conditions for the equivalence of internal and external stability for distributed parameter systems conditions are given on a controlled observed distributed parameter system under which external stability also known as input output stability implies internal stability which is the exponential stability of the underlying semigroup generator it is shown that when the system satisfies a general definition of stabilizability and detectability external stability is equivalent to internal stability unlike the case previous definitions of stabilizability and detectability a system which satisfies these does not necessarily have a spectrum decomposition therefore in order to check internal stability it is often sufficient to check the boundedness of the transfer function in the right half complex plane this is illustrated with a simple example

on stabilization and the existence of coprime factorizations shows that any transfer function matrix whose elements belong to the quotient field of h sub infinity and which is stabilizable has a matrix fraction representation over h sub infinity which is coprime in the sense that a matrix bezout identity can be satisfied

luenberger boundary observer synthesis for sturm liouville systems a static luenberger observer of a system with sturm liouville operator is synthesised with the aid of a boundary control formulation to this aim approximate observability detectability and stability of the system is studied and design results are worked out for a typical biochemical case study

non differentiable variational principles abstract we develop a calculus of variations for functionals which are defined on a set of non differentiable curves we first extend the classical differential calculus in a quantum calculus which allows us to define a complex operator called the scale derivative which is the non differentiable analogue of the classical derivative we then define the notion of extremals for our functionals and obtain a characterization in term of a generalized euler lagrange equation we finally prove that solutions of the schrodinger equation can be obtained as extremals of a non differentiable variational principle leading to an extended hamilton s principle of least action for quantum mechanics we compare this approach with the scale relativity theory of nottale which assumes a fractal structure of space time

admissibility of unbounded control operators for linear systems described by dot x t ax t bu t where a generates a semigroup on the state space x and b is an unbounded operator some necessary as well as some sufficient conditions are given for b to be admissible i e for any t the state x t should be in x and should depend continuously on the input u in l p this approach begins with an axiomatic description of such a system in terms of a functional equation the results are applied to the wave equation on a bounded interval

a connection between state space and doubly coprime fractional representations explicit formulas for a doubly coprime fractional representation of the transfer function of a lumped lti system are given in terms of a stabilizable and detectable state space realization of the transfer function these formulas allow existing computational algorithms to be utilized for the purpose of computing doubly coprime factorizations several additional implications of this result are briefly discussed as are some immediate extensions and future research directions

finite element approximations of compensator design for analytic generators with fully unbounded controls observations an approximation theory leading to a design of a finite dimensional compensator for control systems generated by analytic semigroups is presented the novelty of this paper with respect to other results available in the literature is threefold i it treats fully unbounded control observation operators ii it does not require compactness property of the underlined generator an assumption that is often violated in practice and iii the design of a finite dimensional compensator is based on finite element approximation of the original model rather than on modal eigenfunctions approximations which in turn require the a priori knowledge of the eigenvalues for the system applications of the theory to heat equations and plate equations are provided

dynamic stabilization of an euler bernoulli beam under boundary control and non collocated observation abstract we study the dynamic stabilization of an euler bernoulli beam system using boundary force control at the free end and bending strain observation at the clamped end we construct an infinite dimensional observer to track the state exponentially a proportional output feedback control based on the estimated state is designed the closed loop system is shown to be non dissipative but admits a set of generalized eigenfunctions which forms a riesz basis for the state space as consequences both the spectrum determined growth condition and exponential stability are concluded

fractional differential equations and the schrodinger equation in a previous paper we defined following a previous work of kolvankar and gangal a notion of a derivative 0 a 1 in this article we study a differential equations associated to our fractional calculus we then discuss a fundamental problem concerning the schrodinger equation in the framework of nottale s scale relativity theory

approximation of solutions to fractional integral equation in this paper we shall study a fractional integral equation in an arbitrary banach space x we used the analytic semigroups theory of linear operators and the fixed point method to establish the existence and uniqueness of solutions of the given problem we also prove the existence of global solution the existence and convergence of the faedo galerkin solution to the given problem is also proved in a separable hilbert space with some additional assumptions on the operator a finally we give an example to illustrate the applications of the abstract results

the regularity of the wave equation with partial dirichlet control and colocated observation in this paper we analyze a multidimensional controlled wave equation on a bounded domain subject to partial dirichlet control and colocated observation by means of a partial fourier transform it is shown that the system is well posed and regular in the sense of d salamon and g weiss the corresponding feedthrough operator is found to be the identity operator on the input space

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

dynamic stabilization of regular linear systems we consider a general class of infinite dimensional linear systems called regular linear systems for which convenient representations are known to exist both in time and in frequency domain for this class of systems we investigate the concepts of stabilizability and detectability in particular their invariance under feedback and their relationship to exponential stability we introduce two concepts of dynamic stabilization the first formulated as usual with the plant and the controller connected in feedback and the second with two feedback loops even for finite dimensional systems the second concept stabilization with an internal loop in the controller is more general we argue that the more general concept is the natural one and we derive sufficient conditions under which an observer based stabilizing controller with an internal loop can be constructed

output feedback stabilization of an unstable wave equation we consider the problem of stabilization of a one dimensional wave equation that contains instability at its free end and control on the opposite end in contrast to classical collocated boundary damper feedbacks for the neutrally stable wave equations with one end satisfying a homogeneous boundary condition the controllers and the associated observers designed in the paper are more complex due to the open loop instability of the plant the controller and observer gains are designed using the method of backstepping which results in explicit formulae for the gain functions we prove exponential stability and the existence and uniqueness of classical solutions for the closed loop system we also derive the explicit compensators in frequency domain the results are illustrated with simulations

maximum principle for optimal control of non well posed elliptic differential equations in this paper we shall study two optimal control problems p1 and p2 governed by some semilinear elliptic di erential equations which can admit more than one solution we shall call such systems non well posed systems we obtain the pontryagin maximum principle for problems p1 and p2 in the 6rst problem p1 the cost functional may be non smooth and the set of controls is convex while in the second problem p2 the cost functional is smooth and the set of controls is non convex in both problem p1 and problem p2 we consider some state constraint of integral type due to some practical interests many authors studied optimal control for elliptic di erential equations here we cite 1 10 12 15 however most of these works except for 7 10 13 deal with state equations which are governed by monotone operators and admit a unique solution corresponding to each control in such problems the pontryagin maximum principle was obtained by studying the variations of the state with respect to some perturbations of the control cf 1 2 12 15 in the present work the state equations are governed by non monotone operators and can admit more than one solution we cannot obtain the variations of the state with respect to the control with the same methods in 1 2 12 15 when the state equations admit more than one solution we even do not know how to make the sensitivity analysis of the state with respect to the control

robust stabilizability of normalized coprime factors the infinite dimensional case the problem of robustly stabilizing a linear system subject to h bounded perturbations in the numerator and the denominator of its normalized left coprime factorization is considered for a class of infinite dimensional systems this class has possible unbounded finite rank input and output operators which include many delay and distributed systems the optimal stability margin is expressed in terms of the solutions of the control and filter algebraic riccati equations

finite dimensional compensators for infinite dimensional systems with unbounded input operators this paper contains a design procedure for constructing finite dimensional compensators for a class of infinite dimensional systems with unbounded input operators applications to retarded functional differential systems with delays in the input or the output variable and to partial differential equations with boundary input operators are discussed

a direct approach to compensator design for distributed parameter systems we present a direct approach to finite order compensator design for distributed parameter systems i e one that is not based on reduced order modelling instead we use a parametrization around an initial compensator which displays both controller order and closed loop stability in a convenient way the main result is an existence theorem which holds for a wide class of linear time invariant systems parabolic delay damped hyperbolic the most important assumptions are bounded inputs and outputs finitely many unstable modes completeness of eigenvectors an example is included to illustrate the feasibility of our method for purposes of design

about non differentiable functions we develop the notion of local fractional derivative introduced by kolvankar and gangal it allows a fine study of the local structure of irregular fractal functions using this tool we extend classical theorems of analysis extrema rolle to non differentiable functions in particular we prove a generalized taylor expansion theorem we introduce a new derivative of real order and discuss its properties

exponential stability of coupled beams with dissipative joints a frequency domain approach two examples of coupled euler bernoulli beams with a dissipative joint are considered the joint placements that lead to exponential stability for these systems are characterized the technique used shows input output stability of a related controlled observed system and then shows that in these examples input output stability implies exponential stability in the first example the energy dissipation arises from a discontinuity in the shear at the joint in the second example the energy dissipation arises from a discontinuity in the bending moment at the joint the analysis of this system involves a complete spectrum analysis of the zero dynamics of the associated controlled observed system

transfer functions of regular linear systems part i characterizations of regularity we recall the main facts about the representation of regular linear systems essentially that they can be described by equations of the form x t ax t bu t y t cx t du t like finite dimensional systems but now a b and c are in general unbounded operators regular linear systems are a subclass of abstract linear systems we define transfer functions of abstract linear systems via a generalization of a theorem of foures and segal we prove a formula for the transfer function of a regular linear system which is similar to the formula in finite dimensions the main result is a simple to state but hard to prove necessary and sufficient condition for an abstract linear system to be regular in terms of its transfer function other conditions equivalent to regularity are also obtained the main result is a consequence of a new tauberian theorem which is of independent interest

stability of control systems and graphs of linear systems new conditions for internal stability of a closed loop control system are given in terms of the graphs of the multiplication operators induced by the transfer functions of the plant and the controller these conditions can be given a geometrical interpretation this relates closed loop stability to the minimal angle between the graph space associated with the system and the graph space associated with the controller the maximally stabilizing controller is defined as the controller that maximizes the minimum angle between the graph space associated with the system and the graph space associated with the controller it is shown that this controller can be calculated as a nehari extension of the coprime factors of the system

coprime factorization for regular linear systems mild sufficient conditions are given for the existence of a doubly coprime factorization of the transfer function of a regular linear system as well as formulae for such a factorization the results are illustrated by two examples of delay systems one of which has infinitely many unstable poles

the operator carleson measure criterion for admissibility of control operators for diagonal semigroups on 1 2 abstract suppose a control system is described by a diagonal semigroup on the state space l 2 and by an unbounded control operator b defined on the input space l 2 we formulate a condition called the operator carleson measure criterion and show that this condition is necessary for the admissibility of b if the semigroup is analytic or invertible then the condition is sufficient as well our results extend the ones of ho russell and weiss concerning the case where the input space is one dimensional

convergence analysis of space time jacobi spectral collocation method for solving time fractional schrodinger equations abstract in this paper the space time jacobi spectral collocation method jsc method is used to solve the time fractional nonlinear schr o dinger equations subject to the appropriate initial and boundary conditions at first the considered problem is transformed into the associated system of nonlinear volterra integro partial differential equations pdes with weakly singular kernels by the definition and related properties of fractional derivative and integral operators therefore by collocating the associated system of integro pdes in both of the space and time variables together with approximating the existing integral in the equation using the jacobi gauss type quadrature formula then the problem is reduced to a set of nonlinear algebraic equations we can consider solving the system by some robust iterative solvers in order to support the convergence of the proposed method we provided some numerical examples and calculated their l norm and weighted l2 norm at the end of the article

robustly stabilizing controllers for dissipative infinite dimensional systems with collocated actuators and sensors we derive formulas for normalized doubly coprime factorizations over h for a class of nonexponentially stabilizable systems using these results we solve the problem of robust stabilization with respect to normalized coprime factor perturbations for a class of infinite dimensional systems s a b b d with finite rank inputs and outputs and dissipative operator a these systems are not exponentially stabilizable we give a parameterization of controllers that achieve a given robustness margin

fractional embedding of differential operators and lagrangian systems this paper is a contribution to the general program of embedding theories of dynamical systems following our previous work on the stochastic embedding theory developed with darses c r acad sci ser i math 342 333 2006 preprint ihes 06 27 p 87 2006 we define the fractional embedding of differential operators and ordinary differential equations we construct an operator combining in a symmetric way the left and right riemann liouville fractional derivatives for lagrangian systems our method provides a fractional euler lagrange equation we prove developing the corresponding fractional calculus of variations that such equation can be derived via a fractional least action principle we then obtain naturally a fractional noether theorem and a fractional hamiltonian formulation of fractional lagrangian systems all these constructions are coherents i e the embedding procedure is compatible with the fractional calculus of variations we then extend our results to cover the ostrogradski

lack of time delay robustness for stabilization of a structural acoustics model in this paper we consider a natural robustness question for a model for structural acoustics this model which has been of great interest in recent years is represented by a wave equation in r 2 coupled to a kelvin voigt beam the coupling is natural physically and is represented mathematically by highly unbounded operators we assume that the observation consists of point evaluation of the beam position the beam velocity and the wave velocity we are interested in the effect of arbitrarily small delays in the feedback loop on a controller that uses these observations we show that it is not possible to construct a dynamic stabilizer of a very general form including static feedback such that the stabilization is robust with respect to delays in the feedback loop in order to do this we need to carefully analyze the input to output map finally we relate these results to already existing numerical results obtained for a galerkin approximation of the system

scale calculus and the schrodinger equation this paper is twofold in a first part we extend the classical differential calculus to continuous nondifferentiable functions by developing the notion of scale calculus the scale calculus is based on a new approach of continuous nondifferentiable functions by constructing a one parameter family of differentiable functions f t e such that f t e f t when e goes to zero this led to several new notions as representations fractal functions and e differentiability the basic objects of the scale calculus are left and right quantum operators and the scale operator which generalizes the classical derivative we then discuss some algebraic properties of these operators we define a natural bialgebra called quantum bialgebra associated with them finally we discuss a convenient geometric object associated with our study in a second part we define a first quantization procedure of classical mechanics following the scale relativity theory developed by nottale we obtain a nonlinear schrodinger equation v

well posedness and exact controllability of fourth order schrodinger equation with boundary control and collocated observation in this paper we study the well posedness and exact controllability of a system described by a fourth order schrodinger equation on a bounded domain of mathbb r n n geqslant 2 with boundary control and collocated observation the neumann boundary control problem is first discussed it is shown that the system is well posed in the sense of d salamon this implies the exponential stability of the closed loop system under proportional output feedback control the well posedness result is then generalized to the dirichlet boundary control problem in particular in order to conclude feedback stabilization from well posedness we discuss the exact controllability with the dirichlet boundary control which is similar to the neumann boundary control case in addition we show that both systems are regular in the sense of g weiss and their feedthrough operators are zero

quadratic optimal control of stable well posed linear systems we consider the infinite horizon quadratic cost minimization problem for a stable time invariant well posed linear system in the sense of salamon and weiss and show that it can be reduced to a spectral factorization problem in the control space more precisely we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution if both the system and the spectral factor are regular then the feedback operator can be expressed in terms of the riccati operator and the riccati operator is a positive self adjoint solution of an algebraic riccati equation this riccati equation is similar to the usual algebraic riccati equation but one of its coefficients varies depending on the subspace in which the equation is posed similar results are true for unstable systems as we have proved elsewhere

exact controllability of a class of nonlinear distributed parameter systems using back and forth iterations we investigate the exact controllability of a nonlinear plant described by the equation x t ax t bu t bnn x t t where t 0 here a is the infinitesimal generator of a strongly cont

weak lp stability of a linear semigroup on a hilbert space implies exponential stability george weiss department of theoretical mathematics the weizmann institute rehovot 76100 israel received september 24 1987 we prove that a strongly continuous semigroup of linear operators on a hilbert space is weakly p stable for some pe cl co if and only if the semigroup is exponentially stable as an application we prove that the cauchy problems associated with the semigroup are well posed on the infinite time interval co 0 if and only if the semigroup is exponentially stable

a fully spectral collocation approximation for multi dimensional fractional schrodinger equations a shifted legendre collocation method in two consecutive steps is developed and analyzed to numerically solve one and two dimensional time fractional schrodinger equations tfses subject to initial boundary and non local conditions the first step depends mainly on shifted legendre gauss lobatto collocation sl gl c method for spatial discretization an expansion in a series of shifted legendre polynomials for the approximate solution and its spatial derivatives occurring in the tfse is investigated in addition the legendre gauss lobatto quadrature rule is established to treat the nonlocal conservation conditions thereby the expansion coefficients are then determined by reducing the tfse with its nonlocal conditions to a system of fractional differential equations sfdes for these coefficients the second step is to propose a shifted legendre gauss radau collocation sl gr c scheme for temporal discretization to reduce such system into a system of algebraic equations which is far easier to be solved the proposed collocation scheme both in temporal and spatial discretizations is successfully extended to solve the two dimensional tfse numerical results are carried out to confirm the spectral accuracy and efficiency of the proposed algorithms by selecting relatively limited legendre gauss lobatto and gauss radau collocation nodes we are able to get very accurate approximations demonstrating the utility and high accuracy of the new approach over other numerical methods

infinite dimensional linear systems with unbounded control and observation a functional analytic approach abstract the object of these notes is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators on the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem the implications of the theory for large classes of functional and partial differential equations are discussed in detail keywords representation of infinite dimensional systems semigroups boundary control feedback linear quadratic control

technical communique an exponential stability result for the wave equation we consider a system described by the one dimensional linear wave equation in a bounded domain with appropriate boundary conditions to stabilize this system we propose a dynamic boundary controller applied at the free end of the system the transfer function of the proposed controller is a proper rational function which consists of a strictly positive real function and some poles on the imaginary axis we then show that under some conditions the closed loop system is exponentially stable

low gain control of uncertain regular linear systems it is well known that closing the loop around an exponentially stable finite dimensional linear time invariant plant with square transfer function matrix bg s compensated by a controller of the form k s gamma 0 where k in bbb r and gamma 0 in bbb r mm will result in an exponentially stable closed loop system which achieves tracking of arbitrary constant reference signals provided that i all the eigenvalues of bg 0 gamma 0 have positive real parts and ii the gain parameter k is positive and sufficiently small in this paper we consider a rather general class of infinite dimensional linear systems called regular systems for which convenient representations are known to exist both in time and in frequency domain the purpose of the paper is twofold i we extend the above result to the class of exponentially stable regular systems and ii we show how the parameters k and gamma 0 can be tuned adaptively the resulting adaptive tracking controllers are not based on system identification or parameter estimation algorithms nor is the injection of probing signals required