impulse output rapid stabilization for heat equations abstract the main aim of this paper is to provide a new feedback law for the heat equations in a bounded domain with dirichlet boundary condition two constraints will be compulsory first the controls are active in a subdomain of and at discrete time points second the observations are made in another subdomain and at different discrete time points our strategy consists in linking an observation estimate at one time minimal norm impulse control approximate inverse source problem and rapid output stabilization

the beale kato majda criterion for the 3d magneto hydrodynamics equations we study the blow up criterion of smooth solutions to the 3d mhd equations by means of the littlewood paley decomposition we prove a beale kato majda type blow up criterion of smooth solutions via the vorticity of velocity only namely sup j in mathbb z int 0 t delta j nabla times u infty dt where j is the frequency localization operator in the littlewood paley decomposition

uniform controllability of semidiscrete approximations of parabolic control systems controlling an approximation model of a controllable infinite dimensional linear control system does not necessarily yield a good approximation of the control needed for the continuous model in the present paper under the main assumptions that the discretized semigroup is uniformly analytic and that the control operator is mildly unbounded we prove that the semidiscrete approximation models are uniformly controllable moreover we provide a computationally efficient way to compute the approximation controls an example of application is implemented for the one and two dimensional heat equation with neumann boundary control

local regularity criteria of the 3d navier stokes and related equations abstract in this paper we are concerned with the local regularity of suitable weak solutions to the 3d navier stokes equations and its generalized case with fractional power of the laplacian 3 4 1 in the first part we derive some regularity criteria in terms of the deformation tensor d u for the classical navier stokes equations in the second part for the fractional case we obtain some regularity conditions for suitable weak solutions including the velocity u the gradient of the velocity u the rotation tensor u and the deformation tensor d u this generalizes the results obtained by gustafson et al 2007

generalized mhd equations abstract solutions of the d dimensional generalized mhd gmhd equations t u u u p b b u t b u b b u b are studied in this paper we pay special attention to the impact of the parameters and on the regularity of solutions our investigation is divided into three major cases 1 0 and 0 2 0 and 0 and 3 0 and 0 when 0 and 0 the gmhd equations with any 0 and 0 possess a global weak solution corresponding to any l2 initial data furthermore weak solutions associated with 1 2 d 4 and 1 2 d 4 are actually global classical solutions when their initial data are sufficiently smooth as a special consequence smooth solutions of the 3d gmhd equations with 5 4 and 5 4 do not develop finite time singularities the study of the gmhd equations with 0 and 0 is motivated by their potential applications in magnetic reconnection a local existence result of classical solutions and several global regularity conditions are established for this case these conditions are imposed on either the vorticity u or the current density j b but not both and are weaker than some of current existing ones when 0 and 0 the gmhd equations reduce to the ideal mhd equations it is shown here that the ideal mhd equations admit a unique local solution when the prescribed initial data is in a holder space cr with r 1

uniform controllability properties for space time discretized parabolic equations this article is concerned with the analysis of semi discrete in space and fully discrete approximations of the null controllability and controllability to the trajectories for parabolic equations we propose an abstract setting for space discretizations that potentially encompasses various numerical methods and we study how the controllability problems depend on the discretization parameters for time discretization we use schemes with theta in frac 1 2 1 for the proofs of controllability we rely on the strategy introduced by lebeau and robbiano comm partial differ equ 20 335 356 1995 for the null controllability of the heat equation which is based on a spectral inequality we obtain relaxed uniform observability estimates in both the semi discrete and fully discrete frameworks and associated uniform controllability properties for the practical computation of the control functions we follow j l lions hilbert uniqueness method strategy exploiting the relaxed uniform observability estimate algorithms for the computation of the controls are proposed and analysed in the semi discrete and fully discrete cases additionally we prove an error bound between the fully discrete and the semi discrete control functions this bound is however not uniform with respect to the space discretization the theoretical results are illustrated through numerical experimentations

blowup criterion for viscous baratropic flows with vacuum states we prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth strong solutions for the 3 dimensional 3d barotropic compressible navier stokes equations more precisely if a solution of the 3d barotropic compressible navier stokes equations is initially regular and loses its regularity at some later time then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches our result is the same as ponce s criterion for 3 dimensional incompressible euler equations ponce in commun math phys 98 349 353 1985 in addition initial vacuum states are allowed in our cases

stieltjes polynomials and related quadrature rules in this paper we present a survey of theoretical results some of them new and numerical evidence of others concerning stieltjes polynomials kronrod schemes and their generalizations applied to integrals with a classical nonnegative weight function the use of these schemes in automatic integration routines is briefly outlined

gauss kronrod quadrature formulae for weight functions of bernstein szego type ii we study the kronrod extensions of gaussian quadrature rules whose weight functions on 1 l consist of any one of the four chebyshev weights divided by an arbitrary quadratic polynomial that remains positive on 1 11 we show that in almost all cases these extended gauss kronrod quadrature rules have all the desirable properties kronrod nodes interlacing with gauss nodes all nodes contained in 1 11 and all weights positive and representable by semiexplicit formulas exceptions to these properties occur only for small values of n the number of gauss nodes namely n 3 and are carefully identified the precise degree of exactness of each of these gauss kronrod formulae is determined and shown to grow like 4n rather than 3n as is normally the case our findings are the result of a detailed analysis of the underlying orthogonal polynomials and stieltjes polynomials the paper concludes with a study of the limit case of a linear divisor polynomial in the weight function

modified optimal quadrature extensions optimal extensions of quadrature rules are of importance in the construction of automatic integrators but many sequences fail to exist in usable form the paper considers some techniques for overcoming the problem of inextensibility with a minimal effect on integrating efficiency a modification to the extension procedure proposed recently by begumisa and robinson is shown to be just a special case of the standard theory for quadrature extension some illustrative examples are included

a new proof of the caffarelli kohn nirenberg theorem here we give a self contained new proof of the partial regularity theorems for solutions of incompressible navier stokes equations in three spatial dimensions these results were originally due to scheffer and caffarelli kohn and nirenberg our proof is much more direct and simpler 1998 john wiley sons inc

the linear regulator problem for parabolic systems we present an approximation framework for computation in finite dimensional spaces of riccati operators that can be guaranteed to converge to the riccati operator in feedback controls for abstract evolution systems in a hilbert space it is shown how these results may be used in the linear optimal regulator problem for a large class of parabolic systems

a posteriori error estimates for discontinuous galerkin time stepping method for optimal control problems governed by parabolic equations in this paper we examine the discontinuous galerkin dg finite element approximation to convex distributed optimal control problems governed by linear parabolic equations where the discontinuous finite element method is used for the time discretization and the conforming finite element method is used for the space discretization we derive a posteriori error estimates for both the state and the control approximation assuming only that the underlying mesh in space is nondegenerate for problems with control constraints of obstacle type which are the kind most frequently met in applications further improved error estimates are obtained

impulse and sampled data optimal control of heat equations and error estimates we consider the optimal control problem of minimizing some quadratic functional over all possible solutions of an internally controlled multidimensional heat equation with a periodic terminal state constraint this problem has a unique optimal solution which can be characterized by an optimality system derived from the pontryagin maximum principle we define two approximations of this optimal control problem the first one is an impulse approximation and consists of considering a system of linear heat equations with impulse control the second one is obtained by the sample and hold procedure applied to the control resulting in a sampled data approximation of the controlled heat equation we prove that both problems have a unique optimal solution and we establish precise error estimates for the optimal controls and optimal states of the initial problem with respect to its impulse and sampled data approximations

analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls an optimal control problem for impressed cathodic systems in electrochemistry is studied the control in this problem is the current density on the anode a matching objective functional is considered we first demonstrate the existence and uniqueness of solutions for the governing partial differential equation with a nonlinear boundary condition we then prove the existence of an optimal solution next we derive a necessary condition of optimality and establish an optimality system of equations finally we define a finite element algorithm and derive optimal error estimates

internality of generalized averaged gaussian quadrature rules and truncated variants for modified chebyshev measures of the second kind abstract generalized averaged gaussian quadrature rules associated with some measure and truncated variants of these rules can be used to estimate the error in gaussian quadrature rules however the former quadrature rules may have nodes outside the interval of integration and therefore it may not be possible to apply them when the integrand is defined on the interval of integration only this paper investigates whether generalized averaged gaussian quadrature rules associated with modified chebyshev measures of the second kind and truncated variants of these rules are internal i e if all nodes of these quadrature rules are in the interval of integration

stieltjes polynomials and related quadrature formulae for a class of weight functions consider a nonnegative measure d with support in the interval a b such that the respective orthogonal polynomials above a specific index l satisfy a three term recurrence relation with constant coefficients we show that the corresponding stieltjes polynomials above the index 2l 1 have a very simple and useful representation in terms of the orthogonal polynomials as a result of this the gauss kronrod quadrature formulae for d have all the desirable properties namely the interlacing of nodes their inclusion in the closed interval a b under an additional assumption on d and the positivity of all weights furthermore the interpolatory quadrature formulae based on the zeros of the stieltjes polynomials have positive weights and both of these quadrature formulae have elevated degrees of exactness

bilinear estimates in bmo and the navier stokes equations we prove that the bmo norm of the velocity and the vorticity controls the blow up phenomena of smooth solutions to the navier stokes equations our result is applied to the criterion on uniqueness and regularity of weak solutions in the marginal class

ritz galerkin approximation of the time optimal boundary control problem for parabolic systems with dirichlet boundary conditions this paper deals with ritz galerkin approximations of the following two problems i boundary value problems with l 2 boundary data given in the form of dirichlet boundary conditions ii time optimal control problems for parabolic systems with control acting on the boundary in both cases optimal rates of convergence of approximation are obtained also the specialization is considered to approximating with nonconformal elements which are not required to satisfy boundary conditions corresponding rates of convergence are also proved

maximum of the modulus of kernels in gauss turan quadratures we study the kernels k n s z in the remainder terms r n s f of the gauss turan quadrature formulae tor analytic functions on elliptical contours with foci at 1 when the weight is a generalized chebyshev weight function for the generalized chebyshev weight of the first third kind it is shown that the modulus of the kernel k n s z attains its maximum on the real axis positive real semi axis for each n no no n 0 s it was stated as a conjecture in math comp 72 2003 1855 1872 for the generalized chebyshev weight of the second kind in the case when the number of the nodes n in the corresponding gauss turan quadrature formula is even it is shown that the modulus of the kernel attains its maximum on the imaginary axis for each n no no n 0 p s numerical examples are included

controllability of impulse controlled systems of heat equations coupled by constant matrices abstract this paper studies the approximate and null controllability for impulse controlled systems of heat equations coupled by a pair a b of constant matrices we present a necessary and sufficient condition for the approximate controllability which is exactly kalman s controllability rank condition of a b we prove that when such a system is approximately controllable the approximate controllability over an interval 0 t can be realized by adding controls at arbitrary q a b different control instants 0 1 2 q a b t provided that q a b 1 d a where d a min im a and q a b n we also show that in general such systems are not null controllable

finite element approximations of impulsive optimal control problems for heat equations abstract in this paper we study the priori error analysis for semidiscrete galerkin finite element approximation of the impulsive optimal control problem governed by linear heat equation with control constraint we first derive pontryagin s maximum principle and construct the semidiscrete finite element approximation scheme for our impulsive optimal control problem then the error estimates for optimal control u the related state y and the adjoint state will be obtained finally an efficient algorithm for our numerical experiment is designed which verifies the theoretical results obtained in this paper

error bounds of certain gaussian quadrature formulae we study the kernel of the remainder term of gauss quadrature rules for analytic functions with respect to one class of bernstein szego weight functions the location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated this leads to effective error bounds of the corresponding gauss quadratures

maximum of the modulus of kernels of gaussian quadrature formulae for one class of bernstein szegő weight functions abstract we continue with the study of the kernels k n z in the remainder terms r n f of the gaussian quadrature formulae for analytic functions f inside elliptical contours with foci at 1 and a sum of semi axes 1 the weight function w of bernstein szeg type here is w t w 1 2 t 1 1 t 2 1 4 1 2 t 2 1 t 1 1 1 0 sufficient conditions are found ensuring that the kernel attains its maximum absolute value at the intersection point of the contour with either the real or the imaginary axis this leads to effective error bounds of the corresponding gauss quadratures the quality of the derived bounds is demonstrated by a comparison with other error bounds intended for the same class of integrands

interior regularity criteria for suitable weak solutions of the navier stokes equations we present new interior regularity criteria for suitable weak solutions of the 3 d navier stokes equations a suitable weak solution is regular near an interior point z if either the scaled l p q x t norm of the velocity with 3 p 2 q 2 1 q or the l p q x t norm of the vorticity with 3 p 2 q 3 1 q or the l p q x t norm of the gradient of the vorticity with 3 p 2 q 4 1 q 1 p is sufficiently small near z

an algebraic study of gauss kronrod quadrature formulae for jacobi weight functions we study gauss kronrod quadrature formulae for the jacobi weight function w d t 1 t a 1 t and its special case a f a of the gegenbauer weight function we are interested in delineating regions in the a f plane resp intervals in a for which the quadrature rule has a the interlacing property i e the gauss nodes and the kronrod nodes interlace b all nodes contained in 1 1 c all weights positive d only real nodes not necessarily satisfying a and or b we determine the respective regions numerically for n 1 1 20 4 40 in the gegenbauer case and for n 1 1 10 in the jacobi case where n is the number of gauss nodes algebraic criteria in particular the vanishing of appropriate resultants and discriminants are used to determine the boundaries of the regions identifying properties a and d the regions for properties b and c are found more directly a number of conjectures are suggested by the numerical results finally the gauss kronrod formula for the weight w al 2 is obtained from the one for the weight w ca and similarly the gauss kronrod formula with an odd number of gauss nodes for the weight function w t itlv 1 t2 c is derived from the gauss kronrod formula for the weight w a l y 2

output feedback stabilization for heat equations with sampled data controls abstract in this paper we build up an output feedback law to stabilize a sampled data controlled heat equation with a potential in a bounded domain the feedback law abides the following rules first we divide equally the time interval 0 into infinitely many disjoint time periods and divide each time period into three disjoint subintervals second for each time period we observe a solution over an open subset of in the first subinterval take sample from outputs at one time point of the first subinterval add a time invariant output feedback control over another open subset of in the second subinterval let the equation evolve free in the last subinterval thus the corresponding feedback control is of sampled data our feedback law has the following advantages the sampling period which is the length of the above time period can be arbitrarily taken the feedback law has an explicit expression in terms of the sampling period the behaviors of the norm of the feedback law when the sampling period goes to zero or infinity are clear the construction of the feedback law is based on two kinds of approximate null controllability for heat equations one has time invariant controls while another has impulse controls the studies of the aforementioned controllability with time invariant controls need a new observability inequality for heat equations built up in the current work

positivity of the weights of extended clenshaw curtis quadrature rules we prove that some extended clenshaw curtis quadrature rules have all weights positive we also present extended filippi rules of open type having all weights positive conjectures on the possibility of other positive quadrature rules embedded in clenshaw curtis or filippi rule are suggested

hermite and hermite fejer interpolation for stieltjes polynomials let x 1 x 2 1 2 and p n be the ultraspherical polynomials with respect to x then we denote by e n 1 the stieltjes polynomials with respect to x satisfying in this paper we show uniform convergence of the hermite fejer interpolation polynomials h n 1 and h 2n 1 based on the zeros of the stieltjes polynomials e n 1 and the product e n 1 p n for 0 1 and 0 a 1 2 respectively to prove these results we prove that the lebesgue constants of hermite fejer interpolation operators for the stieltjes polynomials e n 1 and the product e n 1 p n are optimal that is the lebesgue constants h n 1 0 a 1 and h 2n 1 0 a 1 2 have optimal order o 1 in the case of the hermite fejer interpolation polynomials h 2n 1 for 1 2 1 we prove weighted uniform convergence moreover we give some convergence theorems of hermite fejer and hermite interpolation polynomials for 0 1 in weighted lp norms

remarks on a paper by j t beale t kato and a majda we prove that the maximum norm of the deformation tensor controls the possible breakdown of smooth solutions for the 3 dimensional euler equations more precisely the loss of regularity in a local smooth solution of the euler equations implies the growth without bound of the deformation tensor as the critical time approaches equivalently if the deformation tensor remains bounded the existence of a smooth solution is guaranteed

error bounds for gauss turan quadrature formulae of analytic functions we study the kernels of the remainder term rn s f of gauss thran quadrature formulas 11f t w t dt ai vf i v rn s f n n s n0 for classes of analytic functions on elliptical contours with foci at 1 when the weight w is one of the special jacobi weights w t 1 t 1 t 1 2 1 2 s 1 2 1 2 s 1 2 s 1 2 we investigate the location on the contour where the modulus of the kernel attains its maximum value some numerical examples are included

stieltjes polynomials and related quadrature formulae for a class of weight functions ii consider a nonnegative measure d sigma with support in the interval a b such that the respective orthogonal polynomials satisfy a three term recurrence relation with coefficients alpha n left begin array ll alpha e n hbox even alpha o n hbox odd end array right beta n beta hbox for n ge ell where alpha e alpha o beta and ell are specific constants we show that the corresponding stieltjes polynomials above the index 2 ell 1 have a very simple and useful representation in terms of the orthogonal polynomials as a result of this the gauss kronrod quadrature formula for d sigma has all the desirable properties namely the interlacing of nodes their inclusion in the closed interval a b under an additional assumption on d sigma and the positivity of all weights while the formula enjoys an elevated degree of exactness furthermore the interpolatory quadrature formula based on the zeros of the stieltjes polynomials has positive weights and also elevated degree of exactness it turns out that this formula is the anti gaussian formula for d sigma while the resulting averaged gaussian formula coincides with the gauss kronrod formula for this measure moreover we show that the only positive and even measure d sigma on a a for which the gauss kronrod formula is almost of chebyshev type i e it has almost all of its weights equal is the measure d sigma t a 2 t 2 1 2 dt

error bounds for gaussian quadrature of analytic functions for gaussian quadrature rules over a finite interval applied to analytic or meromorphic functions we develop error bounds from contour integral representations of the remainder term as in previous work on the subject we consider both circular and elliptic contours in contrast with earlier work however we attempt to determine exactly where on the contour the kernel of the error functional attains its maximum modulus we succeed in answering this question for a large class of weight distributions including all jacobi weights when the contour is a circle in the more difficult case of elliptic contours we can settle the question for certain special jacobi weight distributions with parameters pm frac 1 2 and we provide empirical results for more general jacobi weights we further point out that the kernel of the error functional at any complex point outside the interval of integration can be evaluated accurately and efficiently by a recursive procedure the same procedure is useful also t

on the continuous dependence with respect to sampling of the linear quadratic regulator problem for distributed parameter systems the convergence of solutions to the discrete or sampled time linear quadratic regulator problem and associated riccati equation for infinite dimensional systems to the solutions to the corresponding continuous time problem and equation as the length of the sampling interval the sampling rate tends toward zero infinity is established both the finite and infinite time horizon problems are studied in the finite time horizon case strong continuity of the operators that define the control system and performance index together with a stability and consistency condition on the sampling scheme are required for the infinite time horizon problem in addition the sampled systems must be stabilizable and detectable uniformly with respect to the sampling rate classes of systems for which this condition can be verified are discussed results of numerical studies involving the control of a heat diffusion equation a hereditary or delay system and a flexible beam are presented and discussed

an approximation theorem for the algebraic riccati equation for an infinite dimensional linear quadratic control problem in hilbert space approximation of the solution of the algebraic riccati operator equation in the strong operator topology is considered under conditions weaker than uniform exponential stability of the approximating systems as an application strong onvergence of the approximating riccati operators in case of a previously developed spline approximation scheme for delay systems is established finally convergence of the transfer functions of the approximating systems is investigated

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

partial regularity of suitable weak solutions to the fractional navier stokes equations in this paper we study the partial regularity of fractional navier stokes equations in mathbb r 3 times 0 infty with 3 4 s 1 we show that the suitable weak solution is regular away from a relatively closed singular set whose 5 4s dimentional hausdorff measure is zero the result is a generalization of the partial regularity for the classical navier stokes system in caffarelli et al commun pure appl math 35 771 831 1982

a unified proof on the partial regularity for suitable weak solutions of non stationary and stationary navier stokes equations abstract the partial regularity of the suitable weak solutions to the navier stokes equations in r n with n 2 3 4 and the stationary navier stokes equations in r n for n 2 3 4 5 6 are investigated in this paper using some elementary observation of these equations together with de giorgi iteration method we present a unified proof on the results of caffarelli kohn and nirenberg 1 struwe 17 dong and du 5 and dong and strain 7 particularly we obtain the partial regularity of the suitable weak solutions to the 4d non stationary navier stokes equations which improves the previous result of 5 where dong and du studied the partial regularity of smooth solutions of the 4d navier stokes equations at the first blow up time

a family of gauss kronrod quadrature formulae etude d une famille de formules de quadrature de gauss kronrad formules explicites des poids