stochastic stability and l1 gain analysis for positive nonlinear semi markov jump systems with time varying delay via t s fuzzy model approach abstract this paper treats the issues of stochastic stability and l 1 gain analysis for continuous time positive fuzzy semi markov jump systems pfsmjss with time varying delay the system under consideration involves semi markov stochastic process related to weibull distribution the practical systems such as lotka volterra population model described by pfsmjss always need to consider the sudden change in the operating process to deal with these problems a sufficient condition for stochastic stability of pfsmjss is established which is dependent of the bound of time varying delay then based on the obtained results l 1 gain performance is analyzed in standard linear programming lp finally lotka volterra population model shows the validity of the conclusions

positive filtering for continuous time positive systems under l 1 performance in this paper we address the positive filtering problem for positive continuous time systems under the l1 induced performance a pair of positive filters with error bounding feature is proposed to estimate the output of positive systems a novel characterisation is first obtained to ensure that the filtering error system is asymptotically stable with a prescribed l1 induced performance then necessary and sufficient conditions for the existence of required filters are presented and the obtained results are expressed in terms of linear programming problems which can be easily checked by standard software finally a numerical example is given to illustrate the effectiveness of the proposed design procedures

max type copositive lyapunov functions for switching positive linear systems the paper aims to expand the stability analysis framework based on copositive lyapunov functions clfs for arbitrarily switching positive systems with continuous or discrete time dynamics the first part focuses on max type clfs it provides an algebraic characterization in terms of weak quasi linear inequalities as well as an existence criterion relying on a concrete construction procedure in terms of perron frobenius eigenstructure the second part explores the connections between max type and two other classes of clfs namely linear and quadratic diagonal new qualitative and quantitative features are revealed for the clfs belonging to the two mentioned classes

stabilization of positive switched linear systems and its application in consensus of multiagent systems in this note we study the stabilization problem of positive switched linear system psls with disturbance by introducing a class of weak common linear copositive lyapunov functions we design time dependent switching rules under which the system is asymptotically stable both continuous time and discrete time psls are taken into consideration and the obtained results are also extended to nonpositive switched systems finally the theoretical results are applied to consensus problem of multiagent systems with switching topologies and disturbance

improved results on stability of continuous time switched positive linear systems in this paper the problems of stability for switched positive linear systems splss under arbitrary switching are investigated in a continuous time context the so called copositive polynomial lyapunov function cplf giving a generalization of copositive types of lyapunov function is first proposed which is formulated in a higher order form of the positive states of the underlying systems it is illustrated in this paper that some classical types of lyapunov functions can be seen as special cases of the proposed cplf then new stability conditions are developed by the new lyapunov function approach it is also proved that the conservativeness of the obtained criteria can be further reduced as the degree of the lyapunov function increases a numerical example is given to demonstrate the effectiveness and less conservativeness of the developed techniques

asymptotic stability and stabilizability of special classes of discrete time positive switched systems abstract in this paper we consider discrete time positive switched systems switching among autonomous subsystems characterized either by monomial matrices or by circulant matrices necessary and sufficient conditions are provided guaranteeing either global uniform asymptotic stability or stabilizability i e the possibility of driving to zero the state trajectory corresponding to any initial state by resorting to some switching sequence such conditions lead to simple algorithms that allow to easily detect under suitable conditions whether a given positive switched system is not stabilizable

stability of continuous time positive switched linear systems a weak common copositive lyapunov functions approach in this paper we study the problem of asymptotic stability of continuous time positive switched linear systems under both arbitrary and restricted switchings it is well known that asymptotic stability under arbitrary switching can be implied by several classes of strong common copositive lyapunov functions clfs i e functions whose derivative along the nontrivial system trajectories is negative however asymptotically stable positive switched systems may not admit strong common clfs the main contribution of this paper is to study the stability problem by requiring only weak common clfs firstly necessary and sufficient conditions are established for asymptotic stability under arbitrary switching among them an easily verifiable graphical stability criterion based on the connectivity of the digraphs associated with the subsystem matrices is proposed secondly we further relax the obtained graphical condition to derive a relaxed weak excitation condition for asymptotic stability under dwell time switching finally two examples are provided to illustrate the effectiveness of our theoretical results

a boundary value problem for nematic liquid crystals with a variable degree of orientation in this paper we represent the degree of microscopic order through a scalar valued function the degree of orientation which vanishes where the fluid becomes isotropic

on the simultaneous diagonal stability of a pair of positive linear systems in this paper we derive a necessary and sufficient condition for the existence of a diagonal common quadratic lyapunov function cqlf for a pair of positive linear time invariant lti systems

nonlinear theory of defects in nematic liquid crystals phase transition and flow phenomena this is a summary of some recent mathematical advances in the theory of defects in nematic liquid crystals it also includes some new results concerning disclinations line defects which have not been published elsewhere

a linear framework on the distributed model predictive control of positive systems abstract this paper investigates the distributed model predictive control dmpc for positive systems with interval and polytopic uncertainties respectively different from the traditional quadratic dmpc a new linear dmpc framework is established for positive systems first a linear cooperative cost function performance index function is introduced for the dmpc of positive systems accordingly some linear constraint conditions are imposed on the systems then a linear co positive lyapunov function is constructed using the linear lyapunov function the dmpc controllers are designed for interval positive systems and polytopic positive systems respectively the controller of each subsystem contains not only the state information of the subsystem but also the state information of other correlated subsystems linear programming based optimization algorithm is addressed to compute the controller parameters to guarantee the feasibility and stability of the systems a cone invariant set is proposed finally two case studies are provided to illustrate the effectiveness of the theoretical findings

performance optimization over positive l cones abstract in this paper we study linear systems with positivity type of constraints first we consider the case where the input to a system is restricted to be in the positive cone of l denoted by l and seek to characterize the system s induced norm from l to l we obtain an exact characterization of this norm which is particularly easy to calculate in the case of lti systems furthermore we consider and solve the model matching problem and show that time varying linear or nonlinear control filtering does not improve the performance with respect to this norm for lti systems in the second part of the paper we consider the case when the output is forced to be in the positive l cone when the input is in this cone we show if internal positivity is sought a dynamic optimal controller offers no advantage over a static one also if the measurement matrix satisfies certain conditions synthesizing an optimal static output feedback controller reduces to a linear program

l1 induced norm and controller synthesis of positive systems in this paper the problem of 1 1 induced controller design for discrete time positive systems is investigated with the use of linear lyapunov function an analytical method to compute the exact value of 1 1 induced norm is first presented then a novel characterization for stability and 1 1 induced performance is proposed based on the characterization a necessary and sufficient condition for the existence of desired controllers is derived and an iterative convex optimization approach is developed to solve the condition in addition the synthesis of the state feedback controller for single input multiple output simo positive systems is investigated for this special case an analytic solution is established to show how the optimal 1 1 induced controller can be designed and some links to the spectral radius of the closed loop systems are provided finally the theoretical results are illustrated through a numerical example

exponential stabilization of switched time varying systems with delays and disturbances abstract this paper deals with exponential stabilization for a class of switched time varying systems by taking time varying delays and nonlinear disturbances into consideration time dependent switching signals have been characterized in terms of metzler matrices such that the resulting system is globally exponentially stable compared with preceding works we introduce a model transformation and an approach without involving the lyapunov krasovskii functional to derive new exponential stability criteria for switched time varying systems under the average dwell time switching numerical examples show that the obtained theoretical results can be applied to some cases not covered by some existing results

on delay independent stability of a class of nonlinear positive time delay systems we present a condition for delay independent stability of a class of nonlinear positive systems this result applies to systems that are not necessarily monotone and extends recent work on cooperative nonlinear systems

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

dynamics of the ericksen leslie equations with general leslie stress ii the compressible isotropic case in this article the non isothermal compressible ericksen leslie system for nematic liquid crystals subject to general leslie stress is considered it is shown that this system is locally well posed within the l q setting and that for initial data close to equilibria points which are identical with the ones for the incompressible situation the solution exists globally moreover any global solution which does not develop singularities converges to an equilibrium in the topology of the natural state manifold note that no structural assumptions on the leslie coefficients are imposed and in particular parodi s relation is not being assumed the results can be viewed as an extension of the studies in hieber and pruss math ann 369 977 996 2017 for the incompressible case to the compressible situation

stabilization of discrete time switched positive linear systems via weak switched linear copositive lyapunov function abstract by introducing a class of weak switched linear copositive lyapunov functions this paper designs new time dependent switching signals guaranteeing asymptotic stability of discrete time switched positive linear systems whose subsystems are only marginally stable our results have advantages over some existing ones in the literature moreover the main results have been extended to the general switched linear time varying system with disturbance numerical examples well demonstrate the superiority of the obtained results

technical communique on stability of a class of switched nonlinear systems this note is focused on the stability analysis for a class of switched nonlinear systems with disturbance input and delay sufficient conditions in terms of linear inequalities are presented such that the switched system is asymptotically stable for arbitrary switching any admissible sector nonlinearities and disturbances and any constant delay we not only drop a condition of one result given in aleksandrov chen platonov and zhang 2011 but also extend the main result to a more general switched nonlinear system with disturbance input and delay

a structure preserving fem for the uniaxially constrained mathbf q tensor model of nematic liquid crystals we consider the one constant landau de gennes model for nematic liquid crystals the order parameter is a traceless tensor field mathbf q which is constrained to be uniaxial mathbf q s mathbf n otimes mathbf n d 1 mathbf i where mathbf n is a director field s in mathbb r is the degree of orientation and d ge 2 is the dimension building on similarities with the one constant ericksen energy we propose a structure preserving finite element method for the computation of equilibrium configurations we prove stability and consistency of the method without regularization and gamma convergence of the discrete energies towards the continuous one as the mesh size goes to zero we design an alternating direction gradient flow algorithm for the solution of the discrete problems and we show that such a scheme decreases the energy monotonically finally we illustrate the method s capabilities by presenting some numerical simulations in two and three dimensions including non orientable line fields

brief paper on linear co positive lyapunov functions for sets of linear positive systems in this paper we derive necessary and sufficient conditions for the existence of a common linear co positive lyapunov function for a finite set of linear positive systems both the state dependent and arbitrary switching cases are considered our results reveal an interesting characterisation of linear stability for the arbitrary switching case namely the existence of such a linear lyapunov function can be related to the requirement that a number of extreme systems are metzler and hurwitz stable examples are given to illustrate the implications of our results

stability analysis and delay control for switched positive linear systems in this paper the problems of stability analysis and delay control for switched positive linear systems splss are studied by using average dwell time adt switching by developing a new class of multiple piecewise continuous linear copositive lyapunov function mplclf approach an improved stability condition for splss is obtained which can obtain a tighter bound on the adt guaranteeing the stability the obtained stability condition covers the existing results on stability analysis for splss since the multiple linear copositive lyapunov function used in the literature can be viewed as a special case of our proposed mplclf and our result can obtain a tighter bound of the adt then a more efficient controller design method over the existing ones is also established for splss with switching delay to guarantee the underlying system to be exponentially stable to illustrate the advantages of our proposed results two numerical examples are finally given

stability of positive linear switched systems on ordered banach spaces abstract we provide sufficient criteria for the stability of positive linear switched systems on ordered banach spaces the switched systems can be generated by finitely many bounded operators in infinite dimensional spaces with a general class of order inducing cones in the discrete time case we assume an appropriate interior point of the cone whereas in the continuous time case an appropriate interior point of the dual cone is sufficient for stability this is an extension of the concept of linear lyapunov functions for positive systems to the setting of infinite dimensional partially ordered spaces we illustrate our results with examples

on nematic liquid crystals with variable degree of orientation energy minimizing static configurations of nematic liquid crystals with variable degree of orientation can be described by a map which minimizes certain degenerate variational integrals this mathematical model was recently posed by j l erickson in this paper we shall prove the existence local uniqueness and global regularity of minimizers we shall also estimate the size of possible defects

brief paper stability of switched positive linear systems with average dwell time switching in this paper the stability analysis problem for a class of switched positive linear systems splss with average dwell time switching is investigated a multiple linear copositive lyapunov function mlclf is first introduced by which the sufficient stability criteria in terms of a set of linear matrix inequalities are given for the underlying systems in both continuous time and discrete time contexts the stability results for the splss under arbitrary switching which have been previously studied in the literature can be easily obtained by reducing mlclf to the common linear copositive lyapunov function used for the system under arbitrary switching those systems finally a numerical example is given to show the effectiveness and advantages of the proposed techniques

a coupled ericksen allen cahn model for liquid crystal droplets abstract we present a model and discretization that couples the ericksen model of liquid crystals with variable degree of orientation to the allen cahn equations with a mass constraint the coupled system models liquid crystal droplets with anisotropic surface tension effects due to the liquid crystal molecular alignment the total energy consists of the ericksen energy phase field allen cahn energy and a weak anchoring energy that couples the liquid crystal to the diffuse interface we describe our discretization of the total energy along with a method to compute minimizers via a discrete gradient flow algorithm which has a strictly monotone energy decreasing property numerical experiments are given in three dimensions that illustrate a wide variety of droplet shapes that result from their interaction with defects

stabilization of switched positive system with impulse and marginally stable subsystems a mode dependent dwell time method abstract in this article the problem of stabilization for the switched positive linear system spls with impulse and marginally stable subsystems is studied in which both the continuous time case and the discrete time case are taken into account by the usage of a piecewise weak linear copositive lyapunov function lclf and the mode dependent dwell time mdt time dependent switching control laws are designed to guarantee the asymptotic stability of the system furthermore based on the comparison principal for positive systems the main results are generalized to the time variant spls two examples are worked out to indicate that our results have advantages over some known results in the literature

stability analysis of positive polynomial fuzzy model based control systems with time delay under imperfect premise matching this paper deals with the stability and positivity analysis of polynomial fuzzy model based pfmb control systems with time delay which is formed by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop under imperfect premise matching to improve the design and realization flexibility the polynomial fuzzy model and the polynomial fuzzy controller are allowed to have their own set of premise membership functions a sum of squares based stability analysis approach using the lyapunov stability theory is employed to investigate the positivity and stability of the pfmb control systems and synthesize the polynomial fuzzy controller in order to relax the stability results we propose two methods first membership functions are considered as symbolic variables in the stability analysis and second the property of the membership functions and the boundary information of the membership functions are considered in the stability analysis a simulation example is given to illustrate the effectiveness of the proposed approach

positive filtering for positive takagi sugeno fuzzy systems under l 1 performance in this paper the positive filtering problem is addressed for positive takagi sugeno t s fuzzy systems under the 1 induced performance to estimate the output of positive t s fuzzy systems error bounding positive filters are constructed a new performance characterization is first established to guarantee the asymptotic stability of the filtering error system with the 1 induced performance moreover sufficient conditions expressed by linear programming problems are derived to design the required filters finally a numerical example is presented to show the effectiveness of the derived theoretical results

on linear copositive lyapunov functions and the stability of switched positive linear systems we consider the problem of common linear copositive lyapunov function existence for positive switched linear systems in particular we present a necessary and sufficient condition for the existence of such a function for switched systems with two constituent linear time invariant systems several applications of this result are also given

stability and positivity of equilibria for subhomogeneous cooperative systems building on recent work on homogeneous cooperative systems we extend results concerning stability of such systems to subhomogeneous systems we also consider subhomogeneous cooperative systems with constant input and relate the global asymptotic stability of the unforced system to the existence and stability of positive equilibria for the system with input

on easily verifiable conditions for the existence of common linear copositive lyapunov functions this technical note addresses the existence of a common linear copositive lyapunov function clclf for continuous time switched positive linear systems two necessary and sufficient conditions for the existence of a clclf are provided one condition shows that the existence of a clclf for a family of n n dimensional metzler matrices can only be determined by n 1 lower dimensional hurwitz matrices the other condition further shows that the existence of a clclf is also determined by n 1 characteristic constants which gives an easily verifiable computation algorithm without checking a large amount of metzler hurwitz matrices

a note on the eigenvectors of perturbed matrices with applications to linear positive systems a result is presented describing the eigenvectors of a perturbed matrix for a class of structured perturbations one motivation for doing so is that positive eigenvectors of nonnegative irreducible matrices are known to induce norms acting much like lyapunov functions for linear positive systems which may help estimate or control transient dynamics the results apply to both discrete and continuous time linear positive systems the theory is illustrated with several examples

a novel methodology to solve the cqlf problem for a finite set of stable second order systems a new methodology to provide conclusive information about the existence non existence of a common quadratic lyapunov function cqlf for a finite set of stable second order systems is presented despite the high complexity of the cqlf problem even in the case of n second order systems the results presented in this paper have a very simple and intuitive theoretical support including topics such as classical intersection of convex sets and properties of convex linear combinations illustrative examples to show the performance of the proposed methodology are provided

asymptotics of minimizers of variational problems involving curl functional in this paper we are concerned with singularly perturbed variational problems involving the curl functional which arise in the mathematical theory of liquid crystals the asymptotic behavior of the minimizers in the singular limiting process is discussed which is closely related to the variational problems for curl functional under various constraints

vibration analysis of membranes and plates by a discrete least squares technique abstract a discrete least squares technique for computing the free transverse vibrations of membranes and plates is presented the proposed technique consists of two steps in the first step a quadratic matrix eigenvalue problem resulting from the minimization of a discrete residual error function is solved if the accuracy of the eigenvalue approximations is not sufficient these initial approximations can be improved by using the gauss newton method numerical results concerning the vibration of the fixed circular membrane fixed circular membrane with circular ridges and clamped circular plate are reported as shown the trial functions need not satisfy any of the prescribed boundary conditions which makes possible the vibration analysis of membranes and plates with complicated shape at the end of the paper a short discussion is given on how to use the proposed technique in solving problems of higher complexity and some possible directions in the development of this technique are outlined

free vibrations of non homogeneous circular and annular membranes a non homogeneous membrane is a membrane that has variable thickness or material density several recent publications deal with the axisymmetric and antisymmetric modes of transverse vibration for a composite doubly connected and solid membrane with constant or variable density in this paper exact solutions for both the axisymmetric and antisymmetric modes of circular and annular membranes with any piecewise polynomial variation of the density are given using a power series solution the dynamic problem is solved exactly using a recurrence relationship up to any accuracy desired by deriving the dynamic stiffness matrix for circular and annular membrane elements many results for linear parabolic and cubic variation of complete and annular membranes are solved using the dynamic stiffness method and presented in the tables

a novel linear second order unconditionally energy stable scheme for a hydrodynamic q tensor model of liquid crystals abstract the hydrodynamic q tensor model has been used for studying flows of liquid crystals and liquid crystal polymers it can be derived from a variational approach together with the generalized onsager principle in which the total energy decreases in time in this paper we develop a novel linear second order semi discrete scheme in time to solve the governing system of equations in the model the scheme is developed following the novel energy quadratization strategy so that it is linear and unconditionally energy stable at the semi discrete level this scheme is further discretized in space using a second order finite difference method and implemented on a gpu for high performance computing the convergence rate in time is established using a mesh refinement test several numerical examples are presented to demonstrate the usefulness of the model and the effectiveness of the numerical scheme in simulating defect dynamics in flows of liquid crystals

on the kamke muller conditions monotonicity and continuity for bi modal piecewise smooth systems abstract we show that the kamke muller conditions for bimodal piecewise smooth systems are equivalent to simple conditions on the vector fields defining the system as a consequence we show that for a specific class of such systems monotonicity is equivalent to continuity furthermore we apply our results to derive a stability condition for piecewise positive linear systems

sign properties of metzler matrices with applications abstract several results about sign properties of metzler matrices are obtained it is first established that checking the sign stability of a metzler sign matrix can be either characterized in terms of the hurwitz stability of the unit sign matrix in the corresponding qualitative class or in terms the negativity of the diagonal elements of the metzler sign matrix and the acyclicity of the associated directed graph similar results are obtained for the case of metzler block matrices and metzler mixed matrices the latter being a class of metzler matrices containing both sign and real type entries the problem of assessing the sign stability of the convex hull of a finite and summable family of metzler matrices is also solved and a necessary and sufficient condition for the existence of common lyapunov functions for all the matrices in the convex hull is obtained the concept of sign stability is then generalized to the concept of ker b sign stability a problem that arises in the analysis of certain jump markov processes a sufficient condition for the ker b sign stability of metzler sign matrices is obtained and formulated using inverses of sign matrices and the concept of l matrices several applications of the results are discussed in the last section

diagonal riccati stability and positive time delay systems abstract we consider a class of algebraic riccati equations arising in the study of positive linear time delay systems we show that this class admits diagonal positive definite solutions this implies that exponentially stable positive linear time delay systems possess lyapunpov krasovskii functionals of a simple quadratic form we also show that for this class of equations the existence of positive definite solutions is equivalent to a simple spectral condition on the coefficient matrices

stability of discrete time positive switched linear systems with stable and marginally stable subsystems abstract in this paper we investigate the stability of discrete time positive switched linear systems the highlight of this work is that each subsystem is only assumed to be stable or marginally stable some sufficient conditions imposed on the subsystem matrices and switching signals are established to guarantee exponential stability of the considered positive switched linear systems an extension to the case of time varying and unbounded delay is also given compared with the continuous time counterparts in the literature our discrete time case is much more complicated involving the transitive connectivity between the subsystems specified by the switching signals

dynamics of the ericksen leslie equations with general leslie stress i the incompressible isotropic case the ericksen leslie model for nematic liquid crystals in a bounded domain with general leslie and isotropic ericksen stress tensor is studied in the case of a non isothermal and incompressible fluid this system is shown to be locally strongly well posed in the l p setting and a dynamical theory is developed the equilibria are identified and shown to be normally stable in particular a local solution extends to a unique global strong solution provided the initial data are close to an equilibrium or the solution is eventually bounded in the topology of the natural state manifold in this case the solution converges exponentially to an equilibrium in the topology of the state manifold the above results are proven without any structural assumptions on the leslie coefficients and in particular without assuming parodi s relation

comparison of discrete singular convolution and generalized differential quadrature for the vibration analysis of rectangular plates this paper presents a comprehensive comparison study between the discrete singular convolution dsc and the well known global method of generalized differential quadrature gdq for vibration analysis so as to enhance the understanding of the dsc algorithm the dsc method is implemented through lagrange s delta sequence kernel dsc lk which utilizes local lagrange polynomials to calculate weighting coefficients whereas the gdq requires global ones moreover it is shown that the treatments of boundary conditions and the use of grid systems are different in the two methods comparison study is carried out on 21 rectangular plates of different combinations of simply supported s clamped c and transversely supported with nonuniform elastic rotational restraint e edges and five rectangular plates of mixed supporting edges some of which with a range of aspect ratios and rotational spring coefficients all the results of the dsc lk agree very well with both those in the literature and newly computed gdq results furthermore it is observed that the dsc lk performs much better for plates vibrating at higher order eigenfrequencies unlike the gdq the dsc lk is numerically stable for problems which require a large number of grid points

the ericksen model of liquid crystals with colloidal and electric effects we present a robust discretization of the ericksen model of liquid crystals with variable degree of orientation coupled with colloidal effects and electric fields the total energy consists of the ericksen energy a weak anchoring or penalized dirichlet energy to model colloids and an electrical energy for a given electric field we describe our special discretization of the total energy along with a method to compute minimizers via a discrete quasi gradient flow algorithm which has a strictly monotone energy decreasing property numerical experiments are given in two and three dimensions to illustrate that the method is able to capture non trivial defect patterns such as the saturn ring defect we conclude with a rigorous proof of the convergence of our discrete energy to the continuous energy

discrete weighted least squares method for the poisson and biharmonic problems on domains with smooth boundary in this article a discrete weighted least squares method for the numerical solution of elliptic partial differential equations exhibiting smooth solution is presented it is shown how to create well conditioned matrices of the resulting system of linear equations using algebraic polynomials carefully selected matching points and weight factors two simple algorithms generating suitable matching points the chebyshev matching points for standard two dimensional domains and the approximate fekete points of sommariva and vianello for general domains are described the efficiency of the presented method is demonstrated by solving the poisson and biharmonic problems with the homogeneous dirichlet boundary conditions defined on circular and annular domains using basis functions in the form satisfying and in the form not satisfying the prescribed boundary conditions

extremal norms for positive linear inclusions abstract for finite dimensional linear semigroups which leave a proper cone invariant it is shown that irreducibility with respect to the cone implies the existence of an extremal norm in case the cone is simplicial a similar statement applies to absolute norms the semigroups under consideration may be generated by discrete time systems continuous time systems or continuous time systems with jumps the existence of extremal norms is used to extend results on the lipschitz continuity of the joint spectral radius beyond the known case of semigroups that are irreducible in the representation theory interpretation of the word

lyapunov functions and duality for convex processes the paper studies convex lyapunov functions for differential and difference inclusions with right hand sides given by convex processes that is by set valued mappings the graphs of which are convex cones convex conjugacy between weak lyapunov functions for such inclusions and lyapunov functions for adjoint inclusions is established asymptotic stability concepts are compared and the existence of convex lyapunov functions for classes of convex processes is shown the relevance of the results for the study of asymptotic controllability or stabilizability and of detectability for linear control systems with conical control and state constraints is underlined

a note on recursive schur complements block hurwitz stability of metzler matrices and related results it is known that the stability of a metzler matrix can be characterized in a routh hurwitz like fashion based on a recursive application of scalar schur complements 1 our objective in this brief note is to show that recently obtained stability conditions are equivalent statements of this result and can be deduced directly therefrom using only elementary results from linear algebra implications of this equivalence are also discussed and several examples are given to illustrate potentially interesting system theoretic applications of this observation

dynamic output feedback control for continuous time interval positive systems under l1 performance this paper is concerned with the design of l1 induced dynamic output feedback controller for continuous time positive systems with interval uncertainties a necessary and sufficient condition for stability and l1 induced performance of positive linear systems is proposed in terms of linear inequalities based on this conditions for the existence of robust dynamic output feedback controllers are established and an iterative convex optimization approach is developed to solve the conditions one illustrative example is provided to show the effectiveness and applicability of the theoretical results

stability and stabilizability criteria for discrete time positive switched systems in this paper we consider the class of discrete time switched systems switching between p autonomous positive subsystems first sufficient conditions for testing stability based on the existence of special classes of common lyapunov functions are investigated and these conditions are mutually related thus proving that if a linear copositive common lyapunov function can be found then a quadratic positive definite common function can be found too and this latter in turn ensures the existence of a quadratic copositive common function secondly stabilizability is introduced and characterized it is shown that if these systems are stabilizable they can be stabilized by means of a periodic switching sequence which asymptotically drives to zero every positive initial state conditions for the existence of state dependent stabilizing switching laws based on the values of a copositive linear quadratic lyapunov function are investigated and mutually related too finally some properties of the patterns of the stabilizing switching sequences are investigated and the relationship between a sufficient condition for stabilizability the existence of a schur convex combination of the subsystem matrices and an equivalent condition for stabilizability the existence of a schur matrix product of the subsystem matrices is explored

a constructive approach to linear lyapunov functions for positive switched systems using collatz wielandt sets we establish the link between linear lyapunov functions for positive switched systems and corresponding collatz wielandt sets this leads to an algorithm to compute a linear lyapunov function whenever a lyapunov function exists

stability and stabilizability of continuous time linear compartmental switched systems in this paper we introduce continuous time linear compartmental switched systems and investigate their stability and stabilizability properties by their nature these systems are always stable necessary and sufficient conditions for asymptotic stability for arbitrary switching functions and sufficient conditions for asymptotic stability under certain dwell time conditions on the switching functions are proposed finally stabilizability is thoroughly investigated and proved to be equivalent to the existence of a hurwitz convex combination of the subsystem matrices a condition that for positive switched systems is only sufficient for stabilizability

stability of diffusively coupled linear systems with an invariant cone abstract this paper concerns a question that frequently occurs in various applications is any diffusive coupling of stable linear systems also stable although it has been known for a long time that this is not the case we shall identify a reasonably diverse class of systems for which it is true

conditions on the stability of a class of second order switched systems for a special class of systems it is shown that the existence of a common quadratic lyapunov function cqlf is necessary and sufficient for the stability of an associated switched system under arbitrary switching furthermore it is shown that the existence of a cqlf for n n 2 subsystems is equivalent to the existence of a cqlf for every pair of subsystems an algorithm is proposed to compute a cqlf for the subsystems when it exists using the left and right eigenvectors of a critical matrix obtained from a matrix pencil

landau de gennes theory of nematic liquid crystals the oseen frank limit and beyond we study global minimizers of a continuum landau de gennes energy functional for nematic liquid crystals in three dimensional domains subject to uniaxial boundary conditions we analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly in w1 2 to a global minimizer predicted by the oseen frank theory for uniaxial nematic liquid crystals with constant order parameter moreover the convergence is uniform in the interior of the domain away from the singularities of the limiting oseen frank global minimizer we obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the landau de gennes global minimizer we also study the interplay between biaxiality and uniaxiality in landau de gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions

stability properties of a class of positive switched systems with rank one difference abstract given a single input continuous time positive system described by a pair a b with a a diagonal matrix we investigate under what conditions there exists a state feedback law u t c x t that makes the resulting controlled system positive and asymptotically stable by this meaning that a b c is metzler and hurwitz in the second part of this note we assume that the state space model switches among different state feedback laws c i i 1 2 p each of them ensuring the positivity and show that the asymptotic stability of this type of switched system is equivalent to the asymptotic stability of all its subsystems while its stabilizability is equivalent to the existence of an asymptotically stable subsystem

stability analysis and control synthesis for positive semi markov jump systems with time varying delay abstract this paper deals with stability analysis and control synthesis for positive semi markov jump systems s mjss with time varying delay in which the stochastic semi markov process related to nonexponential distribution is considered the main motivation for this paper is that the positive condition sometimes needs to be considered in s mjss and the controller design methods in the existing have some conservation to deal with these problems the weak infinitesimal operator is firstly derived from the point of view of probability distribution under the constraint of positive condition then some sufficient conditions for stochastic stability of positive s mjss are established by implying the linear lyapunov krasovskii functional depending on the bound of time varying delay furthermore an improved stabilizing controller is designed via decomposing the controller gain matrix such that the resulting closed loop system is positive and stochastically stable in standard linear programming the advantages of the new framework lie in the following facts 1 the weak infinitesimal operator is derived for s mjss with time varying delay under the constraint of positive condition and 2 the less conservative stabilizing controller is designed to achieve the desired control performance finally three examples one of which is the virus mutation treatment model are given to demonstrate the validity of the main results

stabilization of positive linear systems we consider stabilization of equilibrium points of positive linear systems which are in the interior of the first orthant the existence of an interior equilibrium point implies that the system matrix does not possess eigenvalues in the open right half plane this allows to transform the problem to the stabilization problem of compartmental systems which is known and for which a solution has been proposed already we provide necessary and sufficient conditions to solve the stabilization problem by means of affine state feedback a class of stabilizing feedbacks is given explicitly

decoupled energy stable schemes for phase field models of two phase complex fluids we consider in this paper numerical approximations of phase field models for two phase complex fluids we first reformulate the phase field models derived from an energetic variational formulation into a form which is suitable for numerical approximation and establish their energy laws then we construct two classes stabilized and convex splitting of decoupled time discretization schemes for the coupled nonlinear systems these schemes are unconditionally energy stable and lead to decoupled elliptic equations to solve at each time step furthermore these elliptic equations are linear for the stabilized version stability analysis and ample numerical simulations are presented

l induced output feedback controller synthesis for positive nonlinear systems via t s fuzzy model approach abstract this paper investigates the l induced output feedback controller synthesis problem for positive takagi sugeno t s fuzzy systems for positive t s fuzzy systems a sufficient condition for stability and l induced performance is first put forward using a group of linear inequalities then on the basis of these inequalities the conditions on which static output feedback controllers exist are set up and to solve these conditions a method of iterative convex optimization is provided moreover a complete solution to the synthesis problem of the controller is proposed based on linear programming for positive t s fuzzy systems with single output or input finally two examples are presented to illustrate and verify the effective and applicable methods derived in this paper

rectifiability of line defects in liquid crystals with variable degree of orientation in 2 hardt lin and the author proved that the defect set of minimizers of the modified ericksen energy for nematic liquid crystals consists locally of a finite union of isolated points and holder continuous curves with finitely many crossings in this article we show that each holder continuous curve in the defect set is of finite length hence locally the defect set is rectifiable for the most part the proof closely follows the work of de lellis et al rectifiability and upper minkowski bounds for singularities of harmonic q valued maps arxiv 1612 01813 2016 on harmonic mathcal q valued maps the blow up analysis in alper et al calc var partial differ equ 56 5 128 2017 allows us to simplify the covering arguments in 11 and locally estimate the length of line defects in a geometric fashion

existence of surface smectic states of liquid crystals abstract the landau de gennes model of liquid crystals is a functional acting on wave functions order parameters and vector fields director fields in a specific asymptotic limit of the physical parameters we construct critical points such that the wave function order parameter is localized near the boundary of the domain and we determine a sharp localization of the boundary region where the wave function concentrates furthermore we compute the asymptotics of the energy of such critical points along with a boundary energy that may serve in localizing the director field in physical terms our results prove the existence of a surface smectic state

linear copositive lyapunov functions for continuous time positive switched systems continuous time positive systems switching among p subsystems are introduced and a complete characterization for the existence of a common linear copositive lyapunov function for all the subsystems is provided when the subsystems are obtained by applying different feedback control laws to the same continuous time single input positive system the above characterization leads to a very easy checking procedure

necessary and sufficient conditions for stabilizability of positive lti systems this paper obtains necessary and sufficient conditions for the stabilization of positive lti systems using a vertex algorithmic approach the results of this paper also encompass positive direct output feedback control the topic of the sequential separation principle and a particularly efficient algorithm for a class of positive systems

on the stability of switched positive linear systems it was recently conjectured that the hurwitz stability of the convex hull of a set of metzler matrices is a necessary and sufficient condition for the asymptotic stability of the associated switched linear system under arbitrary switching in this note we show that 1 this conjecture is true for systems constructed from a pair of second order metzler matrices 2 the conjecture is true for systems constructed from an arbitrary finite number of second order metzler matrices and 3 the conjecture is in general false for higher order systems the implications of our results both for the design of switched positive linear systems and for research directions that arise as a result of our work are discussed toward the end of the note