on isoperimetric profiles of algebras abstract isoperimetric profile in algebras was first introduced by gromov in m gromov entropy and isoperimetry for linear and non linear group actions groups geom dyn 2 4 2008 499 593 we study the behavior of the isoperimetric profile under various ring theoretic constructions and its relation with the gelfand kirillov dimension

control effort considerations in the stabilization of uncertain dynamical systems abstract this paper investigates the problem of designing a state feedback control to stabilize an uncertain nonlinear system we focus attention on the amplitude norm of the controller which is used to achieve this end the uncertain system is described by a state equation which contains uncertain parameters which are unknown but bounded a lyapunov function is used to establish the stability of the closed loop system the paper gives necessary and sufficient concitions for the uncertain system to be stabilizable with a given lyapunov function furthermore a procedure is indicated for the construction of the desired feedback control law

brief paper set invariance and performance analysis of linear systems via truncated ellipsoids the objective of this paper is to develop numerically simple and effective methods for system analysis using truncated ellipsoids the paper studies linear time invariant systems subject to persistent disturbance and state constraint the maximal output under a set of initial conditions and the overshoot under a given step input are estimated attempts are also made to detect an invariant set as large as possible within state constraint the results are based on the set invariance condition for the truncated ellipsoid and a characterization of the set where the output possibly reaches a local extrema the analysis problems are converted into optimization problems whose constraints become linear matrix inequalities when a few variables are fixed numerical examples demonstrate the effectiveness of the truncated ellipsoid and the proposed methods

the spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds it is proved that the spaces of index one minimal surfaces and stable constant mean curvature surfaces with genus greater than one in non fixed flat three manifolds are compact in a strong sense given a sequence of any of the above surfaces we can extract a convergent subsequence of both the surfaces and the ambient manifolds in the ck topology these limits preserve the topological type of the surfaces and the affine diffeomorphism class of the ambient manifolds some applications to the isoperimetric problem are given

brief paper non conservative matrix inequality conditions for stability stabilizability of linear differential inclusions this paper shows that the matrix inequality conditions for stability stabilizability of linear differential inclusions derived from two classes of composite quadratic functions are not conservative it is established that the existing stability stabilizability conditions by means of polyhedral functions and based on matrix equalities are equivalent to the matrix inequality conditions this implies that the composite quadratic functions are universal for robust possibly constrained stabilization problems of linear differential inclusions in particular a linear differential inclusion is stable stabilizable with without constraints iff it admits a lyapunov control lyapunov function in these classes examples demonstrate that the polyhedral functions can be much more complex than the composite quadratic functions to confirm the stability stabilizability of the same system

on constrained infinite time linear quadratic optimal control abstract this work presents a solution to the infinite time linear quadratic optimal control itlqoc problem with state and control constraints it is shown that a single finite dimensional convex program of known size can yield this solution properties of the resulting value function with respect to initial conditions are also established and are shoen to be useful in determining the aforementioned problem size an example illustrating the method is finally presented

on the discreteness of the spectrum of the laplacian on noncompact riemannian manifolds necessary and sufficient conditions for the discreteness of the laplacian on a noncompact riemannian manifold m are established in terms of the isocapacitary function of m the relevant capacity takes a different form according to whether m has finite or infinite volume conditions involving the more standard isoperimetric function of m can also be derived but they are only sufficient in general as we demonstrate by concrete examples

a new class of lyapunov functions for the constrained stabilization of linear systems the constrained stabilization of linear uncertain systems is investigated via the set theoretic framework of control lyapunov r functions a novel composition rule allows the design of a composite control lyapunov function with external level set that exactly shapes the maximal controlled invariant set and inner sublevel sets arbitrarily close to any choice of smooth ones generalizing both polyhedral and truncated ellipsoidal control lyapunov functions the feasibility test of the proposed smooth control lyapunov functions can be cast into matrix inequalities conditions the constrained linear quadratic control is addressed as an application

constant mean curvature spheres in rm sol _3 based on work of daniel and mira we prove that for each h0 there exists a unique up to ambient isometry immersed constant mean curvature h sphere s h in rm sol 3 where rm sol 3 is the non abelian solvable 3 dimensional lie group equipped with its usual left invariant metric that makes it into a model space for one of the eight thurston geometries see bonahon or scott for a discussion of the thurston geometries that can occur for compact 3 manifolds since this proof also demonstrates that s h has index one the results in daniel mira imply s h is embedded and the left invariant gauss map of s h is a diffeomorphism one key new result that we prove in order to obtain these results for constant mean curvature spheres is the existence of height estimates for certain constant mean curvature h graphs in rm sol 3 which depend only on any fixed positive lower bound for h by the work in daniel mira the existence of the spheres s h for all h0 implies the existence for all h geq 0 of a complex holomorphic only when h 0 quadratic differential q h defined on all constant mean curvature h surfaces m in rm sol 3 and which for h0 vanishes on an open set of m if and only if m is contained in some left translation of s h furthermore when q h is not identically zero on m it has isolated zeros on m and negative index at these zeros

control sharing and merging control lyapunov functions given two control lyapunov functions clfs a merging is a new clf whose gradient is a positive combination of the gradients of the two parents clfs the merging function is an important trade off since this new function may for instance approximate one of the two parents functions close to the origin while being close to the other far away for nonlinear control affine systems some equivalence properties are shown between the control sharing property i e the existence of a single control law which makes simultaneously negative the lyapunov derivatives of the two given clfs and the existence of merging clfs it is shown that even for linear time invariant systems the control sharing property does not always hold with the remarkable exception of planar systems the class of linear differential inclusions is also discussed and similar equivalence results are presented for this class of systems linear matrix inequalities conditions are provided to guarantee the control sharing property finally a constructive procedure based on the recently considered r functions is defined to merge two smooth positively homogeneous clfs

analysis and synthesis of nonlinear systems with uncertain initial conditions this technical note considers the problem of computing extremal values of the trajectories over a given set of initial conditions as well as finding output controllers minimizing these extremal values under time domain constraints for nonlinear systems it is shown that upper bounds of the sought extremal values as well as candidates of the sought controllers can be computed by solving a one parameter sequence of bilinear matrix inequality bmi optimizations obtained through the square matricial representation smr of polynomials moreover a necessary and sufficient condition is proposed to establish the tightness of the found upper bound in spite of the conservatism introduced by the nonconvexity of bmi optimizations and the chosen degree of the lyapunov function and relaxing polynomials

radial rapid decay property for cocompact lattices abstract we study haagerup inequality for radial functions on uniform lattices in semisimple lie groups with respect to riemannian metrics and in some case to word metrics in particular we extend the swiatkowski valette results to any lattice acting properly and essentially transitively on classical buildings

bounds for eigenfunctions of the laplacian on noncompact riemannian manifolds we deal with eigenvalue problems for the laplacian on noncompact riemannian manifolds m of finite volume sharp conditions ensuring l q m and l infinity m bounds for eigenfunctions are exhibited

a new class of universal lyapunov functions for the control of uncertain linear systems the authors analyze the problem of synthesizing a state feedback control for the class of uncertain continuous time linear systems affected by time varying memoryless parametric uncertainties they consider as candidate lyapunov functions the elements of the class spl sigma sub p sup z which is formed by special homogeneous positive definite functions they show that this class is universal in the sense that a lyapunov function exists if and only if there exists a lyapunov function in spl sigma sub p sup z they prove this result in a constructive way showing that such a lyapunov function can always be obtained by smoothing a polyhedral function for which construction algorithms are available the authors show that unlike the polyhedral lyapunov functions these functions allow us to derive explicit formulas for the stabilizing controller

nonquadratic lyapunov functions for robust control abstract this paper deals with the problem of controlling a linear continuous time system with structured time varying parameter uncertainties and input disturbances with a lyapunov function approach in contrast with most of the previous results in the literature we do not confine our attention to the class of quadratic lyapunov functions conversely the basic motivation of this paper is to determine whether there exist other functions that can be conveniently used as candidate lyapunov functions this question has a positive answer the proposed class is that of polyhedral norms or more generally of polyhedral minkowski functionals we show that the class of these functions is universal in the sense that if the problem of ultimately bounding the state in an assigned convex set via state feedback control can be solved via a lyapunov function and a continuous state feedback compensator then it can be solved via a polyhedral lyapunov function and a possibly different continuous control moreover we show that the control can be piecewise linear a numerical technique for constructing the controller is presented for the case in which the uncertainty constraint sets are polyhedral

stabilization of constrained switched systems via multiple lyapunov r functions abstract this paper investigates the stabilization problem of switched linear systems subject to state constraints input saturation and external disturbance within the set theoretic framework of r function the multiple lyapunov r functions mlrfs are presented and merge both the multiple polyhedral and the multiple quadratic lyapunov functions the advantages of the mlrfs are that the external level sets are designed in accordance to the state constraints and inner level sets arbitrary can be made as close as possible to the smooth one based on the mlrfs method stabilization conditions for the constrained switched system under a gradient based switching law are derived the union of the invariant truncated ellipsoidal sets is provided as an alternative approximation to the domain of attraction of the constrained switched system each truncated ellipsoidal set is the intersection of a polyhedral set and an ellipsoidal set which come from the polyhedral lyapunov function and the quadratic lyapunov function respectively the sub optimality bounds are investigated by a policy iteration two examples are simulated to show the benefits of the proposed strategy

constrained stabilization via smooth lyapunov functions abstract we consider the problem of stabilizing a dynamic system by means of bounded controls we show that the largest domain of attraction can be arbitrarily closely approximated by a smooth domain of attraction for which we provide an analytic expression such an expression allows for the determination of a non linear control law in explicit form

lq control for constrained continuous time systems a design method of lq optimal control law is considered for constrained continuous time systems by introducing singular value decomposition for finite time horizon linear systems the sequence of lq sub optimal control laws which converges to the exact solution is obtained based on quadratic programming it is also shown that the nonlinear sub optimal feedback gain is found by the union of affine state functions

the explicit linear quadratic regulator for constrained systems we present a technique to compute the explicit state feedback solution to both the finite and infinite horizon linear quadratic optimal control problem subject to state and input constraints we show that this closed form solution is piecewise linear and continuous as a practical consequence of the result constrained linear quadratic regulation becomes attractive also for systems with high sampling rates as on line quadratic programming solvers are no more required for the implementation

on quadratic dehn functions we confirm with new examples that solvable groups of high rank are expected to satisfy a polynomial isoperimetric inequality gro93 5a9 to that end we study invariant quasi geodesic foliations in simply connected solvable lie groups endowed with left invariant riemannian metrics whose leaves are isometric to closed subgroups we establish a decomposition theorem which implies upper bounds on the dehn or filling function of loops by disks of the solvable group in terms of the dehn functions of the leaves we obtain examples of metabelian polycyclic groups with exponential growth and quadratic dehn functions we also deduce that the horospheres in sl 4 so 4 which bound an invariant core for sl 4 and that the horospheres which bound an invariant core for hilbert modular groups in open image in new window have quadratic filling functions the main theorem also applies to some solvable lie groups which are not quasi isometric to horospheres in symmetric spaces

constrained regulation of linear continuous time dynamical systems abstract in this paper the linear constrained regulation problem lcrp for continuous time dynamical systems is studied the first part of the paper deals with the existence of linear state feedback control laws that transfer asymptotically to the origin all initial states belonging to a given polyhedral subset of the state space while respecting linear constraints on both state and control vectors then the lcrp is formulated as an optimization problem which can be solved by applying linear programming techniques

a hopf theorem for non constant mean curvature and a conjecture of a d alexandrov we prove a uniqueness theorem for immersed spheres of prescribed non constant mean curvature in homogeneous three manifolds in particular this uniqueness theorem proves a conjecture by a d alexandrov about immersed spheres of prescribed weingarten curvature in mathbb r 3 for the special but important case of prescribed mean curvature as a consequence we extend the classical hopf uniqueness theorem for constant mean curvature spheres to the case of immersed spheres of prescribed antipodally symmetric mean curvature in mathbb r 3

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

shifted tableaux and the projective representations of symmetric groups les groupes de representation de s n classes de conjugaison algebres de clifford et la representation spin de base produits induits pour des representations spin une application caracteristique pour des caracteres de spin tableaux decales et q fonctions de schur les caracteres de spin irreductibles de s n et a n une analogue decalee de la regle lr produits tensoriels interieurs

a fixed point formula of lefschetz type in arakelov geometry iii representations of chevalley schemes and heights of flag varieties we give a new proof of the jantzen sum formula for integral representations of chevalley schemes over spec z except for three exceptional cases this is done by applying the fixed point formula of lefschetz type in arakelov geometry to generalized flag varieties our proof involves the computation of the equivariant ray singer torsion for all equivariant bundles over complex homogeneous spaces furthermore we find several explicit formulae for the global height of any generalized flag variety

spectral distribution and l2 isoperimetric profile of laplace operators on groups we give a formula relating the l2 isoperimetric profile to the spectral distribution of a laplace operator on a finitely generated group we prove the asymptotic stability of the spectral distribution under changes of measures with finite second moment as a consequence we can apply techniques from geometric group theory to estimate the spectral distribution of the laplace operator in terms of the growth and the folner s function of the group this leads to upper bounds on spectral distributions of some non solvable amenable groups and to sharp estimates of the spectral distributions of some solvable groups with exponential growth