on common diagonal lyapunov solutions abstract several recent results regarding common diagonal lyapunov solutions are further explored here the first one attributed to redheffer and revisited by shorten and narendra reduces the diagonal stability of a matrix to common diagonal lyapunov solutions on two matrices of order one less we present a shorter purely matrix theoretic proof of this result along with its extensions the second one concerns two different necessary and sufficient conditions due to oleng narendra and shorten for a pair of 2 2 matrices to share a common diagonal lyapunov solution we show that these two conditions are connected directly to each other

on discrete time diagonal and d stability abstract this paper introduces four discrete time analogs of different types of matrix stability diagonal simultaneous vertex and d stability the last three being defined in terms of a certain associated polytope of matrices the diagonal stability of any vertex of this polytope is shown to imply its simultaneous stability and hence d stability of the vertices it is shown by a counterexample that d stability as in the continuous time case is not equivalent to diagonal stability several important classes for which this equivalence is true are identified it is shown that simultaneous stability of the vertices is equivalent to the simultaneous stability of the whole polytope and it is conjectured that this equivalence holds without the requirement of simultaneity some other conjectures relating the four types of stability are made and it is shown that in the 2 x 2 case all four are equivalent

on the existence of a common quadratic lyapunov function for a rank one difference abstract suppose that a and b are real hurwitz matrices and that their difference a b is rank one then a and b have a common quadratic lyapunov function if and only if the product ab has no real negative eigenvalue this result is due to shorten and narendra who showed that it follows as a consequence of the kalman yacubovich popov lemma and the solution of the lur e problem here we present a new and independent proof based on results from convex analysis and the theory of moments

singularity conditions for the non existence of a common quadratic lyapunov function for pairs of third order linear time invariant dynamic systems abstract the condition that a finite collection of stable matrices a 1 a m has no common quadratic lyapunov function cqlf is formulated as a hierarchy of singularity conditions for block matrices involving a number of unknown parameters these conditions are applied to the case of two stable 3 3 matrices where they are used to derive necessary and sufficient conditions for the non existence of a cqlf

implementation of the hierarchical reference theory for simple one component fluids combining renormalization group theoretical ideas with the integral equation approach to fluid structure and thermodynamics the hierarchical reference theory is known to be successful even in the vicinity of the critical point and for subcritical temperatures we present here a software package independent of earlier programs for the application of this theory to simple fluids composed of particles interacting via spherically symmetrical pair potentials restricting ourselves to hard sphere reference systems using the hard core yukawa potential with z 1 8 sigma for illustration we discuss our implementation and the results it yields paying special attention to the core condition and emphasizing the decoupling assumption s role

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

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

hurwitz stability of metzler matrices in this note a simple method is presented that is both necessary and sufficient for determining whether a given metzler matrix a is hurwitz the method is based on the well known fact that a hurwitz metzler matrix is also diagonally stable by using this fact very simple conditions are derived for the hurwitz stability of a metzler matrix the conditions are stated in terms of the signs of the diagonal entries of a sequence of lower dimensional matrices the efficacy of the conditions is demonstrated by applying them to determine stability in several examples

the α scalar diagonal stability of block matrices abstract we generalize two results kraaijevanger s 1991 characterization of diagonal stability via hadamard products and the block matrix version of the closure of the positive definite matrices under hadamard multiplication we restate our generalizations in terms of p matrices and scalar diagonally stable matrices

on the kalman yakubovich popov lemma for discrete time positive linear systems a novel simple proof and some related results theorems of alternatives on the feasibility of linear matrix inequalities lmis are used in order to provide novel simple proofs for two considered versions of the kalman yakubovich popov kyp lemma for discrete time positive linear systems two different and novel recursive methods to determine whether a positive matrix is or is not schur are obtained as an application of an existing connection between the strict inequality version of the kyp lemma for single input single output siso discrete time positive linear systems and a schur matrix condition examples are included which provide illustration on these recursive methods

on the kalman yakubovich popov lemma the purpose of this note is to present a new elementary proof for the multivariable k y p lemma a minimum of linear algebra and finite dimensional convexity theory is used

on a theorem of redheffer concerning diagonal stability abstract an important problem in system theory concerns determining whether or not a given lti system x ax is diagonally stable more precisely this problem is concerned with determining conditions on a matrix a such that there exists a diagonal matrix d with positive diagonal entries i e a positive diagonal matrix satisfying a t d da q 0 while this problem has attracted much attention over the past half century two results of note stand out i a result based on theorems of the alternative derived by barker berman and plemmons and ii algebraic conditions derived by redheffer this paper is concerned with the second of these conditions our principal contribution is to show that redheffer s result can be obtained from the kalman yacubovich popov lemma we then show that this method of proof leads to natural generalisations of redheffer s result and we use these results to derive new conditions for diagonal and hurwitz stability for special classes of matrices

a note on plurisubharmonic defining functions in mathbb c n let omega subset subset mathbb c n n 3 be a smoothly bounded domain suppose that admits a smooth defining function which is plurisubharmonic on the boundary of then a diederich fornaess exponent can be chosen arbitrarily close to 1 and the closure of admits a stein neighborhood basis

stronger than lyapunov notions of matrix stability or how flowers help solve problems in mathematical ecology abstract persistent patterns of interactions in a multi component system e g intra and inter species relations in a community of n interacting species may imply a number of formalizations as special stronger than lyapunov notions of matrix stability like d stability qualitative stability volterra lyapunov stability and others a variety of these notions each having a certain motivation with regard to uncertainties inherent in model applications constitute a hierarchical topology sometimes very intricate and not yet well understood in a formal space of real n n matrices as visible forms of this hierarchy matrix flowers are suggested where petals correspond to subsets of particular stability kinds whose visible inclusion intersection represent logical implication junction the flowers are constructed under a few simple conventions and in the absence of ready characterizations to draw a petal often poses a challenging mathematical problem whose solution may reveal a new biological knowledge of a general nature particular petals concern the topics of strong and weak interactions in a community key species in its structure diffusion instability in its spatial dynamics where the flower plays a heuristic role in formulating a new problem or and stimulating a new application

a note on plurisubharmonic defining functions in mathbb c 2 let mathbb c 2 be a smoothly bounded domain suppose that admits a smooth defining function which is plurisubharmonic on the boundary of then the diederich fornaess exponent can be chosen arbitrarily close to 1 and the closure of admits a stein neighborhood basis

brief paper lyapunov exponential stability of 1 d linear hyperbolic systems of balance laws explicit boundary dissipative conditions are given for the exponential stability in l 2 norm of one dimensional linear hyperbolic systems of balance laws email protected email protected email protected x 0 over a finite interval when the matrix m is marginally diagonally stable the result is illustrated with an application to boundary feedback stabilisation of open channels represented by linearised saint venant exner equations

h_2 analysis of lti systems via conversion to externally positive systems motivated by recent advances in the study of linear time invariant lti positive systems we explore analysis techniques of general not necessarily positive lti systems using positive system theory even though a positive system is characterized by its peculiar property that its impulse response is nonnegative we often deal with nonnegative impulse responses even in general lti system analysis a typical example is the computation of the h 2 norm where we focus on squared impulse responses to deal with such products of impulse responses in a systematic fashion in this paper we first establish a construction technique of an lti system whose impulse response is given by the product of impulse responses of two different lti systems then as the main result we reduce the h 2 norm computation problem of a general lti system into the l infty induced norm computation problem or l 1 problem in short of a positive system by which we can derive various formulas for the h 2 norm computation

extension of diagonal stability and stabilization for continuous time fractional positive linear systems abstract this paper considers an extension of diagonal stability for continuous fractional positive linear systems fpls with the fractional order 0 1 based on diagonal stability of metzler matrix an extension that involves the combination of a diagonal positive definite matrix and a skew symmetric anti diagonal matrix related a hurwitz and metzler matrix is established combining this extension and the linear matrix inequalities lmis criteria of stability for fractional order systems fos a necessary and sufficient condition for extension diagonal stability of fpls is presented a state feedback controller is given which ensures the stabilization and positivity of the closed loop systems numerical examples are provided to demonstrate the effectiveness and applicability of the proposed methods

common lyapunov solutions for two matrices whose difference has rank one abstract real stable matrices a and b with rank of a b equal to one have a common lyapunov solution if and only if their product ab has no real negative eigenvalue this was proved by shorten and narendra r n shorten k s narendra on common quadratic lyapunov functions for pairs of stable lti systems whose system matrices are in companion form ieee trans automat control 48 4 2003 618 621 whose proof is based on the fundamental results of kalman on lure s problem in this paper we give an alternative proof of this result and its generalization to the general regular inertia case and to the case when the matrices a and b are complex

an iterative method of lyapunov function construction for differential inclusions abstract the problem of choosing the lyapunov function from the class of quadratic forms for differential inclusions is reduced to that of searching saddle points of a suitable function an iterative algorithm is used for this purpose

brief paper diagonal stability of stochastic systems subject to nonlinear disturbances and diagonal h2 norms a known result in the stability theory of stochastic systems with nonlinear lipschitz bounded noise intensity states that the robust stability radius of such a stochastic system is equal to the inverse of the h 2 norm of its noise to output transfer function this paper extends this result to the case where one is interested in the diagonal stability of the system under consideration this problem arises naturally when studying large scale interconnected systems subject to random perturbations as one is often interested in using diagonal or block diagonal lyapunov functions for such plants the main result of the paper is the characterization of the diagonal stochastic stability radius which is similar to the mentioned result for non diagonal stability

symmetric formulation of the s procedure kalman yakubovich popov lemma and their exact losslessness conditions in the robust stability analysis of linear time invariant systems the frequency domain and uncertainty domain of interest play algebraically symmetric roles this paper presents a new formulation of the s procedure and the kyp lemma which emphasizes this symmetry the new formulation provides a novel and unified approach for understanding when the kyp lemma provides an exact lmi test for robust stability the notions of weak and strong mutual losslessness are introduced to characterize lossless s procedure and kyp lemma the new formulation has sufficient flexibility to accommodate some recent extensions of the kyp lemma including the generalized kyp lemma the kyp lemma for nd systems and the diagonal bounded real lemma for internally positive systems using the proposed framework we also provide a lossless scaled small gain test for internally positive systems which gives an alternative proof that the structured singular value for such systems with arbitrary number of scalar uncertainties can be efficiently computed

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

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

common solution to the lyapunov equation for 2 2 complex matrices abstract in this work we solve the problem of a common solution to the lyapunov equation for 2 2 complex matrices we show that necessary and sufficient conditions for the existence of a common solution to the lyapunov equation for 2 2 complex matrices a and b is that matrices a i i b i i and a i i 1 b i i have no negative real eigenvalues for all r we show how these results relate to a special class of 4 4 real matrices

a characterization of lyapunov diagonal stability using hadamard products abstract several new characterizations of lyapunov diagonal stability are presented one of the characterizations is used to derive a simple necessary and sufficient condition for lyapunov diagonal stability of a 3 3 matrix

infinite compressibility states in the hierarchical reference theory of fluids ii numerical evidence continuing our investigation into the hierarchical reference theory of fluids for thermodynamic states of infinite isothermal compressibility t we now turn to the available numerical evidence to elucidate the character of the partial differential equation of the three scenarios identified previously only the assumption of the equations turning stiff when building up the divergence of t allows for a satisfactory interpretation of the data in addition to the asymptotic regime where the arguments of part i directly apply a similar mechanism is identified that gives rise to transient stiffness at intermediate cutoff for low enough temperature heuristic arguments point to a connection between the form of the fourier transform of the perturbational part of the interaction potential and the cutoff where finite difference approximations of the differential equation cease to be applicable and they highlight the rather special standing of the hard core yukawa potential as regards the severity of the computational difficulties

the bounded real lemma for internally positive systems and h infinity structured static state feedback we consider the bounded real lemma for internally positive linear time invariant systems we show that the h norm of such systems can be evaluated by checking the existence of a certain diagonal quadratic storage function taking advantage of this fact the problem of designing a structured static state feedback controller achieving internal stability contractiveness and internal positivity in closed loop becomes convex and tractable

matrix theoretic conditions for the realizability of sliding manifolds systems of the type s ksgn s s rm where sgn is the vector sign function usually determine the realizability of sliding modes in control systems with high frequency gain k a new condition on k is established for the global stability of this system using a nonsmooth liapunov function this condition is used to establish necessary and sufficient stability conditions for second order systems

on structural properties of the lyapunov matrix equation for optimal diagonal solutions we revisit the classical problem of finding a positive diagonal solution p to the lyapunov equation at p pa 0 that minimizes the lyapunov exponent the maximum eigenvalue of at p pa for a isin rnxn with the aim of identifying structural properties of the lyapunov matrix equation at the optimum using eigenvalue sensitivity notions together with optimization machinery we are able to obtain an explicit characterization of the minimum lyapunov exponent that provides such structural insight

on the lyapunov stability of discrete time processes modeled by difference inclusions abstract the subject of this note is the stability in the sense of lyapunov of difference inclusions connection with the convergence of algorithms is also discussed

regularity of canonical operators and nebenhulle hartogs domains abstract we relate the regularity of the bergman projection operator and the canonical solution operator to the nebenhulle of complete hartogs domains

the diederich fornaess index ii for domains of trivial index abstract we study bounded pseudoconvex domains in complex euclidean spaces we find analytical necessary conditions and geometric sufficient conditions for a domain being of trivial diederich fornaess index i e the index equals to 1 we also connect a differential equation to the index this reveals how a topological condition affects the solution of the associated differential equation and consequently obstructs the index being trivial the proofs rely on a new method of study of the complex geometry of the boundary the method was motivated by geometric analysis of riemannian manifolds we also generalize our main theorems under the context of de rham cohomology

stability of simultaneously block triangularisable switched systems with partial state reset abstractwe study the stability of a certain class of switched systems where discontinuous jumps resets on some of the state components are allowed at the switching instants it is known that if all components of the state are available for reset the system can be stabilisable by an adequate choice of resets however this question may have negative answer if there are forbidden state components for reset we give a sufficient condition for the stabilisability of a switched system under arbitrary switching by partial state reset in terms of a block simultaneous triangularisability condition based on this sufficient condition we show that the particular class of systems with partially commuting stable system matrices is stabilisable by partial state reset we also provide an algorithm that allows testing whether a switched system belongs to this particular class of systems

a sufficient condition for compactness of the neumann operator we show that the existence of a certain family of plurisubharmonic functions on a bounded domain in cn implies that the neumann operator associated to the domain is compact our condition generalizes previous work of catlin on compactness of the neumann operator

quadratic lyapunov functions for systems with state dependent switching abstract in this paper we consider the existence of quadratic lyapunov functions for certain types of switched linear systems given a partition of the state space a set of matrices linear dynamics and a matrix valued function a x constructed by associating these matrices with regions of the state space in a manner governed by the partition we ask whether there exists a positive definite symmetric matrix p such that a x t p pa x is negative definite for all x t for planar systems necessary and sufficient conditions are given extensions for higher order systems are also presented

on common α scalar lyapunov solutions abstract we present two new characterizations regarding common scalar solutions for systems of lyapunov matrix inequalities so as to bridge and broaden relevant existing results as applications of such results we further explore the cases involving a set of conformally partitioned block triangular matrices and a set of z matrices respectively to obtain new results concerning these cases which are useful in the areas of control and systems theory and engineering

infinite compressibility states in the hierarchical reference theory of fluids i analytical considerations in its customary formulation for one component fluids the hierarchical reference theory yields a quasilinear partial differential equation pde for an auxiliary quantity f that can be solved even arbitrarily close to the critical point reproduces non trivial scaling laws at the critical singularity and directly locates the binodal without the need for a maxwell construction in the present contribution we present a systematic exploration of the possible types of behavior of the pde for thermodynamic states of diverging isothermal compressibility t as the renormalization group theoretical momentum cutoff approaches zero by purely analytical means we identify three classes of asymptotic solutions compatible with infinite t characterized by uniform or slowly varying bounds on the curvature of f by monotonicity of the build up of diverging t and by stiffness of the pde in part of its domain respectively these scenarios are analzyed and discussed with respect to their numerical properties a seeming contradiction between two of these alternatives and an asymptotic solution derived earlier parola et al phys rev e 48 3321 1993 is easily resolved

robust d stabilization of a polytope of matrices the problem of stabilization of a polytope of matrices in a subregion d r of the complex plane is revisited a new sufficient condition of robust d r stabilization is given it implies the solution of an lmi involving matrix variables constrained by a non linear algebraic relation some lmi relaxations are first proposed then it is shown that a cone complementarity formulation of this condition allows us to associate an efficient iterative numerical procedure which leads to a low computational burden this algorithm is tested on different numerical examples for which existing approaches in control literature fail

on common quadratic lyapunov functions for pairs of stable lti systems whose system matrices are in companion form in this note the problem of determining necessary and sufficient conditions for the existence of a common quadratic lyapunov function for a pair of stable linear time invariant systems whose system matrices a sub 1 and a sub 2 are in companion form is considered it is shown that a necessary and sufficient condition for the existence of such a function is that the matrix product a sub 1 a sub 2 does not have an eigenvalue that is real and negative examples are presented to illustrate the result

stability of switched systems with partial state reset in this note we consider switched systems and switched systems with state reset in particular we focus on the case of partial reset i e where only some state components may undergo the action of a reset first we consider switched systems with pre specified partial reset and investigate under which conditions such systems are stable in a second stage we consider the problem of stabilization by partial reset which consists in finding a suitable partial reset for a given switched system that makes this system stable under arbitrary switching

regularity of the bergman projection on forms and plurisubharmonicity conditions let omega subset subset mathbb c n be a smoothly bounded domain suppose has a defining function such that the sum of any q eigenvalues of its complex hessian is non negative we show that this implies global regularity of the bergman projection b j 1 and the bar partial neumann operator n j acting on 0 j forms for j in q ldots n

on the asymptotic stability of switched homogeneous systems abstract the stability of switched systems generated by the family of autonomous subsystems with homogeneous right hand sides is investigated it is assumed that for each subsystem the proper homogeneous lyapunov function is constructed the sufficient conditions of the existence of the common lyapunov function providing global asymptotic stability of the zero solution for any admissible switching law are obtained in the case where we can not guarantee the existence of a common lyapunov function the classes of switching signals are determined under which the zero solution is locally or globally asymptotically stable it is proved that for any given neighborhood of the origin one can choose a number l 0 dwell time such that if intervals between consecutive switching times are not smaller than l then any solution of the considered system enters this neighborhood in finite time and remains within it thereafter

optimisation and asymptotic stability abstractthe problems of unconstrained optimisation and establishing asymptotic stability have much in common understanding the analogy between these two sheds light on their interconnection and may lead to a number of new results for instance in this paper we provide estimates of the rate of convergence when analysing asymptotic stability of differential equations rather than just ascertain the very fact of stability also standard methods for the design of lyapunov functions e g those having the meaning of the full energy of the system turn out to be unsatisfactory from this point of view and have to be modified these claims are exemplified in the paper by considering the heavy ball method for function minimisation in parallel with the problem of asymptotic stability for the synchronous motor equation

an alternative proof of the barker berman plemmons bbp result on diagonal stability and extensions abstract we revisit the theorem of barker berman and plemmons on the existence of a diagonal quadratic lyapunov function for a stable linear time invariant lti dynamical system g p barker a berman r j plemmons positive diagonal solutions to the lyapunov equations linear and multilinear algebra 5 3 1978 249 256 we use recently derived results to provide an alternative proof of this result and to derive extensions

diagonal riccati stability and applications abstract we consider the question of diagonal riccati stability for a pair of real matrices a b a necessary and sufficient condition for diagonal riccati stability is derived and applications of this to two distinct cases are presented we also describe some motivations for this question arising in the theory of generalised lotka volterra systems

on the uniqueness of the lyapunov scaling factors sufficient conditions are given for a lyapunov diagonally semistable matrix a to have a unique up to scalar multiplication lyapunov scaling factor suppose d is a lyapunov scaling factor of a a necessary and sufficient condition is given for the uniqueness of d the results extend work of hershkowitz and schneider

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

sobolev estimates for the bar partial neumann operator on domains inc n admitting a defining function that is plurisubharmonic on the boundary neumann operator on domains inc n admitting a defining function that is plurisubharmonic on the boundary

global regularity for the neumann operator and bounded plurisubharmonic exhaustion functions abstract for a smooth bounded pseudoconvex domain c n we derive a new sufficient condition for global regularity of the neumann operator that generalizes mcneal s property p the approximately holomorphic vector fields of boas and straube and a condition involving bounded plurisubharmonic exhaustion functions due to kohn

a primal dual probabilistic setting for quadratic stability of uncertain systems abstract this paper introduces a theoretical framework for the development of randomized algorithms for quadratic stability of uncertain systems affected by structured and unstructured uncertainty we propose a general duality setting which deals with a probabilistic version of the classical primal dual problems in particular the probabilistic primal problem is defined so that a linear matrix inequality has a solution with probability one and the probabilistic dual problem is formulated as a solvability problem for a certain matrix integral inequality on the class of positive semidefinite symmetric matrix valued functions the main result of the paper establishes a rigorous equivalence between infeasibility of the primal problem and existence of a solution of the dual problem in the second part of the paper we study various applications of this result in the context of uncertain systems

comparison of the bergman and szego kernels abstract the quotient of the szego and bergman kernels for a smooth bounded pseudoconvex domains in c n is bounded from above by a constant multiple of log p for any p n where is the distance to the boundary for a class of domains that includes those of d angelo finite type and those with plurisubharmonic defining functions the quotient is also bounded from below by a constant multiple of log p for any p 1 moreover for convex domains the quotient is bounded from above and below by constant multiples of

geometric programming for optimal positive linear systems this article studies the parameter tuning problem of positive linear systems for optimizing their stability properties we specifically show that under certain regularity assumptions on the parameterization the problem of finding the minimum cost parameters that achieve a given requirement on a system norm reduces to a geometric program which in turn can be exactly and efficiently solved by convex optimization the flexibility of geometric programming allows the state input and output matrices of the system to simultaneously depend on the parameters to be tuned the class of system norms under consideration includes the h 2 norm h infty norm hankel norm and schatten p norm also the parameter tuning problem for ensuring the robust stability of the system under structural uncertainties is shown to be solved by geometric programming the proposed optimization framework is further extended to delayed positive linear systems where it is shown that the parameter tuning problem jointly constrained by the exponential decay rate the mathcal l 1 gain and the mathcal l infty gain can be solved by convex optimization the assumption on the system parameterization is stated in terms of posynomial functions which form a broad class of functions and thus allow us to deal with various interesting positive linear systems arising from for example dynamical buffer networks and epidemic spreading processes we present numerical examples to illustrate the effectiveness of the proposed optimization framework

lmi approach to linear positive system analysis and synthesis abstract this paper is concerned with the analysis and synthesis of linear positive systems based on linear matrix inequalities lmis we first show that the celebrated perron frobenius theorem can be proved concisely by a duality based argument again by duality we next clarify a necessary and sufficient condition under which a hurwitz stable metzler matrix admits a diagonal lyapunov matrix with some identical diagonal entries as the solution of the lyapunov inequality this new result leads to an alternative proof of the recent result by tanaka and langbort on the existence of a diagonal lyapunov matrix for the lmi characterizing the h performance of continuous time positive systems in addition we further derive a new lmi for the h performance analysis where the variable corresponding to the lyapunov matrix is allowed to be non symmetric we readily extend these results to discrete time positive systems and derive new lmis for the h performance analysis and synthesis we finally illustrate their effectiveness by numerical examples on robust state feedback h controller synthesis for discrete time positive systems affected by parametric uncertainties

common lyapunov functions and gradient algorithms this note is concerned with the problem of finding a quadratic common lyapunov function for a large family of stable linear systems we present gradient iteration algorithms which give deterministic convergence for finite system families and probabilistic convergence for infinite families

the diederich fornaess index i for domains of non trivial index abstract we study bounded pseudoconvex domains in complex euclidean space we define an index associated to the boundary and show this new index is equivalent to the diederich fornaess index defined in 1977 this connects the diederich fornaess index to boundary conditions and refines the levi pseudoconvexity we also prove the worm domain is of index 2 it is the first time that a precise non trivial diederich fornaess index in euclidean spaces is obtained this finding also indicates that the diederich fornaess index is a continuum in 0 1 not a discrete set the ideas of proof involve a new complex geometric analytic technique on the boundary and detailed estimates on differential equations

on the unique solvability of the runge kutta equations we consider the existence of a unique solution to the systems of equations that arise when we apply a runge kutta method to a stiff nonlinear system of differential equationsu f t u withf satisfying a one sided lipschitz condition with constant for any given product h whereh denotes the step size we present algebraic conditions on the runge kutta matrixa which are necessary and sufficient for unique solvability of the equations as a second topic we consider the question whether the solution to the system of equations is stable with respect to perturbations known as bsi stability for this property also necessary and sufficient algebraic conditions ona are presented

the relationship between hadamard and conventional multiplication for positive definite matrices abstract it is known that if a is positive definite hermitian then a a 1 i in the positive semidefinite ordering our principal new result is a converse to this inequality under certain weak regularity assumptions about a function f on the positive definite matrices a f a af a for all positive definite a if and only if f a is a positive multiple of a 1 in addition to the inequality a a 1 i it is known that a a 1 t i and stronger that min a b min ab t for a b positive definite hermitian we also show that min a b min ab and note that min ab and min ab t can be quite different for a b positive definite hermitian we utilize a simple technique for dealing with the hadamard product which relates it to the conventional product and which allows us to give especially simple proofs of the closure of the positive definites under hadamard multiplication and of the inequalities mentioned

diagonal riccati stability and the hadamard product we present an extension of a recent characterisation of diagonal riccati stability and using this extend a result of kraaijevanger on diagonal lyapunov stability to riccati stability of time delay systems we also describe a class of transformations that preserve the property of being diagonally riccati stable and apply these two results to provide novel stability results for classes of time delay systems

a new characterization of simultaneous lyapunov diagonal stability via hadamard products abstract a well known characterization by kraaijevanger 14 for lyapunov diagonal stability states that a real square matrix a is lyapunov diagonally stable if and only if a s is a p matrix for any positive semidefinite s with nonzero diagonal entries this result is extended here to a new characterization involving similar hadamard multiplications for simultaneous lyapunov diagonal stability on a set of matrices among the main ingredients for this extension are a new concept called p sets and a recent result regarding simultaneous lyapunov diagonal stability by berman goldberg and shorten 2

numerical methods of lyapunov function construction and their application to the absolute stability problem the minimax approach is used to study numerical selection of a lyapunov function from a multiparameter class of functions a numerical method is suggested for plotting an absolute stability region in the space of system gains for a control system with several nonstationary nonlinearities

some results on quadratic stability of switched systems with interval uncertainty in this paper we present some results on the quadratic stability of switched systems with uncertain parameters we show that the quadratic stability of a class of switched uncertain systems may be readily verified using simple algebraic conditions

tensor conditions for the existence of a common solution to the lyapunov equation necessary and sufficient conditions for the existence of a common solution to the lyapunov equation for two stable complex matrices are derived these conditions are applied to the cases when a common weak solution to the lyapunov equation exists conditions for the existence of a common solution to the lyapunov equation for two complex 2 2 and two complex 3 3 matrices are derived

an extension of the kyp lemma for the design of state dependent switching systems with uncertainty abstract we consider the problem of designing state dependent linear switching systems subject to parameter uncertainty the switching rule is governed by a fixed partition of the state space and the parameter uncertainty appears in the matrices which determine the resulting dynamics we derive a kyp like lemma that can be used to design such systems in a manner that guarantees quadratic stability for an arbitrary number of independent switching rules each with its own switching region and its own domain of parameter uncertainty examples are given to illustrate the usefulness of our result

semidefinite programming duality and linear time invariant systems several important problems in control theory can be reformulated as semidefinite programming problems i e minimization of a linear objective subject to linear matrix inequality lmi constraints from convex optimization duality theory conditions for infeasibility of the lmis as well as dual optimization problems can be formulated these can in turn be reinterpreted in control or system theoretic terms often yielding new results or new proofs for existing results from control theory we explore such connections for a few problems associated with linear time invariant systems

a kalman yakubovich popov type lemma for systems with certain state dependent constraints in this note a result is presented that may be considered an extension of the classical kalman yakubovich popov kyp lemma motivated by problems in the design of switched systems we wish to infer the existence of a quadratic lyapunov function qlf for a nonlinear system in the case where a matrix defining one system is a rank 1 perturbation of the other and where switching between the systems is orchestrated according to a conic partitioning of the state space r n we show that a necessary and sufficient condition for the existence of a qlf reduces to checking a single constraint on a sum of transfer functions irrespective of problem dimension furthermore we demonstrate that our conditions reduce to the classical kyp lemma when the conic partition of the state space is r n with the transfer function condition reducing to the condition of strict positive realness