parameter dependent lmis in robust analysis characterization of homogeneous polynomially parameter dependent solutions via lmi relaxations this note investigates the robust stability of uncertain linear time invariant systems in polytopic domains by means of parameter dependent linear matrix inequality pd lmi conditions exploiting some algebraic properties provided by the uncertainty representation a systematic procedure to construct a family of finite dimensional lmi relaxations is provided the robust stability is assessed by means of the existence of a lyapunov function more specifically a homogeneous polynomially parameter dependent lyapunov hppdl function of arbitrary degree for a given degree if an hppdl solution exists a sequence of relaxations based on real algebraic properties provides sufficient lmi conditions of increasing precision and constant number of decision variables for the existence of an hppdl function which tend to the necessity alternatively if an hppdl solution of degree exists a sequence of relaxations which increases the number of variables and the number of lmis will provide an hppdl solution of larger degree the method proposed can be applied to determine homogeneous parameter dependent matrix solutions to a wide variety of pd lmis by transforming the infinite dimensional lmi problem described in terms of uncertain parameters belonging to the unit simplex in a sequence of finite dimensional lmi conditions which converges to the necessary conditions for the existence of a homogeneous polynomially parameter dependent solution of arbitrary degree illustrative examples show the efficacy of the proposed conditions when compared with other methods from the literature

a less conservative robust stability test for linear uncertain time delay systems this note concerns with the robust stability of linear uncertain systems with state delay the uncertainty is assumed to be of polytopic type a less conservative delay dependent linear matrix inequality lmi method is presented based on a new lyapunov krasovskii functional the present method incorporates a relaxed parameter dependent technique combined with a recently proposed idea of introducing free weighting matrices when confined to delay free case the present result is also less conservative than existing stability tests using parameter dependent methods numerical examples are given to show the less conservativeness of the results

affine parameter dependent lyapunov functions and real parametric uncertainty this paper presents new tests to analyze the robust stability and or performance of linear systems with uncertain real parameters these tests are extensions of the notions of quadratic stability and performance where the fixed quadratic lyapunov function is replaced by a lyapunov function with affine dependence on the uncertain parameters admittedly with some conservatism the construction of such parameter dependent lyapunov functions can be reduced to a linear matrix inequality lmi problem and hence is numerically tractable these lmi based tests are applicable to constant or time varying uncertain parameters and are less conservative than quadratic stability in the case of slow parametric variations they also avoid the frequency sweep needed in real spl mu analysis and numerical experiments indicate that they often compare favorably with spl mu analysis for time invariant parameter uncertainty

a less conservative lmi condition for the robust stability of discrete time uncertain systems abstract in this paper a less conservative condition for the robust stability of uncertain discrete time linear systems is proposed the uncertain parameters assumed to be time invariant are supposed to belong to convex bounded domains polytope type uncertainty the stability condition is formulated in terms of a set of linear matrix inequalities involving only the vertices of the polytope domain a simple and numerically efficient feasibility test provides a set of lyapunov matrices whose convex combination can be used to assess the stability of any dynamic matrix inside the uncertainty domain examples illustrate the results

a linear programming oriented procedure for quadratic stabilization of uncertain systems abstract this paper gives a new necessary and sufficient condition for linear quadratic stabilization of linear uncertain systems when both the dynamic as well as the control matrix are subject to uncertainty a constructive numerical procedure is defined to check the condition and it furthermore provides a stabilizing linear feedback gain some experiments are presented

new approach to robust cal d stability analysis of linear time invariant systems with polytope bounded uncertainty this note presents a new approach to robust d stability analysis of linear time invariant systems with polytope bounded uncertainty the proposed approach combines sufficient conditions for robust d stability in terms of feasibility problems with linear matrix inequalities lmi constraints and a new polytope partition technique if the initial polytope does not attain the robust d stability sufficient condition the polytope is successively subdivided until all subpolytopes attain the sufficient condition in the case of robustly d stable uncertain system or it is found a subpolytope vertex that does not attain the regional pole placement constraints in the case of an uncertain system that is not robustly d stable it is presented a new general format polytope partition technique that allows the implementation of the proposed approach the efficiency of the proposed approach is verified by means of illustrative examples and three different lmi based analysis formulations

robust absolute stability and nonlinear state feedback stabilization based on polynomial lur e functions this paper provides finite dimensional convex conditions to construct homogeneous polynomially parameter dependent lur e functions which ensure the stability of nonlinear systems with state dependent nonlinearities lying in general sectors and which are affected by uncertain parameters belonging to the unit simplex the proposed conditions are written as linear matrix inequalities parametrized in terms of the degree g of the parameter dependent solution and in terms of the relaxation level d of the inequality constraints based on the algebraic properties of positive matrix polynomials with parameters in the unit simplex as g and d increase progressively less conservative solutions are obtained the results in the paper include as special cases existing conditions for robust stability and for absolute stability analysis a convex solution suitable for the design of robust nonlinear state feedback stabilizing controllers is also provided numerical examples illustrate the efficiency of the proposed conditions c 2008 elsevier ltd all rights reserved

space time tradeoffs for emptiness queries we develop the first nontrivial lower bounds on the complexity of online hyperplane and halfspace emptiness queries our lower bounds apply to a general class of geometric range query data structures called partition graphs informally a partition graph is a directed acyclic graph that describes a recursive decomposition of space we show that any partition graph that supports hyperplane emptiness queries implicitly defines a halfspace range query data structure in the fredman yao semigroup arithmetic model with the same asymptotic space and time bounds thus results of bronnimann chazelle and pach imply that any partition graph of size s that supports hyperplane emptiness queries in time t satisfies the inequality st d omega n log n d d 1 d 1 using different techniques we improve previous lower bounds for hopcroft s problem given a set of points and hyperplanes does any hyperplane contain a point in dimensions four and higher using this offline result we show that for online hyperplane emptiness queries omega n d mbox polylog n space is required to achieve polylogarithmic query time and omega n d 1 d mbox polylog n query time is required if only o n polylog n space is available these two lower bounds are optimal up to polylogarithmic factors for two dimensional queries we obtain an optimal continuous tradeoff st 2 omega n 2 between these two extremes finally using a lifting argument we show that the same lower bounds hold for both offline and online halfspace emptiness queries in mathbb r d d 3 2

robust stability and performance of linear time varying systems in polytopic domains a sufficient condition to assess the stability of linear time varying systems in polytopic domains is given in this paper this condition is written as a feasibility test of linear matrix inequalities expressed at the polytope vertices and taking bounds on the time derivatives of the system parameters into account differently from other techniques in the literature there is no need of gridding procedures on the parameter space neither restrictive assumptions on the uncertainty structure the proposed test can be solved in polynomial time yielding as solution a parameter dependent lyapunov function that assures the system stability moreover the results can be applied to systems with affine parameter uncertainty and can be easily extended to deal with guaranteed cost computation numerical examples show that the proposed condition can lead to less conservative evaluations of the stability domain and of the performance index for this class of systems when compared to the results based on the conventional

lmi characterization of structural and robust stability abstract this paper introduces several stability conditions for a given class of matrices expressed in terms of linear matrix inequalities lmi being thus simply and efficiently computable diagonal and simultaneous stability both characterized by polytopes of matrices are addressed using this approach a method particularly attractive to test a given matrix for d stability is proposed lyapunov parameter dependent functions are built in order to reduce conservativeness of the stability conditions the key idea is to relate hurwitz stability with a positive realness condition

polynomially parameter dependent lyapunov functions for robust stability of polytopic systems an lmi approach in this note robust stability of state space models with respect to real parametric uncertainty is considered specifically a new class of parameter dependent quadratic lyapunov functions for establishing stability of a polytope of matrices is introduced i e the homogeneous polynomially parameter dependent quadratic lyapunov functions hpd qlfs the choice of this class which contains parameter dependent quadratic lyapunov functions whose dependence on the uncertain parameters is expressed as a polynomial homogeneous form is motivated by the property that a polytope of matrices is stable if and only there exists an hpd qlf the main result of the note is a sufficient condition for determining the sought hpd qlf which amounts to solving linear matrix inequalities lmis derived via the complete square matricial representation csmr of homogeneous matricial forms and the lyapunov matrix equation numerical examples are provided to demonstrate the effectiveness of the proposed approach

an lmi condition for the robust stability of uncertain continuous time linear systems a new sufficient condition for the robust stability of continuous time uncertain linear systems with convex bounded uncertainties is proposed in this note the results are based on linear matrix inequalities lmis formulated at the vertices of the uncertainty polytope which provide a parameter dependent lyapunov function that assures the stability of any matrix inside the uncertainty domain with the aid of numerical procedures based on unidimensional search and the lmis feasibility tests a simple and constructive way to compute robust stability domains can be established

geometric retrieval problems a large class of geometric retrieval problems has the following form given a set x of geometric objects preprocess to obtain a data structure d x now use d x to rapidly answer queries on x we say an algorithm for such a problem has worst case space time complexity o f n g n if the space requirement for d x is o f and the locate run time required for each retrieval is o g we show three techniques which can consistently be exploited in solving such problems for instance using our techniques we obtain an o n2 e log n log n log 1 e space time algorithm for the polygon retrieval problem for arbitrarily small e improving on the previous solution having complexity o n7 log n

existence and nonexistence results for positive solutions of an integral boundary value problem abstract this paper deals with the existence and nonexistence of positive solutions for the integral boundary value problem a u b u g t f t u cos 0 u 0 sin 0 u 0 0 1 u d cos 1 u 1 sin 1 u 1 0 1 u d we provide sufficient conditions for the existence of either none or one or more positive solutions of the above problem the main tool used in the proofs of existence results is a fixed point theorem in a cone due to krasnoselskii and zabreiko the results presented substantially unify and extend the existing results in the literature

on sign changing solution for some three point boundary value problems abstract in this paper an existence result of sign changing solution for some three point boundary value problems is obtained by using the fixed point index method

approximating the weight of the euclidean minimum spanning tree in sublinear time we consider the problem of computing the weight of a euclidean minimum spanning tree for a set of n points in mathbb r d we focus on the setting where the input point set is supported by certain basic and commonly used geometric data structures that can provide efficient access to the input in a structured way we present an algorithm that estimates with high probability the weight of a euclidean minimum spanning tree of a set of points to within 1 eps using only widetilde o sqrt n text poly 1 eps queries for constant d the algorithm assumes that the input is supported by a minimal bounding cube enclosing it by orthogonal range queries and by cone approximate nearest neighbor queries

multiple solutions for fourth order boundary value problem abstract in this paper we study the existence and multiplicity of nontrivial solutions for the fourth order two point boundary value problems making use of the theory of fixed point index in cone and leray schauder degree under general conditions on nonlinearity we prove that there exist at least six different nontrivial solutions for the fourth order two point boundary value problems furthermore if the nonlinearity is odd we obtain that there exist at least eight different nontrivial solutions

robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties this paper studies the static output feedback sof control problem of continuous time linear systems with polytopic uncertainties novel lmi conditions with a line search over a scalar variable for designing robust sof controllers are proposed where the uncertain output matrix of the considered system is allowed to be not of full row rank in particular it is shown that the new method can give less or at least the same conservative results than those methods by inserting a matrix equality constraint between system output matrix and lyapunov matrix furthermore the result is extended to the case of h control numerical examples are given to illustrate the effectiveness of the proposed method

an improved lmi condition for robust d stability of uncertain polytopic systems a sufficient condition for robust d stability of linear systems with polytope type uncertainties is proposed the result is based on a linear parameter dependent lyapunov function obtained from the feasibility test of a set of linear matrix inequalities lmis defined at the vertices of the polytope this improved lmi condition encompasses previous results based on additional inequalities as well as results based on extra variables

technical communique further result on reachable set bounding for linear uncertain polytopic systems with interval time varying delays the problem of reachable set estimation of linear uncertain polytopic time varying delay systems subject to bounded peak inputs is studied in this paper the delays considered in this paper are assumed to be non differentiable and vary within an interval where the lower and upper bounds are known based on the lyapunov krasovskii method and delay decomposition technique a sufficient condition for the existence of a ball that bounds the reachable set of the system is proposed in terms of matrix inequalities containing only one scalar which can be solved by using an one dimensional search method and matlab s lmi toolbox and allow us to find the smallest radius a numerical example is given to illustrate the effectiveness of the proposed result

a new lmi condition for robust stability of polynomial matrix polytopes a sufficient condition for checking the robust stability of a polytope of polynomial matrices is proposed in this note a simple feasibility test performed in a convex set of linear matrix inequalities defined at the vertices of the polytope yields sufficient conditions for the robust stability of the entire domain both continuous time left half plane and discrete time stability unit disk are investigated numerical comparisons with quadratic stability results and with another method recently appeared in the literature show that the conditions proposed provide in general less conservative results

on uniform stabilization of discrete time linear parameter varying control systems for discrete time polytopic linear parameter varying systems the uniform exponential stability which is equivalent to the asymptotic stability and includes the quadratic stability as a special case is characterized by the union of an increasing family of linear matrix inequality conditions on the other hand in certain cases nonconservative synthesis of uniformly stabilizing controllers is achieved via a system of finite number of linear matrix inequalities if the controller is allowed to have finite memory of past parameters two such cases are robust state feedback in a relaxed sense against polytopic parameter variations and multiple output injections under polytopic fusion rules

on sign changing solutions for nonlinear operator equations abstract in this paper the existence of sign changing solutions for nonlinear operator equations is discussed by using the topological degree and fixed point index theory the main theorems are some new three solution theorems which are different from the famous amann s and leggett williams three solution theorems as well as the results in f li g han generalization for amann s and leggett williams three solution theorems and applications j math anal appl 298 2004 638 654 these three solutions are all nonzero one of them is positive another is negative and the third one is a sign changing solution furthermore the theoretical results are successfully applied to both integral and differential equations

robust h filter design with past output measurements for uncertain discrete time systems this paper is concerned with the design of robust filters for uncertain discrete time linear systems the uncertainty is assumed to be time invariant belonging to a polytope an upper bound to the h norm of the transfer function from the system input to the filtering error is used as performance criterion the novelty is that the resulting robust filter has order greater than the order of the system being filtered by incorporating past output measurements the design conditions are expressed in terms of convex problems constrained by linear matrix inequalities performance is assessed through parameter dependent lyapunov functions the results generalize existing robust filtering procedures without bringing in any additional conservatism simple numerical examples illustrate the procedure

positive solutions to a system of second order nonlocal boundary value problems abstract in this paper we study the existence of positive solutions to a system of second order nonlocal boundary value problems by using fixed point index theory in a cone our hypotheses imposed on nonlinearities are those which characterize systems of nonlocal boundary value problems and our boundary value conditions are expressed in terms of possibly nonlinear functions of riemann stieltjes integrals thus generalizing and unifying the boundary value conditions in the literature therefore our results cannot be routinely deduced from the ones for single nonlocal problem in the literature

the range 1 query r1q problem we define the range 1 query r1q problem as follows given a d dimensional d 1 input bit matrix a preprocess a so that for any given region mathcal r of a one can efficiently answer queries asking if mathcal r contains a 1 or not we consider both orthogonal and non orthogonal shapes for mathcal r including rectangles axis parallel right triangles certain types of polygons and spheres we provide space efficient deterministic and randomized algorithms with constant query times in constant dimensions for solving the problem in the word ram model the space usage in bits is sublinear linear or near linear in the size of a depending on the algorithm

structure of positive solution sets of semi positone singular boundary value problems abstract in this paper we first obtain some results on the structure of positive solution sets of some differential boundary value problems then by using these results we obtain some existence and multiplicity results for positive solutions of differential boundary value problems the method used to show the main results is that of global bifurcation theories

static output feedback control synthesis for linear systems with time invariant parametric uncertainties this technical note is concerned with the problem of designing robust static output feedback controllers for linear discrete and continuous time systems with time invariant polytopic uncertainties sufficient conditions for static output feedback stabilizing controller designs are given in terms of solutions to a set of linear matrix inequalities furthermore the results are also extended to h2 static output feedback controller designs numerical examples are given to illustrate the effectiveness of the proposed design methods

lower bounds on the complexity of some optimal data structures a technique is presented for deriving lower bounds on the complexity of optimal data structures which permit insertions and deletions of records and queries of the form text query region quad text return quad sum begin subarray l text key r in text region end subarray text value r quad text region in gamma where value r the value associated with a record r lies in a commutative semi group s and gamma denotes a set of regions of the space of possible keys this technique is illustrated with several examples

stability of polytopes of matrices via affine parameter dependent lyapunov functions asymptotically exact lmi conditions abstract this paper investigates necessary and sufficient conditions for the existence of an affine parameter dependent lyapunov function assuring the hurwitz or schur stability of a polytope of matrices a systematic procedure for constructing a family of linear matrix inequalities conditions of increasing precision is given at each step a set of linear matrix inequalities provides sufficient conditions for the existence of the affine parameter dependent lyapunov function necessity is asymptotically attained through a relaxation based on a generalization of polya s theorem to the case of matrix valued functions numerical experiments illustrate the results

existence and nonexistence results of impulsive first order problem with integral boundary condition abstract this paper is concerned with a first order impulsive boundary value problem ibvp for short with integral boundary condition and depending on a parameter we obtain the existence and nonexistence results of positive solutions for the ibvp when the nonlinear term and the impulses satisfy different requirements of superlinearity and sublinearity and the parameter lies in some intervals

analysis and synthesis of robust control systems using linear parameter dependent lyapunov functions this note provides sufficient robust stability conditions for continuous time polytopic systems they are obtained from the frobenius perron theorem applied to the time derivative of a linear parameter dependent lyapunov function and are expressed in terms of linear matrix inequalities lmi they contain as special cases various sufficient stability conditions available in the literature to date as a natural generalization the determination of a guaranteed h2 cost is addressed a new gain parametrization is introduced in order to make possible the state feedback robust control synthesis using parameter dependent lyapunov functions through linear matrix inequalities numerical examples are included for illustration

lower bounds for orthogonal range searching i the reporting case we establish lower bounds on the complexity of orthogonal range reporting in the static case given a collection of n points in d space and a box a 1 b 1 x x a d b d report every point whose i th coordinate lies in a i b i for each i l d the collection of points is fixed once and for all and can be preprocessed the box on the other hand constitutes a query that must be answered online it is shown that on a pointer machine a query time of o k polylog n where k is the number of points to be reported can only be achieved at the expense of o n log n log log n d 1 storage interestingly these bounds are optimal in the pointer machine model but they can be improved ever so slightly on a random access machine in a companion paper we address the related problem of adding up weights assigned to the points in the query box

a new framework for addressing temporal range queries and some preliminary results given a set of n objects each characterized by d attributes specified at m fixed time instances we are interested in the problem of designing space efficient indexing structures such that a class of temporal range search queries can be handled efficiently when m 1 our problem reduces to the d dimensional orthogonal search problem we establish efficient data structures to handle several classes of the general problem our results include a linear size data structure that enables a query time of o log n log m f for one sided queries when d 1 where f is the number of objects satisfying the query a similar result is shown for counting queries we also show that the most general problem can be solved with a polylogarithmic query time using superlinear space data structures

sign changing solutions to second order integral boundary value problems abstract in this paper by using the fixed point index theory and leray schauder degree theory we consider the existence and multiplicity of sign changing solutions to nonlinear second order integral boundary value problem u t f u t for all t 0 1 subject to u 0 0 and u 1 g 0 1 u s d s where f g c r r we obtain some new existence results concerning sign changing solutions by computing hardly eigenvalues and the algebraic multiplicities of the associated linear problem if f and g satisfy certain conditions then this problem has at least six different nontrivial solutions two positive solutions two negative solutions and two sign changing solutions moreover if f and g are also odd then the problem has at least eight different nontrivial solutions which are two positive two negative and four sign changing solutions

dynamic orthogonal range queries in olap we study the problem of pre computing auxillary information to support on line range queries for the sum and max functions on a datacube for a d dimensional datacube with size n in each dimension we propose a dynamic range max data structure with o d s n ld query time o d s n l2dnd l update time and o sd storage where l and s are parameters chosen by the user before the construction of the data structure such that l 1 log n and s is a non negative multiple of n and s n is the functional inverse of ackermann s function moreover the data structure can be initialized in time linear to its size there are three major techniques employed in designing the data structure namely a technique for trading query and update times a technique for trading query time and storage and a technique for extending one dimensional data structures to d dimensional ones our techniques are also applicable to range queries over any semigroup and group operation such as min sum and count

computing closest and farthest points for a query segment in this paper we present an improved algorithm for finding k closest farthest points for a given arbitrary query segment we show how to preprocess a planar set p of n given points in o n 2logn expected time or alternatively in o n 2log 2n deterministic time and a subquadratic space in order to report k closest points to an arbitrary given query line segment in o k log 2nloglogn time here for the first time the data structure that provides polylogarithmic query time and uses subquadratic space is presented we also show an algorithm for reporting the k farthest points from an arbitrary given query line segment

improved lmi representations for the analysis and the design of continuous time systems with polytopic type uncertainty simple modifications to the representation of the bounded real lemma brl in the linear continuous time invariant case are introduced these modifications reduce the overdesign that occurs in the analysis and the design of systems with polytopic type uncertainties a different lyapunov function accompanies each of the vertices of the uncertainty polytope thus eliminating the need for quadratic stability or stabilizability the advantages of these new representations are demonstrated by way of two examples

piecewise linear quadratic optimal control the use of piecewise quadratic cost functions is extended from stability analysis of piecewise linear systems to performance analysis and optimal control lower bounds on the optimal control cost are obtained by semidefinite programming based on the bellman inequality this also gives an approximation to the optimal control law an upper bound to the optimal cost is obtained by another convex optimization problem using the given control law a compact matrix notation is introduced to support the calculations and it is proved that the framework of piecewise linear systems can be used to analyze smooth nonlinear dynamics with arbitrary accuracy

an efficient k nearest neighbors searching algorithm for a query line we present an algorithm for finding k nearest neighbors of a given query line among a set of n points distributed arbitrarily on a two dimensional plane our algorithm requires o n2 time and o n2 log n space to preprocess the given set of points and it answers the query for a given line in o k log n time where k may also be an input at the query time almost a similar technique works for finding k farthest neighbors of a query line keeping the time and space complexities invariant we also show that if k is known at the time of preprocessing the time and space complexities for the preprocessing can be reduced keeping the query times unchanged

multiple sign changing solutions to the sturm liouville boundary value problem with resonance abstract in this paper we study the sturm liouville boundary value problem p x u q x u f x u x 0 1 subject to u 0 u 0 u 1 u 1 0 by constructing a new sobolev space h 1 0 1 we discuss the existence of multiple solutions especially the existence of multiple sign changing solutions to this problem when the nonlinear f is resonant both at 0 and by combining the methods of the morse theory the topological degree and the fixed point index we establish a multiple solutions theorem which guarantees that the problem has at least six nontrivial solutions if this problem has only finitely many solutions then of these solutions there are two positive solutions two negative solutions and two sign changing solutions

lmi conditions for robust stability analysis based on polynomially parameter dependent lyapunov functions abstract the robust stability of uncertain linear systems in polytopic domains is investigated in this paper the main contribution is to provide a systematic procedure for generating sufficient robust stability linear matrix inequality conditions based on homogeneous polynomially parameter dependent lyapunov matrix functions of arbitrary degree on the uncertain parameters the conditions exploit the positivity of the uncertain parameters being constructed in such a way that as the degree of the polynomial increases the number of linear matrix inequalities and free variables increases and the test becomes less conservative if a feasible solution exists for a certain degree the conditions will also be verified for larger degrees for any given degree the feasibility of a set of linear matrix inequalities defined at the vertices of the polytope assures the robust stability both continuous and discrete time uncertain systems are addressed as illustrated by numerical examples

on the complexity of maintaining partial sums let f bf r i s i 0 leqq i n be a file of n records where bf r i are d dimensional vectors and s i are elements of a commutative semigroup s we are interested in the partial sum problem in which queries of the form sum r i leqq bf a s i are to be answered a space time tradeoff t omega log n log m log n n is established for storing a static two dimensional file it will also be shown that for the one dimensional problem any dynamic algorithm must have a worst case time omega n log n log log n in processing a sequence o n insert and query instructions

convergent lmi relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter dependent lyapunov functions abstract this paper investigates the problems of checking robust stability and evaluating robust 2 performance of uncertain continuous time linear systems with time invariant parameters lying in polytopic domains the novelty is the ability to check robust stability by constructing a particular parameter dependent lyapunov function which is a polynomial function of the uncertain system matrices as opposed to a general polynomial function of the uncertain parameter the degree of the polynomial is tied to a certain integer the existence of such lyapunov function can be proved by solving parameter dependent linear matrix inequalities lmis which are guaranteed to be solvable for a sufficiently large yet finite value of whenever the system is robustly stable extensions to guaranteed 2 cost computation are also provided numerical aspects concerning the programming and the evaluations of the proposed tests are discussed and illustrated by examples

lower bounds for orthogonal range searching part ii the arithmetic model lower bounds on the complexity of orthogonal range searching in thestatic case are established specifically we consider the followingdominance search problem given a collection of n weighted points in d space and a query point q compute the cumulative weight ofthe points dominated in all coordinates by q it is assumed that the weights arechosen in a commutative semigroup and that the query time measures onlythe number of arithmetic operations needed to compute the answer it isproved that if m units of storage areavailable then the query time is at least proportional to log n log 2 m n d 1 in both the worst and average cases this lower bound is provably tightfor m o n log n d 1 e and any fixed e 0 a lower bound of o n loglog n d on the time required for executing n inserts and queries is also established author s abstract

new results on output feedback h_ infty control for linear discrete time systems this note investigates the problem of output feedback h control for linear discrete time systems three types of h controllers are considered namely static output feedback controllers dynamic output feedback controllers and observer based output feedback controllers new design conditions for the three type of output feedback controllers are introduced in terms of unified linear matrix inequality lmi representations which guarantee the prescribed h performances of the closed loop systems in contrast to the existing lmi conditions for designing the output feedback h controllers the improvement of the proposed results over the existing ones is shown by theoretical proof and numerical example

robust h performance using lifted polynomial parameter dependent lyapunov functions this paper investigates the robust h performance of time invariant linear uncertain systems where the uncertainty is in polytopic domains robust h is checked by constructing a quadratic parameter dependent lyapunov function the matrix associated with this quadratic lyapunov function is a polynomial function of the uncertain parameters expressed as a particular polynomial matrix involving powers of the dynamic matrix of the system and one symmetric matrix to be determined the degree of this polynomial matrix function is arbitrary finsler s lemma is used to lift the obtained stability conditions into a larger space in which sufficient stability tests can be developed in the form of linear matrix inequalities as increases less conservative h evaluations are obtained both continuous and discrete time systems are investigated numerical examples illustrate the method and compare the present results with similar works in the literature

a parameter dependent lyapunov function for a polytope of matrices a new sufficient condition for a polytope of matrices to be hurwitz stable is presented the stability is a consequence of the existence of a parameter dependent quadratic lyapunov function which is assured by a certain linear constraint for generating extreme matrices of the polytope the condition can be regarded as a duality of the known extreme point result on quadratic stability of matrix polytopes where a fixed quadratic lyapunov function plays the role the obtained results are applied to a polytope of second degree polynomials for illustration

robust h control for polytopic nonlinear control systems this technical note develops a robust control storage function method for solving the h control problem of single input polytopic nonlinear control systems necessary and sufficient conditions for verifying robust control storage functions are provided moreover conditions for the existence of continuous robust h state feedback controllers are derived an explicit formula for constructing continuous robust h controllers is presented finally a numerical example is given for illustration

a new robust d stability condition for real convex polytopic uncertainty abstract the problem of robust d stability analysis with respect to real convex polytopic uncertainties is tackled a new lmi based sufficient condition for the existence of parameter dependent lyapunov functions is proposed this condition generalises previously published conditions numerical comparisons with quadratic stability results as well as previous results based on parameter dependent lyapunov functions illustrate the relevance of this new condition finally this result appears to be promising for robust multi objective performance analysis and control synthesis purposes

lyapunov based exact stability analysis and synthesis for linear single parameter dependent systems we propose a class of polynomially parameter dependent quadratic ppdq lyapunov functions for assessing the stability of single parameter dependent linear time invariant s pdlti systems in a non conservative manner it is shown that the stability of s pdlti systems is equivalent to the existence of a ppdq lyapunov function a bound on the degree of the polynomial dependence of the lyapunov function in the system parameter is given explicitly the resulting stability conditions are expressed in terms of a set of matrix inequalities whose feasibility over a compact and connected set can be cast as a convex finite dimensional optimisation problem extensions of the approach to state feedback controller synthesis are also provided

existence of nontrivial solutions for a nonlinear sturm liouville problem with integral boundary conditions abstract this paper deals with the existence of nontrivial solutions for a nonlinear singular sturm liouville problem with integral boundary conditions

h guaranteed cost computation by means of parameter dependent lyapunov functions a linear matrix inequality approach to compute h guaranteed costs by means of parameter dependent lyapunov functions is presented in this paper the uncertain linear time invariant systems are supposed to belong to convex bounded domains polytope type uncertainty the conditions proposed are less conservative than other parameter dependent based methods from the literature providing better estimates of worst case h norms in both continuous and discrete time systems as illustrated by means of numerical examples

positive solutions of a second order integral boundary value problem in this paper we study the existence of positive solutions of a second order integral boundary value problems for ordinary differential equations our results presented here unify generalize and substantially improve the existing results in the literature moreover it is worthwhile to point out that our method will dispense with constructing a new green function

on sign changing solutions for a second order integral boundary value problem in this paper we obtain the existence of sign changing solutions for nonlinear second order differential equations with integral boundary value conditions by applying a new fixed point theorem in ordered banach spaces with the lattice structure derived by liu and sun our results improve on those in the literature

sampling method for semidefinite programmes with non negative popov function constraints an important class of optimisation problems in control and signal processing involves the constraint that a popov function is non negative on the unit circle or the imaginary axis such a constrain

dynamic planar convex hull operations in near logarithmic amortized time we give a data structure that allows arbitrary insertions and deletions on a planar point set p and supports basic queries on the convex hull of p such as membership and tangent finding updates take o log 1 e n amori tzed time and queries take o log n time each where n is the maximum size of p and e is any fixed positive constant for some advanced queries such as bridge finding both our bounds increase to o log 3 2 n the only previous fully dynamic solution was by overmars and van leeuwen from 1981 and required o log 2 n time per update and o log n time per query

on a model of indexability and its bounds for range queries we develop a theoretical framework to characterize the hardness of indexing data sets on block access memory devices like hard disks we define an indexing workload by a data set and a set of potential queries for a workload we can construct an indexing scheme which is a collection of fixed sized subsets of the data we identify two measures of efficiency for an indexing scheme on a workload storage redundancy r how many times each item in the data set is stored and access overhead a how many times more blocks than necessary does a query retrieve for many interesting families of workloads there exists a trade off between storage redundancy and access overhead given a desired access overhead a there is a minimum redundancy that any indexing scheme must exhibit we prove a lower bound theorem for deriving the minimum redundancy by applying this theorem we show interesting upper and lower bounds and trade offs between a and r in the case of multidimensional range queries and set queries

fundamental connections among the stability conditions using higher order time derivatives of lyapunov functions for the case of linear time invariant systems abstract it has already been recognized that looking for a positive definite lyapunov function such that a high order linear differential inequality with respect to the lyapunov function holds along the trajectories of a nonlinear system can be utilized to assess asymptotic stability when the standard lyapunov approach examining only the first derivative fails in this context the main purpose of this paper is on one hand to theoretically unveil deeper connections among existing stability conditions especially for linear time invariant lti systems and from the other hand to examine the effect of the higher order time derivatives approach on the stability results for uncertain polytopic lti systems in terms of conservativeness to this end new linear matrix inequality lmi stability conditions are derived by generalizing the concept mentioned above and through the development relations among some existing stability conditions are revealed examples illustrate the improvement over the quadratic approach

a less conservative lmi condition for robust cal d stability of polynomial matrix polytopes a projection approach this technical note investigates the problem of checking robust d stability of a polytope of polynomial matrices finsler s projection lemma is taken to derive a new sufficient condition in terms of a linear matrix inequality lmi feasibility problem the basic idea behind this condition is to lift the existing stability condition into the one of larger space by means of finsler s lemma and based on it to introduce additional decision variables examples are given to show that the proposed condition can yield less conservative results than the previous one

establishing stability and instability of matrix hypercubes abstract the problem of establishing stability and instability of a matrix hypercube is considered and some conditions are proposed based on the stability boundary crossing principle and sum of squares relaxations specifically a sufficient and asymptotically necessary condition for the stability is derived which can be checked through convex lmi optimizations with respect to existing approaches that provide nonconservative conditions the contribution consists of a significantly smaller computational burden in some cases indeed even among small systems there are cases in which such approaches cannot be used due to the huge computational burden while the proposed technique can be easily applied moreover a sufficient and asymptotically necessary condition for the instability is proposed which amounts to solving a linear algebra problem once that the condition for the stability has been investigated such a condition has never been proposed in the literature

stability analysis and gain scheduled state feedback control for continuous time systems with bounded parameter variations the problems of robust stability analysis and state feedback control based on gain scheduling for continuous time systems with time varying parameters that have bounded rates of variation and lie inside a polytope are addressed in this article with respect to previous results in the literature two main contributions of the article are i the robust stability analysis conditions are less conservative and demand less computational effort than the existing ones ii the conditions can be extended to cope with the problem of control design for this class of system parameter dependent linear matrix inequality lmi conditions are given for the existence of a parameter dependent lyapunov function quadratic in the state and homogeneous polynomially of arbitrary degree in the parameter assuring robust stability two convex procedures based on lmis exhibiting distinct complexities are proposed to solve the problem of robust stability an extension to deal with the computation of a stabilising parameter dependent state feedback gain for this class of time varying systems is also provided as a sequence of lmis of increasing precision examples illustrate the results including comparisons with other techniques from the literature

analysis and synthesis of robust control systems via parameter dependent lyapunov functions in this paper the problem of robust stability of systems subject to parametric uncertainties is considered sufficient conditions for the existence of parameter dependent lyapunov functions are given in terms of a criterion which is reminiscent of but less conservative than popov s stability criterion an equivalent frequency domain criterion is demonstrated the relative sharpness of the proposed test and existing stability criteria is then discussed the use of parameter dependent lyapunov functions for robust controller synthesis is then considered it is shown that the search for robustly stabilizing controllers may be limited to controllers with the same order as the original plant a possible synthesis procedure and a numerical example are then discussed

constructing levels in arrangements and higher order voronoi diagrams we give simple randomized incremental algorithms for computing the amk level in an arrangement of n lines in the plane or in an arrangement of n planes in reals 3 the expected running time of our algorithms is o nk n alpha n log n for the planar case and o nk2 n log3 n for the three dimensional case both bounds are optimal unless k is very small the algorithm generalizes to computing the amk level in an arrangement of discs or x monotone jordan curves in the plane our approach can also compute the k level this yields a randomized algorithm for computing the order k voronoi diagram of n points in the plane in expected time o k n k log n n log3 n

optimal point location in a monotone subdivision point location often known in graphics as hit detection is one of the fundamental problems of computational geometry in a point location query we want to identify which of a given collection of geometric objects contains a particular point let mathcal s denote a subdivision of the euclidean plane into monotone regions by a straight line graph of m edges in this paper we exhibit a substantial refinement of the technique of lee and preparata siam j comput 6 1977 pp 594 606 for locating a point in mathcal s based on separating chains the new data structure called a layered dag can be built in o m time uses o m storage and makes possible point location in o log m time unlike previous structures that attain these optimal bounds the layered dag can be implemented in a simple and practical way and is extensible to subdivisions with edges more general than straight line segments

lyapunov stability robust analysis and robustness design for linear continuous time systems the linear continuous time systems to be discussed are described by state space models with structured time varying uncertainties first the explicit maximal perturbation bound for maintaining quadratic lyapunov stability of the closed loop systems is presented then a robust design method is proposed given an open loop nominal system with information of the uncertainty structure the proposed robust design method produces a linear constant output feedback control law which maximizes the perturbation bound for maintaining quadratic lyapunov stability both the optimal results for robust analysis and robustness design can be obtained by using an efficient numerical algorithm e g minimax of matlab with analytic gradients given in this paper examples are investigated in some detail to show the improvements and advantages of the proposed methods over previous ones