on the stability radius of a generalized state space system the concept of distance to instability of a system matrix is generalized to system pencils which arise in descriptor semistate systems difficulties arise in the case of singular systems because the pencil can be made unstable by an infinitesimal perturbation it is necessary to measure the distance subject to restricted or structured perturbations in this paper a suitable measure for the stability radius of a generalized state space system is defined and a computable expression for the distance to instability is derived for regular pencils of index less than or equal to one for systems which are strongly controllable it is shown that this measure is related to the sensitivity of the poles of the system over all feedback matrices assigning the poles

robustness of uncertain descriptor systems this paper considers a class of descriptor systems with perturbations it derives sufficient conditions for the uncertain systems to be impulse free regular and or asymptotically stable the problems of the impulse controllability and impulse observability and the so called robust impulsive controller ric of the uncertain system are considered numerical examples are presented to illustrate the results

decay estimates for the wave equation in two dimensions abstract we establish strichartz estimates both reversed and some direct ones pointwise decay estimates and weighted decay estimates for the linear wave equation in dimension two with an almost scaling critical potential in the case when there is no resonance or eigenvalue at the edge of the spectrum we also prove some simple applications

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

generalized strichartz estimates and scattering for 3d zakharov system we obtain scattering for the 3d zakharov system with non radial small data in the energy space with angular regularity of degree one the main ingredient is a generalized strichartz estimate for the schrodinger equation in the space of l2 angular integrability

controllability of generalized linear time invariant systems this note deals with the problem of c controllability of generalized systems a new criterion for determining c controllability of generalized systems is established with controllability established it is shown that it is always possible to transfer the system from the initial position to any target by a control vector valued function whose elements are polynomials of bounded degree the polynomial coefficients are obtained as a solution to a set of linear equations

generalized strichartz inequalities for the wave equation strichartz inequalities for the wave equation u f are estimates of the solution u of the cauchy problem for that equation in the form of space time integral norms in terms of similar norms of the inhomogeneity f and of suitable norms of the initial data

on existence and scattering with minimal regularity for semilinear wave equations abstract we prove existence and scattering results for semilinear wave equations with low regularity data we also determine the minimal regularity that is needed to ensure local existence and well posedness and we give counterexamples to well posedness more specifically we show that equations of the type u u p with initial data u u t in r n 1 r n have a local solution if p n and we construct counterexamples if p n the existence results rely on mixed norm space time estimates of strichartz type

l2 concentration of blow up solutions for two coupled nonlinear schrodinger equations abstract in this paper we consider the blow up solutions of cauchy problem for two coupled nonlinear schrodinger equations the l 2 concentration and blow up rate for radially symmetric blow up solutions are obtained

solvability controllability and observability of continuous descriptor systems in this paper we investigate the properties of the continuous descriptor system e dot x t ax t bu t 0 leqt leqb where e a and b are complex and possibly singular matrices and u t is a complex function differentiable sufficiently many times the traditional approach to such systems is to separate the state equations from the algebraic equations however such algorithms usually destroy the natural physically based sparsity and structure of the original system therefore we consider descriptor systems in their original form such systems possess numerous properties not shared by the well known state variable systems first we relate classical theories of matrix pencils to the solvability of descriptor systems then we extend the concepts of reachability controllability and observability of state variable systems to descriptor systems and describe the set of reachable states for descriptor systems

analysis and applications of robust nonsingularity problem using the structured singular value this note uses the structured singular value technique to solve the robust nonsingularity analysis problem for matrices with unstructured and or quadratically coupled structured uncertainties with the ready spl mu analysis tools a new approach is proposed for finding a set of uncertainties within which the perturbed matrix keeps nonsingularity applications and examples in robust control theory are given to show the usefulness and the practicality of the proposed approach

uniqueness of minimal blow up solutions to nonlinear schrodinger system abstract in this paper we are concerned with the finite time blow up solutions to the following n coupled nonlinear schrodinger system in r 2 0 1 i t j j k 1 n a j k k 2 j 0 where n 2 and the coefficient matrix a satisfies a j k a k j 0 we study the properties of the blow up solutions which obtain the minimal l 2 norm and prove that such solution is unique up to the symmetries of the system

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

some remarks on strichartz estimates for homogeneous wave equation abstract we give several remarks on strichartz estimates for homogeneous wave equation with special attention to the cases of l x estimates radial solutions and initial data from the inhomogeneous sobolev spaces in particular we give the failure of the endpoint estimate l t 4 l x for n 2

necessary and sufficient conditions for the controllability of linear interval descriptor systems abstract in this paper we investigate a class of uncertain linear descriptor systems with interval plants and give necessary and or sufficient conditions for controllability examples are also given to illustrate the results the corresponding results for the dual observability robustness problems are straightforward extensions

endpoint strichartz estimates we prove an abstract strichartz estimate which implies previously unknown endpoint strichartz estimates for the wave equation in dimension n 4 and the schrodinger equation in dimension n 3 three other applications are discussed local existence for a nonlinear wave equation and strichartz type estimates for more general dispersive equations and for the kinetic transport equation

analysis of stability robustness for generalized state space systems with structured perturbations abstract a new approach is proposed to analyze the stability robustness of generalized state space systems with structured perturbations the presented method is computationally simple to use and can easily be calculated by computer as far as we are aware this paper seems to be the first one to solve the robust stability problems for generalized state space systems with structured uncertainties the robust stability problem of generalized state space systems is more complicated than that of regular state space systems because it needs consideration of not only stability robustness but also system regularity and impulse elimination the latter two ones need not be considered in regular state space systems

remark on the strichartz estimates in the radial case abstract we prove up to an endpoint the full radial strichartz estimates for the schrodinger type equation in the radial case our results improve the one obtained in guo and wang 2011 2 the main ingredient is that we exploit some orthogonality to overcome the logarithmic divergence

least energy solitary waves for a system of nonlinear schrodinger equations in mathbb r n in this paper we consider systems of coupled schrodinger equations which appear in nonlinear optics the problem has been considered mostly in the one dimensional case here we make a rigorous study of the existence of least energy standing waves solitons in higher dimensions we give conditions on the parameters of the system under which it possesses a solution with least energy among all multi component solutions conditions under which the system does not have positive solutions and the associated energy functional cannot be minimized on the natural set where the solutions lie

uniqueness for stochastic non linear wave equations abstract we prove uniqueness of solutions of the equation u t t u f u g u w u 0 u 0 u t 0 v 0 on r d in the class of processes with paths in h 1 here u 0 v 0 is a random variable in h 1 l 2 w is a spatially homogeneous wiener process with finite spectral measure and f g are locally lipschitz functions of subcritical growth

robust controllability assessment via semi definite programming abstract this paper presents a new framework for systematic assessment of the controllability of uncertain linear time invariant lti systems the objective is to evaluate controllability of an uncertain system with norm bounded perturbation over the entire uncertain region the method is based on a singular value minimization problem over the entire complex plane to solve the problem first a necessary and sufficient condition is proposed to avert the difficulties of griding over the complex plane and to verify the controllability of directed and undirected networks in a single step secondly the results are utilized to formulate the problem as two lyapunov based linear matrix inequalities lmis for undirected networks and four lmis for directed ones the proposed approach is subsequently extended to evaluate the maximum guaranteed distance to uncontrollability of a system through a quasi convex optimization problem whose solution is guaranteed to be globally optimal by duality analogous results are established for robust observability of directed and undirected networks the proposed framework is implemented efficiently using lmi tools and provides a fast and reliable tool for the assessment of robust controllability and by duality robust observability it is also extendable to robust control system design problems such as control node selection for uncertain systems

concentration of blow up solutions for the coupled schrodinger equations in r2 abstract we prove a concentration result for blow up solutions of the coupled schrodinger equations with non spherically symmetric initial data in h 1 r 2 using the profile decomposition technique and assuming a conjecture of sirakov is valid which was proved by wei and yao and ikoma for some cases we could remove the radial restriction on the initial data and improve the concentration rate from lu and liu 2011 9

counterexamples to strichartz estimates for the kinetic transport equation based on besicovitch sets abstract we show that the strichartz estimates for the kinetic transport equation containing l x norms in the spatial variables x fail on the characteristic functions of besicovitch sets

regular articlegeneralized strichartz inequalities for the wave equation we make a synthetic exposition of the generalized strichartz inequalities for the wave equation obtained in 6 together with the limiting cases recently obtained in 13 with as simple proofs as possible the proofs combine stationary phase estimates dyadic decompositions the hardy littlewood inequality or the holder inequality in time and abstract duality and interpolation arguments

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

global existence for a coupled system of schrodinger equations with power type nonlinearities in this manuscript we consider the cauchy problem for a schrodinger system with power type nonlinearitiesi tuj uj k 1majk uk p uj p 2uj 0 uj x 0 j0 x where uj rn r c j0 rn c for j 1 2 m and ajk akj are positive real numbers global existence for the cauchy problem is established for a certain range of p a sharp form of a vector valued gagliardo nirenberg inequality is deduced which yields the minimal embedding constant for the inequality using this minimal embedding constant global existence for small initial data is shown for the critical case p 1 2 n finite time blow up as well as stability of solutions in the critical case is discussed

on the boundary strichartz estimates for wave and schrodinger equations abstract we consider the l t 2 l x r estimates for the solutions to the wave and schrodinger equations in high dimensions for the homogeneous estimates we show l t 2 l x estimates fail at the critical regularity in high dimensions by using stable levy process in r d moreover we show that some spherically averaged l t 2 l x estimate holds at the critical regularity as a by product we obtain strichartz estimates with angular smoothing effect for the inhomogeneous estimates we prove double l t 2 type estimates

the stability robustness of generalized eigenvalues the concept of stability radius is generalized to matrix pairs a matrix pair is said to be stable if its generalized eigenvalues are located in the open left half of the complex plane the stability radius of a matrix pair a b is defined to be the norm of the smallest perturbation delta a such that a delta a b is unstable the purpose is to estimate the stability radius of a given matrix pair depending on whether the matrices under consideration are complex or real the problem can be classified into two cases the complex case is easy and a complete solution is provided the real case is more difficult and only a partial solution is given

fractal strichartz estimate for the wave equation abstract we consider strichartz estimates for the wave equation with respect to general measures which satisfy certain growth conditions in r 3 1 we obtain the sharp estimate and in higher dimensions improve the previous results

strichartz estimates for dirichlet wave equations in two dimensions with applications we establish the strauss conjecture for nontrapping obstacles when the spatial dimension n is two as pointed out in 7 this case is more subtle than n 3 or 4 due to the fact that the arguments of the first two authors 11 burq 1 and metcalfe 9 showing that local strichartz estimates for obstactles imply global ones require that the sobolev index equal 1 2 when n 2 we overcome this difficulty by interpolating between energy estimates 0 and ones for 1 2 that are generalizations of minkowski space estimates of fang and the third author 4 5 the second author 12 and sterbenz 14

ground state of n coupled nonlinear schrodinger equations in mathbb r n n leq 3 we establish some general theorems for the existence and nonexistence of ground state solutions of steady state n coupled nonlinear schrodinger equations the sign of coupling constants ij s is crucial for the existence of ground state solutions when all ij s are positive and the matrix is positively definite there exists a ground state solution which is radially symmetric however if all ij s are negative or one of ij s is negative and the matrix is positively definite there is no ground state solution furthermore we find a bound state solution which is non radially symmetric when n 3

singular systems a new approach in the time domain a new approach in the time domain is developed for the study of singular linear systems of the form e dot x ax bu y cx with e singular central to the approach is the fundamental triple alpha e a 1 e alphae a 1 b c with a real number satisfying det alpha e a neq 0 controllability and observability matrices are expressed in terms of the fundamental triple new tests for impulse controllability and impulse observability are also established feedback control problems including pole placement decoupling and disturbance localization are studied by use of a modified proportional and derivative feedback of the state in the form of u f alpha x dot x v

endpoint estimates and global existence for the nonlinear dirac equation with potential abstract we prove endpoint estimates with angular regularity for the wave and dirac equations perturbed with a small potential the estimates are applied to prove global existence for the cubic dirac equation perturbed with a small potential for small initial h 1 data with additional angular regularity this implies in particular global existence in the critical energy space h 1 for small radial data

robust c controllability and or c observability for uncertain descriptor systems with interval perturbations in all matrices for descriptor systems of the form ex spl dot t ax t bu t y t cx t existing results are mostly for systems with uncertainties in the a matrix only the case where the e matrix is perturbed renders its robust analysis quite involved and has seldom been touched by researchers this paper will show that if e a b and g are all interval matrices the robust controllability and or observability problems can be solved in terms of the structured singular value p of certain fixed matrices necessary and sufficient conditions are given using the structured singular value spl mu by changing the problem into a robust nonsingularity problem for a class of uncertain matrices