on the characteristic polynomial of a random unitary matrix we present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial z of a random n n unitary matrix as n first we show that evaluated at a finite set of distinct points is asymptotically a collection of i i d complex normal random variables this leads to a refinement of a recent central limit theorem due to keating and snaith and also explains the covariance structure of the eigenvalue counting function next we obtain a central limit theorem for ln z in a sobolev space of generalised functions on the unit circle in this limiting regime lower order terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting gaussian measure large deviations results for ln z a evaluated at a finite set of distinct points can be obtained for for higher order scalings we obtain large deviations results for ln z a evaluated at a single point there is a phase transition at a ln n which only applies to negative deviations of the real part reflecting a switch from global to local conspiracy

stability of a mach configuration we prove the stability of a mach configuration which occurs in shock reflection off an obstacle or shock interaction in compressible flow the compressible flow is described by a full steady euler system of gas dynamics the unperturbed mach configuration is composed of three straight shock lines and a slip line carrying contact discontinuity among four regions divided by these four lines in the neighborhood of the intersection two are supersonic regions and other two are subsonic regions we prove that if the constant states in the supersonic regions are slightly perturbed then the structure of the whole configuration holds while the other two shock fronts and the slip line as well as the flow field in the subsonic regions are also slightly perturbed such a conclusion asserts the existence and stability of the general mach configuration in shock theory in order to prove the result we reduce the problem to a free boundary value problem where two unknown shock fronts are free boundaries while the slip line is transformed to a fixed line by a lagrange transformation in the region where the solution is to be determined we have to deal with an elliptic hyperbolic composed system by decoupling this system and combining the technique for both hyperbolic equations and elliptic equations we establish the required estimates which are crucial in the proof of the existence of a solution to the free boundary value problem 2005 wiley periodicals inc

negative moments of characteristic polynomials of random goe matrices and singularity dominated strong fluctuations we calculate the negative integer moments of the regularized characteristic polynomials of n n random matrices taken from the gaussian orthogonal ensemble goe in the limit as n the results agree nontrivially with a recent conjecture of berry and keating motivated by techniques developed in the theory of singularity dominated strong fluctuations this is the first example where nontrivial predictions obtained using these techniques have been proved

compressible flow and transonic shock in a diverging nozzle the paper is devoted to the study of compressible flows and transonic shocks in diverging nozzles in the framework of the full compressible euler system consider a nozzle having a shape as a diverging truncated sector with generic opening angle if the upstream flow at the entrance is supersonic and is near to an axial symmetric flow and if all parameters of the upstream flow and the receiver pressure at the exit are suitably assigned then a transonic shock appears in the nozzle to determine the transonic shock and the flow in the nozzle leads to a free boundary value problem for a nonlinear partial differential equation we prove that the receiver pressure can uniquely determine the location of the transonic shock as well as the flow behind the shock such a conclusion was conjectured by courant and friedrichs and is confirmed theoretically in this paper for the divergent nozzles

e h type mach configuration and its stability this paper is devoted to the study of the classification and stability of mach configurations in the mach reflection of a shock wave by a rigid wall the flow behind the reflected shock front can be supersonic or subsonic correspondingly the mach configuration can be classified as e e type and e h type in this paper we proved the stability of e h type mach configurations provided the reflected shock is weak the result is the complement of the stability of e e type mach configurations in our earlier work

a general result on the stabilization of linear systems using bounded controls we present two constructions of controllers that globally stabilize linear systems subject to control saturation we allow essentially arbitrary saturation functions the only conditions imposed on the system are the obvious necessary ones namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition hold one of the constructions is in terms of a neural network type one hidden layer architecture while the other one is in terms of cascades of linear maps and saturations

the transonic shock in a nozzle 2 d and 3 d complete euler systems abstract in this paper we study a transonic shock problem for the euler flows through a class of 2 d or 3 d nozzles the nozzle is assumed to be symmetric in the diverging or converging part if the supersonic incoming flow is symmetric near the divergent or convergent part of the nozzle then as indicated in section 147 of r courant k o friedrichs supersonic flow and shock waves interscience publ new york 1948 there exist two constant pressures p 1 and p 2 with p 1 p 2 such that for given constant exit pressure p e p 1 p 2 a symmetric transonic shock exists uniquely in the nozzle and the position and the strength of the shock are completely determined by p e moreover it is shown in this paper that such a transonic shock solution is unique under the restriction that the shock goes through the fixed point at the wall in the multidimensional setting furthermore we establish the global existence stability and the long time asymptotic behavior of an unsteady symmetric transonic shock under the exit pressure p e when the initial unsteady shock lies in the symmetric diverging part of the 2 d or 3 d nozzle on the other hand it is shown that an unsteady symmetric transonic shock is structurally unstable in a global in time sense if it lies in the symmetric converging part of the nozzle

stability of transonic shocks in steady supersonic flow past multidimensional wedges abstract we are concerned with the stability of multidimensional m d transonic shocks in steady supersonic flow past multidimensional wedges one of our motivations is that the global stability issue for the m d case is much more sensitive than that for the 2 d case which requires more careful rigorous mathematical analysis in this paper we develop a nonlinear approach and employ it to establish the stability of weak shock solutions containing a transonic shock front for potential flow with respect to the m d perturbation of the wedge boundary in appropriate function spaces to achieve this we first formulate the stability problem as a free boundary problem for nonlinear elliptic equations then we introduce the partial hodograph transformation to reduce the free boundary problem into a fixed boundary value problem near a background solution with fully nonlinear boundary conditions for second order nonlinear elliptic equations in an unbounded domain to solve this reduced problem we linearize the nonlinear problem on the background shock solution and then after solving this linearized elliptic problem develop a nonlinear iteration scheme that is proved to be contractive

stability of transonic shocks in supersonic flow past a wedge abstract in this paper we study the stability of transonic shocks in steady supersonic flow past a wedge we take the potential flow equation as the mathematical model to describe the compressible flow it is known that in generic case such a problem admits two possible location of shock connecting the flow ahead it and behind it they can be distinguished as supersonic supersonic shock and supersonic subsonic shock or transonic shock both these possible shocks satisfy the rankine hugoniot conditions and entropy condition in this paper we prove that the transonic shock is also stable under perturbation of the coming flow provided the pressure at infinity is well controlled

on global transonic shocks for the steady supersonic euler flows past sharp 2 d wedges abstract in this paper under certain downstream pressure condition at infinity we study the globally stable transonic shock problem for the perturbed steady supersonic euler flow past an infinitely long 2 d wedge with a sharp angle as described in the book of courant and friedrichs r courant k o friedrichs supersonic flow and shock waves interscience new york 1948 pages 317 318 when a supersonic flow hits a sharp wedge it follows from the rankine hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the edge of the sharp wedge in terms of the different pressure states in the downstream region which correspond to the supersonic shock and the transonic shock respectively it has frequently been stated that the strong shock is unstable and that therefore only the weak shock could occur however a convincing proof of this instability has apparently never been given the aim of this paper is to understand this open problem more concretely we will establish the global existence and stability of a transonic shock solution for 2 d full euler system when the downstream pressure at infinity is suitably given meanwhile the asymptotic state of the downstream subsonic solution is determined

a reduced order framework applied to linear systems with constrained controls a major issue in the control of dynamical systems is the integration of both technological constraints and some dynamic performance requirements in the design of the control system the authors show in this work that it is possible to solve a class of constrained control problems of linear systems by using a reduced order system obtained by the projection of the trajectories of the original system onto a subspace associated with the undesirable open loop eigenvalues the class of regulation schemes considered uses full state feedback to guarantee that any trajectory emanating from a given polyhedral set of admissible initial states remains in that set this set of admissible states is said to be positively invariant with respect to the closed loop system the authors also address the important issues of numerical stability and complexity of the computations

linear systems with state and control constraints the theory and application of maximal output admissible sets the initial state of an unforced linear system is output admissible with respect to a constraint set y if the resulting output function satisfies the pointwise in time condition y t in y t or 0 the set of all possible such initial conditions is the maximal output admissible set o sub infinity the properties of o sub infinity and its characterization are investigated in the discrete time case it is generally possible to represent o sub infinity or a close approximation of it by a finite number of functional inequalities practical algorithms for generating the functions are described in the continuous time case simple representations of the maximal output admissible set are not available however it is shown that the discrete time results may be used to obtain approximate representations

feedback control of linear discrete time systems under state and control constraints in this paper the problem of stabilizing linear discrete time systems under state and control linear constraints is studied based on the concept of positive invariance existence conditions of linear state feedback control laws respecting both the constraints are established these conditions are then translated into an algorithm of linear programming

output feedback robust stabilization of uncertain linear systems with saturating controls an lmi approach the problem of robust controller design is addressed for an uncertain linear system subject to control saturation no assumption is made concerning open loop stability and no a priori information is available regarding the domain of stability a saturating linear output feedback law and a safe set of initial conditions are determined using a heuristic based on iterative lmi relaxation procedures a readily implementable algorithm based on standard numerical techniques is described and illustrated on two numerical examples

regulator problem for linear systems with constraints on control and its increment or rate this paper discusses the problem of constraints on both control and its rate or increment for linear systems in state space form in both the continuous and discrete time domains necessary and sufficient conditions are derived for autonomous linear systems with constrained state increment or rate for the continuous time case such that the system evolves respecting incremental or rate constraints a pole assignment technique is then used to solve the inverse problem giving stabilizing state feedback controllers that respect non symmetrical constraints on both control and its increment or rate an illustrative example shows the application of the method on the double integrator problem

brief polynomial approach to the control of siso periodic systems subject to input constraint the problem of stabilization of single input single output discrete time periodic systems subject to input constraint is discussed in this paper precisely the initial state of the system is assumed to belong to a given polyhedron and the scalar input is constrained to be bounded it is shown how this problem can be posed in terms of the polynomial description of the system the solution can be found by using a linear programming optimization algorithm

a proof of existence of perturbed steady transonic shocks via a free boundary problem we prove the existence of a solution of a free boundary problem for the transonic small disturbance equation the free boundary is the position of a transonic shock dividing two regions of smooth flow assuming inviscid irrotational flow as modeled by the transonic small disturbance equation the equation is hyperbolic upstream where the flow is supersonic and elliptic in the downstream subsonic region to study the stability of a uniform planar transonic shock we consider perturbation by a steady c1 upstream disturbance if the upstream disturbance is small in a c1 sense then there is a steady solution in which the downstream flow and the transonic shock are holder continuous perturbations of the uniform configuration this result provides a new use of inviscid perturbation techniques to demonstrate in two dimensions the stability of transonic shock waves of the type that appear for example over the wing of an airplane along an airfoil or as bow shocks in a flow with a supersonic free stream velocity 2000 john wiley sons inc

supersonic flow onto a solid wedge we consider the problem of two dimensional supersonicflow onto a solid wedge or equivalently in a concave corner formed by two solid walls for mild corners there are two possible steady state solutions one with a strong and one with a weak shock emanating from the corner the weak shock is observed in supersonic flights a longstanding natural conjecture is that the strong shock is unstable in some sense we resolve this issue by showing that a sharp wedge will eventually produce weak shocks at the tip when accelerated to a supersonic speed more precisely we prove that for upstream state as initial data in the entire domain the timedependent solution is self similar with a weak shock at the tip of the wedge we construct analytic solutions for self similar potential flow both isothermal and isentropic with arbitrary 1 in the process of constructing the self similar solution we develop a large number of theoretical tools for these elliptic regions these tools allow us to establish large data results rather than a small perturbation we show that the wave pattern persists as long as the weak shock is supersonic supersonic when this is no longer true numerics show a physical change of behavior in addition we obtain rather detailed information about the elliptic region including analyticity as well as bounds for velocity components and shock tangents c 2007 wiley periodicals inc

stability of the e h type regular shock refraction abstract in this paper the refraction of shocks on the interface for 2 d steady compressible flow is discussed particularly the e h type regular refraction is defined and its global stability of the wave structure is proved the 2 d steady potential flow equation is employed to describe the motion of the fluid the stability of the e h type regular refraction can be reduced to a free boundary problem of a nonlinear mixed type equation in an unbounded domain the corresponding linearized problem is a generalized tricomi problem of the lavrentiev bitsadze mixed type equation and it can be reduced to a nonlocal boundary value problem of an elliptic system the later is finally solved by establishing the bijection of a corresponding nonlocal operator in a weighted holder space the result obtained in this paper develops the theory of nonlinear mixed type equations and gives an application to a significant physical problem

a unified framework for the study of anti windup designs we present a unified framework for the study of linear time invariant lti systems subject to control input nonlinearities the framework is based on the following two step design paradigm design the linear controller ignoring control input nonlinearities and then add anti windup bumpless transfer awbt compensation to minimize the adverse eflects of any control input nonlinearities on closed loop performance the resulting awbt compensation is applicable to multivariable controllers of arbitrary structure and order all known lti anti windup and or bumpless transfer compensation schemes are shown to be special cases of this framework it is shown how this framework can handle standard issues such as the analysis of stability and performance with or without uncertainties in the plant model the actual analysis of stability and performance and robustness issues are problems in their own right and hence not detailed here the main result is the unification of existing schemes for awbt compensation under a general framework

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

a free boundary problem for a quasi linear degenerate elliptic equation regular reflection of weak shocks we prove the existence of a solution to the weak regular reflection problem for the unsteady transonic small disturbance utsd model for shock reflection by a wedge in weak regular reflection the state immediately behind the reflected shock is supersonic and constant the flow becomes subsonic further downstream the equation in self similar coordinates is degenerate at the sonic line the reflected shock becomes transonic and begins to curve there its position is the solution to a free boundary problem for the degenerate equation using the rankine hugoniot conditions along the reflected shock we derive an evolution equation for the transonic shock and an oblique derivative boundary condition at the unknown shock position by regularizing the degenerate problem we construct uniform bounds we apply local compactness arguments to extract a limit that solves the problem the solution is smooth in the interior and continuous up to the degenerate boundary this work completes a stage in our program to construct self similar solutions of two dimensional riemann problems in a series of papers we developed techniques for solving the degenerate elliptic equations that arise in self similar reductions of hyperbolic conservation laws in other papers especially in joint work with gary lieberman we developed techniques for solving free boundary problems of the type that arise from rankine hugoniot relations for the first time in this paper we combine these approaches and show that they are compatible although our construction is limited to a finite part of the unbounded subsonic region it suggests that this approach has the potential to solve a variety of problems in weak shock reflection including mach and von neumann reflection in the utsd equation and the analogous problems for the unsteady full potential equation 2002 john wiley sons inc

a set induced norm approach to the robust control of constrained systems most realistic control problems involve both some type of time domain constraints and model uncertainty however the majority of controller design procedures currently available focus only on one aspect of the problem with only a handful of methods capable of simultaneously addressing both issues recently it has been proposed to address this class of problems by using a constrained robustness measure generated by a constraint set induced operator norm to assess the stability properties of a family of systems in this paper the properties of this constrained robustness measure are explored and the theoretical framework is extended to include control as well as state constraints these results are applied to the problem of designing fixed order stabilizing feedback controllers for systems subject to structured parametric model uncertainty and time domain constraints

a class of invariant regulators for the discrete time linear constrained regulation problem abstract stable dynamic systems admit positively invariant domains associated to their lyapunov functions conversely some domains can be made positively invariant for systems with state feedback controllers designed in such a way that some associated non negative definite functions are bound to decrease in particular this approach can be used to establish conditions on the gain matrix for linear constrained regulation problems lcrp we construct fixed and variable regulators easy to compute through linear programming for a class of constrained linear systems

existence and stability of perturbed transonic shocks for compressible steady potential flows abstract we establish an existence theorem for perturbed transonic steady potential flows where the boundary of the subsonic region consists of a transonic shock and thus the governing equation becomes a free boundary problem we establish gradient estimates and use the fixed point theorem of caristi s which does not require a compactness of the map to establish the existence result we provide new techniques to handle the gradients and show that the flow is strictly elliptic inside of the subsonic region

random matrix theory and ζ 1 2 it we study the characteristic polynomials z u of matrices u in the circular unitary ensemble cue of random matrix theory exact expressions for any matrix size n are derived for the moments of z and z z and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log z as n in the limit we show that these two distributions are independent and gaussian costin and lebowitz 15 previously found the gaussian limit distribution for im log z using a different approach and our result for the cumulants proves a conjecture made by them in this case we also calculate the leading order n asymptotics of the moments of z and z z these cue results are then compared with what is known about the riemann zeta function s on its critical line re s 1 2 assuming the riemann hypothesis equating the mean density of the non trivial zeros of the zeta function at a height t up the critical line with the mean density of the matrix eigenvalues gives a connection between n and t invoking this connection our cue results coincide with a theorem of selberg for the value distribution of log 1 2 it in the limit t they are also in close agreement with numerical data computed by odlyzko 29 for large but finite t this leads us to a conjecture for the moments of 1 2 it finally we generalize our random matrix results to the circular orthogonal coe and circular symplectic cse ensembles

universal results for correlations of characteristic polynomials riemann hilbert approach we prove that general correlation functions of both ratios and products of characteristic polynomials of hermitian random matrices are governed by integrable kernels of three different types a those constructed from orthogonal polynomials b those constructed from cauchy transforms of the same orthogonal polynomials and finally c those constructed from both orthogonal polynomials and their cauchy transforms these kernels are related with the riemann hilbert problem for orthogonal polynomials for the correlation functions we obtain exact expressions in the form of determinants of these kernels derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via the deift zhou steepest descent stationary phase method for riemann hilbert problems and in particular to find negative moments of characteristic polynomials this reveals the universal parts of the correlation functions and moments of characteristic polynomials for an arbitrary invariant ensemble of 2 symmetry class

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

universal twinkling exponents for spectral fluctuations associated with mixed chaology for systems that are neither fully integrable nor fully chaotic bifurcations of periodic orbits give rise to semiclassically emergent singularities in the fluctuating part n fl of the energy level counting function the bifurcations dominate the spectral moments m m n fl 2 m in the limit 0 we argue that m m const vm and calculate the twinkling exponents v m as the result of a competition between bifurcations with different codimensions and repetition numbers

steady transonic shocks and free boundary problems in infinite cylinders for the euler equations we establish the existence and stability of multidimensional transonic shocks hyperbolic elliptic shocks for the euler equations for steady compressible potential fluids in infinite cylinders the euler equations consisting of the conservation law of mass and the bernoulli law for velocity can be written as a second order nonlinear equation of mixed elliptic hyperbolic type for the velocity potential the transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem the free boundary is the location of the multidimensional transonic shock which divides two regions of c1 flow in the infinite cylinder and the equation is hyperbolic in the upstream region where the c1 perturbed flow is supersonic we develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase and the free boundary is c1 provided that the hyperbolic phase is close in c1 to a uniform flow we further prove that if the steady perturbation of the hyperbolic phase is c2 the free boundary is c2 and stable under the steady perturbation 2003 wiley periodicals inc

on invariant polyhedra of continuous time linear systems this note presents some conditions of existence of positively invariant polyhedra for linear continuous time systems these conditions are first described algebraically then interpreted on the basis of the system eigenstructure then a simple state feedback placement method is described for solving some linear regulation problems under constraints

mach configuration in pseudo stationary compressible flow when a shock front in unsteady compressible flow is moving along an inclined ramp the incident shock will be reflected by the ramp as it moves forward in the process of reflection various reflected wave pattern may occur depending on the slope of the ramp and the strength of the incident shock 14 23 this problem is one of the most important multidimensional prototype problems in gas dynamics generally if the angle of inclination is larger than a critical value then the reflected shock starting from the intersection of the incident shock with the ramp forms a smooth bubble which expands when time goes on such a reflection is called regular reflection on the other hand if the angle of inclination is small then the intersection of the incident shock and the reflected shock will not touch the ramp and an additional shock front called mach stem appears which connects the ramp with the intersection of the incident shock and the reflected shock in this case three shock fronts meet at one point and form a triple shock configuration an important fact is that the structure solely containing three shock fronts separating the neighborhood of the triple intersection into three zones with different continuous states does not exist 14 27 many physical experiments indicate that there is a slip line issuing from the triple intersection such a local wave pattern is called mach configuration or mach structure correspondingly the whole process of reflection is called mach reflection because of the complexity of the whole picture of mach reflection it is natural and necessary to start work on getting a clear understanding of the local mach configuration mach reflection also occurs in various other cases for compressible flow for instance in the study on oblique shock reflection in stationary compressible flow von neumann indicated that see 14 25 if the incident shock hit a wall with an angle larger than a critical value then the reflection must be mach reflection many physical experiments approximate analysis and numerical computations verified the appearance of mach configuration and a tremendous amount of literature is devoted to study such problems see 3 4 11 19 20 24 and the references therein however for the rigorous proof on the stability of mach configuration we can only mention a related result on stationary flow in 13

semi global exponential stabilization of linear systems subject to input saturation via linear feedbacks abstract it is known that a linear time invariant system subject to input saturation can be globally asymptotically stabilized if it has no eigenvalues with positive real parts it is also shown by fuller 1997 and sussmann and yang 1991 that in general one must use nonlinear control laws and only some special cases can be handled by linear control laws in this paper we show the existence of linear state feedback and or output feedback control laws for semi global exponential stabilization rather than global asymptotic stabilization of such systems we explicitly construct linear static state feedback laws and or linear dynamic output feedback laws that semi globally exponentially stabilize the given system our results complement the negative result of fuller 1977 and sussmann and yang 1991

polyhedral regions of local stability for linear discrete time systems with saturating controls the study and the determination of polyhedral regions of local stability for linear systems subject to control saturation is addressed the analysis of the nonlinear behavior of the closed loop saturated system is made by dividing the state space in regions of saturation inside each of these regions the system evolution can be represented by a linear system with an additive disturbance from this representation a necessary and sufficient condition relative to the contractivity of a given convex compact polyhedral set is stated consequently the polyhedral set can be associated with a lyapunov function and the local asymptotic stability of the saturated closed loop system inside the set is guaranteed furthermore it is shown how in some particular cases the compactness condition can be relaxed in order to ensure the asymptotic stability in unbounded polyhedra finally an application of the contractivity conditions is presented in order to determine local asymptotic stability regions for the closed loop saturated system

transonic shocks for the full compressible euler system in a general two dimensional de laval nozzle in this paper we study the transonic shock problem for the full compressible euler system in a general two dimensional de laval nozzle as proposed in courant and friedrichs supersonic flow and shock waves interscience new york 1948 given the appropriately large exit pressure pe x if the upstream flow is still supersonic behind the throat of the nozzle then at a certain place in the diverging part of the nozzle a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes pe x we solve this problem completely for a general class of de laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible euler system the problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic elliptic system one of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted holder space with low regularities for a second order quasilinear elliptic equation with a free parameter the position of the shock curve at one wall of the nozzle and non local terms involving the trace on the shock of the first order derivatives of the unknown function

existence of stationary supersonic flows past a pointed body in this paper we study the mathematical aspects of the stationary supersonic flow past a non axisymmetric curved pointed body the flow is described by a steady potential flow equation which is a quasilinear hyperbolic equation of second order we prove the local existence of the solution to this problem with a pointed shock attached at the tip of the pointed body provided the pointed body is a perturbation of a circular cone and the vertex angle of the approximate cone of the pointed body is less than a critical value the solution is smooth in between the shock and the surface of the body consequently such a structure of flow near the tip of the pointed body and its stability is verified mathematically

singularity dominated strong fluctuations for some random matrix averages the circular and jacobi ensembles of random matrices have their eigenvalue support on the unit circle of the complex plane and the interval 0 1 of the real line respectively the averaged value of the modulus of the corresponding characteristic polynomial raised to the power 2 diverges for 2 1 at points approaching the eigenvalue support using the theory of generalized hypergeometric functions based on jack polynomials the functional form of the leading asymptotic behaviour is established rigorously in the circular ensemble case this confirms a conjecture of berry and keating

eigenstructure assignment for state constrained linear continuous time systems abstract linear state constraints define a polyhedron in the state space of a dynamical system if this polyhedron can be made positively invariant by state feedback the constraints are satisfied all along the trajectory for any initial point in this polyhedron in the context of continuous time linear systems some algebraic positive invariance conditions are established for unbounded symmetrical polyhedra the existence of a solution to the corresponding set of equations and inequalities depends on the structural properties of an associated input output system in particular the number of zeros of this system should be appropriate and they should be stable if these conditions are satisfied the problem can easily be solved by eigenvalue and eigenvector assignment

relatively optimal control and its linear implementation motivated by the fact that determining a feedback solution for the optimal control problem under constraints is a hard task we introduce the concept of relative optimality roughly optimality for a specific nominal plant initial condition we consider a generic discrete time finite horizon constrained optimal control problem for linear systems and we seek for a state feedback possibly dynamic controller as a fundamental requirement we do not admit preactions or controller state initialization based on the plant initial state and we assume our controller to be time invariant in particular we do not consider controllers simply achieved by the feedforward and tracking of the optimal trajectory a relatively optimal control is a stabilizing controller such that if initialized at its zero state produces the optimal constrained trajectory for the nominal initial condition of the plant we show that one of such controllers is linear dead beat and its order is equal to the length of the horizon minus the plant order thus of complexity which is known a priori some additional features such as the assignment of the compensator poles to achieve strong stabilization are proposed we show that by means of the proposed approach we can face several problems such as optimal point to point operations optimal impulse response and optimal tracking

brief control of linear systems subject to input constraints a polynomial approach a polynomial approach is pursued for locally stabilizing discrete time linear systems subject to input constraints using the youla kucera parametrization and geometric properties of polyhedra and ellipsoids the problem of simultaneously deriving a stabilizing controller and the corresponding stability region is cast into standard convex optimization problems solved by linear second order cone and semidefinite programming key topics are touched on such as stabilization of multi input multi output plants or maximization of the size of the stability domain readily implementable algorithms are described

the riemann zeros and eigenvalue asymptotics comparison between formulae for the counting functions of the heights tn of the riemann zeros and of semiclassical quantum eigenvalues en suggests that the tn are eigenvalues of an unknown hermitean operator h obtained by quantizing a classical dynamical system with hamiltonian hcl many features of hcl are provided by the analogy for example the riemann dynamics should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers statistics of the tn have a similar structure to those of the semiclassical en in particular they display random matrix universality at short range and nonuniversal behaviour over longer ranges very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues the riemann siegel formula for the zeta function is described in detail its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations we speculate that the riemann dynamics is related to the trajectories generated by the classical hamiltonian hcl xp

on the instability problem of a 3 d transonic oblique shock wave abstract in this paper we are concerned with the instability problem of a 3 d transonic oblique shock wave for the steady supersonic flow past an infinitely long sharp wedge the flow is assumed to be isentropic and irrotational it was indicated on p 317 of 7 that if a steady supersonic flow comes from minus infinity and hits a sharp symmetric wedge then it follows from the rankine hugoniot conditions and the physical entropy condition that there possibly appears a weak shock or a strong shock attached at the edge of the sharp wedge which corresponds to a supersonic shock or a transonic shock respectively the question arises which of the two actually occurs it has frequently been stated that the strong one is unstable and that therefore only the weak one could occur however a convincing proof of this instability has apparently never been given the aim of this paper is to understand such a longstanding open question we will show that the attached 3 d transonic oblique shock problem is overdetermined with respect to the periodic perturbation which implies that the 3 d transonic shock is unstable in general

steady supersonic flow past an almost straight wedge with large vertex angle abstract this paper studies the problem on the steady supersonic flow at the constant speed past an almost straight wedge with a piecewise smooth boundary it is well known that if each vertex angle of the straight wedge is less than an extreme angle determined by the shock polar the shock wave is attached to the tip of the wedge and constant states on both side of the shock are supersonic this paper is devoted to generalizing this result under the hypotheses that each vertex angle is less than the extreme angle and the total variation of tangent angle along each edge is sufficiently small a sequence of approximate solutions constructed by a modified glimm scheme is proved to be convergent to a global weak solution of the steady problem a sequence of the corresponding approximate leading shock fronts issuing from the tip is shown to be convergent to the leading shock front of the obtained solution the regularity of the leading shock front is established and the asymptotic behaviour of the obtained solution at infinity is also studied

lmis for constrained polynomial interpolation with application in trajectory planning we consider an open loop trajectory planning problem for linear systems with bound constraints originating from saturations or physical limitations using an algebraic approach and results on positive polynomials we show that this control problem can be cast into a constrained polynomial interpolation problem admitting a convex linear matrix inequality lmi formulation

global shock waves for the supersonic flow past a perturbed cone we prove the global existence of a shock wave for the stationary supersonic gas flow past an infinite curved and symmetric cone the flow is governed by the potential equation as well as the boundary conditions on the shock and the surface of the body it is shown that the solution to this problem exists globally in the whole space with a pointed shock attached at the tip of the cone and tends to a self similar solution under some suitable conditions our analysis is based on a global uniform weighted energy estimate for the linearized problem combining this with the local existence result of chen li 1 we establish the global existence and decay rate of the solution to the nonlinear problem

nonlinear stability of a self similar 3 dimensional gas flow we show that the 3 dimensional supersonic gas flow past an infinite cone is nonlinear staple upon the perturbation of the obstacle the perturbed flow exists globally in space and tends to the self similar flow downstream there is a thin layer of concentration of vorticities and entropy variation our analysis is based on an approximation scheme using local self similar solutions as building blocks this enables us to obtain global estimates of the nonlinear interactions of waves needed for the stability analysis

control of linear systems subject to time domain constraints with polynomial pole placement and lmis this note focuses on the control of continuous time linear systems subject to time domain constraints input amplitude limitation output overshoot on closed loop signals using recent results on positive polynomials it is shown that finding a youla kuc spl breve era polynomial parameter of fixed degree hence a controller of fixed order such that time domain constraints are satisfied amounts to solving a convex linear matrix inequality lmi optimization problem as soon as distinct strictly negative closed loop poles are assigned by pole placement proceeding this way time domain constraints are handled by an appropriate choice of the closed loop zeros

steady supersonic flow past a curved cone abstract this paper studies the steady supersonic flow past a lipschitz curved cone under the assumptions that the cone has an opening angle less than a critical value and has sufficiently small total variation of the tangent of the perturbation and that the mach number of incoming flow is sufficiently large the global weak solution is constructed via glimm scheme for 1 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

clusters of near degenerate levels dominate negative moments of spectral determinants the negative moments of spectral determinants n 1 e i en k diverge when 0 as k for a spectrum with equally distributed levels the exponent k k 1 for random matrix ensembles with parameters 1 orthogonal 2 unitary 4 symplectic we argue that the divergences for each k are determined by competitions between near degenerate level clusters whose sizes depend on k and we conjecture that k int k 1 1 k 1 int k 1 1 for poisson distributed levels unrestricted clustering leads to the divergence of the moments increasing with the number n of levels in the interval considered and k n k 1

on the solutions of the position dependent effective mass schrodinger equation of a nonlinear oscillator related with the isotonic oscillator we study exact solutions of the position dependent effective mass schrodinger equation by considering the new solvable nonlinear oscillator that relates to the isotonic oscillator through the method of point canonical transformations we construct exactly solvable potentials by choosing physically important position dependent mass distributions we also provide the energy spectrum of the bound states and the wavefunctions of the solvable potentials

stability of reflection and refraction of shocks on interface abstract in this paper we study the stability of the nonlinear wave structure caused by the attack of an incident shock on an interface of two different kinds of media the attack will produce a reflected wave and a refracted wave and also let the interface deflected in this paper we will mainly study the case when the reflected wave is a shock and the flow between the reflected wave and the refracted shock is relatively subsonic our result indicates that the wave structure and the flow field for the reflection refraction problem in this case is conditionally stable to describe the motion of the fluid we use the inviscid euler system as the mathematical model the reflection refraction problem can be reduced to a free boundary value problem where the unknown reflected shock and refracted shock are free boundaries and the deflected interface is also to be determined in the proof of the existence and the stability of the corresponding wave structure we apply the lagrange transformation to fix the interface and the decoupling technique to decouple the elliptic hyperbolic composite system in its principal part meanwhile some efficient weighted sobolev estimates are established to derive the existence for corresponding nonlinear problems

existence and stability of supersonic euler flows past lipschitz wedges it is well known that when the vertex angle of a straight wedge is less than the critical angle there exists a shock front emanating from the wedge vertex so that the constant states on both sides of the shock front are supersonic since the shock front at the vertex is usually strong especially when the vertex angle of the wedge is large then a global flow is physically required to be governed by the isentropic or adiabatic euler equations in this paper we systematically study two dimensional steady supersonic euler i e nonpotential flows past lipschitz wedges and establish the existence and stability of supersonic euler flows when the total variation of the tangent angle functions along the wedge boundaries is suitably small we develop a modified glimm difference scheme and identify a glimm type functional by naturally incorporating the lipschitz wedge boundary and the strong shock front and by tracing the interaction not only between the boundary and weak waves but also between the strong shock front and weak waves to obtain the required bv estimates these estimates are then employed to establish the convergence of both approximate solutions to a global entropy solution and corresponding approximate strong shock fronts emanating from the vertex to the strong shock front of the entropy solution the regularity of strong shock fronts emanating from the wedge vertex and the asymptotic stability of entropy solutions in the flow direction are also established