homogenization of fully nonlinear uniformly elliptic and parabolic partial differential equations in stationary ergodic media we study the homogenization of fully nonlinear uniformly elliptic and parabolic second order partial differential equations in stationary ergodic media c 2004 wiley periodicals inc

rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media we establish a logarithmic type rate of convergence for the homogenization of fully nonlinear uniformly elliptic second order pde in strongly mixing media with similar i e logarithmic decorrelation rate the proof consists of two major steps the first which is actually the only place in the paper where probability plays a role establishes the rate for special quadratic data using the methodology developed by the authors and wang to study the homogenization of nonlinear uniformly elliptic pde in general stationary ergodic random media the second is a general argument based on the new notion of viscosity solutions which is introduced in this paper that shows that rates known for quadratic can be extended to general data as an application of this we also obtain here rates of convergence for the homogenization in periodic and almost periodic environments the former is algebraic while the latter depends on the particular equation

a computational algorithm for pole assignment of linear single input systems an efficient computational algorithm for pole assignment of linear multiinput systems is proposed a preliminary stage of the algorithm is a reduction of the system matrices into orthogonal canonical form the gain matrix elements are then found by orthogonal transformation of the closed loop system matrix into upper quasi triangular form whose diagonal blocks correspond to the desired poles the algorithm is numerically stable the computed gain matrix being exact for a system with slightly perturbed matrices it works equally well with real and complex distinct and multiple poles and is applicable to ill conditioned and high order problems

an algorithm for the multiinput eigenvalue assignment problem a very simple and inexpensive algorithm is presented for pole placement in the multiinput case the algorithm consists of orthogonal reduction to a block hessenberg form and a simple linear recursion it yields a matrix f such that a bf has any specified set of eigenvalues whenever the system is controllable it is extremely easy to program on a computer the algorithm is not a robust pole placement algorithm but appears to give comparable results in most well conditioned cases at a fraction of the cost it is a direct noniterative algorithm and no eigenvalues or singular values are computed the algorithm does not need any complex arithmetic even when complex conjugate eigenvalues need to be assigned

magnetoelectroelastic analysis of an arbitrary shape inclusion undergoing eigenfields and remote loadings abstract within the framework of the linear theory of magnetoelectroelasticity the problem of an arbitrary shape inclusion in an entire plane under remotely uniform in plane electromagnetic and anti plane mechanical loadings is investigated in this paper the inclusion is undergoing eigenfields and has different magnetoelectroelastic moduli as that of the matrix using complex variable and faber series method the complex potential for each component can be expressed in series form with unknown coefficients the continuity conditions of the interfaces are used to build up a set of linear equations to determine the unknown coefficients through solving these linear equations one can obtain the complex potentials both inside the inclusion and in the matrix numerical results are provided and shown that different combinations of piezoelectric piezomagnetic exhibit significant magnetoelectromechanical coupling effect that is not present in single phase piezoelectric or piezomagnetic material applying appropriate remote loadings and eigenfields can effectively reduce stress level and increase magnetoelectric response around the inclusion

the boundary due terms in the green operator of inclusion patterns from distant to contact and to connected situations using radon transforms illustration for spheroid alignments in isotropic media abstract we examine the boundary due components in the mean modified green operator integral green operator for short of an inclusion pattern in distant contact and contact connection transitions the direct rt and inverse irt radon transforms which allow specification of the different contributions to the mean green operator of the pattern in simple geometrical terms are used the already well documented case of axially symmetric alignments of equidistant identical oblate spheroids in an infinite matrix of isotropic elastic like or dielectric like properties is treated up to infinite alignments and for any aspect ratio from unity spheres to infinitesimal platelets simple closed forms for this mean green operator and for its different parts are newly obtained these closed forms allow an easy parametric study of the operator variations in terms of the alignment characteristics from distant to contact situations from contact to connection of the inclusions the changes in the green operator s contributions are pointed what provides relevant operator forms for the connected patterns these results are of interest in problems where phase percolation connectivity inversions or co continuity are implied

on the absence of eshelby property for non ellipsoidal inclusions it is shown that the eshelby property does not hold for any inclusion bounded by a polynomial surface of higher than the second degree or any inclusion bounded by a non convex surface inclusions bounded by segments of two or more different surfaces are also precluded the absence of the eshelby property for non ellipsoidal inclusions is then discussed

comparison of two state observer design algorithms there are two contributions of this note first it clarifies and unifies the two design approaches which will generate the general state observer and the identity state observer respectively especially for the first approach even a very recent paper failed to clarify it second this note shows that due to computational error the general state observer will have an estimation error while the identity state observer will have an error in its desired poles

higher order convergence rates in theory of homogenization iii viscous hamilton jacobi equations abstract in this paper we establish the higher order convergence rates in periodic homogenization of viscous hamilton jacobi equations which is convex and grows quadratically in the gradient variable we observe that although the nonlinear structure governs the first order approximation the nonlinear effect is absorbed as an external source term of a linear equation in the second and higher order approximation moreover we find that the geometric shape of the initial data has to be chosen carefully according to the effective hamiltonian in order to achieve the higher order convergence rates

an algorithm to assign eigenvalues in a hessenberg matrix single input case we propose a conceptually simple algorithm to assign eigenvalues in a hessenberg matrix the method is based on the evaluation of a simple recursive relation and it is extremely easy to program on a computer numerical experiments compare favorably to those of other methods recently proposed for the problem

an algorithm for the multiinput pole assignment problem abstract we present an algorithm for the multiinput pole assignment problem this algorithm is based on basic linear operations such as matrix vector multiplications resolution of upper triangular systems of smaller sizes than the original problem and resolution of systems with multiple right hand sides for which efficient and parallel algorithms have been developed furthermore those independent systems can be computed in parallel the accuracy of the algorithm has been checked on some numerical examples

higher order convergence rates in theory of homogenization equations of non divergence form we establish higher order convergence rates in the theory of periodic homogenization of both linear and fully nonlinear uniformly elliptic equations of non divergence form the rates are achieved by involving higher order correctors which fix the errors occurring both in the interior and on the boundary layer of our physical domain the proof is based on a viscosity method and a new regularity theory which captures the stability of the correctors with respect to the shape of our limit profile

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

the perturbed test function method for viscosity solutions of nonlinear pde the method of viscosity solutions for nonlinear partial differential equations pdes justifies passages to limits by in effect using the maximum principle to convert to the corresponding limit problem for smooth test functions we describe in this paper a perturbed test function device which entails various modifications of the test functions by lower order correctors applications include homogenisation for quasilinear elliptic pdes and approximation of quasilinear parabolic pdes by systems of hamilton jacobi equations

higher order convergence rates in theory of homogenization ii oscillatory initial data abstract we establish higher order convergence rates in periodic homogenization of fully nonlinear uniformly parabolic cauchy problems accompanied with rapidly oscillating initial data such result is new even for linear problems here we construct higher order initial layer and interior correctors which describe the oscillatory behavior near the initial and interior time zone of the domain to construct higher order correctors we develop a regularity theory in macroscopic scales and prove an exponential decay estimate for initial layer correctors the higher order expansion requires an iteration process successively correcting the initial layer then the interior this leads to a more complicated asymptotic expansion as compared to the non oscillating data case and this complexity is present even in the linear case a notable observation for fully nonlinear operators is that even if the given operator is space time periodic the interior correctors become aperiodic in the time variable as we proceed with the iteration process moreover each interior corrector of higher order is paired with a space time periodic version and the difference between the two decays exponentially fast with time

state feedback design using reduced order nonsquare descriptions a pole placement technique is proposed for computing state feedback in multi input linear systems the algorithm has good numerical behavior and utilizes nonsquare pencils and the notion of decomposability the fact that the method deals with reduced order pencils and exploits the schur hessenberg form improves the conditioning of the problem

on the finite transmission zero assignment problem abstract in this paper the problem of finite transmission zero assignment is examined by squaring a system from the outputs to the inputs in particular we study this problem for two cases namely for state accessible and partially state accessible systems we show that the problem of finite transmission zero assignment problem for state accessible systems is equivalent to a pole placement problem with state feedback on a reduced order subspace which always has a solution in the case of partially state accessible systems we show that the finite transmission zero assignment problem is equivalent to a pole placement problem with output feedback on a reduced order subspace which generically has a solution for the case when p m n for both cases we exploit the block hessenberg form of the system and the concept of sylvester equations in order to solve this problem in a computationally efficient manner

h sub infinity control by state feedback and fast algorithms for the computation of optimal h sub infinity norms the suboptimality of some parameter for h sub infinity optimization by dynamic state feedback is characterized in terms of the solvability of riccati inequalities this is done without restricting the finite zero structure of the plant if there are no system zeros on the imaginary axis the h sub infinity problem can be treated in a complete and satisfactory way explicit characterizations optimum to be achieved are provided and a closed formula for the optimal value is derived in terms of the h sub infinity norm of some fixed transfer matrix if the optimum is not attained any sequence of controllers of bounded size which is constructed to approach the infimal norm must necessarily be high gain a globally and quadratically convergent algorithm to compute the optimal value is proposed this algorithm is generalized to the h sub infinity optimization problem by measurement feedback

comments on new techniques for the design of observers this note is to point out that by fully using the freedom of the observer equation rather than the arbitrary choice suggested in the paper significantly better results can be derived with applications and solutions to many important problems

a new robust stability measure for state feedback systems abstract this brief paper proposes the use of sensitivities of each individual system pole to measure robust stability of state feedback systems compared with the existing measures of this kind m 1 and m 2 this measure is much easier to work with than m 1 while at least as accurate and is definitely more accurate than m 2 the upper and lower bounds of this measure are also derived and proved to be the same as those of m 1 this measure can be maximized numerically using existing explicit and systematic procedures some analytical guidelines which relate explicitly and directly the state feedback system s individual pole selection performance and decoupling to this new measure are also available and presented

viscosity solutions methods for singular perturbations in deterministic and stochastic control viscosity solutions methods are used to pass to the limit in some penalization problems for first order and second order degenerate parabolic hamilton jacobi bellman equations this characterizes the limit of the value functions of singularly perturbed optimal control problems for deterministic systems and for controlled degenerate diffusions the results apply to cases where the usual order reduction method does not give the correct limit and to systems with fast state variables depending nonlinearly on the control some connections with ergodic control and periodic homogenization are discussed

effective hamiltonians and averaging for hamiltonian dynamics i this paper building upon ideas of mather moser fathi e and others applies pde partial differential equation methods to understand the structure of certain hamiltonian flows the main point is that the cell or corrector pde introduced and solved in a weak sense by lions papanicolaou and varadhan in their study of periodic homogenization for hamilton jacobi equations formally induces a canonical change of variables in terms of which the dynamics are trivial we investigate to what extent this observation can be made rigorous in the case that the hamiltonian is strictly convex in the momenta given that the relevant pde does not usually in fact admit a smooth solution

the generalized eigenstructure problem in linear system theory the algebraic theory of linear time invariant systems has been studied in large detail during the past few decades and numerous computational algorithms have been developed to solve problems arising in this context in this paper the numerical aspects of a certain class of such algorithms dealing with what the author calls generalized eigenstructure problems are discussed some new and or modified algorithms are presented both the nmnerical stability of the algorithms and the conditioning of the problems they solve are analyzed using numerical criteria

using the radon transform to solve inclusion problems in elasticity abstract from the fourier transform method the modified green operator integral over a bounded domain in an infinite elastic medium takes on each domain point the form of a weighted average over an angular distribution of a single elementary operator the radon transform provides a geometric definition of the weight function characteristic of the domain shape in terms of the domain intersections with all planes passing through the point it allows a geometrically more meaningful analytical resolution of the general inclusion problem in an infinite medium of general elasticity symmetry the inclusion being any bounded domain possibly made of groups or distributions of inclusions the method is also likely to provide insights in the related problem of effective moduli estimates for heterogeneous microstructures the determination of the weight functions characteristics of the involved inclusional domain shapes is therefore a key step of the resolution the mean values of these weight functions being of first order interest here it is exemplified on the case of cuboidal domain shapes that for material morphologies involving shapes of hardly accessible exact mean weight functions one can make use of approximate conveniently analytical expressions to remain more accurate than using ellipsoidal approximations of the shapes

output feedback eigenstructure assignment using two sylvester equations the eigenstructure assignment problem with output feedback is studied for systems satisfying the condition p m n the main tool used is the concept of c a b invariance and two coupled sylvester equations the solution of which leads to the computation of an output stabilizing feedback a computationally efficient algorithm for the solution of these two coupled equations which leads to the computation of a desired output feedback is presented

a new approach for computing the state feedback gains of multivariable systems this note presents some new results in linear systems first the relationship between the polynomial matrix description and the state space representation of multivariable systems is clarified then we show that once such a relationship is determined the coprime matrix fraction description can be easily computed and we can further develop a closed form formula to solve the pole assignment problem of multivariable systems such formula can be thought of as an extension of the ackermann s formula for mimo systems thus this note potentially gives us a clearer insight into linear systems from the theoretical viewpoint

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

solutions to the polya szego conjecture and the weak eshelby conjecture eshelby showed that if an inclusion is of elliptic or ellipsoidal shape then for any uniform elastic loading the field inside the inclusion is uniform he then conjectured that the converse is true that is that if the field inside an inclusion is uniform for all uniform loadings then the inclusion is of elliptic or ellipsoidal shape we call this the weak eshelby conjecture in this paper we prove this conjecture in three dimensions in two dimensions a stronger conjecture which we call the strong eshelby conjecture has been proved if the field inside an inclusion is uniform for a single uniform loading then the inclusion is of elliptic shape we give an alternative proof of eshelby s conjecture in two dimensions using a hodographic transformation as a consequence of the weak eshelby s conjecture we prove in two and three dimensions a conjecture of polya and szego on the isoperimetric inequalities for the polarization tensors pts the polya szego conjecture asserts that the inclusion whose electrical pt has the minimal trace takes the shape of a disk or a ball

an algorithm for computing state feedback in multiinput linear systems a new algorithm is proposed in this note to compute the state feedback for arbitrary pole placement in multiinput linear systems the algorithm has good numerical behavior it also allows the user to adjust the eigenstructure of the resulting feedback system and therefore improve the condition of the problem and computation

analytical mean green operators eshelby tensors for patterns of coaxial finite long or flat cylinders in isotropic matrices abstract recently an analytical solution has been provided for mean and axial green operators go of circular cylindrical inclusions with finite length in 3d isotropic elastic like and dielectric like materials from long rods to truly penny shaped ones this result provides a situation for examining the commonly used shape spheroidal approximation of same aspect ratio as well as possibly more appropriate alternatives for cases when such a spheroidal approximation would remain needed we propose and compare an operator spheroidal approximation in terms of best matching go which is exactly operator identical for dielectric like properties and nearly identical for elastic like ones next a simple analytical solution is explicated for the mean isotropic go of all patterns of coaxial circular long or flat cylinders with same radii obtained from solving the coaxial cylinder pair interaction problem these patterns can be seen as either fragmented cylinders or as cylinder alignments the obtained go solution form is further shown to formally hold for similar patterns of domains having more general shapes the established general property of such patterns is that the interaction part of their mean go can be expressed as a linear combination of the mean go of all the pattern sub domains the mean go solution for one such pattern is further proved to be valid as well for the domains of the double inclusion type obtained in connecting the pattern elements by their outer envelop of zero thickness thanks to the go blindness to connectivity differences this go equivalency is finally used to define a global operator spheroidal approximation in terms of best matching go for any such coaxial cylinder pattern effective properties for an isotropic matrix reinforced by congruent cylinders or patterns are estimated within a mean field approximation context

effects of using observers on stabilization of uncertain linear systems the effects of observers on robust linear feedback controllers are studied sufficient conditions are obtained that guarantee full recovery of the allowable uncertainty bounds attainable by full state feedback the effects of the resulting high gain observers on the disturbance rejection bounds are also studied it is shown that full recovery of the uncertainty bound leads to possible large degradation in disturbance rejection however if there is only additive plant disturbance and no measurement disturbance this degradation can be prevented

a complete analytical solution to the equation ta ft lc and its applications in this note a complete analytical and restriction free solution with complete and explicit freedom of the matrix equation ta ft lc is proposed here a c is given and is observable and f is in the jordan form with arbitrary given eigenvalues this solution appears to be new because it can be applied directly to obtain significantly better solutions to the following three basic design problems 1 2 d system eigenvalue assignment 2 function observer design and 3 state feedback eigenstructure design as shown in this note

on the modified green operator integral for polygonal polyhedral and other non ellipsoidal inclusions abstract a strange particularity of polyhedral inclusions and of fibres of regular polygonal cross sections has been recently stressed in the literature for respectively fully or transversally isotropic elasticity of the embedding material they have a mean and a central green operator integral both equal to the uniform one of respectively the sphere or the cylindrical fibre in using the radon transform rt method this particularity is here shown to be shared by much larger shape types in the same limits of material elasticity symmetry as a subcase even more shape types fulfill the similar particularity for material linear properties of second rank characteristic tensor such as thermal conductivity magnetic or dielectric properties when calculated using the rt method the modified green operator integral at any interior point of a bounded domain inclusion takes the form of a weighted average over an angular distribution of a single elementary operator the weight function is geometrically defined from the characteristic function of the domain and the four rank or second rank elementary operator depends on the material linear property of concern the rt method simply shows that the noticed particularity is due to matching symmetry between the inclusion shape through the weight function and the material property through the elementary operator the general geometrical characteristics of the inclusion shapes belonging to these sphere class and cylindrical fibre class are specified and some remarkable shapes of these classes are commented

a schur method for pole assignment in this paper a new numerical algorithm for pole assignment of linear time invariant systems is presented the proposed algorithm uses reliable numerical techniques based on the schur form of the state matrix and on the use of orthogonal transformations only the bad eigenvalues of the system are modified by a sequential pole shifting procedure and the resulted gains in the feedback matrix are minimized

computational design techniques for reduced order compensators the authors propose two computational techniques for the pole placement problem using reduced order compensators computationally efficient algorithms are presented for the observer based compensator and the brasch pearson compensator the key tool for these algorithms is the concept of two coupled sylvester equations the solutions of which completely characterize the desired reduced order compensators

an algorithm for pole assignment of time invariant linear systems an algorithm is suggested for the computation of a linear state feedback for a single input system such that the resultant closed loop system has specified eigenvalues the algorithm is more efficient than many comparable techniques and has some desirable numerical properties it is closely related to the qr algorithm for the eigenproblem and extends to descriptor systems