relaxation of three solenoidal wells and characterization of extremal three phase h measures we fully characterize quasiconvex hulls for three arbitrary solenoidal divergence free wells in dimension three with this aim we establish weak lower semicontinuity of certain functionals with integrands restricted to generic two dimensional planes and convex in up to three rank 2 directions within the planes within the framework of the theory of compensated compactness the latter represents an example when the differential constraints fail the constant rank condition but nevertheless the so called convexity still implies lower semicontinuity and fancyscript a quasiconvexity which essentially means that rank 2 convexity implies s quasiconvexity that is quasiconvexity in the sense of the divergence free differential constraints on the planes the proof employs a version of muller s estimates of haar wavelet projections in terms of the riesz transform the above semicontinuity result is then applied to the three solenoidal wells problem via analogs of sverak s nontrivial quasiconvex functions and connectedness properties of the rank 2 envelopes as another application of the semicontinuity result we obtain a geometric result of a more general nature characterization of certain extremal three point h measures for three phase mixtures of three characteristic functions in dimension three we also discuss the applicability of the results to problems with other differential constrains in particular to three linear elastic wells and further generalizations

improved bounds for the overall properties of a nonlinear composite dielectric the construction of generalized hashin shtrikman bounds for nonlinear composite problems relies on the introduction of a comparison material in recent work a nonlinear comparison medium has been used however this requires detailed knowledge of the properties of the trial fields that are employed the fields used have the property of bounded mean oscillation and this enables the size of the penalty exacted by using a nonlinear comparison material to be controlled some recent results concerning riesz transforms and the beurling operator are used here to reduce the effect of the penalty and hence to generate improved bounds in addition an exact solution is established for a particular class of composites

h measures a new approach for studying homogenisation oscillations and concentration effects in partial differential equations new mathematical objects called h measures are introduced for studying oscillations and concentration effects in partial differential equations applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach

on the relation of a three well energy this paper studies the relaxation of multi well non convex energies in the context of infinitesimal geometrically linear elasticity and the analogous problem for gradient fields a dual variational principle of hashin shtrikman type is developed for the relaxed energy under the assumption that the original energy function is a difference of a quadratic function and of another convex function in the particular case of linear elastic wells sharing the same elastic modulus this reduces to the approach of r v kohn and leads to the need to minimize an energy functional with respect to matrix measures on the unit sphere the h measures which are subject to certain constraints it is shown that for the three well linearized energy the minimizing measure subject to these constraints can be chosen as a sum of no more than three dirac masses a subclass of such three point measures realizable by microstructures is described it is demonstrated by means of examples that the minimizing measures do fall within this subclass in some cases thereby providing an exact value for the three well non convex relaxed energy more generally the minimization algorithm leads to an improved lower bound for the relaxed energy if the three phases are pairwise compatible the relaxed energy is a convexification of the original energy

variational second order estimates for nonlinear composites it is shown here that the approximate second order method of ponte castaneda for estimating the effective behaviour of nonlinear composite materials can be given a variational status this is accomplished by means of a slight generalization of the variational structure of talbot and willis consisting of the replacement of the infimum over the polarization field by the more general corresponding stationary operation since the relevant functions are not convex in general this replacement provides additional flexibility that can be exploited advantageously the net result is a stationary principle from which stationary variational estimates can be generated under certain hypotheses on the choice of the linear comparison composite it is shown that the resulting estimates are identical to the second order estimates thus showing that these second order estimates are indeed variational in character although only of the stationary variety the connection of these second order estimates to earlier bounds obtained from related extremum variational principles is also explored

robust redesign of adaptive control effects of unmodeled high frequency dynamics on stability and performance of adaptive control schemes are analyzed in the regulation problem global stability properties are no longer guaranteed but a region of attraction exists for exact adaptive regulation the dependence of the region of attraction on unmodeled parasitics is examined first then the general case of model reference adaptive control is considered in which parasitics can destroy stability and boundedness properties a modified adaptive law is proposed guaranteeing the existence of a region of attraction from which all signals converge to a residual set which contains the equilibrium for exact tracking the size of this set depends on design parameters the frequency range of parasitics and the reference input signal characteristics

on effective resistivity and related parameters for periodic checkerboard composites an approach for finding the effective properties of two dimensional periodic rectangular checkerboard media is presented it is demonstrated that one can take a relatively simple model problem an appropriate system of joined quarter planes and that after appropriate conformal transformation one can use this to solve for either periodic or doubly periodic structures illustrative calculations involving three phase media are presented that in limiting cases recover earlier analyses useful asymptotic relations are also given

controllability observability and discrete time markovian jump linear quadratic control this paper is concerned with the controllability and observability of discrete time linear systems that possess randomly jumping parameters described by finite stale markov processes and the relationship between these properties and the solution of the infinite time jump linear quadratic jlq optimal control problem the solution of the markovian jlq problem with finite or infinite time horizons is known necessary and sufficient conditions for the existence of optimal constant control laws that lead to finite optimal expected costs as the time horizon becomes infinite are also known sufficient conditions for these steady state control laws to stabilize the controlled system are also available chizeck et al 1986 these conditions are not easy to test however various definitions of controllability and observability for stochastic systems exist in the literature these definitions are unfortunately not related to the steady state jlq control problem in a manner that is analogous to the role of deter

correctors and field fluctuations for the p x laplacian with rough exponents we provide a corrector theory for the strong approximation of fields inside composites made from two materials with different power law behavior the correctors are used to develop bounds on the local singularity strength for gradient fields inside micro structured media the bounds are multi scale in nature and can be used to measure the amplification of applied macroscopic fields by the micro structure

upper and lower bounds for the overall properties of a nonlinear composite dielectric ii periodic microgeometry the study of the effective properties of a nonlinear composite dielectric begun in the companion to this paper is continued in the context of a simple cubic array of inclusions embedded in a matrix use of a nonlinear comparison material in the nonlinear hashin shtrikman variational principles permits the generation of both upper and lower bounds for any composite it is straightforward to apply the new approach to the simple composite considered here as the trial electric field which is substituted into the hashin shtrikman variational principle is finite everywhere the results obtained indicate the likely improvement over the simple classical bounds that can be obtained using the hashin shtrikman principles

rigidity and lack of rigidity for solenoidal matrix fields the problem of finding a curl free matrix valued field e with values in an assigned set of matrices has received considerable attention given a bounded connected open set and a compact set of m n matrices in this paper we establish existence or non existence results for the following problem find b l m n such that div b 0 in in the sense of distributions under the constraint that b x almost everywhere in we consider the case of a b rank a b n and we establish non existence both for the case of exact solutions described above and for the case of approximate solutions described in 1 we also prove existence of approximate solutions for a suitably chosen triple a 1 a 2 a 3 of matrices with rank a i a j n i j and i j 1 2 3 we give examples when the differential constraints are of a different type and present some applications to composites

bounds and estimates for the overall plastic behaviour of composites with strain gradient effects the macroscopic plastic response is estimated for a composite with each phase satisfying a strain gradient constitutive description in a j 2 deformation theory of strain gradient plasticity the strain energy function is assumed to depend on both the von mises strain invariant epsilon e and on a curvature invariant chi e for a general coupling between epsilon e and chi e a nonlinear variational principle is formulated generalizing that of ponte castaneda 1992 j mech phys solids 40 1757 the minimum principle is used to derive bounds and estimates for the overall plastic response of statistically homogeneous and isotropic strain gradient composites an essential ingredient of the strategy is the use of previous results of the authors 1994 j mech phys solids 42 1851 for a linear comparison composite for a particular coupling between epsilon e and chi e suggested earlier by fleck hutchinson 1993 the hashin shtrikman upper bounds and lower estimates are calculated for a power law hardening solid including the rigid perfectly plastic limit the results demonstrate a size effect

new bounds on the effective thermal conductivity of n phase materials based on the perturbation solution to the effective thermal conductivity problem for an n component material we derive new third and fourth order bounds for the effective thermal conductivity of the composite the new third order bounds are accurate to third order in epsilon a epsilon b where epsilon a is the thermal conductivity of phase a and require a total of frac 1 2 n 1 2 third order correlation parameters in their evaluation for two phase composites these bounds require only one geometrical parameter and are identical to beran s bounds for n geqslant 3 the third order bounds use the same information as beran s bounds but always more restrictive than beran s bounds the new fourth order bounds are accurate to fourth order in epsilon a epsilon b and require an additional frac 1 4 n 2 n 1 2 fourth order correlation parameters for n 2 only two parameters are needed and the fourth order bounds are shown to be more restrictive than the third order beran s bounds and the second order hashin shtrikman s bounds the application of the third order bounds to a spherical cell material of miller is illustrated

bounds which incorporate morphological information for a nonlinear composite dielectric the nonlinear hashin shtrikman variational structure relies on the introduction of a comparison material in recent work a nonlinear comparison composite has been employed this has enabled both upper and lower bounds to be found that depend on the three point statistics of the microstructure through a parameter introduced by g w milton at least one of the bounds requires an estimate of the size of the trial field to be known and even if the best estimate available is used and the milton parameter is known for a particular microstructure the bounds can still be widely separated o p bruno has obtained bounds for a linear composite consisting of a matrix containing spherical inclusions where the inclusions are coated with the matrix phase the bounds remain non trivial even when the properties of the inclusions tend to either zero or infinity these bounds are used to improve the bounds for a nonlinear composite comprising either a nonlinear matrix containing linear inclusions or a linear matrix containing nonlinear inclusions for the microgeometry considered here it is possible to construct a bounded trial field and a simple example involving this approach is also described

dielectric breakdown optimal bounds we study dielectric breakdown for composites made of two isotropic phases we show that sachs s bound is optimal this simple example is used to illustrate a variational principle which departs from the traditional one we also derive the usual variational principle by elementary means without appealing to the technology of convex duality

some simple explicit bounds for the overall behaviour of nonlinear composites abstract variational expressions developed over the last few years provide bounds for the overall energy functions of a range of nonlinear composite materials the evaluation of a bound requires the solution of a system of nonlinear algebraic equations and this generally involves a computation there are however certain simple composites comprising a nonlinear matrix containing either rigid inclusions or cavities for which very simple explicit formulae can be given these formulae are displayed here at a level of generality greater than in any previous presentation the energy density function of the matrix is arbitrary and the microgeometry of the composite appears through an expression which bounds the energy of a linear composite with the same geometry to the extent that such linear bounds can be developed making allowance for any amount of statistical information on the composite the new nonlinear bounds collect this new results at the level of employing bounds of hashin shtrikman type for the linear problem are given for an incompressible matrix reinforced by aligned rigid platelets or weakened by aligned cracks in the course of the work a recently derived formula more general than any available previously is presented and developed explicitly for any two phase composite

theory and design of robust direct and indirect adaptive control schemes a general approach for designing and the theory for analysing robust direct and indirect adaptive control schemes for continuous time plants is presented the design approach involves the development of a general robust adaptive law and the use of the certainty equivalence principle to combine it with robust model reference and pole placement control structures the global stability properties and robustness of the developed adaptive control schemes are established by using a general theory which relates the properties of signals in the mean sense over intervals of time the developed theory and design approach are used to analyse and compare the robustness properties and performance of a wide class of robust adaptive laws which employ a dead zone fixed e1 and a switching modification as well as their variations

homogenization and optimal bounds in linear elasticity in their celebrated paper of 1963 hashin shtrikman 1 addressed the problem of determining optimal bounds for the bulk and shear moduli of a statistically isotropic elastic composite material with arbitrary isotropic phase geometry they derived a set of bounds for these moduli in the physically meaningful case of three dimensional elasticity in the case of a two phase composite let k 1 1 and k 2 2 respectively denote the bulk and shear moduli for the first and the second phase let k denote their analogues for the composite and let 0 stand for the volume fraction of the first phase in the composite under the ordering restriction that both

robust adaptive controller with zero residual tracking errors the adaptive control of a plant whose dominant part has transfer function with relative degree n ast 1 has been considered in the presence of parasitics and disturbances a new adaptive law is proposed which guarantees the existence of a large region of attraction from which all signals are bounded and the tracking error converges to a small residual set in contrast to the adaptive law used in 1 2 the new adaptive law guarantees a smaller residual set for the tracking error which reduces to zero when the parasitics and disturbances disappear

a variational approach to the overall sink strength of a nonlinear lossy composite medium an extension of the hashin shtrikman variational structure to nonlinear problems initiated by j r willis j appl mech 50 1202 1209 1983 and developed generally by d r s talbot j r willis ima j appl math 35 39 54 1985 is applied to the problem of homogenization of the semilinear steady state diffusion equation nabla 2c partial w partial c c x k x 0 where the function tilde w is convex in c but varies randomly with position x a strict definition for an overall function w which defines the homogenized medium is given and its properties are discussed bounds for tilde w are then developed from variational principles classical energy principles yield bounds analogous to the voigt and reuss bounds of elasticity theory while the new hashin shtrikman structure yields tighter bounds which are sensitive to the two point statistics of the medium explicit results are presented for a particular two phase composite medium for which one phase is linear so that w is quadratic whereas the second has a quadratic nonlinearity so that w is cubic

a new model reference robust adaptive controller in the presence of unmodeled dynamics and bounded disturbances abstract this paper presents a new model reference robust adaptive controller which does not require a priori knowledge of the norm desired controller parameter vector the algorithm is designed according to a reduced order plant which is assumed to be of minimum phase and of an arbitrary relative degree in the adaptive law the output error plays a dual role in the adjustment of the control parameter vector in the absence of unmodeled dynamics and disturbances the tracking error is shown to be zero when reference input is persistently exciting