robust control in the gap a state space solution in the presence of a single input delay a solution is provided to the problem of sub optimal robust design in the gap for a system with a single input lag the problem is shown to reduce to two matrix algebraic riccati equations that solve an allied linear quadratic gaussian h sub 2 problem and a parameterized differential matrix riccati equation the analysis intertwines finite dimensional delay equations and distributed abstract evolution models the parameterized family of suboptimal compensators is presented in terms of a neutral equation model

on spl hscr sub spl infin control for dead time systems a mixed sensitivity spl hscr sub spl infin problem is solved for dead time systems it is shown that for a given bound on the spl hscr sub spl infin norm causal stabilizing controllers exist that achieve this bound if and only if a related finite dimensional riccati equation has a solution with a certain nonsingularity property in the case of zero time delay the riccati equation is a standard riccati equation and the nonsingularity condition is that the solution be nonnegative definite for nonzero time delay the nonsingularity condition is more involved but still allows us to obtain controllers all suboptimal controllers are parameterized and the central controller is shown to be a feedback interconnection of a finite dimensional system and a finite memory system both of which can be implemented some spl hscr sub spl infin problems are rewritten as pure rational spl hscr sub spl infin problems using a smith predictor parameterization of the controller

a monte carlo algorithm for lattice ribbons a lattice ribbon is a connected sequence of plaquettes subject to certain selfavoidance conditions the ribbon can be closed to form an object which is topologically either a cylinder or a mobius band depending on whether its surface is orientable or nonorientable we describe a grand canonical monte carlo algorithm for generating a sample of these ribbons prove that the associated markov chain is ergodic and present and discuss numerical results about the dimensions and entanglement complexity of the ribbons

higher integrability of iterated operators on differential forms abstract in this paper we first prove the local higher integrability and higher order imbedding theorems for the iterated operators defined on differential forms then we prove the global higher integrability and higher order imbedding inequalities for these operators finally we demonstrate applications of the main results by examples

formulas on preview and delayed h infty control a generalized hinfin control problem which covers preview and delayed control strategies is discussed based on a state space approach by introducing a hamiltonian matrix which is associated with a delay free generalized plant the analytic solution to the corresponding operator riccati equation is newly established based on the result obtained here the hinfin control problem is solved and for typical control problems e g hinfin and linear quadratic lq control for multiple input delay systems hinfin preview control some interpretations are provided on the resulting control system

theory of differential forms on manifolds in this paper we establish a hodge type decomposition for the lp space of differential forms on closed i e compact oriented smooth rieman nian manifolds critical to the proof of this result is establishing an lp es timate which contains as a special case the l2 result referred to by morrey as gaffney s inequality this inequality helps us show the equivalence of the usual definition of sobolev space with a more geometric formulation which we provide in the case of differential forms on manifolds we also prove the lp boundedness of green s operator which we use in developing the lp theory of the hodge decomposition for the calculus of variations we rigorously verify that the spaces of exact and coexact forms are closed in the lp norm for nonlinear analysis we demonstrate the existence and uniqueness of a solution to the a harmonic equation

a caccioppoli inequality and partial regularity in the calculus of variations a new proof is given of the partial regularity of minimisers under the principal assumptions of uniform strict quasiconvexity and polynomial growth the proof is based on a caccioppoli inequality and an indirect blow up argument this avoids the technical complications of a direct argument

dirac harmonic equations for differential forms abstract in this paper we introduce the dirac harmonic equation for differential forms and prove some basic norm inequalities including the caccioppoli type inequality poincare type inequality and the weak reverse holder inequality for the solutions of this kind of differential equations

robust controller design for delay systems in the gap metric a robust stabilization problem in the gap metric is discussed for a system with delays in control the authors derive a design procedure of robust controllers based on finite dimensional ricatti equations and it is shown that the resulting controller has a structure of observer based predictive action by employing completing the square argument a game theoretic interpretation is also provided on the trade off between the initial uncertainties and attenuating the disturbance caused by plant uncertainties

brief h performance of preview control systems performance of the preview control is investigated in terms of h criterion in the infinite horizon time setting the limitation of h performance in preview control action is characterized with the hamiltonian matrix by employing the property of hamiltonian matrix the solvability and analytic solution for the preview full information control problem are clarified based on an auxiliary introduced matrix riccati equation the property of resulting control law is illustrated with a numerical example

local and global norm comparison theorems for solutions to the nonhomogeneous a harmonic equation abstract we establish local and global norm inequalities for solutions of the nonhomogeneous a harmonic equation a x g d u h d v for differential forms as applications of these inequalities we prove the sobolev poincare type imbedding theorems and obtain l p estimates for the gradient operator and the homotopy operator t from the banach space l s d l to the sobolev space w 1 s d l 1 l 1 2 n these results can be used to study both qualitative and quantitative properties of solutions of the a harmonic equations and the related differential systems

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

h sup spl infin control and estimation with preview part i matrix are solutions in continuous time preview control and fixed lag smoothing allow a noncausal component in the controller estimator time domain variational analysis is used in a reduction to an open loop differential game leading to a complete necessary and sufficient characterization of suboptimal values and an explicit state space design in terms of a parameterized nonstandard algebraic matrix riccati equation in a general continuous time linear system setting the solution offers insight into the appropriate structure of the associated hamiltonian where the state and co state are not the usual state of the original dynamic system and that of its adjoint rather the state and co state are selected to capture the respective lumped effects of initial data and future input selection in the allied game

the monotonic property of ls μ averaging domains and weighted weak reverse holder inequality abstract we first prove a local weighted weak reverse holder inequality for a harmonic tensors then we study the monotonic property of newly introduced l s averaging domains which can be viewed as an application of the local weighted reverse holder inequality in l s averaging domains by applying the local result we also obtain some similar results in other kinds of domains

global weighted inequalities for operators and harmonic forms on manifolds abstract in this paper both the local and global weighted sobolev poincare imbedding inequalities and poincare inequalities for the composition t g are established where t is the homotopy operator and g is green s operator applied to a harmonic forms on manifolds

weighted imbedding theorems in the space of differential forms abstract we prove a r weighted imbedding theorems for differential forms these results can be used to study the weighted norms of the homotopy operator t from the banach space l p d l to the sobolev space w 1 p d l 1 l 1 2 n and to establish the basic weighted l p estimates for differential forms

control of systems with i o delay via reduction to a one block problem in this paper the standard four block h sup spl infin control problem for systems with a single delay in the feedback loop is studied a simple procedure of the reduction of the problem to an equivalent one block problem having particularly simple structure is proposed the one block problem is then solved by the j spectral factorization approach resulting in the so called dead time compensator dtc form of the controller the advantages of the proposed procedure are its simplicity intuitively clear derivation of the dtc form of the h sup spl infin controller and extensibility to the multiple delay case

robust stabilization of uncertain time delay systems via combined internal external approach a robust stabilization problem is discussed for time delay systems by introducing an indefinite riccati type operator equation for time delay systems a design method for a stabilizing control law robust against parametric uncertainties which occur in lumped parameter elements and in all types of delay element is obtained in the derivation of the control law a combined approach is taken an internal approach based on a lyapunov type operator equation is combined with an external approach based on the small gain theorem this enables the uncertainties of time delay systems to be dealt with in a unified way

ar λ weighted integral inequalities for a harmonic tensors abstract in this paper we prove the ar weighted caccioppoli type inequality and weak reverse holder inequality for a harmonic tensors we also obtain the ar weighted hardy littlewood inequality for conjugate a harmonic tensors these inequalities can be considered as extensions of the classical results

on standard h sub spl infin control of processes with a single delay this note presents a frequency domain method to solve the standard h sub spl infin control problem for processes with a single delay for a given bound on the closed loop h sub spl infin norm there exist proper stabilizing controllers that achieve this bound if and only if both the corresponding delay free h sub spl infin problem and an extended nehari problem with a delay or a one block problem are all solvable the solvability of the extended nehari problem or the one block problem is equivalent to the nonsingularity of a delay dependent matrix the solvability conditions of the standard h sub spl infin control problem with a delay are formulated in terms of the existence of solutions to two delay independent algebraic riccati equations and a delay dependent nonsingularity property all suboptimal controllers solving the three problems are respectively parameterized as a structure incorporating a modified smith predictor

quadratic control of evolution equations with delays in control the quadratic cost problem of evolution equations with delays in control is considered a semigroup model which involves no explicit delays in control but contains an unbounded control operator is introduced with the aid of a family of approximating systems it is shown that the optimal feedback control and the minimum cost are characterized by the solution of a riccati equation three examples are given to illustrate the theory the filtering problem of evolution equations with observation delays is also solved through the duality relation

endpoint strichartz estimates and global solutions for the nonlinear dirac equation abstract we prove endpoint strichartz estimates for the klein gordon and wave equations in mixed norms on the polar coordinates in three spatial dimensions as an application global wellposedness of the nonlinear dirac equation is shown for small data in the energy class with some regularity assumption for the angular variable

robust stabilization in the gap metric controller design for distributed plants the problem of robustness in the gap metric for infinite dimensional systems is considered the problem of computing the optimal controller and the optimal robustness radius for a class of systems whose normalized coprime factors have elements which are h sub infinity functions with continuous boundary values is studied the underlying hankel and related operators which are important in the gap optimization problem are studied and relations between their singular values and vectors are established a computational approach to the optimal robustness problem is developed for single input single output systems whose transfer function is an inner function in h sub infinity times a rational function the procedure is applied to a general first order delay system and a closed form formula is obtained for the optimal controller the frequency response plots of the compensated system for various values of time delay are examined

linear quadratic optimal control of hereditary differential systems infinite dimensional riccati equations and numerical approximations recent theory of infinite dimensional riccati equations is applied to the linear quadratic optimal control problem for hereditary differential systems and it is shown that for most such problems the operator solutions of the riccati equations are of trace class i e nuclear with special attention to trace norm convergence an abstract approximation theory is developed and applied to a particular approximation scheme numerical examples are given problems on both finite and infinite time intervals are studied for both the hereditary system and the approximating systems in the infinite time problem characteristic equations are derived for the closed loop eigenvalues and formulas for the corresponding eigenvectors are given

optimal robustness in the gap metric the application of the gap metric to robust stabilization of feedback systems is considered in particular a solution to the problem of robustness optimization in the gap metric is presented the problem of robust stabilization under simultaneous plant controller perturbations is addressed and the least amount of combined plant controller uncertainty measured by the gap metric that can cause instability of a nominally stable feedback system is determined included are a detailed summary of the main properties of the gap metric and the introduction of a dual metric called the t gap metric a key contribution of this study is to show that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations this settles the question as to whether maximizing allowable coprime factor uncertainty corresponds to tolerating the largest ball of uncertainty in a well defined metric

generalized poincare inequalities for solutions to the a harmonic equation in certain domains abstract we first prove local versions of the poincare inequality for solutions to the a harmonic equation then as applications of the local results we obtain the global versions of the poincare inequality for solutions to the a harmonic equation in l s 0 averaging domains and l s averaging domains respectively these results can be considered as generalizations of the classical poincare theorem

the linear quadratic control problem for infinite dimensional systems with unbounded input and output operators this paper establishes a general semigroup framework for solving quadratic control problems with infinite dimensional state space and unbounded input and output operators

robust stabilizability of normalized coprime factors the infinite dimensional case the problem of robustly stabilizing a linear system subject to h bounded perturbations in the numerator and the denominator of its normalized left coprime factorization is considered for a class of infinite dimensional systems this class has possible unbounded finite rank input and output operators which include many delay and distributed systems the optimal stability margin is expressed in terms of the solutions of the control and filter algebraic riccati equations

poincare inequalities with the radon measure for differential forms we establish the local and global poincare inequalities with the radon measure for the solutions to the nonlinear elliptic partial differential equation for differential forms

linking numbers contacts and mutual inductances of a random set of closed curves we establish here a new general result of integral geometry concerning closed rigid curves of arbitrary shapes ine3 and their linking numbersi it generalizes by a different method the interesting integral property ofi2 found recently by pohl and extended by des cloizeaux and ball for two curves we considern closed curves linked successively to each other and forming a ring the cyclic product of their linking numbers is integrated over the group of rigid motions of the curves this integral is shown to factorize over a special algebra of characteristic functions each curve possesses two such intrinsic functions the same algebra is shown to describe a larger class of integral geometry properties a new theorem is established for a family of displacement integrals involving linking numbers contact angles and mutual inductances of the set ofn curves

lipschitz and bmo norm inequalities for operators abstract in this paper we obtain lipschitz norm inequalities and bmo norm inequalities for operators applied to differential forms satisfying the nonhomogeneous a harmonic equation

inequalities for green s operator with lipschitz and bmo norms in this paper we obtain lipschitz norm and bmo norm inequalities for green s operator applied to differential forms we also establish norm comparison theorems for the bmo norm and the lipschitz norm of green s operator

closed loop properties of the infinite time linear quadratic optimal regulator for systems with delays the infinite time linear quadratic optimal control problem for systems with delays is discussed from the viewpoint of closed loop properties the kalman equation and the circle condition for the optimal closed loop control law are derived and the optimal closed loop poles are characterized via the hamiltonian matrix

poincare type inequalities with lp logl α norms for green s operator in this paper we first establish local poincare type inequalities with l p logl a norms for green s operator applied to the solutions of the nonhomogeneous a harmonic equation then we extend the local results to global versions in the l f m domain

the linear quadratic optimal control problem with delays in state and control variables a state space approach this paper generalizes the results of vinter and kwong to a control operator with a more general delay structure than the one in a ichikawa and to delay systems with both finite and infinite memories the linear quadratic optimal control problem over both finite and infinite time horizons is treated several forms of associated abstract riccati differential and algebraic equations are presented the properties of the kernel of the solution are studied and a coupled set of matrix ordinary and partial differential equations is obtained

weighted integral inequalities for solutions of the a harmonic equation we prove the parametric versions of ar weighted integral inequalities for differential forms satisfying the a harmonic equation these results can be considered as extensions of the classical inequalities for sobolev functions

robust stabilization of delay systems by approximation of coprime factors given a delay system with transfer function g s h2 s h1 s where h1 s 01npi s e is and h2 s 02nqi s e is with 0 0 1 n1 0 0 n2 the pi being polynomials of degree i and i 0for i 0 and the qi polynomials of degree di 0 for each i the robust stabilization of a class of perturbed coprime factors of this system is considered asymptotic estimates are obtained based on recent results on the approximation and stabilization of normalized coprime factors an explicit formula is given for the normalized coprime factor stability margin for the case of a multivariable transfer function of the form g s e str s with r rational

rigid curves at random positions and linking numbers a property of the square of the linking number of two closed rigid curves randomly displaced in a three dimensional space has been recently found by w pohl here this result is reproduced and generalized this new approach is quite different and uses a simple fourier transformation

linking numbers of closed manifolds at random in ℝ n inductances and contacts we give here new results of topology and integral geometry concerning the gauss linking number i of closed manifolds inn dimensional space the rigid manifolds have arbitrary shapes and dimensions and are statistically at random positions in n generalizing pohl s work for two closed manifoldsc 1 r c 2 s of respective dimensionsr ands with 0 r n 1 andr s 1 n we consider the kinematic linking integral i of the square linking number i ofc 1 r andc 2 s over the group of euclidean motions of one manifold translationsx rotationso introducing a new tensorial method and using group theory we show quite generally thati num fact int limits 0 infty d varrho mathcal a 1 varrho mathcal a 2 varrho delta r s mathcal b 1 varrho mathcal b 2 varrho where is a length variable and wherea 1 2 are characteristic functions associated with the manifoldc only we study functionsa and of a manifoldc r of dimensionr in all cases 0 r n 1 a always exists a 0 givesc s area whereas int limits 0 infty mathcal a varrho d varrho equals the interior volume of a hypersurfacec is found to exist and not to vanish only if 2 dimc 1 n andn 3 4q 3 7 11 a and are explicitly calculated for segments andr spheress r as an application the topological excluded volume of a gas of nonlinked spheress r moving in 2r 1 is calculated we generalize ton manifoldsc 1 n linked successively to each other and forming a ring the cyclic product of their linking numbers is integrated over the group of motions of the manifolds it is shown to factorize completely in fourier space with special algebraic rules over the set of 2n characteristic functionsa associated with thec s the same algebra of characteristic functions is shown to describe a larger class of topology and electromagnetism properties a new theorem is given for a family of euclidean group integrals involving the random linking numbers mutual inductances and contact distributions ofn manifolds

dirac structures and their composition on hilbert spaces dirac structures appear naturally in the study of certain classes of physical models described by partial differential equations and they can be regarded as the underlying power conserving structures we study these structures and their properties from an operator theoretic point of view in particular we find necessary and sufficient conditions for the composition of two dirac structures to be a dirac structure and we show that they can be seen as lagrangian hyper maximal neutral subspaces of krein spaces moreover special emphasis is laid on dirac structures associated with operator colligations it turns out that this class of dirac structures is linked to boundary triplets and that this class is closed under composition

two weight imbedding inequalities for solutions to the a harmonic equation abstract we obtain the two weight imbedding inequalities for solutions to the a harmonic equation and explore their applications to the k quasiregular mappings in r n these results can be used to study the properties of the homotopy operator t l p d l w 1 p d l 1 l 1 2 n and to establish the weighted l p estimate for solutions to the a harmonic equation

construction of strict lyapunov krasovskii functionals for time varying time delay systems abstract for stability analysis of time varying time delay systems it is known that time derivatives time shifts in the discrete time setting of the lyapunov krasovskii functionals lkfs for both internal stability and input to state stability iss can be indefinite which can improve the resulting stability conditions in some cases however the non strict lkfs may be insufficient and inconvenient to use in practice in this paper based on the non strict iss lkfs for time varying time delay systems and with the help of uniformly asymptotically stable uas and uniformly exponentially bounded ueb scalar functions three classes of strict iss lkfs are constructed such that their time derivatives are strictly negative definite along the trajectories of the considered system a general construction of strict iss lkfs by using the positive definite and uniformly bounded solution to a scalar lyapunov differential equation is established which includes the proposed three classes of strict iss lkfs as special cases the approaches are also extended to deal with a lyapunov inequality whose right hand side is indefinite and nonlinear both continuous time and discrete time systems are considered the effectiveness of the proposed methods is illustrated by some examples borrowed from the literature

global poincare inequalities for green s operator applied to the solutions of the nonhomogeneous a harmonic equation both local and global a r weighted poincare inequalities for green s operator applied to the solutions of the nonhomogeneous a harmonic equation in john domains are established as applications of the poincare inequalities for green s operator the a r weighted sobolev imbedding inequalities for green s operator in john domains are also proved

ar ω weighted inequalities for a harmonic tensors and related operators abstract we first prove the local a r weighted poincare inequality for a harmonic tensors on domain and the analogue for the projection operator then we prove the global a r weighted poincare inequality for a harmonic tensors finally using the global a r weighted inequality established for a harmonic tensors we obtain the global a r weighted poincare inequality for the projection operator

estimates of weighted integrals for differential forms in this paper we obtain some weighted integral estimates for differential forms which are generalizations of the poincare inequality the caccioppoli type estimate and the weak reverse holder inequality respectively these results can be used to study the integrability of differential forms and to estimate the integrals for differential forms

linking probabilities of off lattice self avoiding polygons and the effects of excluded volume we evaluate numerically the probability of linking i e the probability of a given pair of self avoiding polygons saps being entangled and forming a nontrivial link type l in the simulation we generate pairs of saps of n spherical segments of radius rd such that they have no overlaps among the segments and each of the saps has the trivial knot type we evaluate the probability of a self avoiding pair of saps forming a given link type l for various link types with fixed distance r between the centers of mass of the two saps we define normalized distance r by where denotes the square root of the mean square radius of gyration of sap of the trivial knot 01 we introduce formulae expressing the linking probability as a function of normalized distance r which gives good fitting curves with respect to 2 values we also investigate the dependence of linking probabilities on the excluded volume parameter rd and the number of segments n quite interestingly the graph of linking probability versus normalized distance r shows no n dependence at a particular value of the excluded volume parameter rd 0 2

l φ μ averaging domains and poincare inequalities with orlicz norms abstract we characterize l averaging domains using the whitney covers and the quasihyperbolic metric and study the invariance of l averaging domains under some mappings as applications of the l averaging domains we prove the poincare inequality with orlicz norms for solutions of the non homogeneous a harmonic equation in l averaging domains

global estimates for compositions of operators applied to differential forms in this paper we obtain some global l p estimates for compositions of operators applied to differential forms satisfying the a harmonic equation these estimates indicate that the operators tdg and tg are bounded

random linking of lattice polygons we study the linking probability of polygons on the simple cubic lattice in particular we consider two polygons each having n edges confined to a cube of side l and ask for the linking probability as a function of n and l we also consider other situations in which the polygons are restricted to be not too far apart but not necessarily confined to a cube we prove several rigorous results and use monte carlo methods to address some questions which we are unable to answer rigorously an interesting feature is that the linking probability is a function of l nv where v is the exponent characterizing the radius of gyration of a polygon

h sup spl infin control and estimation with preview part ii fixed size are solutions in discrete time h sup spl infin preview control and fixed lag smoothing problems are solved in general discrete time linear systems via a reduction to equivalent open loop differential games to prevent high order riccati equations found in some solutions the state of the hamilton jacobi system resides in a quotient space of an auxiliary extended state space system the dimension of that auxiliary space is equal to the state space dimension of the original system ignoring the delay

higher integrability of green s operator and homotopy operator abstract we prove the higher integrability and higher order imbedding theorems of the composition of homotopy operator and green s operator we also study analogues of the homotopy operator and green s operator respectively

finite rank criteria for h infty control of infinite dimensional systems this paper studies the hinfin control of infinite dimensional systems whose transfer matrices are expressible as a series connection of a rational transfer matrix and a scalar possibly irrational inner function this class of systems is adequate for describing many control problems in practice when weighting functions are rational and plants have at most finitely many unstable modes we show that this problem can be reduced to solving two matrix valued riccati equations and an additional rank condition furthermore the obtained controller structure is characterized by the inner function and controllers for the finite dimensional part this result provides us with a solution that is easily implementable without much prior knowledge on infinite dimensional control theory a numerical example is given to illustrate the result

on the central controller characterizations via differential games and leqg control problems abstract we provide a game theoretic interpretation to the central controller which plays a central role in characterizing all controllers achieving disturbance attenuation in the sense of the h norm in the imperfect observation case we also establish a direct relation between the central controller and the stationary leqg controller for continuous time systems

some examples of conjugatep harmonic differential forms abstract there has been new interest developing in the study of the hodge theory and p harmonic differential forms in recent years largely pertaining to applications in quasi conformal analysis electromagnetic theory relativity theory mathematical physics nonlinear potential theory etc in this paper we develop an effective method to find conjugate p harmonic differential forms in r n by applying hodge theory in calculations with the p harmonic equation as applications of the method we obtain some interesting examples of conjugate p harmonic differential forms