adaptive control of linear systems with markov perturbations the stochastic model considered is a linear jump diffusion process x for which the coefficients and the jump processes depend on a markov chain z with finite state space first we study the optimal filtering and control problem for these systems with non gaussian initial conditions given noisy observations of the state x and perfect measurements of z we derive a new sufficient condition which ensures the existence and the uniqueness of the solution of the nonlinear stochastic differential equations satisfied by the output of the filter we study a quadratic control problem and show that the separation principle holds next we investigate an adaptive control problem for a state process x defined by a linear diffusion for which the coefficients depend on a markov chain the processes x and z being observed in independent white noises suboptimal estimates for the process x z and approximate control law are investigated for a large class of probability distributions of the initial state asymptotic properties of these filters and this control law are obtained upper bounds for the corresponding error are given

stability of peakons the peakons are peaked solitary wave solutions of a certain nonlinear dispersive equation that is a model in shallow water theory and the theory of hyperelastic rods we give a very simple proof of the orbital stability of the peakons in the h1 norm 2000 john wiley sons inc

the periodic cauchy problem for a two component non isospectral cubic camassa holm system abstract in this paper we study the periodic cauchy problem for a two component non isospectral cubic camassa holm system which includes the fokas olver rosenau qiao forq or modified camassa holm mch equation and the two component mch system as two special cases the system is integrable in the sense of possessing a non isospectral lax pair with the spectrum depending on time t and admits multi peakon solutions in an explicit form furthermore we establish the local well posedness for the system in the besov space b 2 r s t with s 3 2 1 r where the key ingredients include the friedrichs regularization method the littlewood paley decomposition theory and the transport theory in besov spaces then we derive a precise blow up criteria which is dependent of the parameters t and t moreover by the intrinsic structure of the system we obtain a new blow up result for strong solutions with sufficient conditions on the initial data and parameters the entire proof procedure relies upon a newly derived transport equation which is involved in nonlocal velocity term along the characteristic curves

revised adomian decomposition method for solving systems of ordinary and fractional differential equations a modification of the adomian decomposition method applied to systems of linear nonlinear ordinary and fractional differential equations which yields a series solution with accelerated convergence has been presented illustrative examples have been given

multipeakons and the classical moment problem abstract classical results of stieltjes are used to obtain explicit formulas for the peakon antipeakon solutions of the camassa holm equation the closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem it is shown that collisions occur only in peakon antipeakon pairs and the details of the collisions are analyzed using results from the moment problem a sharp result on the steepening of the slope at the time of collision is given asymptotic formulas are given and the scattering shifts are calculated explicitly

solving a singular system of two nonlinear odes abstract in this paper a new method is given in order to solve a singular system of two nonlinear ordinary differential equations having multiple solutions and we may give a special solution by imposing an additional condition representation of the exact solution is given in the form of series in the reproducing kernel space its approximate solution is obtained by truncating the series the error of the approximate solution is monotone decreasing in the sense of w 2 0 1 and the intrinsic merit of the method given in the paper lies in its speedy convergence some numerical examples are studied to demonstrate the accuracy of the present method

revised adomian decomposition method for solving a system of nonlinear equations a modification of adomian decomposition method is suggested and used for solving a system of nonlinear equations which yields a series solution with accelerated convergence illustrative examples have been presented to demonstrate the method and the results obtained are compared with those derived from the standard adomian decomposition method

filters for estimating markov modulated poisson processes and image based tracking we present algorithms for state and parameter estimation of markov modulated poisson processes mmpp we first derive finite dimensional innovations and zakai filters for various statistics of a mmpp using these filters a filter based expectation maximization algorithm is derived for computing maximum likelihood parameter estimates finally we present an application of the techniques in image based tracking by estimating the state of a markovian jump linear system

solution of a system of volterra integral equations of the first kind by adomian method in this article we consider linear and non linear systems of integral equations of the first kind the problem of existence and uniqueness are considered the adomian decomposition method is being used to solve these linear and non linear systems of volterra integral equations of the first kind some examples are prepared to show the efficiency and simplicity of the method

phase plane analysis and traveling wave solution of third order nonlinear singular problems arising in thin film evolution a nonlinear third order singular ordinary differential equation is a model for a viscous fluid which drains over a wet surface as a thin film down a vertical wall here we discuss the physical processes of a thin liquid film evolution the classical thin film equation is being derived and the series solution is obtained by employing the adomian decomposition method adm in this paper a new way for the phase planes analysis and traveling wave solution by adm is presented the proposed algorithm is implemented using mathematical software mathematica 6 0

the interacting multiple model algorithm for systems with markovian switching coefficients an important problem in filtering for linear systems with markovian switching coefficients dynamic multiple model systems is the management of hypotheses which is necessary to limit the computational requirements a novel approach to hypotheses merging is presented for this problem the novelty lies in the timing of hypotheses merging when applied to the problem of filtering for a linear system with markovian coefficients the method is an elegant way to derive the interacting multiple model imm algorithm evaluation of the imm algorithm shows that it performs well at a relatively low computational load these results imply a significant change in the state of the art of approximate bayesian filtering for systems with markovian coefficients

exact finite dimensional filters for maximum likelihood parameter estimation of continuous time linear gaussian systems in this paper we derive a new class of finite dimensional filters for integrals and stochastic integrals of moments of the state for continuous time linear gaussian systems apart from being of significant mathematical interest these new filters can be used with the expectation maximization em algorithm to yield maximum likelihood estimates of the model parameters

solution of the system of ordinary differential equations by adomian decomposition method in this article we use adomian decomposition method which is a well known method for solving functional equations now a days to solve systems of differential equations of the first order and an ordinary differential equation of any order by converting it into a system of differential of the order one theoretical considerations are being discussed and convergence of the method for theses systems is addressed some examples are presented to show the ability of the method for linear and non linear systems of differential equations

continuous dependence and estimates of eigenvalues for periodic generalized camassa holm equations abstract in this paper we are concerned with the spectral problem y 1 y 2 x 1 2 1 2 m 1 2 m 1 2 y 1 y 2 for the periodic generalized camassa holm equations the first aim is to study the structure of eigenvalues and prove the continuous dependence of eigenvalues on potentials we prove that as nonlinear functionals of potentials eigenvalues are continuous in potentials with respect to the weak topologies in the lebesgue space l 1 0 t the second aim is to find the estimates of the eigenvalues when the l 1 norm of potentials is given based on an application of trace formulas

an application of pfaffians to multipeakons of the novikov equation and the finite toda lattice of bkp type abstract the novikov equation is an integrable analogue of the camassa holm equation with a cubic rather than quadratic nonlinear term both these equations support a special family of weak solutions called multipeakon solutions in this paper an approach involving pfaffians is applied to study multipeakons of the novikov equation first we show that the novikov peakon odes describe an isospectral flow on the manifold cut out by certain pfaffian identities then a link between the novikov peakons and the finite toda lattice of bkp type b toda lattice is established based on the use of pfaffians finally certain generalizations of the novikov equation and the finite b toda lattice are proposed together with special solutions to our knowledge it is the first time that the peakon problem is interpreted in terms of pfaffians

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

a detector based approach for the h_ 2 control of markov jump linear systems with partial information in this paper we study the h 2 control for discrete time markov jump linear systems mjls with partial information we consider the case in which we do not have access to the markov jump parameter but instead there is a detector that emits signals which provides information on this parameter a salient feature of our formulation is that it encompasses for instance the cases with perfect information no information and cluster observations of the markov parameter which were previously analyzed in the markov jump control literature the goal is to derive a feedback linear control using the information provided by the detector in order to stochastically stabilize the closed loop system we present two lyapunov like equations for the stochastic stability of the system in addition we show that a linear matrix inequalities lmi formulation can be obtained in order to design a stochastically stabilizing feedback control in the sequel we deal with the h 2 control problem and we show that again an lmi optimization problem can be formulated in order to design a stochastically stabilizing feedback control with guaranteed h 2 cost we also present two special cases one of them always satisfied for the limit case in which the detector provides perfect information on the markov parameter and the bernoulli jump case under which lmi conditions become necessary and sufficient for the stochastic stabilizability of the system and the lmi optimization problems provide the optimal h 2 cost for the bernoulli jump case we show that our formulation generalizes previous ones the case with convex polytopic uncertainty on the parameters of the system and on the transition probability matrix is also considered the paper is concluded with some numerical examples

exact filters for doubly stochastic ar models with conditionally poisson observations the authors derive exact filters for the state of a doubly stochastic auto regressive ar process with parameters which vary according to a nonlinear function of a gauss markov process the observations consist of a discrete time poisson process with rate a positive function of the gauss markov process the dimension of the sufficient statistic increases linearly with the number of observed events

quadratic control with partial information for discrete time jump systems with the markov chain in a general borel space this paper deals with the finite horizon quadratic optimal control problem of discrete time markov jump linear systems mjls considering the case in which the markov chain takes values in a general borel space s it is assumed that the controller has access to an output variable as well as the jump parameter the goal is to design a dynamic markov jump controller such that the closed loop system minimizes the quadratic functional cost of the system over a finite horizon period of time it is shown that an optimal controller can be obtained from two s coupled difference riccati equations one associated to a filtering problem and the other one associated to a control problem in which the state variable is fully available by s coupled it is meant that the difference riccati equations are coupled via an integral over s this result which can be seen as a separation principle for discrete time mjls generalizes previous ones that were restricted to the markov chain taking values in a finite set

dynamics of director fields an asymptotic equation for weakly nonlinear hyperbolic waves governed by variational principles is derived and analyzed the equation is used to study a nonlinear instability in the director field of a nematic liquid crystal it is shown that smooth solutions of the asymptotic equation break down in finite time also constructed are weak solutions of the equation that are continuous despite the fact that their spatial derivative blows up

analysis of adomian decomposition applied to a third order ordinary differential equation from thin film flow abstract the adomian decomposition method is applied to the third order ordinary differential equation y y k obtained by considering a travelling wave solution admitted by a generalized thin film equation or for thin film flow down a vertical wall the adomian decomposition method leads to a power series approximation to the solution of y y k we show that the domain of convergence of the adomian decomposition solution is dependent on k we truncate the adomian decomposition solution at a suitable order to ensure that the truncated solution satisfies the contact line condition y 0 at some point x x we then determine the contact angle where tan d y d x at x x and plot the variation of contact angle with increasing k

robust kalman filtering for discrete time markovian jump systems with parameter uncertainty in this paper we examine the problem of robust kalman filtering for a class of linear uncertain discrete time systems with markovian jump parameters the underlying system is subjected to norm bounded time varying uncertainties in the state and measurement equations first stochastic quadratic stability of the system is studied then a linear state estimator is constructed such that the estimation error covariance is guaranteed to lie within certain bound for all admissible uncertainties the solution is given in terms of two coupled algebraic riccati equations

a generalized nonisospectral camassa holm equation and its multipeakon solutions abstract motivated by the paper of beals sattinger and szmigielski 2000 3 we propose an extension of the camassa holm equation which also admits the multipeakon solutions the novel aspect is that our approach is mainly based on classic determinant technique furthermore the proposed equation is shown to possess a nonisospectral lax pair

moment modification multipeakons and nonisospectral generalizations abstract firstly a formal correspondence is established between the camassa holm ch equation and a two component modified ch or called sqq equation according to the method of moment modification for multipeakon formulae secondly based on the generalized nonisospectral ch equation in chang et al 2014 14 and the interlacing multipeakons of the two component modified ch equation in chang et al 2016 15 we propose a new generalized two component modified ch equation with two parameters which possesses a nonisospectral lax pair the proposed equation still admits multipeakon solutions of explicit and closed form sufficient conditions for global existence of solutions are given and two concrete examples with certain interesting phenomenon are presented last of all as a by product a generalized nonisospectral modified ch equation is deduced together with its lax pair

stochastic stability of ito differential equations with semi markovian jump parameters in this note the problem of stochastic stability for linear systems with jump parameters being semi markovian rather than full markovian is further investigated in particular the system under consideration is described by ito type nonlinear stochastic differential equations with phase type semi markovian jump parameters stochastic stability conditions are presented

an image based filter for discrete time markovian jump linear systems abstract a discrete time filtering problem for hybrid systems is studied in which the state process is observed in white noise and the random jump process called the mode or the regime is observed by a point process the optimal state and mode estimator when only the mode measurements are used is studied some properties of this filter are then established apart from their theoretical interest these results have an interesting practical implication for the design of image based tracking systems since they lead to an easily implementable suboptimal algorithm two illustrative examples are given to show the performance of the proposed approach

kalman filtering for continuous time uncertain systems with markovian jumping parameters studies the problem of kalman filtering for a class of uncertain linear continuous time systems with markovian jumping parameters the system under consideration is subjected to time varying norm bounded parameter uncertainties in the state and measurement equations stochastic quadratic stability of the above system is analyzed a state estimator is designed such that the covariance of the estimation error is guaranteed to be within a certain bound for all admissible uncertainties which is in terms of solutions of two sets of coupled algebraic riccati equations

multipeakons of a two component modified camassa holm equation and the relation with the finite kac van moerbeke lattice abstract a spectral and the inverse spectral problem are studied for the two component modified camassa holm type for measures associated to interlacing peaks it is shown that the spectral problem is equivalent to an inhomogeneous string problem with dirichlet neumann boundary conditions the inverse problem is solved by stieltjes s continued fraction expansion leading to an explicit construction of peakon solutions sufficient conditions for global existence of solutions are given the large time asymptotics reveals that asymptotically peakons pair up to form bound states moving with constant speeds the peakon flow is shown to project to one of the isospectral flows of the finite kac van moerbeke lattice

an alternate algorithm for computing adomian polynomials in special cases in this article we introduce an alternate algorithm for computing adomian polynomials and present some examples to show the simplicity and efficiency of the new method

solving a multi order fractional differential equation using adomian decomposition abstract an algorithm has been developed to convert the multi order fractional differential equation d y t f t y t d 1 y t d n y t y k 0 c k k 0 m where m d denotes caputo fractional derivative of order into a system of fractional differential equations further adomian decomposition method has been employed to solve the system of fractional differential equations some illustrative examples are presented

a new finite difference predictor corrector method for fractional differential equations abstract we present a new finite difference predictor corrector method l1 pcm to solve nonlinear fractional differential equations fdes along with its error and stability analysis the method is further extended for systems of fdes the proposed method is applied to fractional version of chaotic system introduced by bhalekar and daftardar gejji to explore its rich dynamics the proposed method is accurate time efficient and performs well even for very small values of the order of the derivatives

exact finite dimensional filters for doubly stochastic auto regressive processes in this paper we derive exact finite dimensional recursive filters for a class of doubly stochastic auto regressive ar models we assume that the parameters of the doubly stochastic ar process vary according to a nonlinear polynomial function of a gaussian state space process apart from being of mathematical interest these finite dimensional filters have potential applications in time series analysis and image enhanced tracking of maneuvering targets

k symplectic structures the basic properties of k symplectic structures are introduced and developed in analogy with the well known symplectic differential geometry examples of such structures related to k symplectic lie algebras are given

filtering s coupled algebraic riccati equations for discrete time markov jump systems abstract this paper deals with a set of s coupled algebraic riccati equations that arises in the study of filtering of discrete time linear jump systems with the markov chain in a general borel space s by s coupled it is meant that the algebraic riccati equations are coupled via an integral over s conditions for the existence and uniqueness of a positive semi definite solution to the filtering s coupled algebraic riccati equations are obtained in terms of the concepts of stochastic detectability and stochastic stabilizability this result is then applied to solve the infinite horizon minimum mean square linear markov jump filtering problem the obtained results generalize previous ones in the literature which considered only the case of the markov chain taking values in a finite state space

brief optimal filtering of doubly stochastic auto regressive processes in this paper exact finite dimensional filters are derived for a class of doubly stochastic auto regressive models the parameters of the doubly stochastic auto regressive process vary according to a nonlinear function of a gauss markov process we develop a difference equation for the evolution of an unnormalized conditional density related to the state of the doubly stochastic auto regressive process we then give a characterization of the general solution followed by examples for which the state of the filter is determined by a finite number of sufficient statistics these new finite dimensional filters build upon the discrete time kalman filter

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

picard iteration algorithm combined with gauss seidel technique for initial value problems abstract in this paper we introduce and implement a relatively new improvement for the well known picard s method for studying linear and nonlinear systems of ordinary differential equations in this method the solution takes the form of a rapidly convergent series with easily computable components a mathematica computer algebra system mathematica 5 1 is used to introduce a code which generates the series different illustrative examples have been considered as well as different comparison criterion between the proposed method and modern reliable and conventional methods the deeply mathematical roots of the method fixed point theorem and the correspondence with the theory of algebraic systems makes it is very interesting theoretical considerations are being discussed and convergence of the method is addressed other extensions and applications for further work are mentioned

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

image enhanced multiple model tracking we consider tracking algorithms for manoeuvring targets when the observations include extra information on the current operating mode of the target obtained from an image sensor the target is modelled as a markov jump linear system and the image based observations form a discrete time point process we derive the optimal minimum mean square error filtered estimate which intrinsically fuses the image based and primary observations this optimal filter is computationally prohibitive but provides the basis for a clear understanding of various suboptimal approaches we propose the image enhanced imm filter as a practical alternative which retains many desirable properties of the optimal filter and outperforms existing image enhanced tracking algorithms over a broad range of operating scenarios

optimal control of jump linear gaussian systems this paper investigates the problem of controlling a discrete time linear system with jump parameters a review of the literature is presented as well as a development of the application of dynamic programming to this class of control problems dynamic programming has been applied by many researchers and it was observed that no closed form analytical solution could be constructed because of the dual aspects of the controller the main contribution of the present work is an algorithm suitable for computer implementation for the optimal dual control the construction of the algorithm is based on transforming the dynamic programming relations into a space of sufficient statistics and using a finite dimensional optimization procedure to obtain the optimal control as a function of the statistics this is achieved by first developing a suitable recursive realization of a filter which generates the sufficient statistics for the problem and then embedding this filter into the dynamic programming equations

an iterative method for solving nonlinear functional equations an iterative method for solving nonlinear functional equations viz nonlinear volterra integral equations algebraic equations and systems of ordinary differential equation nonlinear algebraic equations and fractional differential equations has been discussed

determinantal identities gauss schur cauchy sylvester kronecker jacobi binet laplace muir and cayley abstract we give a common concise derivation of some important determinantal identities attributed to the mathematicians in the title we also give a formal treatment of determinantal identities of the minors of a matrix

solving a system of nonlinear fractional differential equations using adomian decomposition adomian decomposition method has been employed to obtain solutions of a system of nonlinear fractional differential equations dxiyi x ni x y1 yn yi k 0 cki 0 k i 1 i n and dxi denotes caputo fractional derivative some examples are solved as illustrations using symbolic computation

a computational method for solution of the prey and predator problem in this article a mathematical model of the problem of prey and predator being presented and adomian decomposition method is employed to compute an approximation to the solution of the system of nonlinear volterra differential equations governing on the problem some plots for the population of the prey and predator versus time are presented to illustrate the solution

discrete time point process filter for mode estimation a discrete time model of an image based observation channel using point processes is presented in the context of target tracking improvement a multidimensional point process filter was developed to effectively estimate the actual mode of the target in the presence of the clutter the structure and updating properties of the filter have been studied the sample behavior and the robustness with respect to modeling parameters are illustrated by simulation examples