on the connection between controllability and stabilizability of linear systems with structural uncertain parameters abstract in this paper we investigate the problem of robust stabilization of interval systems by state feedback it is shown under certain conditions that robust stabilizability is equivalent to controllability invariance

stabilization of uncertain systems via linear control this note considers the problem of stabilizing a linear dynamical system whose state equation includes a time varying uncertain parameter vector q cdot given the dynamics dot x t a q t x t b q t u t and a bounding set q for the values q t the objective is to choose a control law u t p x t guaranteeing uniform asymptotic stability for all admissible variations of q cdot our results differ from previous work in one fundamental way that is we show that when working with linear controllers it is possible to dispense with all assumptions on b cdot which have been made by previous authors e g see 1 9 this elimination of hypotheses on b cdot is accomplished roughly as follows the system sigma underline underline delta a q b q is shown to be equivalent to another system sigma underline underline delta a q b as far as stabilization is concerned since b is a constant matrix independent of q the desired result is readily obtained

on the number of integral graphs abstract we show that at most a 2 c n 3 2 proportion of graphs on n vertices have integral spectrum this improves on previous results of ahmadi alon blake and shparlinski 2009 who showed that the proportion of such graphs was exponentially small

global subsonic and subsonic sonic flows through infinitely long axially symmetric nozzles abstract in this paper we establish existence of global subsonic and subsonic sonic flows through infinitely long axially symmetric nozzles by combining variational method various elliptic estimates and a compensated compactness method more precisely it is shown that there exist global subsonic flows in nozzles for incoming mass flux less than a critical value moreover uniformly subsonic flows always approach to uniform flows at far fields when nozzle boundaries tend to be flat at far fields and flow angles for axially symmetric flows are uniformly bounded away from 2 finally when the incoming mass flux tends to the critical value subsonic sonic flows exist globally in nozzles in the weak sense by using angle estimate in conjunction with a compensated compactness framework

inverse littlewood offord problems and the singularity of random symmetric matrices let mn denote a random symmetric n by n matrix whose upper diagonal entries are iid bernoulli random variables which take value 1 and 1 with probability 1 2 improving the earlier result by costello tao and vu 4 we show that mn is nonsingular with probability 1 o n c for any positive constant c the proof uses an inverse littlewood offord result for quadratic forms which is of interest of its own

transonic shock in a nozzle i 2d case in this paper we establish the existence and uniqueness of a transonic shock for the steady flow through a general two dimensional nozzle with variable sections the flow is governed by the inviscid potential equation and is supersonic upstream has no flow boundary conditions on the nozzle walls and an appropriate boundary condition at the exit of the exhaust section the transonic shock is a free boundary dividing two regions of c 1 1 0 flow in the nozzle the potential equation is hyperbolic upstream where the flow is supersonic and elliptic in the downstream subsonic region in particular our results show that there exists a solution to the corresponding free boundary problem such that the equation is always subsonic in the downstream region of the nozzle when the pressure in the exit of the exhaustion section is appropriately larger than that in the entry this problem is motivated by the conjecture of courant and friedrichs on the transonic phenomena in a nozzle 10 furthermore the stability of the transonic shock is also proven when the upstream supersonic flow is a small steady perturbation for the uniform supersonic flow and the corresponding pressure at the exit has a small perturbation the main ingredients of our analysis are a generalized hodograph transformation and multiplier methods for elliptic equation with mixed boundary conditions and comer singularities

quadratic stabilizability of a new class of linear systems with structural independent time varying uncertainty this paper investigates the problem of designing a linear state feedback control to stabilize a new class of single input uncertain linear dynamical systems uncertain parameters in the system matrices are time varying and bounded in given compact sets we first provide a concept called new standard system where some of the entries are required to be negative sign invariant and sign invariant and each entry varies independently in an arbitrarily large range then for a class of new standard systems we derive a necessary and sufficient condition under which a system can be quadratically stabilized by a linear control for all admissible variations of uncertainties the result extends the main result in wei 1990 ieee transactions on automatic control 35 3 268 277

two dimensional subsonic euler flows past a wall or a symmetric body the existence and uniqueness of two dimensional steady compressible euler flows past a wall or a symmetric body are established more precisely given positive convex horizontal velocity in the upstream there exists a critical value rho rm cr such that if the incoming density in the upstream is larger than rho rm cr then there exists a subsonic flow past a wall furthermore rho rm cr is critical in the sense that there is no such subsonic flow if the density of the incoming flow is less than rho rm cr the subsonic flows possess large vorticity and positive horizontal velocity above the wall except at the corner points on the boundary moreover the existence and uniqueness of a two dimensional subsonic euler flow past a symmetric body are also obtained when the incoming velocity field is a general small perturbation of a constant velocity field and the density of the incoming flow is larger than a critical value the asymptotic behavior of the flows is obtained with the aid of some integral estimates for the difference between the velocity field and its far field states

the conormal derivative problem for elliptic equations of variational type on considere des solutions du probleme qu 0 dans nu 0 sur ou est un domaine borne dans r n et q et n sont definis par qu x div a x u x du x b x u x du x x nu x a x u x du x x x x

quadratic stabilizability of multi input linear systems with structural independent time varying uncertainties this paper investigates the problem of designing a linear state feedback control to stabilize a class of multi input linear dynamical systems we first show that to ensure a stabilizable system some entries of the system matrices must be sign invariant we then derive a sufficient condition under which a system can he quadratically stabilized by a linear control

on two dimensional sonic subsonic flow a compensated compactness framework is established for sonic subsonic approximate solutions to the two dimensional euler equations for steady irrotational flows that may contain stagnation points only crude estimates are required for establishing compactness it follows that the set of subsonic irrotational solutions to the euler equations is compact thus flows with sonic points over an obstacle such as an airfoil may be realized as limits of sequences of strictly subsonic flows furthermore sonic subsonic flows may be constructed from approximate solutions the compactness framework is then extended to self similar solutions of the euler equations for unsteady irrotational flows

subsonic sonic limit of approximate solutions to multidimensional steady euler equations a compactness framework is established for approximate solutions to subsonic sonic flows governed by the steady full euler equations for compressible fluids in arbitrary dimension the existing compactness frameworks for the two dimensional irrotational case do not directly apply for the steady full euler equations in higher dimensions the new compactness framework we develop applies for both non homentropic and rotational flows one of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass balance and the vorticity along with the bernoulli law and the entropy relation through a more delicate analysis on the phase space as direct applications we establish two existence theorems for multidimensional subsonic sonic full euler flows through infinitely long nozzles

quadratic stabilizability of linear systems with structural independent time varying uncertainties the author investigates the problem of designing a linear state feedback control to stabilize a class of single input uncertain linear dynamical systems the systems under consideration contain time varying uncertain parameters whose values are unknown but bounded in given compact sets the method used to establish asymptotical stability of the closed loop system obtained when the feedback control is applied involves the use of a quadratic lyapunov function the author first shows that to ensure a stabilizable system some entries of the system matrices must be sign invariant he then derives necessary and sufficient conditions under which a system can be quadratically stabilized by a linear control for all admissible variations of uncertainties the conditions show that all uncertainties can only enter the system matrices in such a way as to form a particular geometrical pattern called an antisymmetric stepwise configuration for the systems satisfying the stabilizability conditions a computational control design procedure is also provided and illustrated via an example

on a weak solution for a transonic flow problem on essaie d etablir l existence de solutions faibles pour un ecoulement stationnaire bidimensionnel irrotationnel compressible non visqueux dans un canal de longueur infinie ou autour d un profil aerodynamique la vitesse de l ecoulement etant donnee a l infini

multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type we are concerned with the existence and stability of multidimensional transonic shocks for the euler equations for steady potential compressible fluids the euler equations consisting of the conservation law of mass and the bernoulli law for the velocity can be written as the following second order nonlinear equation of mixed elliptic hyperbolic type for the velocity potential r r 1 1 div d d 0 where the density function q is

steady subsonic ideal flows through an infinitely long nozzle with large vorticity in this paper the existence uniqueness and far field behavior of a class of subsonic flows with large vorticity for the steady euler equations in infinitely long nozzles are established more precisely for any given convex horizontal velocity of incoming flow in the upstream there exists a critical value mcr if the mass flux is larger than mcr then there exists a unique smooth subsonic euler flow through the infinitely long nozzle this well posedness result is proved by a new observation for the method developed in xie and xin siam j math anal 42 751 784 2010 to deal with the euler system furthermore the optimal convergence rates of the subsonic flows at far fields are obtained via the maximum principle and an elaborate choice of the comparison functions

existence of steady subsonic euler flows through infinitely long periodic nozzles abstract in this paper we study the global existence of steady subsonic euler flows through infinitely long nozzles which are periodic in x 1 direction with the period l it is shown that when the variation of bernoulli function at some given section is small and mass flux is in a suitable regime there exists a unique global subsonic flow in the nozzle furthermore the flow is also periodic in x 1 direction with the period l if in particular the bernoulli function is a constant we also get the existence of subsonic sonic flows when the mass flux takes the critical value

on a degenerate free boundary problem and continuous subsonic sonic flows in a convergent nozzle this paper concerns the well posedness of a boundary value problem for a quasilinear second order elliptic equation which is degenerate on a free boundary such problems arise when studying continuous subsonic sonic flows in a convergent nozzle with straight solid walls it is shown that for a given inlet being a perturbation of an arc centered at the vertex of the nozzle and a given incoming mass flux belonging to an open interval depending only on the adiabatic exponent and the length of the arc there is a unique continuous subsonic sonic flow from the given inlet with the angle of the velocity orthogonal to the inlet and the given incoming mass flux furthermore the sonic curve of this continuous subsonic sonic flow is a free boundary where the flow is singular in the sense that while the speed is c1 2 holder continuous at the sonic state the acceleration blows up at the sonic state

three dimensional steady subsonic euler flows in bounded nozzles abstract the existence and uniqueness of three dimensional steady subsonic euler flows in rectangular nozzles were obtained when prescribing normal component of momentum at both the entrance and exit if in addition the normal component of the voriticity and the variation of bernoulli s function at the entrance are both zero then there exists a unique subsonic potential flow when the magnitude of the normal component of the momentum is less than a critical number as the magnitude of the normal component of the momentum approaches the critical number the associated flows converge to a subsonic sonic flow furthermore when the normal component of vorticity and the variation of bernoulli function are both small the existence and uniqueness of subsonic euler flows with non zero vorticity are established the proof of these results is based on a new formulation for the euler system a priori estimate for nonlinear elliptic equations with nonlinear boundary conditions detailed study for a linear div curl system and delicate estimate for the transport equations

steady euler flows with large vorticity and characteristic discontinuities in arbitrary infinitely long nozzles abstract we establish the existence and uniqueness of smooth solutions with large vorticity and weak solutions with vortex sheets entropy waves for the steady euler equations for both compressible and incompressible fluids in arbitrary infinitely long nozzles we first develop a new approach to establish the existence of smooth solutions without assumptions on the sign of the second derivatives of the horizontal velocity or the bernoulli and entropy functions at the inlet for the smooth case then the existence for the smooth case can be applied to construct approximate solutions to establish the existence of weak solutions with vortex sheets entropy waves by nonlinear arguments this is the first result on the global existence of solutions of the multidimensional steady compressible full euler equations with free boundaries which are not necessarily small perturbations of piecewise constant background solutions the subsonic sonic limit of the solutions is also shown finally through the incompressible limit we establish the existence and uniqueness of incompressible euler flows in arbitrary infinitely long nozzles for both the smooth solutions with large vorticity and the weak solutions with vortex sheets the methods and techniques developed here will be useful for solving other problems involving similar difficulties

subsonic flows in a multi dimensional nozzle in this paper we study the global subsonic irrotational flows in a multi dimensional n 2 infinitely long nozzle with variable cross sections the flow is described by the inviscid potential equation which is a second order quasilinear elliptic equation when the flow is subsonic first we prove the existence of the global uniformly subsonic flow in a general infinitely long nozzle for arbitrary dimension with sufficiently small incoming mass flux and obtain the uniqueness of the global uniformly subsonic flow then we show that there exists a critical value of the incoming mass flux such that a global uniformly subsonic flow exists uniquely provided that the incoming mass flux is less than the critical value this gives a positive answer to the problem of bers on global subsonic irrotational flows in infinitely long nozzles for arbitrary dimension bers in surveys in applied mathematics vol 3 wiley new york 1958 finally under suitable asymptotic assumptions of the nozzle we obtain the asymptotic behavior of the subsonic flow in far fields by means of a blow up argument the main ingredients of our analysis are methods of calculus of variations the moser iteration techniques for the potential equation and a blow up argument for infinitely long nozzles

a riccati equation approach to the stabilization of uncertain linear systems this paper presents a method for designing a feedback control law to stabilize a class of uncertain linear systems the systems under consideration contain uncertain parameters whose values are known only to within a given compact bounding set furthermore these uncertain parameters may be time varying the method used to establish asymptotic stability of the closed loop system obtained when the feedback control is applied involves the use of a quadratic lyapunov function the main contribution of this paper involves the development of a computationally feasible algorithm for the construction of a suitable quadratic lyapunov function once the lyapunov function has been obtained it is used to construct the stabilizing feedback control law the fundamental idea behind the algorithm presented involves constructing an upper bound for the lyapunov derivative corresponding to the closed loop system this upper bound is a quadratic form by using this upper bounding procedure a suitable lyapunov function can be found by solving a certain matrix riccati equation another major contribution of this paper is the identification of classes of systems for which the success of the algorithm is both necessary and sufficient for the existence of a suitable quadratic lyapunov function

on stabilization of a new class of linear time invariant interval systems via constant state feedback control the paper is concerned with stabilization problem of a new class of linear time invariant uncertain systems via linear state feedback control we derive a necessary and sufficient condition that extends a previous result in terms of system form

graphs with integral spectrum abstract it is shown that only a fraction of 2 n of the graphs on n vertices have an integral spectrum although there are several explicit constructions of such graphs no upper bound for their number has been known graphs of this type play an important role in quantum networks supporting the so called perfect state transfer

existence and optimal convergence rates of multi dimensional subsonic potential flows through an infinitely long nozzle with an obstacle inside in this paper the well posedness and optimal convergence rates of subsonic irrotational flows through a three dimensional infinitely long nozzle with a smooth obstacle inside are established more precisely the global existence and uniqueness of the uniformly subsonic flow are obtained via variational formulation as long as the incoming mass flux is less than a critical value furthermore with the aid of delicate choice of weight functions we prove the optimal convergence rates of the flow at far fields via weighted energy estimates and nash moser iteration