linear hamiltonian eigenvalue problems and characterizes the eigenvalues assumptions are given among them the new notion of strict controllability of a discrete system that imply isolatedness and lower boundedness of the eigenvalues due to the quite general assumptions discrete sturm liouville eigenvalue problems of higher order are included in the presented theory

optimal control of two level quantum systems we study the manipulation of two level quantum systems this research is motivated by the design of quantum mechanical logic gates which perform prescribed logic operations on a two level quantum system a quantum bit we consider the problem of driving the evolution operator to a desired state while minimizing an energy type cost mathematically this problem translates into an optimal control problem for systems varying on the lie group of special unitary matrices of dimension two with cost that is quadratic in the control we develop a comprehensive theory of optimal control for two level quantum systems this includes in particular a classification of normal and abnormal extremals and a proof of regularity of the optimal control functions the impact of the results of the paper on nuclear magnetic resonance experiments and quantum computation is discussed

sufficient conditions for dynamic state feedback linearization sufficient conditions are given for the existence of a dynamic state feedback compensator for a multi input nonlinear system such that the closed loop system is transformable into a linear controllable one by an extended state space change of coordinates an example shows that the conditions are not necessary necessary conditions are also given which are shown to be sufficient when the number of states minus the number of controls is equal to one several examples illustrate how the sufficient conditions obtained lead to the design of the dynamic compensator

flatness based control of a single qubit gate this work considers the open loop control problem of steering a two level quantum system from any initial to any final condition the model of this system evolves on the state space having two inputs that correspond to the complex amplitude of a resonant laser field a symmetry preserving flat output is constructed using a fully geometric construction and quaternion computations simulation results of this flatness based open loop control are provided

a necessary condition for dynamic feedback linearization abstract a necessary condition for dynamic feedback linearizability of control systems x dot f x u in which the controls enter nonlinearly is derived for each x the image of the function f x needs to allow a ruling i e a foliation by straight lines

flatness and defect of non linear systems introductory theory and examples we introduce flat systems which are equivalent to linear ones via a special type of feedback called endogenous their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of kalman s controllability the distance to flatness is measured by a non negative integer the defect we utilize differential algebra where flatness and defect are best defined without distinguishing between input state output and other variables many realistic classes of examples are flat we treat two popular ones the crane and the car with n trailers the motion planning of which is obtained via elementary properties of plane curves the three non flat examples the simple double and variable length pendulums are borrowed from non linear physics a high frequency control strategy is proposed such that the averaged systems become flat

a strong contractivity property for semigroups generated by differential operators frequently nonconservative semigroups generated by partial differential operators in l2 p rk have the property that initial conditions which are large at ixl oo become immediately small at infinity for all t 0 this property is related to the rate of decay of eigenfunctions of the differential operator in this paper this phenomenon is investigated for a large class of differential operators of second and higher order new estimates on the rate of decay of the eigenfunctions are included which are related in special cases to those of agmon o introduction frequently in nonconservative abstract cauchy problems singular behavior is suppressed we deal with a phenomenon in this paper which is best described as the suppression of singularities at infinity suppose that the abstract cauchy problem is formulated in l2 p rk where p is a bounded positive measurable function such that p x approaches zero as ixl approaches infinity the phenomenon we study here is that when u 0 e l2 p rk with norm ii sii irk if x l2 p x dx 1 2 u t e l2 l p rk for all t 0 hence while u 0 snay be allowed to grow rapidly at infinity depending on p u t must decay rapidly at infinity for all t 0 that this behavior occurs at all is rather surprising in this paper it will be shown to occur frequently when the infinitesimal generator is an ordinary or partial differential operator with positive coefficients in the discrete spectrum case although we study this phenomenon as a mathematical one it is also related to such areas as heat conduction and nonconservative quantum mechanics for example in heat conduction it deals with the case where an initial temperature which is initially large at infinity becomes immediately small at infinity the physical interpretation for this is that the equation represents the temperature of a twodimensional thin slab with variable thermal conductivity surrounded by a medium at constant temperature with variable rates of conduction via newton s law of cooling to the surrounding medium under hypotheses of the theorems this conduction approaches perfect conduction at ixl x hence this instantaneous decrease in temperature might be expected nevertheless it seems reasonable that there is an initial growth rate at infinity determined by p beyond which we would not expect the phenomenon to occur a good deal of the paper is involved with the computation of p from the coefficients of the infinitesimal generator p turns out to decay very rapidly at infinity received by the editors july 16 1986 and in revised form march 11 1987 presented to the international conference on differential equations and mathematical physics birmingham alabama march 3 8 1986 sponsored by the nsf uab and ua 1980 mathematie s ks ubjeet 8la ssifie ltion 1985 revi sion primary 47f05 47d05 35p20 35k30 r 1988 americarl mathematical society 0002 9947 88 1 00 25 per page

spectral theory of left definite difference operators this paper is concerned with the spectral theory for the second order left definite difference boundary value problems existence of eigenvalues of boundary value problems is proved numbers of their eigenvalues are calculated and fundamental spectral results are obtained

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

left definite theory for second order differential operators with mixed boundary conditions abstract left definite theory of regular self adjoint operators in a sobolev space was developed a few years ago when the boundary conditions were separated the separation being necessary in order to properly define the sobolev inner product we show how this may be extended when the boundary conditions are not separated but when evaluations at both ends are mixed together there are essentially three cases which arise first when a coefficient matrix is nonsingular second when it is singular but not 0 third when it is 0 the middle case does not arise under separated conditions

sturm liouville eigenvalue problems on time scales for sturm liouville eigenvalue problems on time scales with separated boundary conditions we give an oscillation theorem and establish rayleigh s principle our results not only unifly the corresponding theories for differential and difference equations but are also new in the discrete case

a lie backlund approach to equivalence and flatness of nonlinear systems a new system equivalence relation using the framework of differential geometry of jets and prolongations of infinite order is studied in this setting two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives this is a lie backlund isomorphism the authors prove that although the state dimension is not preserved the number of input channels is kept fixed they also prove that a lie backlund isomorphism can be realized by an endogenous feedback the differentially flat nonlinear systems introduced by the authors 1992 via differential algebraic techniques are generalized and the new notion of orbitally flat systems is defined they correspond to systems which are equivalent to a trivial one with time preservation or not the endogenous linearizing feedback is explicitly computed in the case of the vtol aircraft to track given reference trajectories with stability