closed loop identification for the best asymptotic performance of adaptive robust control discrete linear systems subjected to both additive and parametric deterministic disturbances are considered the certainty equivalence principle is used for the design of adaptive control two different estimation algorithms are proposed to meet different control objectives in the regulation problem with the worst case limsup performance index

some qualitative properties for the total variation flow we prove the existence of a finite extinction time for the solutions of the dirichlet problem for the total variation flow for the neumann problem we prove that the solutions reach the average of its initial datum in finite time the asymptotic profile of the solutions of the dirichlet problem is also studied it is shown that the profiles are nonzero solutions of an eigenvalue type problem that seems to be unexplored in the previous literature the propagation of the support is analyzed in the radial case showing a behaviour entirely different to the case of the problem associated with the p laplacian operator finally the study of the radially symmetric case allows us to point out other qualitative properties that are peculiar of this special class of quasilinear equations

a new modification of the rojo method for solving symmetric circulant five diagonal systems of linear equations abstract a new effective and stable modification of the rojo method 1 for solving of real symmetric circulant five diagonal systems of linear equations is proposed this special kind of system appears in many applications spline approximation difference solution of partial differential equations etc the method presented in this paper a very efficient and stable method

adaptive stabilization of parameter affine minimum phase plants under lipschitz uncertainty the maximum capability of feedback control for discrete time systems under a nonparametric lipschitz uncertainty was first established in xie and guo 2000 for the simplest dynamical control system it was shown that the necessary and sufficient condition for the adaptive stabilizability is of the form l 3 2 2 where l is the lipschitz constant of the uncertainty this result was extended in huang and guo 2012 to a basic class of scalar discrete time minimum phase control systems under an additional parametric uncertainty and the adaptive stabilization was achieved with the use of an impracticable infinite memory feedback based on a brute force search over a sufficiently fine grid in the prior set of unknown parameters in the present paper the adaptive stabilization problem is considered for a subclass of parameter affine minimum phase systems the solution of the problem is based on new recurrent objective inequalities enabling to estimate and validate the closed loop system online and to formulate a simple projection type parameter estimation algorithm instead of the brute force search used in huang and guo 2012 the computational tractability of the proposed finite memory stabilizing feedback is illustrated via simulations

non symbolic algorithms for the inversion of tridiagonal matrices a representation for the entries of the inverse of general tridiagonal matrices is based on the determinants of their principal submatrices it enables us to introduce through the linear recurrence relations satisfied by such determinants a simple algorithm for the entries of the inverse of any tridiagonal nonsingular matrix reduced as well as unreduced the numerical approach is preserved here without invoking the symbolic computation for tridiagonal diagonally dominant matrices a scaling transformation on the recurrences allows us to give another algorithm to avoid overflow and underflow

total variation flow and sign fast diffusion in one dimension abstract we consider the dynamics of the total variation flow tvf u t div d u d u and of the sign fast diffusion equation sfde u t sign u in one spatial dimension we find the explicit dynamic and sharp asymptotic behaviour for the tvf and we deduce the one for the sfde by an explicit correspondence between the two equations

on feedback capability in a class of nonlinearly parameterized uncertain systems in this technical note we will investigate the maximum capability and limitations of the feedback mechanism in globally stabilizing a basic class of discrete time nonlinearly parameterized dynamical systems with multiple unknown parameters both possibility and impossibility theorems together with a fairly complete characterization on the capability of feedback will be presented it will be seen that to characterize the feedback capability the growth rates of the sensitivity functions of the nonlinear dynamics with respect to the uncertain parameters play a crucial role and a suitable decomposition of the family of the nonlinearly parameterized functions in question turns out to be necessary

the dirichlet problem for the total variation flow suppose that is an open bounded domain with a lipschitz boundary the purpose of this chapter is to study the dirichlet problem left begin gathered frac partial u partial t div left frac du left du right right in q left 0 infty right times omega hfill u left t x right phi left x right on s left 0 infty right times partial omega hfill u left 0 x right u 0 left x right in x in omega hfill end gathered right 5 1 where u0 l1 and l1 this evolution equation is related to the gradient descent method used to solve the problem begin gathered minimize int omega left du right int omega f udx int partial omega left u phi right d mathcal h n 1 hfill u in bv left omega right hfill end gathered 2 where f l1 l existence for this variational problem was proved in 118 theorem 1 4

multigrid methods for symmetric positive definite block toeplitz matrices with nonnegative generating functions in this paper we introduce a generalized multigrid method for solving linear systems bf t n m bf x bf b where bf t n m in re nm times nm is a symmetric block toeplitz matrix with symmetric toeplitz blocks we use a special choice of the projection operator whose coefficients simply depend on the generating functions associated with the proposed class of matrices this choice leads to iterative methods with convergence rates independent of the euclidean condition number kappa 2 bf t n m and of the dimension of the involved matrices the total arithmetic cost is o nm log nm for dense matrices and o nm for band matrices with band blocks in the pram model only o log nm parallel steps are required this algorithm is therefore competitive with the preconditioned conjugate gradient methods proposed by chan and jin siam j sci statist comput 13 1992 pp 1218 1235 ku and kuo siam j sci statist comput 13 1992 pp 948 966 di benedetto siam j sci comput 17 1996 and serra bit 34 1994 pp 579 594 for dense matrices and improves those results for band block toeplitz matrices with band toeplitz blocks

brief paper on feedback capability for a class of semiparametric uncertain systems the main purpose of this paper is to understand and characterize the maximum capability of the feedback mechanism for a basic class of scalar discrete time semiparametric minimum phase control systems where both parametric and nonparametric uncertainties are included we will demonstrate that the necessary and sufficient condition to stabilize the class of systems is l 32 2 where l is the lipschitz constant describing the size of the uncertainty of the nonparametric part this critical value is the same as that first obtained in xie and guo 2000b for a class of purely nonparametric systems which shows that the capability of feedback is not influenced by the parameterized uncertainty in the systems as long as the corresponding parametric nonlinear function has a linear growth rate while the necessity proof is directly obtainable from xie and guo 2000b the main task of this paper is to prove the sufficiency part

on the stable equilibrium points of gradient systems this paper studies the relations between the local minima of a cost function f and the stable equilibria of the gradient descent flow of f in particular it is shown that under the assumption that f is real analytic local minimality is necessary and sufficient for stability under the weaker assumption that f is indefinitely continuously differentiable local minimality is neither necessary nor sufficient for stability

computing the extremal positive definite solutions of a matrix equation an efficient and numerically stable implementation of a known algorithm is suggested for finding the extremal positive definite solutions of the matrix equation x a x 1 a i if such solutions exist the convergence rate is analyzed a new algorithm that avoids matrix inversion is presented numerical examples are given to illustrate the effectiveness of the algorithms

a family of functional inequalities łojasiewicz inequalities and displacement convex functions abstract for displacement convex functionals in the probability space equipped with the monge kantorovich metric we prove the equivalence between the gradient and functional type ojasiewicz inequalities we also discuss the more general case of convex functions and we provide a general convergence theorem for the corresponding gradient dynamics specialising our results to the boltzmann entropy we recover otto villani s theorem asserting the equivalence between logarithmic sobolev and talagrand s inequalities the choice of power type entropies shows a new equivalence between gagliardo nirenberg inequality and a nonlinear talagrand inequality some nonconvex results and other types of equivalences are discussed

a new method for solving symmetric circulant tridiagonal systems of linear equations in this paper a new method for computing the solution of a linear system having a symmetric circulant tridiagonal matrix is presented this special kind of system appears in many applications after an appropriate partition of the system and the elimination of the last unknown we apply the woodbury formula to arrive at a very efficient and stable method which is quite competitive with gaussian elimination and with the modified double sweep method

calculus and heat flow in metric measure spaces and applications to spaces with ricci bounds from below this paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces x mathsf d mathfrak m our main results are a general study of the relations between the hopf lax semigroup and hamilton jacobi equation in metric spaces x d the equivalence of the heat flow in l 2 x mathfrak m generated by a suitable dirichlet energy and the wasserstein gradient flow of the relative entropy functional mathrm ent mathfrak m in the space of probability measures open image in new window the proof of density in energy of lipschitz functions in the sobolev space w 1 2 x mathsf d mathfrak m a fine and very general analysis of the differentiability properties of a large class of kantorovich potentials in connection with the optimal transport problem is the fourth achievement of the paper our results apply in particular to spaces satisfying ricci curvature bounds in the sense of lott and villani ann math 169 903 991 2009 and sturm acta math 196 65 131 2006 and acta math 196 133 177 2006 and require neither the doubling property nor the validity of the local poincare inequality

a qr method for computing the singular values via semiseparable matrices the standard procedure to compute the singular value decomposition of a dense matrix first reduces it into a bidiagonal one by means of orthogonal transformations once the bidiagonal matrix has been computed the qr method is applied to reduce the latter matrix into a diagonal one in this paper we propose a new method for computing the singular value decomposition of a real matrix in a first phase an algorithm for reducing the matrix a into an upper triangular semiseparable matrix by means of orthogonal transformations is described a remarkable feature of this phase is that depending on the distribution of the singular values after few steps of the reduction the largest singular values are already computed with a precision depending on the gaps between the singular values an implicit qr method for upper triangular semiseparable matrices is derived and applied to the latter matrix for computing its singular values the numerical tests show that the proposed method can compete with the standard method using an intermediate bidiagonal matrix for computing the singular values of a matrix

iterative solution of two matrix equations we study iterative methods for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a q and x a x 1 a q where q is hermitian positive definite general convergence results are given for the basic fixed point iteration for both equations newton s method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation numerical results are reported to illustrate the convergence behaviour of various algorithms

a new critical theorem for adaptive nonlinear stabilization it is fairly well known that there are fundamental differences between adaptive control of continuous time and discrete time nonlinear systems in fact even for the seemingly simple single input single output control system y t email protected 1f y t u t w t 1 with a scalar unknown parameter q 1 and noise disturbance w t and with a known function f having possible nonlinear growth rate characterized by f x email protected x b with b 1 the necessary and sufficient condition for the system to be globally stabilizable by adaptive feedback is b 4 this was first found and proved by guo 1997 for the gaussian white noise case and then proved by li and xie 2006 for the bounded noise case subsequently a number of other types of critical values and impossibility theorems on the maximum capability of adaptive feedback were also found mainly for systems with known control parameter as in the above model in this paper we will study the above basic model again but with additional unknown control parameter q 2 i e u t is replaced by q 2u t in the above model interestingly it turns out that the system is globally stabilizable if and only ifb 3 this is a new critical theorem for adaptive nonlinear stabilization which has meaningful implications for the control of more general uncertain nonlinear systems

using the sherman morrison woodbury inversion formula for a fast solution of tridiagonal block toeplitz systems abstract a fast numerical algorithm for solving systems of linear equations with tridiagonal block toeplitz matrices is presented the algorithm is based on a preliminary factorization of the generating quadratic matrix polynomial associated with the toeplitz matrix followed by the sherman morrison woodbury inversion formula and solution of two bidiagonal and one diagonal block toeplitz systems tight estimates of the condition numbers are provided for the matrix system and the main matrix systems generated during the preliminary factorization the emphasis is put on rigorous stability analysis to rounding errors of the sherman morrison woodbury inversion numerical experiments are provided to illustrate the theory

necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation x a x 1a q abstract we consider the problem of when the matrix equation x a x 1 a q has a positive definite solution here q is positive definite we study both the real and the complex case this equation plays a crucial role in solving a special case of the discrete time riccati equation we present both necessary and sufficient conditions for its solvability this result is obtained by using an analytic factorization approach moreover we present algebraic recursive algorithms to compute the largest and smallest the solution of the equation respectively finally we discuss the number of solutions

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations abstract we study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation u t div a x u f x u 0 on a bounded domain subject to dirichlet boundary and to initial conditions the data are supposed to satisfy suitable regularity and growth conditions our approach to the convergence result and decay estimate is based on the ojasiewicz simon gradient inequality which in the case of the semilinear heat equation is known to give optimal decay estimates the abstract results and their applications are discussed also in the framework of orlicz sobolev spaces

a simple unified approach to some convergence theorems of l simon abstract we give a new and easier proof for the convergences result of l simon 15 concerning global and bounded solutions of a heat equation with analytic nonlinearity by constructing a new liapunov functional for the elliptic equation we are able to prove the second convergence theorem of 15 in the same way as the first theorem the results are more general and can be applied to other problems such as fourth order equation and vector valued systems of gradient type

some algorithms for solving special tridiagonal block toeplitz linear systems this paper is focused on different methods and algorithms for solving tridiagonal block toeplitz systems of linear equations we consider the el sayed method ph d thesis 1996 for such systems and propose several modifications that lead to different algorithms which we discuss in detail our algorithms are then compared with some classical techniques as far as implementation time is concerned number of operations and storage comments and conclusions for computing efficiency of the proposed new algorithms are given numerical experiments corroborating the theoretical results are also presented

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form