extension of the berlekamp massey algorithm to n dimensions abstract we present an algorithm for finding a minimal set of linear recurring relations which are valid for a given n dimensional array over any field where the minimality is defined with respect to the partial order over the n dimensional lattice the algorithm is an extension of our two dimensional version of the berlekamp massey algorithm to more than two dimensions the n dimensional theory is based on more general concepts which can be reduced into those of the two dimensional theory in the previous paper in a typical case the resulting set of polynomials characterizing the minimal linear recurring relations proves to be a groebner basis of the ideal defined by the array and consequently the structure of an n dimensional linear feedback shift register with the minimum number of storage devices which can generate the array is determined by it

exact boundary controllability of vibrating non classical euler bernoulli micro scale beams abstract this study investigates the exact controllability problem for a vibrating non classical euler bernoulli micro beam whose governing partial differential equation pde of motion is derived based on the non classical continuum mechanics in this paper it is proved that via boundary controls it is possible to obtain exact controllability which consists of driving the vibrating system to rest in finite time this control objective is achieved based on the pde model of the system which causes that spillover instabilities do not occur

decoding binary 2 d cyclic codes by the 2 d berlekamp massey algorithm a method of decoding two dimensional 2 d cyclic codes by applying the 2 d berlekamp massey algorithm is proposed to explain this decoding method the author introduces a subclass of 2 d cyclic codes which are called 2 d bch codes due to their similarity with bch codes it is shown that there are some short 2 d cyclic codes with a better cost parameter value the merit of the approach is verified by showing several simple examples of 2 d cyclic codes

controllability cost of conservative systems resolvent condition and transmutation abstract this article concerns the exact controllability of unitary groups on hilbert spaces with unbounded control operator it provides a necessary and sufficient condition not involving time which blends a resolvent estimate and an observability inequality by the transmutation of controls in some time l for the corresponding second order conservative system it is proved that the cost of controls in time t for the unitary group grows at most like exp l 2 t as t tends to 0 in the application to the cost of fast controls for the schrodinger equation l is the length of the longest ray of geometric optics which does not intersect the control region this article also provides observability resolvent estimates implying fast smoothing effect controllability at low cost and underscores that the controllability cost of a system is not changed by taking its tensor product with a conservative system

infinite dimensional luenberger like observers for a rotating body beam system the paper proposes luenberger like observers for a rotating body beam system the latter is described by a partial differential equation and its dynamic is governed a unitary group the observers proposed in the paper are able to reconstitute the dynamic evolution of the beam profile by measuring the moment force on the boundary only meanwhile we propose a reliable numerical scheme based on a finite element method in order to simulate the observation system and the observers the efficiency of our proposed scheme and the performances of the observers are illustrated with numerical simulation results

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

finding recursions for multidimensional arrays we produce a concise description of sakata s algorithm for finding recursion relations that are valid for an n dimensional array the new analysis extends to limit points which are excluded by previous authors we show that under a natural hypothesis the algorithm becomes stationary below each limit point and in particular constructs a full set of recursion relations for a recursive array at some finite point the method used to justify the algorithm stresses the duality between the fundamental extension operation of sakata and the s polynomial operation of buchberger

exact boundary controllability results for a multilayer rao nakra sandwich beam we study the boundary controllability problem for a multilayer rao nakra sandwich beam this beam model consists of a rayleigh beam coupled with a number of wave equations we consider all combinations of clamped and hinged boundary conditions with the control applied to either the moment or the rotation angle at an end of the beam we prove that exact controllability holds provided the damping parameter is sufficiently small in the undamped case exact controllability holds without any restriction on the parameters in the system in each case optimal control time is obtained in the space of optimal regularity for l 2 0 t controls a key step in the proof of our main result is the proof of uniqueness of the zero solution of the eigensystem with the homogeneous boundary conditions together with zero boundary observation

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

controllability and stability of a second order hyperbolic system with collocated sensor actuator a second order hyperbolic system with collocated sensor actuator is considered the semigroup generation is shown for the closed loop system under the feedback of a generic unbounded observation operator the equivalence between the exponential stability of the closed loop system and exact controllability of the open loop system is established in the general framework of well posed linear systems finally the conditions are weakened for the diagonal semigroups with finite dimensional inputs example of beam equation is presented to display the application

exponential stability of an abstract nondissipative linear system in this paper we consider an abstract linear system with perturbation of the form frac dy dt ay varepsilon by on a hilbert space cal h where a is skew adjoint b is bounded and varepsilon is a positive parameter motivated by a work of freitas and zuazua on the one dimensional wave equation with indefinite viscous damping p freitas and e zuazua j differential equations 132 1996 pp 338 352 we obtain a sufficient condition for exponential stability of the above system when b is not a dissipative operator we also obtain a hautus type criterion for exact controllability of system a g where g is a bounded linear operator from another hilbert space to cal h our result about the stability is then applied to establish the exponential stability of several elastic systems with indefinite viscous damping as well as the exponential stabilization of the elastic systems with noncolocated observation and control

algebraic geometry codes the theory of error correcting codes derived from curves in an algebraic geometry was initiated by the work of goppa as generalizations of bose chaudhuri hocquenghem bch reed solomon rs and goppa codes the development of the theory has received intense consideration since that time and the purpose of the paper is to review this work elements of the theory of algebraic curves at a level sufficient to understand the code constructions and decoding algorithms are introduced code constructions from particular classes of curves including the klein quartic elliptic and hyperelliptic curves and hermitian curves are presented decoding algorithms for these classes of codes and others are considered the construction of classes of asymptotically good codes using modular curves is also discussed

a general necessary condition for exact observability suppose a generates an exponentially stable strongly continuous semigroup on the hilbert space x y is another hilbert space and c d a rightarrow y is an admissible observation operator for this semigroup this means that to any initial state in x we can associate an output function in l 2 0 infty y this paper gives a necessary condition for the exact observability of the system defined by a and c this condition called bf e is related to the hautus lemma from finite dimensional systems theory it is an estimate in terms of the operators a and c alone in particular it makes no reference to the semigroup this paper shows that bf e implies approximate observability and if a is bounded it implies exact observability this paper conjectures that bf e is in fact equivalent to exact observability the paper also shows that for diagonal semigroups bf e takes on a very simple form and applies the results to sequences of complex exponential functions

efficient forney functions for decoding ag codes using a forney formula to solve for the error magnitudes in decoding algebraic geometric ag codes requires producing functions spl sigma sub p which are 0 at all but one point p of the variety of the error locator ideal the best such function is produced here in a reasonably efficient way from a lex grobner basis this lex basis is in turn produced efficiently from a weighted grevlex basis by using the fglm algorithm these two steps essentially complete the efficient decoding scheme based on a forney formula started in the author s previous work see ibid vol 42 p 1263 8 1996

exact controllability of the euler bernoulli equation with boundary controls for displacement and moment abstract we consider the euler bernoulli problem 1 1 in the solution w t with boundary controls g1 and g2 acting in the dirichlet traces for w and w respectively we show two main exact controllability results both in the spaces of maximal regularity of the solution at t t see i lasiecka and r triggiani regularity theory for a class of euler bernoulli equations a cosine operator approach boll unione mat ital 7 3 b 1989 and both without geometrical conditions on the open bounded domain except for smoothness of its boundary one with g1 l2 0 t l2 and g2 h1 0 t l2 for any t 0 arbitrarily short and one with g1 h01 0 t l2 and g2 l2 0 t l2 for all t 0 sufficiently large an interpolation result between these two cases is also presented a direct approach is given based on two main steps first by means of an operator model i lasiecka and r triggiani regularity theory for a class of euler bernoulli equations a cosine operator approach boll unione mat ital 7 3 b 1989 for problem 1 1 and a functional analytic approach the question of exact controllability is shown to be equivalent to an a priori inequality for the corresponding homogeneous problem next this a priori inequality is proved to hold true by means of multiplier techniques these are inspired by recent progress on the maximal regularity exact controllability and uniform stabilization questions for second order hyperbolic equations

algebraic geometric codes and multidimensional cyclic codes a unified theory and algorithms for decoding using grobner bases it is proved that any algebraic geometric ag code can be expressed as a cross section of an extended multidimensional cyclic code both ag codes and multidimensional cyclic codes are described by a unified theory of linear block codes defined over point sets ag codes are defined over the points of an algebraic curve and an m dimensional cyclic code is defined over the points in m dimensional space the power of the unified theory is in its description of decoding techniques using grobner bases in order to fit an ag code into this theory a change of coordinates must be applied to the curve over which the code is defined so that the curve is in special position for curves in special position all computations can be performed with polynomials and this also makes it possible to use the theory of grobner bases next a transform is defined for ag codes which generalizes the discrete fourier transform the transform is also related to a grobner basis and is useful in setting up the decoding problem in the decoding problem a key step is finding a grobner basis for an error locator ideal for ag codes multidimensional cyclic codes and indeed any cross section of an extended multidimensional cyclic code sakata s algorithm can be used to find linear recursion relations which hold on the syndrome array in this general context the authors give a self contained and simplified presentation of sakata s algorithm and present a general framework for decoding algorithms for this family of codes in which the use of sakata s algorithm is supplemented by a procedure for extending the syndrome array

architectures for syndrome generation and error location search in the decoding of hermitian codes when a hermitian curve is put in a special position with respect to its infinite rational point the x y coordinates of all its finite rational points have regular algebraic properties based on these properties the use of horner s rule and the mechanism of chien search in the decoding of reed solomon codes can be extended to render architectures for syndrome generation and error location search in the decoding of codes constructed from the hermitian curve

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

exact controllability and stabilization of a vibrating string with an interior point mass in this article we examine the problems of boundary control and stabilization for a one dimensional wave equation with interior point masses we show that singularities in waves are smoothed one order as they cross a point mass thus in the case of one interior point mass with e g l 2 dirichlet control at the left end the most general reachable space from 0 that one can expect is l 2 times h 1 to the left of the mass and h 1 times l 2 to the right of the mass we show that this is in fact the optimal result modulo certain compatibility conditions several related results for both control and stabilization of such systems are also given

exact boundary controllability of a microbeam model abstract in this paper we study the exact controllability property of a microbeam model by means of a single boundary control we use the multiplier method together with the controllability observability duality to obtain a time t 0 such that the corresponding linear equation is exactly controllable provided that the control time is greater than t 0 our exact controllability result improves the one obtained in vatankhah et al 2014 20 which uses six boundary controls instead of a single one

the stabilization of a one dimensional wave equation by boundary feedback with noncollocated observation this note addresses the stabilization of a one dimensional wave equation with control at one end and noncollocated observation at another end a simple exponentially convergent observer is constructed the dynamical stabilizing boundary output feedback is designed via the observed state while the closed loop system is nondissipative we show its exponential stability using riesz basis approach

decoding algebraic geometric codes up to the designed minimum distance a simple decoding procedure for algebraic geometric codes c sub omega d g is presented this decoding procedure is a generalization of peterson s decoding procedure for the bch codes it can be used to correct any d 1 2 or fewer errors with complexity o n sup 3 where d is the designed minimum distance of the algebraic geometric code and n is the codelength

effect of sampling rate on the conjugate gradient method applied to signal extrapolation a different response to changes in sampling rate has been observed for conjugate gradients as opposed to svd singular value decomposition applied to signal extrapolation the conjugate gradient solution is described in terms of the svd which leads to an explanation of the behavior of the conjugate gradient method in this context formulating the conjugate gradient method as a spectral filtering method and analyzing it in terms of the spectral filter functions generated as the iteration progresses have made it possible to explain the difference in the effect of sampling rate changes on the conjugate gradients approach to signal extrapolation as compared to the truncated svd approach discussed by sullivan and liu 1984

on the decoding of algebraic geometric codes this paper provides a survey of the existing literature on the decoding of algebraic geometric codes definitions theorems and cross references will be given we show what has been done discuss what still has to be done and pose some open problems

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

second order observers for second order distributed parameter systems in r2 this note addresses observer design for second order distributed parameter systems in r2 particularly second order distributed parameter systems without distributed damping are studied based on finite number of measurements exponentially stable observer is designed the existence uniqueness and stability of solutions of the observers are based on semigroup theory

decoding geometric goppa codes using an extra place decoding geometric goppa codes can be reduced to solving the key congruence of a received word in an affine ring if the codelength is smaller than the number of rational points on the curve then this method can correct up to 1 2 d l 2 s errors where d is the designed minimum distance of the code and s is the clifford defect the affine ring with respect to a place p is the set of all rational functions which have no poles except at p and it is somehow similar to a polynomial ring for a special kind of geometric goppa code namely c sub omega d mp the decoding algorithm is reduced to solving the key equation in the affine ring which can be carried out by the subresultant sequence in the affine ring with complexity o n sup 3 where n is the length of codewords

controllability of the kirchhoff system for beams as a limit of the mindlin timoshenko system we consider the dynamical one dimensional mindlin timoshenko system for beams we analyze how its controllability properties depend on the modulus of elasticity in shear k in particular we prove that the exact boundary controllability property of the kirchhoff system may be obtained as a singular limit as k rightarrow infty of the partial controllability of projections over a sharp subspace of solutions generated by the eigenfunctions that converge as k rightarrow infty towards the spectrum of the kirchhoff system

backstepping observers for a class of parabolic pdes in this paper we design exponentially convergent observers for a class of parabolic partial integro differential equations p i des with only boundary sensing available the problem is posed as a problem of designing an invertible coordinate transformation of the observer error system into an exponentially stable target system observer gain output injection function is shown to satisfy a well posed hyperbolic pde that is closely related to the hyperbolic pde governing backstepping control gain for the state feedback problem for several physically relevant problems the observer gains are obtained in closed form the observer gains are then used for an output feedback design in both collocated and anti collocated setting of sensor and actuator the order of the resulting compensator can be substantially lowered without affecting stability explicit solutions of a closed loop system are found in particular cases

a spectral approach for the exact observability of infinite dimensional systems with skew adjoint generator abstract let a be a possibly unbounded skew adjoint operator on the hilbert space x with compact resolvent let c be a bounded operator from d a to another hilbert space y we consider the system governed by the state equation z t az t with the output y t cz t we characterize the exact observability of this system only in terms of c and of the spectral elements of the operator a the starting point in the proof of this result is a hautus type test recently obtained in burq and zworski j amer soc 17 2004 443 471 and miller j funct anal 218 2 2005 425 444 we then apply this result to various systems governed by partial differential equations with observation on the boundary of the domain the schrodinger equation the bernoulli euler plate equation and the wave equation in a square are considered for the plate and schrodinger equations the main novelty brought in by our results is that we prove the exact boundary observability for an arbitrarily small observed part of the boundary this is done by combining our spectral observability test to a theorem of beurling on nonharmonic fourier series and to a new number theoretic result on shifted squares

admissibility and exact observability of observation operators for micro beam model time and frequency domain approaches this study focuses on the exact observability of a non classical euler bernoulli micro beam equation this non classical model was derived based on the strain gradient elasticity theory which is intended to explain the phenomenon of size effect at the micron scale spectral properties of the corresponding state operator are studied an asymptotic expression for eigenvalues is calculated and eigenfunctions are analyzed in order to check the necessary conditions for the exact observability of the system by examining the eigenfunctions it is shown that among non collocated boundary outputs only measurement of the non classical moment at the root of the beam yields an admissible observation operator and also defines an exactly observable system an alternative proof based on the multiplier method which is commonly employed in the literature on the observability and controllability of infinite dimensional dynamical systems is presented to provide a comparison between the time and frequency domain approaches

energy decay of a microbeam model with a locally distributed nonlinear feedback control abstract in this paper we address the problem of internal stabilization of the deflection of a microbeam which is modeled by a sixth order hyperbolic equation employing multiplier techniques and an integral inequality we prove that a locally distributed nonlinear feedback control forces the energy associated to the deflection to decay exponentially or polynomially to zero as a consequence of this the deflection goes to the rest position as the time goes to infinity