controllability of linear single input positive discrete time systems linear positive systems are utilized in various scientific areas such as economic modelling behavioural science stochastic processes etc a recent study has focused on some sufficient conditions for the controllability of these systems a full characterization is provided of the controllability for discrete time single input positive systems necessary and sufficient conditions are derived using a digraph theoretic approach the main results show that strong constraints are to be imposed on the digraph associated with the pair a b to ensure the controllability of the system

minimal strongly eventually positive realization for a class of externally positive systems the minimal positive realization of externally positive systems is still an open problem due to various restrictions caused by the nonnegativity constraint on the state space matrices in this paper the minimal strongly eventually positive realization which relaxes the nonnegativity constraint on the state space model and only requires that the state trajectory is nonnegative after a certain number of steps is proposed for an externally positive system it is shown that a discrete time transfer function with a simple strictly dominant root is externally positive if and only if it has a minimal strongly eventually positive realization a constructive proof is provided the continuous time counterpart is also addressed by transforming it into a corresponding discrete time case

a lower bound on the dimension of minimal positive realizations for discrete time systems abstract in some applications one is interested in having a state space realization with nonnegative matrices positive realization of a given transfer function and it is known that such a realization may have a dimension strictly larger than the order of the transfer function itself moreover in most cases it is desirable to have a realization with minimal dimension unfortunately it is not known to date how to determine in general the minimum dimension of a positive realization and only lower and upper bounds to it are available this letter provides an upper bound on the dimension of a minimal positive realization for transfer functions with simple poles this is a considerable improvement on an earlier upper bound in which only transfer functions with real poles were considered

minimal positive realizations of transfer functions with nonnegative multiple poles this note concerns a particular case of the minimality problem in positive system theory a standard result in linear system theory states that any nth order rational transfer function of a discrete time invariant linear single input single output siso system admits a realization of order n in some applications however one is restricted to realizations with nonnegative entries i e a positive system and it is known that this restriction may force the order n of realizations to be strictly larger than n a general solution to the minimality problem i e determining the smallest possible value of n is not known in this note we consider the case of transfer functions with nonnegative multiple poles and give sufficient conditions for the existence of positive realizations of order n n with the help of our results we also give an improvement of an existing result in positive system theory

a note on the existence of positive realizations the problem of existence of positive realizations for a class of continuous and discrete time systems is addressed and solved it is shown that the existence of a minimal positive realization can be decided on the basis of two sub matrices of the infinite hankel matrix associated with the transfer function to be realized applications of this result to the problem of existence and construction of non minimal realizations are also discussed

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

a note on discrete time positive realizations abstract a discrete time positive linear system has the property that any nonnegative initial state and any nonnegative input produces a nonnegative trajectory in the state space and output for all time such systems are r n reachable if every state of the nonnegative orthant r n is reachable from origin with a nonnegative input function in a finite time interval a simple necessary and sufficient condition for r n realizability i e for the existence of a nonnegative realization that is also r n reachable is stated the realization procedure is straightforward

reachability observability and realizability of continuous time positive systems this paper discusses reachability observability and realizability of single input single output linear time invariant systems in which state variables and or input output functions are restricted to be nonnegative to reflect physical constraints frequently encountered in real systems we define a set reachable from the origin with nonnegative inputs and also a set observable with nonnegative outputs we investigate geometrical structures of the sets through convex analysis and a duality relation between them is established next we consider positive realization of a given transfer function using the reachable set and the observable set we give a necessary and sufficient condition for positive realizability an example is given to demonstrate that a positive realizable transfer function does not in general have a jointly controllable and observable positive realization

on the relationship between the classical dicke jaynes cummings gaudin model and the nonlinear schrodinger equation in this paper the relationship between the classical dicke jaynes cummings gaudin djcg model and the nonlinear schrodinger nls equation is studied it is shown that the classical djcg model is equivalent to a stationary nls equation moreover the standard nls equation can be solved by the classical djcg model and a suitably chosen higher order flow further it is also shown that classical djcg model can be transformed into the classical gaudin spin model in an external magnetic field through a deformation of lax matrix finally the separated variables are constructed on the common level sets of casimir functions and the generalized action angle coordinates are introduced via the hamilton jacobi equation

non uniform dependence on initial data for the two component novikov system considered herein is the initial value problem for the two component novikov system it is shown that the solution map of this problem is not uniformly continuous in sobolev spaces hs hs 1 for s 5 2 on the line based on the well posedness result and the lifespan for this problem the method of approximate solutions is utilized

an example of how positivity may force realizations of large dimension abstract a standard result of linear system theory states that an siso proper rational transfer function of degree n always has a realization of dimension n in some applications one is interested in having a realization with nonnegative entries and it is known that when the dominant poles display a specific pattern forcing nonnegativity leads to a system which is not jointly reachable and observable in this paper we show that the minimal dimension of a positive realization may be large even in the case of a single dominant pole more precisely we provide a family of transfer functions each of which is of degree n 3 such that for any integer n 3 the corresponding member of the family admits a minimal positive realization of state space dimension not smaller than n

primes in several classes of the positive matrices the classification of primes in the semi ring of the positive matrices is of interest to control and system theory a few examples of primes in the positive matrices are known in this paper results on the classification of primes in the positive matrices in the doubly stochastic matrices and in the doubly stochastic circulants are presented

on the global existence and wave breaking criteria for the two component camassa holm system considered herein is a two component camassa holm system modeling shallow water waves moving over a linear shear flow a wave breaking criterion for strong solutions is determined in the lowest sobolev space hs s 32 by using the localization analysis in the transport equation theory moreover an improved result of global solutions with only a nonzero initial profile of the free surface component of the system is established in this sobolev space hs

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

local well posedness and blow up criteria for a two component novikov system in the critical besov space abstract in this paper we mainly investigate the cauchy problem of a two component novikov system we first prove the local well posedness of the system in besov spaces b p r s 1 b p r s with p r 1 s max 1 1 p 3 2 by using the littlewood paley theory and transport equations theory then by virtue of logarithmic interpolation inequalities and the osgood lemma we establish the local well posedness of the system in the critical besov space b 2 1 1 2 b 2 1 3 2 moreover we present two blow up criteria for the system by making use of the conservation laws

on the blow up rate and the blow up set of breaking waves for a shallow water equation we consider the problem of the development of singularities for classical solutions to a new periodic shallow water equation blow up can occur only in the form of wave breaking i e the solution remains bounded but its slope becomes unbounded in finite time a quite detailed description of the wave breaking phenomenon is given there is at least a point in general depending on time where the slope becomes infinite exactly at breaking time the precise blow up rate is established and for a large class of initial data we also determine the blow up set

minimal positive continuous time realizations of positive response maps abstract positive nonlinear continuous time local realizations of positive response maps are studied the realizations are minimal if they are positively reachable and positively observable necessary and sufficient conditions for a response map to have a minimal positive local realization are provided it is also shown that a minimal positive local realization has the minimal dimension among all positive local realizations of the given response map and that it is unique up to an isomorphism of systems

positive n periodic descriptor control systems this paper establishes the relationship between the positive n periodic descriptor system in discrete time and its associated invariant systems reachability and controllability properties of these kinds of systems are analyzed transmission of structural properties from the n periodic system to their associated invariant systems is studied finally some comments on the stability property are made

integrable peakon equations with cubic nonlinearity we present a new integrable partial differential equation found by vladimir novikov like the camassa holm and degasperis procesi equations this new equation admits peaked soliton peakon solutions but it has nonlinear terms that are cubic rather than quadratic we give a matrix lax pair for v novikov s equation and show how it is related by a reciprocal transformation to a negative flow in the sawada kotera hierarchy infinitely many conserved quantities are found as well as a bi hamiltonian structure the latter is used to obtain the hamiltonian form of the finite dimensional system for the interaction of n peakons and the two body dynamics n 2 is explicitly integrated finally all of this is compared with some analogous results for another cubic peakon equation derived by zhijun qiao

nonnegative realizations of matrix transfer functions abstract various aspects of the nonnegative finite dimensional realizability of time invariant discrete linear systems are considered a new proof of the basic result of the nonnegative realizability of a primitive scalar valued transfer function with nonnegative impulse response function is given an algorithm for establishing whether a scalar valued transfer function with nonnegative impulse response has a nonnegative realization is presented the main result characterizes the nonnegative realizability of a scalar valued transfer function with the help of primitive transfer functions and is extended to the general case of matrix valued transfer functions then conditions for the existence of some special nonnegative realizations of transfer functions are presented e g where the middle main matrix is irreducible strictly positive or primitive

necessary conditions for positive realizability of continuous time linear systems abstract this paper deals with the positive realization problem the problem is to find from a given transfer function a state equation in which state variables and the output take nonnegative values whenever initial states and inputs are nonnegative necessary conditions are investigated and a new one is given together with some related results

from the special 2 1 toda lattice to the kadomtsev petviashvili equation the nonlinearization of the eigenvalue problems associated with the toda hierarchy and the coupled korteweg de vries kdv hierarchy leads to an integrable symplectic map s and an integrable hamiltonian system h0 respectively it is proved that s and h0 have the same integrals hk the quasi periodic solution of the 2 1 dimensional kadomtsev petviashvili equation is split into three hamiltonian systems h0 h1 h2 while that of the special 2 1 dimensional toda equation is separated into h0 h1 plus the discrete flow generated by the symplectic map s a clear evolution picture of various flows is given through the window of abel jacobi coordinates the explicit theta function solutions are obtained by resorting to this separation technique

an efficient algorithm for positive realizations we observe that successive applications of known results from the theory of positive systems lead to an it efficient general algorithm for positive realizations of transfer functions we give two examples to illustrate the algorithm one of which complements an earlier result of cite large finally we improve a lower bound of cite mn2 to indicate that the algorithm is indeed efficient in general

classical yang baxter equations and quantum integrable systems quantum integrable models associated with nondegenerate solutions of classical yang baxter equations related to the simple lie algebras are investigated these models are diagonalized for rational and trigonometric solutions in the cases of sl n gl n o n and sp n algebras the analogy with the quantum inverse scattering method is demonstrated

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

positive nonlinear realizations of response maps abstract the problem of realization of a positive response map as a positive initialized system on a finite dimensional euclidean state space is investigated the dynamical part of the system is described by a delta differential equation on an arbitrary time scale this incorporates continuous and discrete time systems the main result states necessary and sufficient conditions for existence of positive realization of a particular class the criterion is expressed in the language of skew differential global universes

on the cauchy problem for a two component b family system abstract this paper is concerned with the cauchy problem for a two component b family system we prove that the solution map of the cauchy problem of the b family system is not uniformly continuous in h s r h s 1 r s 5 2

brief identification of positive linear systems with poisson output transformation positive systems are systems in which the input state output variables are always positive since they represent quantities we propose an identification procedure for a class of positive linear systems

positive realizations of nonlinear input output discrete time systems abstract the problem of realization of a positive nonlinear input output discrete time system as a positive state space system is investigated the input output system is given by a difference equation of a higher order a necessary and sufficient condition for existence of positive and regularly positively observable realization is provided the state space variables of the realization are found as independent generators of some positive universe associated to the input output system only the single input single output case is studied and all functions describing the systems and their properties are assumed to be continuous

a tutorial on the positive realization problem this paper is a tutorial on the positive realization problem that is the problem of finding a positive state space representation of a given transfer function and characterizing existence and minimality of such representation this problem goes back to the 1950s and was first related to the identifiability problem for hidden markov models then to the determination of internal structures for compartmental systems and later embedded in the more general framework of positive systems theory within this framework developing some ideas sprang in the 1960s during the 1980s the positive realization problem was reformulated in terms of a geometric condition which was recently exploited as a tool for finding the solution to the existence problem and providing partial answers to the minimality problem in this paper the reader is carried through the key ideas which have proved to be useful in order to tackle this problem in order to illustrate the main results contributions and open problems several motivating examples and a comprehensive bibliography on positive systems organized by topics are provided

positive realizations of linear systems abstract this paper deals with the realization problem of linear systems in which a sign pattern is imposed on the realization a b c t this problem arises in the context of positive linear systems i e systems in which state variables and the output take nonnegative values whenever initial states and inputs are nonnegative using a unitary framework for continuous time and discrete time systems we derive necessary and sufficient conditions for positive realizability by means of convex analysis moreover the complete solution in the case of a transfer function of mcmillan degree n 3 is obtained

a new approach to the internally positive representation of linear mimo systems the problem of representing linear systems through combination of positive systems is relevant when signal processing schemes such as filters state observers or control laws are to be implemented using positive technologies such as charge routing networks and fiber optic filters this problem well investigated in the siso case can be recasted into the more general problem of internally positive representation ipr of systems this paper presents a methodology for the construction of such iprs for mimo systems based on a suitable convex positive representation of complex vectors and matrices the stability properties of the iprs are investigated in depth achieving the important result that any stable system admits a stable ipr of finite dimension a main algorithm and three variants all based on the proposed methodology are presented for the construction of stable iprs all of them are straightforward and are characterized by a very low computational cost the first and second may require a large state space dimension to provide a stable ipr while the third and the fourth are aimed to provide stable iprs of reduced order

a counterexample on dynamical systems described by linear difference inequalities abstract in this note we will start a preliminary investigation of discrete time systems described by a finite number of linear inequalities as such it can be seen as an extension of willems 1991 behavioral approach of linear systems

positive and compartmental systems when dealing with compartmental systems an important question is given an experiment i e an input output sequence and supposing there is no error in the data is the sequence compatible with the compartmental assumption if the process under analysis is linear then the previous question is obviously equivalent to asking whether a given transfer function is that of a compartmental system in this note we provide an answer to the latter question giving necessary and sufficient conditions for a transfer function to be that of a compartmental system of some finite order i e number of compartments another problem tackled in this note originates from the observation that in many cases one wants to determine the number of compartments involved in the process in this note we report a step toward the solution of this fundamental problem by proving necessary and sufficient conditions for a given third order transfer function with real poles to be that of a compartmental system with three compartments

on the existence of a positive realization the positive realization problem for linear systems is to find conditions for a given transfer function with nonnegative impulse response to have a realization such that the resulting system is a positive system recently it has been shown that under a mild assumption on the long term behaviour of the impulse response this problem is related to the maximum modulus poles only in this paper necessary and sufficient conditions for positive realizability of discrete time systems are given they show that also nondominant poles play a role in the most general case positive realizability conditions for the continuous time case are also given

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

about nonnegative realizations we will state sufficient conditions such that a discrete time time invariant finite dimensional system with a nonnegative impulse response has a nonnegative finite dimensional realization the conditions are in terms of the minimal polynomial

phase type distributions and representations some results and open problems for system theory in this paper we consider phase type distributions these distributions correspond to the random hitting time of an absorbing markov chain they are used for modelling various random times in particular those which appear in manufacturing systems as processing times times to failure repair times etc the markovian nature of these distributions allows the use of very efficient matrix based computer methods for performance evaluation in this paper we give a system theory oriented introduction to phase type distributions we concentrate mainly on the representation problem which consists of finding a markov chain associated with some phase type distribution this is a realization problem in the sense of system theory with a lot of links with the classical linear system theory but also with a number of constraints which make the problem harder but more interesting indeed this problem has strong connections with the positive realization problem in control theory the paper recalls known results gives s

minimal eventually positive realizations of externally positive systems it is a well known fact that externally positive linear systems may fail to have a minimal positive realization in order to investigate these cases we introduce the notion of minimal eventually positive realization for which the state update matrix becomes positive after a certain power eventually positive realizations capture the idea that in the impulse response of an externally positive system the state of a minimal realization may fail to be positive but only transiently as a consequence we show that in discrete time it is possible to use downsampling to obtain minimal positive realizations matching decimated sequences of markov coefficients of the impulse response in continuous time instead if the sampling time is chosen sufficiently long a minimal eventually positive realization leads always to a sampled realization which is minimal and positive

driving variable realizations and the non negative realization problem for controllable behaviours non negative linear systems traditionally investigated within the state space framework have been recently analysed within the behavioural setting in a couple of recent papers j w nieuwenhuis when to call a linear system nonnegative linear algebra its appl 281 1998 pp 43 58 m e valcher nonnegative linear systems in the behavioural approach the autonomous case linear algebra and its appl 2000 319 pp 147 162 several definitions and results about non negative behaviours and in particular about non negative autonomous behaviours have been derived moreover the non negative realization problem for autonomous behaviours has been fully explored in m e valcher non negative realization of autonomous systems in the behavioral approach siam j on control and opt 2001 40 pp 540 556 thus deriving an extended set of necessary and sufficient conditions for an autonomous behaviour to admit a non negative realization here we focus our attention on the non negative re

wave breaking for a modified two component camassa holm system abstract in this paper we establish sufficient conditions on the initial data to guarantee blow up phenomenon for the modified two component camassa holm mch2 system

complex form reduction and lie poisson structure for the nonlinearized spectral problem of the heisenberg hierarchy abstract in this paper the relation between the different restricted systems associated with the heisenberg magnetic hm equation is studied by the reduction procedure with the help of a lie group homomorphism of su 2 into so 3 the euler rodriguez type parameters are introduced to generate new finite dimensional integrable system it has shown that the resulting system which is the nonlinearized spectral problem of hm hierarchy on c2n is a hamiltonian system in complex form further poisson reduction and lie poisson structure are derived by the method of invariants the reduced system is found to be a hamiltonian system on the orbit space c2n tn r3n coinciding with the nonlinearized lenard spectral problem moreover the fully reduced systems on the leaves of the symplectic foliation are also given specifically the reduction extended to the common level set of the complex cones is the usual 2 2 nonlinearized spectral problem finally the integrability of the system with lie poisson structure is proven by making use of the so 3 symmetry

bounds on the size of minimal nonnegative realizations for discrete time lti systems nonnegative linear time invariant lti state space models are characterized by the property that their state variables and outputs remain nonnegative whenever the initial state and the inputs are nonnegative necessary and sufficient conditions for the existence of a nonnegative state space realization of a given discrete time dt impulse response were obtained recently in this letter we look at minimal nonnegative realizations of dt impulse responses and develop bounds on the number of state variables that are necessary

singular solutions of a modified two component camassa holm equation the camassa holm ch equation is a well known integrable equation describing the velocity dynamics of shallow water waves this equation exhibits spontaneous emergence of singular solutions peakons from smooth initial conditions the ch equation has been recently extended to a two component integrable system ch2 which includes both velocity and density variables in the dynamics although possessing peakon solutions in the velocity the ch2 equation does not admit singular solutions in the density profile we modify the ch2 system to allow a dependence on the average density as well as the pointwise density the modified ch2 system mch2 does admit peakon solutions in the velocity and average density we analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data numerical results for the mch2 system are given and compared with the pure ch2 case these numerics show that the modification in the mch2 system to introduce the average density has little short time effect on the emergent dynamical properties however an analytical and numerical study of pairwise peakon interactions for the mch2 system shows a different asymptotic feature namely besides the expected soliton scattering behavior seen in overtaking and head on peakon collisions the mch2 system also allows the phase shift of the peakon collision to diverge in certain parameter regimes

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

order bound for the realization of a combination of positive filters in a problem on the realization of digital filters initiated by gersho and gopinath we extend and complete a remarkable result of benvenuti farina and anderson on decomposing the transfer function t z of an arbitrary linear asymptotically stable discrete time invariant single input single output system as a difference t z t1 z t2 z of two positive asymptotically stable linear systems we give an easy to compute algorithm to handle the general problem in particular also the case of transfer functions t z with multiple poles which was left open in a previous paper one of the appearing positive asymptotically stable systems is always one dimensional while the other has dimension depending on the order and in the case of nonreal poles also on the location of the poles of t z the appearing dimension is seen to be minimal in some cases and it can always be calculated before carrying out the realization

a 2 1 dimensional derivative toda equation in the context of the kaup newell spectral problem a 2 1 dimensional derivative toda equation is derived from the lax triad composed of the kaup newell the negative kaup newell and a discrete spectral problem two integrable hamiltonian systems and an integrable symplectic map are derived from the lax triad they are straightened out in the jacobi variety of the associated hyperelliptic curve a finite genus solution is obtained through a technique based on the riemann jacobi inversion theorem besides explicit solutions are calculated for other associated integrable models including the mkp equation and the nkn equation

a construction method for positive realizations with an order bound abstract this paper presents an efficient construction method to address the positive realization problem with an order bound for primitive transfer functions with multiple real or complex conjugate pairs of poles using polyhedral cone concepts through a down sampling step the non primitive transfer function is decomposed into a sum of primitive transfer functions a construction method for the positive realization of a primitive transfer function with multiple poles is proposed by using a regular polygon in the complex plane a sufficient condition for the existence of the positive realization with an order bound is analyzed numerical examples are provided to demonstrate the efficiency of the proposed method for obtaining positive realizations

on the reachable set for third order linear discrete time systems with positive control the case of complex eigenvalues abstract in this paper we study the geometrical properties of the set of reachable states of a single input third order discrete time linear system with positive controls this set is a cone and we give a complete geometrical characterization of this set when the system has a complex conjugate pair of eigenvalues more in detail we give necessary and sufficient conditions for properness and polyhedrality of the cone and provide the number of its edges in terms of eigenvalue locations

generalizations of the camassa holm equation we classify generalized camassa holm type equations which possess infinite hierarchies of higher symmetries we show that the obtained equations can be treated as negative flows of integrable quasi linear scalar evolution equations of orders 2 3 and 5 we present the corresponding lax representations or linearization transformations for these equations some of the obtained equations seem to be new

excitability stability and sign of equilibria in positive linear systems abstract the relationship between sign and stability of equilibria in positive linear systems is investigated in this paper in particular it is proved that the well known result stability and irreducibility imply strict positivity can be better specified by introducing the notion of excitability in fact stability excitability and strict positivity of the equilibria are three perfectly balanced properties since any one of them is implied by the two others

identifiability of hidden markov information sources and their minimum degrees of freedom if only a function of the state in a finite state markov chain is observed then the stochastic process is no longer markovian in general this type of information source is found widely and the basic problem of its identifiability remains open that is the problem of showing when two different markov chains generate the same stochastic process the identifiability problem is completely solved by linear algebra where a block structure of a markov transition matrix plays a fundamental role and from which the minimum degree of freedom for a source is revealed

nonnegative linear systems in the behavioral approach the autonomous case nonnegative linear systems which have traditionally been investigated within the state space framework are in this paper introduced and analyzed by means of the behavioral approach starting from certain definitions and results presented in a recent paper j w nieuwenhuis systems control lett 1 1982 283 we have explored the general autonomous case by deriving an extended set of necessary and sufficient conditions for an autonomous behavior to be nonnegative in the scalar case in particular necessary conditions for nonnegativity which refer to the set of zeros of any polynomial involved in the kernel description of the behavior have been provided

positive input reachability and controllability of positive systems abstract we study controllability and reachability for discrete linear control systems x k 1 ax k bu k in which the state vector x k the control vector u k the n n matrix a and the n m matrix b have nonnegative entries for an unconstrained system controllability is equivalent to reachability from the zero state in n steps furthermore a b is controllable if and only if the matrix c n b ab a 2 b a n 1 b has rank n we show that these properties are not equivalent for positive systems this has important implications for control strategy the timing of control inputs and the size and structure of the control matrix b are much more critical the zero nonzero pattern of c n plays a crucial role we illustrate these fundamental differences with models of a two species reversible chemical reaction and with two pharmacokinetic models of drug distribution in the human body

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

internally positive representation of a class of continuous time systems the problem of realizing arbitrary transfer functions as combinations of positive filters or more in general of representing arbitrary systems by means of internally positive representations iprs has been widely investigated in the discrete time framework an ipr is a positive state space representation endowed with input state and output transformations that realizes arbitrary input state output dynamics this technical note investigates the problem of the ipr construction for continuous time systems and proposes a technique that provides stable iprs for a particular class of systems

minimal positive realizations of transfer functions with real poles a standard result of linear system theory states that a single input single output siso rational n th order transfer function always has a state space realization of the same order in some applications one is interested in having a realization with nonnegative entries i e a positive system and it is known that such constraints may lead to a minimal order positive realization of order much greater than the transfer function order n in this technical note necessary and sufficient conditions for a third order transfer function with distinct real poles to have a third order positive realization are given these conditions are expressed in terms of lower bounds for the first three samples of the impulse response and therefore are very easy to check this result is an extension of a previous result for transfer functions with distinct real positive poles

controllability of positive linear discrete time systems the class of positive linear discrete time systems is considered reachability sets of such systems denned on cones are studied and some of their properties established reachability null controllability and controllability criteria are obtained in forms that are easier to implement than the corresponding criteria for linear discrete time systems of general type some classes of positive linear discrete time systems are investigated and their controllability properties are established

classical bethe ansatz and normal forms in an integrable version of the dicke model abstract the dicke jaynes cummings gaudin model describes a collection of n spins coupled to a harmonic oscillator it is known to be integrable so one can define a moment map which associates to each point in phase space the list of values of the n 1 conserved hamiltonians we identify all the critical points of this map and we compute the corresponding quadratic normal forms using the lax matrix representation of the model the normal coordinates are constructed by a procedure which appears as a classical version of the bethe ansatz used to solve the quantum model we show that only elliptic or focus focus singularities are present in this model which provides an interesting example of a symplectic toric action with singularities to explore these we study in detail the degeneracies of the spectral curves for the n 1 and n 2 cases this gives a complete picture for the image of the momentum map imm and the associated bifurcation diagram for n 2 we found in particular some lines of rank 1 which lie for one part on the boundary of the imm where they behave like an edge separating two faces and which go for another part inside the imm

non uniform dependence and well posedness for the hyperelastic rod equation abstract it is shown that the solution map for the hyperelastic rod equation is not uniformly continuous on bounded sets of sobolev spaces with exponent greater than 3 2 in the periodic case and greater than 1 in the non periodic case the proof is based on the method of approximate solutions and well posedness estimates for the solution and its lifespan

controllability and reachability criteria for discrete time positive systems discrete time positive systems have been the object of widespread interest in the literature in the last decade particular attention has been devoted to the analysis and characterization of the different notions of reachability and controllability which in this context make their appearance in this paper the properties of ordinary and essential controllability and reachability are analysed based on a graph theoretic approach we provide practical criteria to verify whether a given positive system f g is endowed with these properties together with canonical forms for describing reachable controllable pairs

on the reachable set for third order linear discrete time systems with positive control abstract in this paper we study the geometrical properties of the set of reachable states of a single input third order discrete time linear system with positive controls this set is a cone and we give a complete geometrical characterization of this set when the system has all real eigenvalues more in detail we give necessary and sufficient conditions for properness and polyhedrality of the cone and provide the number of its edges in terms of eigenvalue locations moreover we provide necessary and sufficient conditions for finite time reachability of every reachable state and characterize the minimum number of steps needed to reach every state in terms of eigenvalue locations

global existence and blow up phenomena for an integrable two component camassa holm shallow water system abstract this paper is concerned with global existence and blow up phenomena for an integrable two component camassa holm shallow water system a new global existence result and several new blow up results of strong solutions to the system are presented our obtained results for the system are sharp and improve considerably earlier results

non uniform dependence for a modified camassa holm system this paper is concerned with the non uniform dependence on initial data for a modified camassa holm system we prove that the solution map of the cauchy problem of the camassa holm system is not uniformly continuous in hs r s 1 moreover we obtain the similar result for the initial boundary value problem for the camassa holm system

from nonlinear schrodinger hierarchy to some 2 1 dimensional nonlinear pseudodifferential equations the poisson structure on cn rn is introduced to give the hamiltonian system associated with a spectral problem which yields the nonlinear schrodinger nls hierarchy the hamiltonian system is proven to be liouville integrable some 2 1 dimensional equations including nls equation kadomtesev petviashvili i kpi equation coupled kpi equation and modified kadomtesev petviashvili mkp equation are decomposed into hamilton flows via the nls hierarchy the algebraic curve abel jacobi coordinates and riemann jacobi inversion are used to obtain the algebrogeometric solutions of these equations