improved least squares identification an improved recursive least squares algorithm for parameter estimation is presented which includes on off criteria to prevent parameter drift during periods of low excitation a variable forgetting factor which maintains the trace of the covariance matrix at a user specified value data preprocessing and normalization to improve numerical accuracy scaling of the regressor vector to minimize the condition number of the covariance matrix plus independent estimation of the mean values of the input output data which can be used to eliminate errors due to d c bias or slowly drifting elements in the regressor vector the algorithm can also include parameter projection to constrain the estimates to a priori specified regions and retains the formal properties such as convergence of a true weighted least squares algorithm the proposed algorithm is compared with other modifications suggested in the literature and its advantages are demonstrated by a simulated example

bounded disturbance adaptive control with exponential and selective data weightings a method of combination of selective and exponential data weighting algorithms for parameter identification is proposed bounded disturbances in systems with unknown statistical properties are considered the proposed method ensures a stable control system in addition it does not need the condition of persistent excitation when the predicted output error is small

adjoint equations and analysis of complex systems application to virus infection modelling recent development of applied mathematics is characterized by ever increasing attempts to apply the modelling and computational approaches across various areas of the life sciences the need for a rigorous analysis of the complex system dynamics in immunology has been recognized since more than three decades ago the aim of the present paper is to draw attention to the method of adjoint equations the methodology enables to obtain information about physical processes and examine the sensitivity of complex dynamical systems this provides a basis for a better understanding of the causal relationships between the immune system s performance and its parameters and helps to improve the experimental design in the solution of applied problems we show how the adjoint equations can be used to explain the changes in hepatitis b virus infection dynamics between individual patients

adaptive control for time variant processes abstract this paper establishes conditions of global stability for adaptive predictive control systems apcs when applied to a general class of multivariate stochastic time variant stable inverse processes with time delays these conditions rely on assumed convergence properties of the apcs estimation scheme therefore the design of stable apcs or research into apcs stability proofs for time variant processes proceeds from this perspective with the design and analysis of estimation schemes which may satisfy the required convergence properties using this approach an asymptotically stable apcs is derived for a class of time variant processes with a finite number of bounded changes in their parameter values and a perturbation vector with a finite l2 norm the methodology used to derive the convergent estimation scheme has a general scope and is conceptually simple

numerical solution by lmms of stiff delay differential systems modelling an immune response we consider the application of linear multistep methods lmms for the numerical solution of initial value problem for stiff delay differential equations ddes with several constant delays which are used in mathematical modelling of immune response for the approximation of delayed variables the nordsieck s interpolation technique providing an interpolation procedure consistent with the underlying linear multistep formula is used an analysis of the convergence for a variable stepsize and structure of the asymptotic expansion of global error for a fixed stepsize is presented some absolute stability characteristics of the method are examined implementation details of the code difsub dde being a modification of the gear s difsub are given finally an efficiency of the code developed for solution of stiff ddes over a wide range of tolerances is illustrated on biomedical application model

asymptotic expansions of the global error of fixed stepsize methods in his fundamental paper on general fixed stepsize methods skeel 6 studied convergence properties but left the existence of asymptotic expansions as an open problem in this paper we give a complete answer to this question for the special cases of one step and linear multistep methods our proof is shorter than the published ones asymptotic expansions are the theoretical base for extrapolation methods

a compact finite difference scheme for the fractional sub diffusion equations in this paper a compact finite difference scheme for the fractional sub diffusion equations is derived after a transformation of the original problem the l1 discretization is applied for the time fractional part and fourth order accuracy compact approximation for the second order space derivative the unique solvability of the difference solution is discussed the stability and convergence of the finite difference scheme in maximum norm are proved using the energy method where a new inner product is introduced for the theoretical analysis the technique is quite novel and different from previous analytical methods finally a numerical example is provided to show the effectiveness and accuracy of the method

a class of second order difference approximations for solving space fractional diffusion equations a class of second order approximations called the weighted and shifted grunwald difference wsgd operators are proposed for riemann liouville fractional derivatives with their effective applications to numerically solving space fractional diffusion equations in one and two dimensions the stability and convergence of our difference schemes for space fractional diffusion equations with constant coefficients in one and two dimensions are theoretically established several numerical examples are implemented to test the efficiency of the numerical schemes and confirm the convergence order and the numerical results for variable coefficients problem are also presented

compact difference scheme for the fractional sub diffusion equation with neumann boundary conditions an effective finite difference scheme is considered for solving the time fractional sub diffusion equation with neumann boundary conditions a difference scheme combining the compact difference approach the spatial discretization and l 1 approximation for the caputo fractional derivative is proposed and analyzed although the spatial approximation order at the neumann boundary is one order lower than that for interior mesh points the unconditional stability and the global convergence order o t 2 a h 4 in discrete l 2 norm of the compact difference scheme are proved rigorously where t is the temporal grid size and h is the spatial grid size numerical experiments are included to support the theoretical results and comparison with the related works are presented to show the effectiveness of our method

unconditionally stable difference methods for delay partial differential equations this paper is concerned with the numerical solution of parabolic partial differential equations with time delay we focus in particular on the delay dependent stability analysis of difference methods that use a non constrained mesh i e the time step size is not required to be a submultiple of the delay we prove that the fully discrete system unconditionally preserves the delay dependent asymptotic stability of the linear test problem under consideration when the following discretization is used a variant of the classical second order central differences to approximate the diffusion operator a linear interpolation to approximate the delay argument and finally the trapezoidal rule or the second order backward differentiation formula to discretize the time derivative we end the paper with some numerical experiments that confirm the theoretical results

numerical approximation of distributed order reaction diffusion equations in this paper an implicit scheme for the numerical approximation of the distributed order time fractional reaction diffusion equation with a nonlinear source term is presented the stability and the convergence order of the numerical scheme are analysed and illustrated through some numerical examples

order stars and stability for delay differential equations we consider runge kutta methods applied to delay differential equations y t ay t by t 1 with real a and b if the numerical solution tends to zero whenever the exact solution does the method is called tau 0 stable using the theory of order stars we characterize high order symmetric methods with this property in particular we prove that all gauss methods are tau 0 stable furthermore we present sufficient conditions and we give evidence that also the radau methods are tau 0 stable we conclude this article with some comments on the case where a andb are complex numbers

numerical solution of distributed order fractional differential equations in this paper a method for the numerical solution of distributed order fdes fractional differential equations of a general form is presented the method applies to both linear and nonlinear equations the caputo type fractional derivative is employed the distributed order fde is approximated with a multi term fde which is then solved by adjusting appropriately the numerical method developed for multi term fdes by katsikadelis several example equations are solved and the response of mechanical systems described by such equations is studied the convergence and the accuracy of the method for linear and nonlinear equations are demonstrated through well corroborated numerical results

a stable adaptive predictive control system an adaptive predictive control system apcs is described and a formal proof of global stability is presented for the case of multi variable stable in verse time invariant processes in the presence of bounded unmeasured disturbances plus process and measurement noise global stability is proven in the sense that the process input output vector remains bounded and the control error is minimized the proof of stability is based on certain properties of the aposteriori estimation error and convergence properties of the estimated parameters reference is also made to successful experimental and simulation applications of apcs

an explicit finite difference method and a new von neumann type stability analysis for fractional diffusion equations a numerical method for solving the fractional diffusion equation which could also be easily extended to other fractional partial differential equations is considered in this paper we combine the forward time centered space ftcs method well known for the numerical integration of ordinary diffusion equations with the grunwald letnikov discretization of the riemann liouville derivative to obtain an explicit ftcs scheme for solving the fractional diffusion equation the stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well known von neumann method for nonfractional partial differential equations the analytical stability bounds are in excellent agreement with numerical test a comparison between exact analytical solutions and numerical predictions is made

numerical solution of retarded initial value problems local and global error and stepsize control retarded initial value problems are routinely replaced by an initial value problem of ordinary differential equations along with an appropriate interpolation scheme hence one can control the global error of the modified problem but not directly the actual global error of the original problem in this paper we give an estimate for the actual global error in terms of controllable quantities further we show that the notion of local error as inherited from the theory of ordinary differential equations must be generalized for retarded problems along with the new definition we are led to developing a reliable basis for a step selection scheme

numerical treatment of delay differential equations by hermite interpolation a class of numerical methods for the treatment of delay differential equations is developed these methods are based on the wellknown runge kutta fehlberg methods the retarded argument is approximated by an appropriate multipoint hermite interpolation the inherent jump discontinuities in the various derivatives of the solution are considered automatically problems with piecewise continuous right hand side and initial function are treated too real life problems are used for the numerical test and a comparison with other methods published in literature

a globally stable apcs in the presence of bounded noises and disturbances this paper presents a globally stable adaptive predictive control system apcs for discrete linear processes with pure time delays in the presence of bounded measurement and process noise plus unmeasured disturbances the main contribution is the proof of stability in the context of discrete adaptive control schemes this proof is based on the minimization of the a posteriori estimation error and the convergence of the estimated parameters

delay dependent stability of high order runge kutta methods this paper is concerned with the study of the delay dependent stability of runge kutta methods for delay differential equations first a new sufficient and necessary condition is given for the asymptotic stability of analytical solution then based on this condition we establish a relationship between 0 stability and the boundary locus of the stability region of numerical methods for ordinary differential equations consequently a class of high order runge kutta methods are proved to be 0 stable in particular the 0 stability of the radau iia methods is proved

the stability of difference formulas for delay differential equations a new simple geometric technique is presented for analyzing the stability of difference formulas for the model delay differential equation y t py t qy t 8 where p and q are complex constants and the delay 8 is a positive constant the technique is based on the argument principle and directly relates the region of absolute stability for ordinary differential equations corresponding to the py t term with the region corresponding to the delay term qy t 8 a sufficient condition for stability is that these regions be disjoint the technique is used to show that for each a stable a a stable or stiffly stable linear multistep formula for ordinary differential equations there is a correspond ing linear multistep formula for delay differential equations with analogous stability properties the analogy does not extend however to a stable one step formulas

the stability of variable stepsize nordsieck methods conditions are given for the stability of multistep methods that vary the stepsize using the interpolation technique of nordsieck if the stepsize selection function is variation bounded then these methods are stable alternatively there exist constants a b c depending on the method and satisfying a leqq b 1 c such that if the stepsize ratio h n 1 h n satisfies either 0 h n 1 h n leqq a or b h n 1 h n leqq c then the method is stable as the frequency of stepsize changing decreases the methods become more stable in the sense that the intervals defined by a b and c are larger tables and graphs are given for the adams and backward differentiation methods indicating how stability changes with frequency of changing stepsize in particular it is shown that k 1 value adams formulas k leqq 6 are stable for 0 h n 1 h n leqq c that is it is shown that stability is maintained whenever the stepsize is decreased by an arb

brief paper stability analysis of adaptively controlled systems subject to bounded disturbances an approach for studying stability of adaptively controlled systems subject to bounded disturbances with unknown statistics is given a shorter proof of some results established for minimum phase systems is also given new results are established for nonminimum phase systems in each case the uniform boundedness of all signals is proven under a condition that is discussed examples of adaptive schemes that can be treated with this approach are given

new solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation a physical mathematical approach to anomalous diffusion is based on a generalized diffusion equation containing derivatives of fractional order in this paper an anomalous subdiffusion equation asub de is considered a new implicit numerical method inm and two solution techniques for improving the order of convergence of the inm for solving the asub de are proposed the stability and convergence of the inm are investigated by the energy method some numerical examples are given the numerical results demonstrate the effectiveness of theoretical analysis these methods and supporting theoretical results can also be applied to other fractional integro differential equations and higher dimensional problems

robust design of adaptive control systems using conic sector theory abstract this paper presents a design approach for discrete adaptive control systems which provides a quantitative measure of the effect of design alternatives such as i adaptive gain ii model order and iii sampling rate on stability in the presence of unmodeled plant dynamics the proposed method based on the conic conditions developed by ortega et al 1985 ieee trans aut control ac 30 1179 1187 is illustrated using a benchmark example the results demonstrate that the sector conditions permit design tradeoffs to be made such that stability is maintained despite the model plant mismatch

compact alternating direction implicit scheme for the two dimensional fractional diffusion wave equation in this paper we consider the numerical method for solving the two dimensional fractional diffusion wave equation with a time fractional derivative of order alpha 1 alpha 2 a difference scheme combining the compact difference approach for spatial discretization and the alternating direction implicit adi method in the time stepping is proposed and analyzed the unconditional stability and h 1 norm convergence of the scheme are proved rigorously the convergence order is mathcal o tau 3 alpha h 1 4 h 4 2 where tau is the temporal grid size and h 1 h 2 are spatial grid sizes in the x and y directions respectively in addition a crank nicolson adi scheme is presented and the corresponding error estimates are also established numerical results are presented to support our theoretical analysis and indicate that the compact adi scheme reduces the storage requirement and cpu time successfully

local linearization method for numerical integration of delay differential equations in this paper a new approach for the numerical computation of delay differential equations ddes is introduced the essential idea consists of obtaining numerical integrators that use a code expressly developed for linear ddes in contrast with the conventional approach of using a code for ordinary differential equations specifically two numerical schemes of this new class of integrators are proposed and their numerical viability analyzed it includes the estimation of the convergence rate the evaluation of the computational cost of the schemes and a simulation study it is proved that these one step explicit integrators converge uniformly with order two to the solution of nonlinear ddes and are able to integrate stiff equations in a satisfactory way with low computational cost

asymptotic stability barriers for natural runge kutta processes for delay equations this paper investigates the asymptotic stability properties of a class of numerical methods for delay differential equations ddes the so called natural runge kutta methods we first examine the behavior of these methods when applied to the neutral model equation y t a y t b y t 1 c y t 1 with a b c in mathbb r we also consider the case when a b c in mathbb c and provide a suitable geometric characterization of their asymptotic stability regions then by means of the obtained results in conjunction with those given in n guglielmi and e hairer ima j numer anal 21 2001 pp 439 450 we are able to give a final answer concerning the possible preservation of asymptotic stability of the considered class of methods when applied to systems of linear ddes of the form y t l y t m y t 1 with l m in mathbb r d times d d 1 we are interested here in methods that produce stable numerical solutions for all values of the parameters a b and c in the first equation and l and m in the second one for which the exact solution tends to zero to this aim we direct our attention to a novel stability setting recently introduced and investigated for the scalar nonneutral case see n guglielmi numer math 77 1997 pp 467 485 n guglielmi ima j numer anal 18 1998 pp 399 418 n guglielmi and e hairer numer math 83 1999 pp 371 383 n guglielmi and e hairer ima j numer anal 21 2001 pp 439 450 s maset numer math 87 2000 pp 355 371

some high order difference schemes for the distributed order differential equations two difference schemes are derived for both one dimensional and two dimensional distributed order differential equations it is proved that the schemes are unconditionally stable and convergent in an l 1 l norm with the convergence orders o 2 h 2 2 and o 2 h 4 4 respectively where h and are the step sizes in time space and distributed order variables several numerical examples are given to confirm the theoretical results

a high order schema for the numerical solution of the fractional ordinary differential equations in this paper we present a general technique to construct high order schemes for the numerical solution of the fractional ordinary differential equations fodes this technique is based on the so called block by block approach which is a common method for the integral equations in our approach the classical block by block approach is improved in order to avoiding the coupling of the unknown solutions at each block step with an exception in the first two steps while preserving the good stability property of the block by block schemes by using this new approach we are able to construct a high order schema for fodes of the order a a 0 the stability and convergence of the schema is rigorously established we prove that the numerical solution converges to the exact solution with order 3 a for 0 1 a series of numerical examples are provided to support the theoretical claims

stable discrete time adaptive control in the presence of unmodeled dynamics results obtained previously on the stability of adaptive predictive control systems are generalized by reducing the requirements for stability to a single condition that is applicable to linear discrete processes with disturbances and unmodeled plant dynamics the proposed design approach uses a normalized parameter estimation scheme which permits a formal proof that the modeling errors can be treated as a bounded disturbance and a parameter adaptation stopping criterion to guarantee global stability

compact difference scheme for distributed order time fractional diffusion wave equation on bounded domains in this paper we derive and analyse a compact difference scheme for a distributed order time fractional diffusion wave equation this equation is approximated by a multi term fractional diffusion wave equation which is then solved by a compact difference scheme the unique solvability of the difference solution is discussed using the discrete energy method we prove the compact difference scheme is unconditionally stable and convergent finally numerical results are presented to support our theoretical analysis