sl 3 r as the group of symmetry transformations for all one dimensional linear systems ii realizations of the lie algebra the two dimensional space time realizations of the lie algebra of sl 3 r are obtained when the group acts as the maximal point symmetry group of any given one dimensional newtonian linear system it is shown that these realizations are isomorphic with the realization of the lie algebra of the projective group in the plane next an active point of view is introduced and sl 3 r is interpreted as a group of mappings that transform one admissible world line of the system into another thus a new mechanical realization of the sl 3 r algebra comes to the fore some miscellaneous examples are included

new results on the synthesis of pid controllers this paper considers the problem of stabilizing a first order plant with dead time using a proportional integral derivative pid controller using a version of the hermite biehler theorem that is applicable to quasi polynomials the complete set of stabilizing pid parameters is determined for both open loop stable and unstable plants the range of admissible proportional gains is first determined in closed form for each proportional gain in this range the stabilizing set in the space of the integral and derivative gains is shown to be either a trapezoid a triangle or a quadrilateral for the case of an open loop unstable plant a necessary and sufficient condition on the time delay is determined for the existence of stabilizing pid controllers

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

boundary feedback stabilization for an unstable time fractional reaction diffusion equation in this paper we consider boundary feedback stabilization for unstable time fractional reaction diffusion equations new state feedback controls with actuation on one end are designed by the backstepping method for both dirichlet and neumann boundary controls by the riesz basis approach and the fractional lyapunov method we prove the existence and uniqueness and the mittag leffler stability for the closed loop systems for both cases the observers and the observer based output feedback are designed to stabilize the systems

restriction of fourier transforms to curves an endpoint estimate with affine arclength measure consider the fourier restriction operators associated to curves in r d 3 we prove for various classes of curves the endpoint restricted strong type estimate with respect to affine arclength measure on the curve an essential ingredient is an interpolation result for multilinear operators with symmetries acting on sequences of vector valued functions

symmetries of nonlinear differential equations and linearisation a non linear ordinary differential equation is linearisable if it possesses sl 3 r symmetry the conditions under which the abelian two dimensional subalgebras of sl 3 r are sufficient for linearisation are established

vibration control of an euler bernoulli beam under unknown spatiotemporally varying disturbance in this article the vibration suppression of an euler bernoulli beam system is considered by using the adaptive boundary control technique the dynamics of the beam are represented by a partial differential equation and four ordinary differential equations involving functions of both space and time by using lyapunov synthesis the robust boundary control with a disturbance observer is first proposed to suppress the vibration and attenuate the effect of the external disturbances to compensate for both the system parametric uncertainties and the disturbances uncertainties the adaptive boundary control is developed with the proposed boundary control the state of the beam system is proven to be uniformly ultimately bounded and converge to a small neighbourhood of zero by appropriately choosing the design parameters the effectiveness of the proposed control is successfully verified by numerical simulations

exceptional properties of second and third order ordinary differential equations of maximal symmetry the riccati transformation is used in the reduction of order of second and third order ordinary differential equations of maximal symmetry the sl 2 r subalgebra is preserved under this transformation the riccati transformation is itself associated with the symmetry that is annihilated in the reduction of order the solution symmetries and the intrinsically contact symmetries become nonlocal symmetries under the riccati transformation we investigate the fate and origins of the contact symmetries arising from the riccati transformation the exceptional properties of the second and third order equations of maximal symmetry are indicated in the context of generalised symmetries we express the solution symmetries contact symmetries and the sl 2 r elements in terms of a jacobian we show that a basis for the solution set of equations of maximal symmetry is given in terms of the solution set of a second order ordinary differential equation

finite point transformations and linearisation of x f t x the usefulness of finite point transformations is emphasised as a systematic tool for studying the eventual linearisation of ordinary second order differential equations

an equation admitting infinite true contact transformations abstract the linear wave equation is shown to possess the unique property that if w n is a true contact transformation admitted by the wave equation i e w n is not linear in the first derivatives of the dependent variable then so is n w n we comment of the physical implications

lie algebras associated with scalar second order ordinary differential equations second order ordinary differential equations are classified according to their lie algebra of point symmetries the existence of these symmetries provides a way to solve the equations or to transform them to simpler forms canonical forms of generators for equations with three point symmetries are established it is further shown that an equation cannot have exactly r 4 5 6 7 point symmetries representative s of equivalence class es of equations possessing s 1 2 3 8 point symmetry generator s are then obtained

feedback and feedforward output regulation of bounded uniformly continuous signals for infinite dimensional systems this article presents a state space output regulation theory for linear infinite dimensional systems with bounded control observation and feedthrough operators in which the reference and disturbance signals are bounded and uniformly continuous output regulation problems are posed and solved for both feedforward and error feedback controllers furthermore under certain assumptions necessary conditions are obtained for those controllers which solve these output regulation problems one consequence of these necessary conditions is a generalization of the well known internal model principle a key feature in our theory is that the exogenous system generating the reference and disturbance signals is infinite dimensional

optimal control with stochastic pde constraints and uncertain controls abstract the optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed the lagrangian that defines the problem is postulated in terms of stochastic functions with the control function possibly decomposed into an unknown deterministic component and a known zero mean stochastic component the extra freedom provided by the stochastic dimension in defining cost functionals is explored demonstrating the scope for controlling statistical aspects of the system response one shot stochastic finite element methods are used to find approximate solutions to control problems it is shown that applying the stochastic collocation finite element method to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional thereby forgoing the primary advantage of the collocation method over the stochastic galerkin method for the considered problem the application of the presented methods is demonstrated through a number of numerical examples the presented framework is sufficiently general to also consider a class of inverse problems and numerical examples of this type are also presented

uniform estimates for the local restriction of the fourier transform to curves we prove sharp estimates with respect to the affine arclength measure for the restriction of the fourier transform to a class of curves in r that includes curves of finite type this measure possesses certain invariance and mitigation properties which are important in establishing uniform results

uniform estimates for fourier restriction to polynomial curves in ℝ d we prove uniform l p to l q bounds for fourier restriction to polynomial curves in bbb r d with affine arclength measure in the conjectured range

well posedness and regularity of naghdi s shell equation under boundary control and observation abstract a system of naghdi s shell equation with dirichlet boundary control and collocated observation is considered results on the associated nonhomogeneous boundary value problem are presented based on these results it is shown that the system is well posed in the sense of d salamon and regular in the sense of g weiss the expression of the corresponding feedthrough operator is explicitly found by means of riemannian geometric method and partial fourier transform these properties make this partial differential control system parallel in many ways finite dimensional ones in the general framework of well posed and regular infinite dimensional systems

brief robust control of parabolic pde systems with time dependent spatial domains we synthesize robust nonlinear static output feedback controllers for systems of quasi linear parabolic partial differential equations with time dependent spatial domains and uncertain variables the controllers are successfully applied to a typical diffusion reaction process with moving boundary and uncertainty

tracking control of the uncertain heat and wave equation via power fractional and sliding mode techniques in the present paper preliminary results towards the generalization to the infinite dimensional setting of some well known robust finite dimensional control algorithms are illustrated more specifically we deal with the tracking problem for some classes of linear uncertain infinite dimensional systems evolving in hilbert spaces we design distributed variable structure stabilizers that are shown to be effective in the presence of external disturbances the main focus of the present paper is on the rejection of nonvanishing external disturbances the generalization to the infinite dimensional setting of the well known finite dimensional controllers namely the power fractional controller s p bhat and d s bernstein siam j control optim 38 2000 pp 751 766 and two second order sliding mode control algorithms the twisting and supertwisting algorithms l fridman and a levant higher order sliding modes as a natural phenomenon in control theory in robust control via variable structure and lyapunov techniques springer verlag berlin 1996 pp 107 133 a levant internat j control 58 1993 pp 1247 1263 is the main contribution of the present investigation first the distributed twisting control algorithm is developed to address the asymptotic state tracking of the perturbed wave equation next the finite time state tracking of the unperturbed heat equation is provided by means of a distributed power fractional controller finally the distributed supertwisting controller is suggested to address the asymptotic state tracking of the heat equation in spite of the presence of persistent disturbances constructive proofs of stability are developed via the lyapunov functional technique which leads to simple tuning rules for the controller parameters simulation results are discussed to verify the effectiveness of the proposed schemes

sl 3 r as the group of symmetry transformations for all one dimensional linear systems the converse problem of similarity analysis is solved in general for the finite symmetry transformations of any inhomogeneous ordinary linear differential equation of the second order x f2 t x f1 t x f0 t the eight parameter realizations of the symmetry group are obtained in the form f 1p2 f where f stands for transformations of t x that depend exclusively on the fundamental solutions of the equation and where p2 is an arbitrary projective transformation in the plane thus it is shown that the full point symmetry group corresponds to sl 3 r indeed without recourse to the lie algebra also a technique is obtained for calculating the finite point symmetry realization of sl 3 r for any given one dimensional linear system some miscellaneous examples are given

adaptive boundary control for unstable parabolic pdes part i lyapunov design we develop adaptive controllers for parabolic partial differential equations pdes controlled from a boundary and containing unknown destabilizing parameters affecting the interior of the domain these are the first adaptive controllers for unstable pdes without relative degree limitations open loop stability assumptions or domain wide actuation it is the first necessary step towards developing adaptive controllers for physical systems such as fluid thermal and chemical dynamics where actuation can be only applied non intrusively the dynamics are unstable and the parameters such as the reynolds rayleigh prandtl or peclet numbers are unknown because they vary with operating conditions our method builds upon our explicitly parametrized control formulae to avoid solving riccati or bezout equations at each time step most of the designs we present are state feedback but we present two benchmark designs with output feedback which have infinite relative degree

universal l p improving for averages along polynomial curves in low dimensions we prove sharp lp lq estimates for averaging operators along general polynomial curves in two and three dimensions these operators are translation invariant given by convolution with the so called affine arclength measure of the curve and we obtain universal bounds over the class of curves given by polynomials of bounded degree our method relies on a geometric inequality for general vector polynomials together with a combinatorial argument due to m christ almost sharp lorentz space estimates are obtained as well

contact symmetries of the harmonic oscillator the symmetry group of contact transformations for the one dimensional harmonic oscillator is determined its generators contain two arbitrary functions i e it is an infinite parameter lie group

on convergence of the nonlinear active disturbance rejection control for mimo systems in this paper the global and semiglobal convergence of the nonlinear active distur bance rejection control adrc for a class of multi input multi output nonlinear systems with large uncertainty that comes from both dynamical modeling and external disturbance are proved as a result a class of linear systems with external disturbance that can be dealt with by the adrc is classified the adrc is then compared both analytically and numerically to the well known in ternal model principle a number of illustrative examples are presented to show the efficiency and advantage of the adrc in dealing with unknown dynamics and in achieving fast tracking with lower overstriking

restriction of the fourier transform to some complex curves abstract the purpose of this paper is to prove a fourier restriction estimate for certain 2 dimensional surfaces in r 2 d d 3 these surfaces are defined by a complex curve z of simple type which is given by a mapping of the form z z z z 2 z d 1 z where z is an analytic function on a domain c this is regarded as a real mapping z x y x y from r 2 to r 2 d our results cover the case z z n for any nonnegative integer n in all dimensions d 3 the main result is a uniform estimate valid when d 3 where z may be taken to be an arbitrary polynomial of degree at most n it is uniform in the sense that the operator norm is independent of the coefficients of the polynomial these results are analogues of the uniform restricted strong type estimates in 5 valid for polynomial curves of simple type and some other classes of curves in r d d 3

parameter estimation and non collocated adaptive stabilization for a wave equation subject to general boundary harmonic disturbance this paper is concerned with the parameter estimation and asymptotic stabilization of a 1 d wave equation that is subject to general harmonic disturbances at the controlled end and suffers from instability at the other end first we design an adaptive observer in terms of measured position and velocity we then adopt the backstepping method for infinite dimensional systems to design an observer based output feedback law the resulting closed loop system is shown to be asymptotically stable and the estimates of the parameters converge to the unknown parameters

error bounds in differentiation of noisy signals by high gain observers the error in estimating the derivative s of a noisy signal by using a high gain observer is studied and quantified the error is bounded in terms of the infinity norms of the noise and a derivative of the signal the error bound is independent of the frequency or derivatives of the noise guidelines are presented for the observer gain design when it is used for on line differentiation analytical and simulation results are presented

on the restriction of the fourier transform to curves endpoint results and the degenerate case for smooth curves f in rn with certain curvature properties it is shown that the composition of the fourier transform in rn followed by restriction to r defines a bounded operator from lp rn to lq f for certain p q the curvature hypotheses are the weakest under which this could hold and p is optimal for a range of q in the proofs the problem is reduced to the estimation of certain multilinear operators generalizing fractional integrals and they are treated by means of rearrangement inequalities and interpolation between simple endpoint estimates

active disturbance rejection control for rejecting boundary disturbance from multidimensional kirchhoff plate via boundary control in this paper an algorithm is developed to reject time and spatially varying boundary disturbances from a multidimensional kirchhoff plate via boundary control the disturbance and control input are assumed to be matched the active disturbance rejection control approach is adopted for developing the algorithm a state feedback scheme is designed to estimate the disturbance based on an infinite number of ordinary differential equations obtained from the original multidimensional system using infinitely many time dependent test functions the proposed control law cancels the disturbance using its estimated value all subsystems in the closed loop are shown to be asymptotically stable simulation results are presented to validate the theoretical conclusions and to exhibit the reduction in the peaking phenomenon due to the use of time varying gains instead of constant high gains

fourier restriction to polynomial curves i a geometric inequality we prove a fourier restriction result for general polynomial curves in bbb r d measuring the fourier restriction with respect to the affine arclength measure of the curve we obtain a universal estimate for the class of all polynomial curves of bounded degree our method relies on establishing a geometric inequality for general polynomial curves which is of interest in its own right applications of this geometric inequality to other problems in euclidean harmonic analysis have recently been established

uniform estimates for the x ray transform restricted to polynomial curves abstract we establish near optimal mixed norm estimates for the x ray transform restricted to polynomial curves with a weight that is a power of the affine arclength the bounds that we establish depend only on the spatial dimension and the degree of the polynomial some of our results are new even in the well curved case

dynamics of transversely vibrating beams using four engineering theories abstract in this paper the full development and analysis of four models for the transversely vibrating uniform beam are presented the four theories are the euler bernoulli rayleigh shear and timoshenko first a brief history of the development of each beam model is presented second the equation of motion for each model and the expressions for boundary conditions are obtained using hamilton s variational principle third the frequency equations are obtained for four sets of end conditions free free clamped clamped hinged hinged and clamped free the roots of the frequency equations are presented in terms of normalized wave numbers the normalized wave numbers for the other six sets of end conditions are obtained using the analysis of symmetric and antisymmetric modes fourth the orthogonality conditions of the eigenfunctions or mode shape and the procedure to obtain the forced response using the method of eigenfunction expansion is presented finally a numerical example is shown for a non slender beam to signify the differences among the four beam models

brief paper sliding mode boundary control of a parabolic pde system with parameter variations and boundary uncertainties this paper considers the stabilization problem of a one dimensional unstable heat conduction system rod modeled by a parabolic partial differential equation pde powered with a dirichlet type actuator from one of the boundaries by applying the volterra integral transformation a stabilizing boundary control law is obtained to achieve exponential stability in the ideal situation when there are no system uncertainties the associated lyapunov function is used for designing an infinite dimensional sliding manifold on which the system exhibits the same type of stability and robustness against certain types of parameter variations and boundary disturbances it is observed that the relative degree of the chosen sliding function with respect to the boundary control input is zero a continuous control law satisfying the reaching condition is obtained by passing a discontinuous signum signal through an integrator

the active disturbance rejection and sliding mode control approach to the stabilization of the euler bernoulli beam equation with boundary input disturbance in this paper we are concerned with the boundary feedback stabilization of a one dimensional euler bernoulli beam equation with the external disturbance flowing to the control end the active disturbance rejection control adrc and sliding mode control smc are adopted in investigation by the adrc approach the disturbance is estimated through an extended state observer and canceled online by the approximated one in the closed loop it is shown that the external disturbance can be attenuated in the sense that the resulting closed loop system under the extended state feedback tends to any arbitrary given vicinity of zero as the time goes to infinity in the second part we use the smc to reject the disturbance by removing the condition in adrc that the derivative of the disturbance is supposed to be bounded the existence and uniqueness of the solution for the closed loop via smc are proved and the monotonicity of the reaching condition is presented without the differentiation of the sliding mode function for which it may not always exist for the weak solution of the closed loop system the numerical simulations validate the effectiveness of both methods

contact symmetry algebras of scalar second order ordinary differential equations there is a natural link between contact symmetries symmetries that leave invariant the first order contact form and first integrals of a given second order ordinary differential equation it will be shown that the contact symmetry algebra of a general second order ordinary differential equation is infinite dimensional and it is generated by the functionally independent first integrals of the equation moreover the contact symmetry algebras of second order equations admitting one two three and eight point symmetries are obtained

on the convergence of an extended state observer for nonlinear systems with uncertainty abstract the extended state observer first proposed by jingqing han in j q han a class of extended state observers for uncertain systems control decis 10 1 1995 85 88 in chinese is the key link toward the active disturbance rejection control that is taking off as a technology after numerous successful applications in engineering unfortunately there is no rigorous proof of convergence to date in this paper we attempt to tackle this long unsolved extraordinary problem the main idea is to transform the error equation of objective system with its extended state observer into a asymptotical stable system with a small disturbance for which the effect of total disturbance error is eliminated by the high gain

on the reduction methods for ordinary differential equations as tandard method based on the use of differential invariants of a lie group g e nables us to reduce any ordinary differential equation invariant under the action of g w e show that this method is applicable to vector fields more general than those associated with lie symmetries we characterize all such vector fields and study their relationship with nonlocal symmetries and symmetries govinder k s and leach p g l 1995 j phys a math gen 28 5349 59 muriel c and romero l 2001 ima j appl math 66 111 25

restriction of fourier transforms to curves and related oscillatory integrals we prove sharp endpoint results for the fourier restriction operator associated to nondegenerate curves in bbb r d d ge 3 and related estimates for oscillatory integral operators moreover for some larger classes of curves in bbb r d we obtain sharp uniform l p to l q bounds with respect to affine arclength measure thereby resolving a problem of drury and marshall

generating functions and existence of contact symmetries of third order scalar ordinary differential equations analyzing the structure of generating functions associated to the contact symmetry condition of third order scalar ordinary differential equations constraints guaranteeing that the ode does not admit contact symmetries are obtained further inspecting the subalgebras generated by contact symmetries having a specific form an alternative procedure to generate third order equations with irreducible contact symmetry algebra sp 4 is developed

internal model based tracking and disturbance rejection for stable well posed systems in this paper we solve the tracking and disturbance rejection problem for infinite dimensional linear systems with reference and disturbance signals that are finite superpositions of sinusoids we explore two approaches both based on the internal model principle in the first approach we use a low gain controller and here our results are a partial extension of results by hamalainen and pohjolainen in their papers the plant is required to have an exponentially stable transfer function in the callier desoer algebra while in this paper we only require the plant to be well posed and exponentially stable these conditions are sufficiently unrestrictive to be verifiable for many partial differential equations in more than one space variable our second approach concerns the case when the second component of the plant transfer function from control input to tracking error is positive in this case we identify a very simple stabilizing controller which is again an internal model but which does not require low gain we apply our results to two problems involving systems modeled by partial differential equations the problem of rejecting external noise in a model for structure acoustics interactions and a similar problem for two coupled beams

restriction estimates for space curves with respect to general measures in this paper we consider adjoint restriction estimates for space curves with respect to general measures and obtain optimal estimates when the curves satisfy a finite type condition the argument here is new in that it doesn t rely on the offspring curve method which has been extensively used in the previous works our work was inspired by the recent argument due to bourgain and guth which was used to deduce linear restriction estimates from multilinear estimates for hypersurfaces

boundary second order sliding mode control of an uncertain heat process with unbounded matched perturbation the primary concern of the present paper is the regulation of an uncertain heat process with collocated boundary sensing and actuation the underlying heat process is governed by an uncertain parabolic partial differential equation pde with neumann boundary conditions it exhibits an unknown constant diffusivity parameter and it is affected by a smooth boundary disturbance which is not available for measurements and which is possibly unbounded in magnitude the proposed robust synthesis is formed by the linear feedback design and by the twisting second order sliding mode control algorithm suitably combined and re worked in the infinite dimensional setting a non standard lyapunov functional is invoked to establish the global asymptotic stability in a sobolev space involving spatial state derivatives of the same order as that of the plant equation the stability proof is accompanied by a set of simple tuning rules for the controller parameters the effectiveness of the developed control scheme is supported by simulation results

endpoint lp lq bounds for integration along certain polynomial curves we establish strong type endpoint lp rd lq rd bounds for the operator given by convolution with affine arclength measure on polynomial curves for d 4 the bounds established depend only on the dimension d and the degree of the polynomial

first integrals and reduction of a class of nonlinear higher order ordinary differential equations abstract we propose a method for constructing first integrals of higher order ordinary differential equations in particular third fourth and fifth order equations of the form x n h x x n 1 x are considered the relation of the proposed method to local and nonlocal symmetries are discussed

approximate tracking and disturbance rejection for stable infinite dimensional systems using sampled data low gain control in this paper we solve tracking and disturbance rejection problems for stable infinite dimensional systems using a simple low gain controller suggested by the internal model principle for stable discrete time systems it is shown that the application of a low gain controller depending on only one gain parameter leads to a stable closed loop system which asymptotically tracks reference signals r of the form r k sum j 1 n lambda j k text mathfrakr r j where text mathfrakr r j in mathbb c p and lambda j in mathbb c with lambda j 1 for j 1 ldots n the closed loop system also rejects disturbance signals which are asymptotically of this form the discrete time result is used to derive results on approximate tracking and disturbance rejection for a large class of infinite dimensional sampled data feedback systems with reference signals which are finite sums of sinusoids and disturbance signals which are asymptotic to finite sums of sinusoids the results are given for both input output systems and state space systems

a note on the construction of the ermakov lewis invariant we use noether symmetries to discuss the source of the ermakov lewis invariant and show that it comes from a generalized symmetry this symmetry is a lie symmetry of the invariant furthermore we use liouville s theorem to construct two integrals of the ermakov system and compute the poisson brackets of all the integrals of the system to see which ones are in involution with each other interesting observations are made

maximal estimates for the bilinear spherical averages and the bilinear bochner riesz operators abstract we study the maximal estimates for the bilinear spherical average and the bilinear bochner riesz operator first we obtain l p l q l r estimates for the bilinear spherical maximal function on the optimal range thus we settle the problem which was previously considered by geba greenleaf iosevich palsson and sawyer later barrionevo grafakos d he honzik and oliveira and recently heo hong and yang secondly we consider l p l q l r estimates for the maximal bilinear bochner riesz operators and improve the previous known results for the purpose we draw a connection between the maximal estimates and the square function estimates for the classical bochner riesz operators