the spectral norm of a nonnegative matrix abstract given a matrix a a ij define a a ij let 2 denote the spectral norm we show that for any matrix a a 2 min r 1 b c 1 c b c a and show that under mild conditions the minimizers in 1 are essentially unique and are related to the left and right singular vectors of a in a simple way we also show that a 2 a 2 and determine the case of equality

generalized controller canonical form for linear and nonlinear dynamics a generalized controller canonical form for linear and nonlinear dynamics is proposed it is obtained using the theorem of the primitive element from differential algebra the derivation of the controller form does not apply to multivariable constant linear systems with the transformations all nonlinear dynamics can be exactly linearized by dynamic feedback the main departure from standard theory is that transformations may depend on input derivatives once differential algebraic tools are introduced the proofs of the results are easy

isometries for the induced c norm on square matrices and some related results abstract let fn be the linear space of column vectors with n coordinates over f r or c denote by gp n the group of n n generalized permutation matrices i e matrices with exactly one nonzero entry with magnitude 1 in each row and column given a nonzero vector c c1 cn t fn define the c norm of x x1 xn t fn by x c max c t p x p gp n and the induced c norm of a fn n by a c max a y c y c 1 we characterize the isometries for induced c norms to achieve our goal we study the geometric properties of fn n and obtain some inequalities related to the induced c norm as by products these results are of independent interest

a weak majorization involving the matrices a b and ab abstract let a and b be n n positive definite matrices and let the eigenvalues of a b and ab be arranged in decreasing order then for all r 0 i k n i r ab i k n i r a b for k 1 n furthermore it is shown that i k n i r a b i k n i r ab for k 1 n is a limiting case of the weak majorization above which settles a conjecture of bapat and johnson

lieb functions and m positivity of norms abstract some properties of 2 or 3 positive maps on matrix spaces are proved where special attention is given to norms it is shown that for a locally bounded function the definition of a lieb function can be simplified somewhat unitarily invariant norms are characterized by a certain monotonicity property the set of 2 positive numerical radii is described and an example of a 3 positive norm is obtained

a pseudospectral method for the optimal control of constrained feedback linearizable systems we consider the optimal control of feedback linearizable dynamical systems subject to mixed state and control constraints in general a linearizing feedback control does not minimize the cost function such problems arise frequently in astronautical applications where stringent performance requirements demand optimality over feedback linearizing controls in this paper we consider a pseudospectral ps method to compute optimal controls we prove that a sequence of solutions to the ps discretized constrained problem converges to the optimal solution of the continuous time optimal control problem under mild and numerically verifiable conditions the spectral coefficients of the state trajectories provide a practical method to verify the convergence of the computed solution the proposed ideas are illustrated by several numerical examples

holder type norm inequalities for schur products of matrices abstract for a unitarily invariant norm on m n and p 1 we define a p by a p 1 p then p is again a unitarily invariant norm we give holder type inequalities for schur products of the form a b 0 p0 a 1 p1 b 2 p2 as a corollary we settle in a stronger form a conjecture of marcus et al on submultiplicativity of a unitarily invariant norm with respect to schur multiplication we prove also holder type inequalities for operator radii w of the form w a b w a w b

inequalities for c s seminorms and lieb functions abstract let m n be the space of n n complex matrices a seminorm on m n is said to be a c s seminorm if a a aa for all a m n and a b whenever a b and b a are positive semidefinite if is any nontrivial c s seminorm on m n we show that a is a unitarily invariant norm on m n which permits many known inequalities for unitarily invariant norms to be generalized to the setting of c s seminorms we prove a new inequality for c s seminorms that includes as special cases inequalities of bhatia et al for unitarily invariant norms finally we observe that every c s seminorm belongs to the larger class of lieb functions and we prove some new inequalities for this larger class

a norm compression inequality for block partitioned positive semidefinite matrices abstract let a be a positive semidefinite matrix block partitioned as a b c c d where b and d are square blocks we prove the following inequalities for the schatten q norm q a q q 2 q 2 c q q b q q d q q 1 q 2 and a q q 2 q 2 c q q b q q d q q 2 q we show that these bounds obey a strong sharpness condition when the blocks are of size at least 2 2 and b q d q c q finally our bounds can be extended to symmetric partitionings into larger numbers of blocks for a aij a q q i a ii q q 2 q 2 i j a ij q q 1 q 2 while for 2 q the inequality is reversed

a cauchy schwarz inequality for operators with applications abstract for any unitarily invariant norm on hilbert space operators it is shown that for all operators a b x and positive real numbers r we have a xb r 2 aa x r xbb r some consequences are then discussed a simple proof is given for the fact that for positive operators a b the function spr a t b t 1 t is monotone in t on the positive half line

dual periodicity in l 1 norm minimisation problems the topic of this paper is the discrete time l1 norm minimisation problem with convolution constraints we find primal initial conditions for which the dual optimal solution is periodic periodicity of the dual optimal solution implies satisfaction of a simple linear recurrence relation by the primal optimal solution

inequalities involving unitarily invariant norms and operator monotone functions let be a unitarily invariant norm on matrices for matrices a b x with a b positive semidefinite and x arbitrary we prove that the function t atxb1 t r a1 txbt r is convex on 0 1 for each r 0 this convexity result interpolates the matrix cauchy schwarz inequality a1 2xb1 2 r 2 ax r xb r due to r bhatia and c davis linear algebra appl 223 224 1995 119 and also it generalizes a w marshall and i olkin s pacific j math 15 1965 241 result that the condition number as a s is increasing in s 0 we prove that if f t is a nonnegative operator monotone function on 0 and is a normalized unitarily invariant norm then f x f x for every matrix x the special case when f t tr 0 r 1 is used to consider the monotonicity of p ap bp 1 p as well as p ap bp 1 p furthermore we obtain some norm inequalities of holder and minkowski types related to the expression a p b p 1 p for example comparisons are made between c a d b and a p b p 1 p c q d q 1 q where p 1 q 1 1

unification of the arithmetic geometric mean and holder inequalities for unitarily invariant norms abstract in this paper we obtain an inequality for unitarily invariant norms which unifies the arithmetic geometric mean inequality and the holder inequality for unitarily invariant norms

a quantitative version of the observation that the hadamard product is a principal submatrix of the kronecker product abstract the hadamard and kronecker products of two n m matrices a b are related by a b p t 1 a b p 2 where p 1 p 2 are partial permutation matrices after establishing several properties of the p matrices this relationship is employed to demonstrate how a simplified theory of the hadamard product can be developed during this process the well known result a b a b aa bb is extended to a b a b 1 2 aa bb ab ba aa bb showing an inherent link between the hadamard product and conventional product of two matrices this leads to a sharper bound on the spectral norm of a b a b 1 2 a 2 b 2 ab 2 1 2 a b and an improvement on the weak majorization of a b 2 a b w 1 2 2 a 2 b 2 ab w 2 a 2 b for a real non singular matrix x and invertible diagonal matrices d e the spectral condition number is shown to be if scaled bounded below as follows dxe 2 x x t 2 x x t 2 1 2 x x t 1 for a 0 we have i a 2 1 2 i a 2 a a i a 2 and a 1 2 a 1 2 2 1 2 i a a 1 a a 1 when a 0 the latter inequality is compared to styan s inequality a a 1 1 2 i a a 1 when a is a correlation matrix and is shown to possess stronger properties of ordering finally the relationship a b p t 1 a b p 2 is applied to determine conditions of singularity of certain orderings of the hadamard products of matrices

isometries for unitarily invariant norms a survey of linear isometries for unitarily invariant norms on real or complex rectangular matrices is given which includes some latest development on the topic a result on isometries for unitarily invariant norms without the linearity assumption is presented related results and problems are discussed

the eigenvalues range of a class of matrices and some applications in cauchy schwarz inequality and iterative methods abstract this paper discusses the range of the eigenvalues of a class of matrices by using the eigenvalues range of a class of matrices an extension of the inner product type cauchy schwarz inequality is obtained the convergence proof of the least squares based iterative algorithm for solving the coupled sylvester matrix equations is given and the best convergence factor is determined moreover by using the eigenvalues range of this class of matrices an iterative algorithm for solving linear matrix equation is established three numerical examples are offered to illustrate the effectiveness of the results suggested in this paper

flatness and defect of non linear systems introductory theory and examples we introduce flat systems which are equivalent to linear ones via a special type of feedback called endogenous their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of kalman s controllability the distance to flatness is measured by a non negative integer the defect we utilize differential algebra where flatness and defect are best defined without distinguishing between input state output and other variables many realistic classes of examples are flat we treat two popular ones the crane and the car with n trailers the motion planning of which is obtained via elementary properties of plane curves the three non flat examples the simple double and variable length pendulums are borrowed from non linear physics a high frequency control strategy is proposed such that the averaged systems become flat

on g invariant norms abstract a result of r mathias and horn cf linear algebra appl 142 1990 63 on the representation of the unitarily invariant norm is extended in the context of eaton triples and of real semisimple lie algebras the representation is related to a function criteria for being a norm is given in terms of and the longest element of the underlying finite reflection group in particular for the real simple lie algebras is a norm on its cartan subspace for a given nonzero if and only if 0 where 0 is the longest element of the weyl group this is the case if the weyl group contains id some related results are obtained

l 1 optimal feedback controllers for mimo discrete time systems the problem considered in this paper is the design of a closed loop system consisting of a mimo discrete time plant and compensator in such a way that the system is internally stable and optimally tracks all persistent bounded inputs the solution consists of two parts first the calculation of the minimum value of the performance index a weighted transfer function which is done by solving a linear programming problem and second construction of the optimal transfer function by solving a set of linear equations it is shown that in general the discrete time problem will have a rational solution and thus appears to be of considerable practical significance

optimal rejection of persistent disturbances robust stability and mixed sensitivity minimization the problem of optimal disturbance rejection of bounded persistent disturbances is solved in the general nonsquare case the minimum value of the objective function can be obtained by solving a semi infinite linear programming problem and an iterative procedure for obtaining approximate solutions is introduced application of the l sup 1 optimal problem to robustness is discussed a mixed sensitivity problem is formulated and shown to guarantee good disturbance rejection in the presence of plant perturbations

pseudospectral methods for optimal motion planning of differentially flat systems the article presents some preliminary results on combining two new ideas from nonlinear control theory and dynamic optimization we show that the computational framework facilitated by pseudospectral methods applies quite naturally and easily to fliess implicit state variable representation of dynamical systems the optimal motion planning problem for differentially flat systems is equivalent to a classic bolza problem of the calculus of variations we exploit the notion that derivatives of flat outputs given in terms of lagrange polynomials at legendre gauss lobatto points can be quickly computed using pseudospectral differentiation matrices additionally the legendre pseudospectral method approximates integrals by gauss type quadrature rules the application of this method to the two dimensional crane model reveals how differential flatness may be readily exploited

monotonicity of unitarily invariant norms abstract any matrix unitarily invariant norm gives rise to a symmetric gauge function of the singular values of its matrix argument but the dependency on the singular values is not equally weighted among them in the sense that the norm may not increase with some of the singular values under sufficiently small increases while it always increases when some other singular values increase no matter how tiny the changes are this paper introduces and characterizes argument dependent classifications of unitarily invariant norms and in particular the class in which a unitarily invariant norm increases with each of the first k largest singular values

the pseudospectral legendre method for discretizing optimal control problems this paper presents a computational technique for optimal control problems including state and control inequality constraints the technique is based on spectral collocation methods used in the solution of differential equations the system dynamics are collocated at legendre gauss lobatto points the derivative x spl dot t of the state x t is approximated by the analytic derivative of the corresponding interpolating polynomial state and control inequality constraints are collocated at legendre gauss lobatto nodes the integral involved in the definition of the performance index is discretized based on the gauss lobatto quadrature rule the optimal control problem is thereby converted into a mathematical programming program thus existing well developed optimization algorithms may be used to solve the transformed problem the method is easy to implement capable of handling various types of constraints and yields very accurate results illustrative examples are included to demonstrate the capability of the proposed method and a comparison is made with existing methods in the literature

optimal rejection of persistent bounded disturbances in this paper we formulate the problem of optimal disturbance rejection in the case where the disturbance is generated as the output of a stable system in response to an input which is assumed to be of unit amplitude but is otherwise arbitrary the objective is to choose a controller that minimizes the maximum amplitude of the plant output in response to such a disturbance mathematically this corresponds to requiring uniformly good disturbance rejection over all time since the problem of optimal tracking is equivalent to that of optimal disturbance rejection if a feedback controller is used see 7 sect 5 6 the theory presented here can also be used to design optimal controllers that achieve uniformly good tracking over all time rather than a tracking error whose l 2 norm is small as is the case with the currently popular h infty theory the present theory is a natural counterpart to the existing theory of optimal disturbance rejection the so called h infty theory which is based on the assumption that the disturbance to be rejected is generated by a stable system whose input is square integrable and has unit energy it is shown that the problem studied here has quite different features from its predecessor complete solutions to the problem are given in several important cases including those where the plant is minimum phase or when it has only a single unstable zero in other cases procedures are given for obtaining bounds on the solution and for obtaining suboptimal controllers

schur complements and matrix inequalities in the lowner ordering abstract the purpose of this paper is to present a matrix inequality on the kronecker product that unifies the proofs of many existing matrix inequalities in the lowner partial ordering on the sum ordinary product and hadamard schur product schur complements serve as the basic tool

a lie backlund approach to equivalence and flatness of nonlinear systems a new system equivalence relation using the framework of differential geometry of jets and prolongations of infinite order is studied in this setting two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives this is a lie backlund isomorphism the authors prove that although the state dimension is not preserved the number of input channels is kept fixed they also prove that a lie backlund isomorphism can be realized by an endogenous feedback the differentially flat nonlinear systems introduced by the authors 1992 via differential algebraic techniques are generalized and the new notion of orbitally flat systems is defined they correspond to systems which are equivalent to a trivial one with time preservation or not the endogenous linearizing feedback is explicitly computed in the case of the vtol aircraft to track given reference trajectories with stability

cauchy schwarz inequalities associated with positive semidefinite matrices abstract using a quasilinear representation for unitarily invariant norms we prove a basic inequality let a l x x m be positive semidefinite where x m m n then x p 2 l p m p for all p 0 and all unitarily invariant norms we show how several inequalities of cauchy schwarz type follow from this bound and obtain a partial analog of our results for l p norms