a random euler scheme for caratheodory differential equations we study a random euler scheme for the approximation of caratheodory differential equations and give a precise error analysis in particular we show that under weak assumptions this approximation scheme obtains the same rate of convergence as the classical monte carlo method for integration problems

moving mesh methods for problems with blow up in this paper we consider the numerical solution of pdes with blow up for which scaling invariance plays a natural role in describing the underlying solution structures it is a challenging numerical problem to capture the qualitative behaviour in the blow up region and the use of nonuniform meshes is essential we consider moving mesh methods for which the mesh is determined using so called moving mesh partial differential equations mmpdes specifically the underlying pde and the mmpde are solved for the blow up solution and the computational mesh simultaneously motivated by the desire for the mmpde to preserve the scaling invariance of the underlying problem we study the effect of different choices of mmpdes and monitor functions it is shown that for suitable ones the mmpde solution evolves towards a moving mesh which close to the blow up point automatically places the mesh points in such a manner that the ignition kernel which is well known to be a natural coordinate in describing the behaviour of blow up approaches a constant as t to t the blow up time several numerical examples are given to verify the theory for these mmpde methods and to illustrate their efficacy

an asymptotic and numerical description of self similar blow up in quasilinear parabolic equations we study the blow up behaviour of two reaction diffusion problems with a quasilinear degenerate diffusion and a superlinear reaction we show that in each case the blow up is self similar in contrast to the linear diffusion limit of each in which the diffusion is only approximately self similar we then investigate the limit of the self similar behaviour and describe the transition from a stable manifold blow up behaviour for quasilinear diffusion to a centre manifold one for the linear diffusion

semidiscretization in time of nonlinear parabolic equations with blowup of the solution in this paper the author considers the first boundary value problem for the nonlinear equation u t delta u m alpha u m in omega a smooth bounded domain in mathbb r n with the zero lateral boundary condition and with a positive initial condition m is supposed to be larger than one and alpha positive a scheme for the discretization in time of that problem is proposed it is proven that if the exact solution blows up in a finite time it is the same for the numerical solution estimates of the blow up time are obtained the stability of the method and the convergence for a class of initial conditions is proved

nonexistence for the kassoy problem in dimensions 1 and 2 modele d ignition pour l explosion thermique a grande energie d activation d un combustible solide

refined asymptotics for the blowup of ut δu up this work is concerned with positive blowing up solutions of the semilinear heat equation ut u up in rn our main contribution is a sort of center manifold analysis for the equation in similarity variables leading to refined asymptotics for u in a backward space time parabola near any blowup point we also explore a connection between the asymptotics of u and the local geometry of the blowup set

closed form boundary state feedbacks for a class of 1 d partial integro differential equations in this paper a problem of boundary stabilization of a class of linear parabolic partial integro differential equations p i des in one dimension is considered using the method of backstepping avoiding spatial discretization required in previous efforts the problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic p i de the kernel p i de is then converted into an equivalent integral equation and by applying the method of successive approximations the equation s well posedness and the kernel s smoothness are established it is shown how to extend this approach to design optimally stabilizing controllers an adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller and the performance bounds are derived for a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed loop solutions are found in closed form a numerical scheme for the kernel p i de is proposed its numerical effort compares favorably with that associated with operator riccati equations

optimal adaptive solution of piecewise regular systems of ivps with unknown switching hypersurface we study the solution of piecewise regular systems of ivps with a switching hypersurface defined by an unknown event function our aim is to design an algorithm that is not based on heuristic steps it does not use information on the event function and preserves the order known for regular systems the algorithm is based on the adaptive mesh which differs from the equidistant mesh only by a single point we provide a rigorous that is independent of heuristic arguments analysis of its error and cost

moving mesh methods based on moving mesh partial differential equations several versions of a moving mesh method are developed based on a mesh spatial smoothing technique and on the moving mesh pdes derived in a previous paper these versions are quite simple and easy to program they are applied to three bench mark one dimensional problems which show different solution behaviour the numerical results clearly demonstrate that the present methods are capable of accurately tracking rapid spatial and temporal transitions

output feedback boundary control of a heat pde sandwiched between two odes we present designs for exponential stabilization of an ordinary differential equation ode heat partial differential equation pde ode coupled system where the control actuation only acts in one ode the combination of pde backstepping and ode backstepping is employed in a state feedback control law and in an observer that estimates pde and two ode states using only one pde boundary measurement based on the state feedback control law and the observer the output feedback control law is then proposed the exponential stability of the closed loop system and the boundedness and exponential convergence of the control law are proved via lyapunov analysis finally numerical simulations validate the effectiveness of this method for the sandwiched system

output feedback control of general linear heterodirectional hyperbolic pde ode systems with spatially varying coefficients this paper presents a backstepping solution for the output feedback control of general linear heterodirectional hyperbolic pde ode systems with spatially varying coefficients thereby the

the state feedback regulator problem for regular linear systems this paper is about the state feedback regulator problem for infinite dimensional linear systems the plant assumed to be an exponentially stable regular linear system is driven by a linear possibly infinite dimensional exosystem via a disturbance signal the exosystem has its spectrum in the closed right half plane and also generates the reference signal for the plant output the regulator problem is to design a controller that while guaranteeing the stability of the closed loop system without the exosystem drives the tracking error to zero a particular version of this problem is the state feedback regulator problem in which the states of the exosystem and the plant are known to the controller under suitable assumptions we show that the latter problem is solvable if and only if a pair of algebraic equations called the regulator equations is solvable we derive conditions in terms of the transfer function of the plant and eigenvalues of the exosystem for the solvability of the regulator equations three examples illustrating the theory are presented

a structure preserving scheme for the kolmogorov fokker planck equation in this paper we introduce a numerical scheme which preserves the behavior of solutions to the kolmogorov equation as time tends to infinity the method presented is based on a self similar change of variables technique to transform the kolmogorov equation into a new form such that the problem of designing structure preserving schemes for the original equation amounts to building a standard scheme for the transformed equation this transformation also has the added benefit of allowing for an exact operator splitting scheme whereas in the original form a standard operator splitting was only second order finally we verify the preservation of long time behavior through numerical simulations

boundedness and blowup for nonlinear degenerate parabolic equations the author deals with the quasilinear parabolic equation ut u fi g u u bu fi 1 f u ru with dirichlet boundary conditions in a bounded domain where f and g are lower order terms he shows that under suitable conditions on f and g whether the solution is bounded or blows up in a flnite time depends only on the flrst eigenvalue of i in with dirichlet boundary condition for some special cases the result is sharp

a rescaling algorithm for the numerical calculation of blowing up solutions on presente un algorithme a invariance d echelle pour le calcul numerique des solutions a singularites explosives des equations d evolution non lineaires

output regulation for a class of linear boundary controlled first order hyperbolic pide systems abstract this manuscript addresses the output regulation problem for a class of scalar boundary controlled first order hyperbolic partial integro differential equation pide systems with fredholm integrals in particular with the advantage of the backstepping approach simple structure systems can be obtained such that regulator equations for the state feedback regulator design are analyzed and solved in backstepping coordinates moreover the finite time output regulation is achieved in the observer based output feedback regulator design it is not necessary that the outputs to be controlled belong to the available output measurements and these outputs can be distributed point wise and or boundary in nature while the boundary placed measurements are used for regulator design for the observer gains design a transformation of the ode pde system into an ode pde cascade is considered it is also shown that the separation principle holds for the output feedback regulator design and the exponential output regulation is realized for the resulting stable closed loop system finally the output regulation results are illustrated with two numerical simulations a korteweg de vries like equation and a pde ode interconnected system

the power of adaption for approximating functions with singularities consider approximating functions based on a finite number of their samples we show that adaptive algorithms are much more powerful than nonadaptive ones when dealing with piecewise smooth functions more specifically let f 1 r be the class of scalar functions f 0 t r whose derivatives of order up to r are continuous at any point except for one unknown singular point we provide an adaptive algorithm a n ad that uses at most n samples of f and whose worst case l p error 1 p oo with respect to reasonable function classes f 1 r f 1 r is proportional to n r on the other hand the worst case error of any nonadaptive algorithm that uses n samples is at best proportional to n 1 p the restriction to only one singularity is necessary for superiority of adaption in the worst case setting fortunately adaption regains its power in the asymptotic setting even for a very general class f r consisting of piecewise c r smooth functions each having a finite number of singular points for any f f r our adaptive algorithm approximates f with error converging to zero at least as fast as n r we also prove that the rate of convergence for nonadaptive methods cannot be better than n 1 p i e is much slower the results mentioned above do not hold if the errors are measured in the l norm since no algorithm produces small l errors for functions with unknown discontinuities however we strongly believe that the l norm is inappropriate when dealing with singular functions and that the skorohod metric should be used instead we show that our adaptive algorithm retains its positive properties when the approximation error is measured in the skorohod metric that is the worst case error with respect to f r 1 equals n r and the convergence in the asymptotic setting for f r is n numerical results confirm the theoretical properties of our algorithms

stability and spectra of blow up in problems with quasi linear gradient diffusivity we study a nonlinear diffusion equation which is invariant under a stretching group of transformations and which reduces to a linear diffusion equation in the limit of 0 we show that if 0 then the equation has solutions which form singularities which have a self similar profile by considering a nonlinear eigenvalue problem the existence and stability of the self similar profiles is discussed the existence of approximately self similar behaviour is also considered for evolution profiles with several maxima and minima this behaviour is compared to similar behaviour for the linear diffusion problem the paper uses a combination of techniques from analysis and the theory of dynamical systems

a finite dimensional robust controller for systems in the cd algebra robust multivariable controllers for stable infinite dimensional systems in the callier desoer cd algebra are discussed in particular the following robust regulation problem is solved given reference and disturbance signals which are linear combinations of signals of the form t sup j sin spl omega sub k t spl phi sub k j spl ges 0 k 0 spl middot spl middot spl middot n find a low order finite dimensional controller so that the outputs asymptotically track the reference signals asymptotically reject the disturbance signals and the closed loop system is stable and robust with respect to a class of perturbations in the plant the proposed controller consists of a positive scalar gain spl epsi and certain polynomial matrices k sub k s for k 0 spl middot spl middot spl middot n the main result of the paper shows that the matrices k sub k s must satisfy certain stability conditions involving the values of the plant transfer function only at the reference and disturbance signal frequencies spl omega sub k for k 0 spl middot spl middot spl middot n the controller has unstable poles on the imaginary axis the behavior of these poles as a function of the scalar parameter spl epsi in the form of a puiseux series is given

finite dimensional regulators for a class of regular hyperbolic pde systems abstractin this paper the output regulation problem is addressed for a class of linear hyperbolic infinite dimensional systems with spatially varying coefficients modelling a large class of convection dominated transport reaction systems in particular distributed parameter systems with bounded input and unbounded output operators are considered first we demonstrate a general conclusion about the exponential stability of the considered system by relating the stability to the solution of an associated differential equation based on the assumption that the hyperbolic system satisfies the exponential stability conditions the main manuscript contribution is the development of two novel finite dimensional regulators output and error feedback regulators such that the controlled output of the plant tracks a reference signal generated by a known signal process exosystem in order to guarantee the feasibility of the proposed regulators the solvability of the corresponding sylvester equations is discusse

robust multivariable pi controller for infinite dimensional systems a robust multivariable controller is introduced for a class of distributed parameter systems the system to be controlled is given as dot x ax bu y cx in a banach space the purpose of the control which is based on the measurement y is to stabilize and regulate the system so that y t rightarrow y r as t rightarrow infty where y r is a constant reference vector under the assumptions that operator a generates a holomorphic stable semigroup b is linear and bounded c is linear and a bounded and the input and output spaces are of the same dimension a necessary and sufficient condition is found for the existence of a robust multivariable controller this controller appears to be a multivariable pi controller also a simple necessary criterion for the existence of a decentralized controller is derived the tuning of the controller is discussed and it is shown that the i part of the controller can be tuned on the basis of step responses without exact knowledge of the system s parameters the presented theory is then used as an example to control the temperature profile of a bar with the dirichlet boundary conditions

optimal global approximation of sdes with time irregular coefficients in asymptotic setting we investigate strong approximation of solutions of scalar stochastic differential equations sdes with irregular coefficients in przybylowicz 2015 23 an approximation of solutions of sdes at a single point is considered such kind of approximation is also called a one point approximation comparing to that article we are interested here in a global reconstruction of trajectories of the solutions of sdes in a whole interval of existence we assume that a drift coefficient a 0 t r r is globally lipschitz continuous with respect to a space variable but only measurable with respect to a time variable a diffusion coefficient b 0 t r is only piecewise holder continuous with holder exponent 0 1 the algorithm and results concerning lower bounds from przybylowicz 2015 23 cannot be applied for this problem and therefore we develop a suitable new technique in order to approximate solutions of sdes under such assumptions we define a discrete type randomized euler scheme we provide the error analysis of the algorithm showing that its error is o n min 1 2 moreover we prove that roughly speaking the error of an arbitrary algorithm for fixed a and b that uses n values of the diffusion coefficient cannot converge to zero faster than n min 1 2 as n hence the proposed version of the randomized euler scheme achieves the established best rate of convergence

the internal model principle for systems with unbounded control and observation in this paper the theory of robust output regulation of distributed parameter systems with infinite dimensional exosystems is extended for plants with unbounded control and observation as the main result we present the internal model principle for linear infinite dimensional systems with unbounded input and output operators we do this for two different definitions of an internal model found in the literature namely the p copy internal model and the mathcal g conditions we also introduce a new way of defining an internal model for infinite dimensional systems the theoretic results are illustrated with an example where we consider robust output tracking for a one dimensional heat equation with boundary control and pointwise measurements

boundary control of coupled reaction diffusion processes with constant parameters the problem of boundary stabilization is considered for some classes of coupled parabolic linear pdes of the reaction diffusion type with reference to n coupled equations each one equipped with a scalar boundary control input a state feedback law is designed with actuation at only one end of the domain and exponential stability of the closed loop system is proven the treatment is addressed separately for the case in which all processes have the same diffusivity and for the more challenging scenario where each process has its own diffusivity and a different solution approach has to be taken the backstepping method is used for controller design and particularly the kernel matrix of the transformation is derived in explicit form of series of bessel like matrix functions by using the method of successive approximations to solve the corresponding pde thus the proposed control laws become available in explicit form additionally the stabilization of an underactuated system of two coupled reaction diffusion processes is tackled under the restriction that only a scalar boundary input is available capabilities of the proposed synthesis and its effectiveness are supported by numerical studies made for three coupled systems with distinct diffusivity parameters and for underactuated linearized dimensionless temperature concentration dynamics of a tubular chemical reactor controlled through a boundary at low fluid superficial velocities when convection terms become negligible

global existence and blowup solutions for quasilinear parabolic equations the authors discuss the quasilinear parabolic equation ut g u u h u u f u with u 0 u x 0 x i ff g and h are polynomials with proper degrees and proper coefficients they show that the blowup property only depends on the first eigenvalue of in with dirichlet boundary condition for a special case they obtain a sharp result

backstepping design of robust state feedback regulators for linear 2 times 2 hyperbolic systems this technical note deals with the robust output regulation problem for boundary controlled linear 2 times 2 hyperbolic systems thereby the output to be controlled can be located at one of the boundaries or is a pointwise in domain output by utilizing the internal model principle a state feedback regulator is determined for a finite dimensional signal model describing the exogenous signals this amounts to stabilizing a hyperbolic pde ode cascade coupled at an intermediate point for this a systematic backstepping method is developed to determine the stabilizing state feedback controller the solvability of the considered output regulation problem can easily be verified on the basis of the nominal plant transfer behaviour for non destabilizing model uncertainties the robustness of the achieved output regulation is verified an uncertain hyperbolic system with an in domain output illustrates the results of the technical note

output regulation for coupled linear parabolic pides abstract this paper presents a backstepping based solution of the output regulation problem for coupled parabolic partial integro differential equations pides with spatially varying coefficients and distinct diffusion coefficients the considered setup assumes in domain as well as boundary disturbances while the output to be controlled can be defined distributed in domain pointwise in domain or at the boundaries and need not be measured by assuming a finite dimensional signal model which may also be non diagonalizable a systematic solution of the output regulation problem is presented by making use of observer based feedforward control existence conditions for the corresponding regulator are formulated in terms of the plant transfer behaviour for the resulting closed loop system exponential stability with a prescribed decay rate is verified the regulator design for two unstable coupled parabolic pides demonstrates the results of the paper

backstepping based adaptive error feedback regulator design for one dimensional reaction diffusion equation abstract in this paper we consider the error feedback regulator problem efrp for one dimensional reaction diffusion equation with unknown harmonic boundary disturbance where the regulated output is anti collocated with the control some auxiliary systems are constructed in order to make the control and the disturbance become collocated or make the measured tracking error become the output then the error feedback adaptive servomechanism is presented in which the parameters of the disturbance and the reference signals are estimated our design is based on the motion planning and the backstepping approaches in addition we give a brief description that our approach is applicable to the heat equation with distributed disturbance it is shown that the proposed adaptive control law regulates the tracking error to zero and keeps the states of all the internal loops bounded

internal model based tracking and disturbance rejection for stable well posed systems in this paper we solve the tracking and disturbance rejection problem for infinite dimensional linear systems with reference and disturbance signals that are finite superpositions of sinusoids we explore two approaches both based on the internal model principle in the first approach we use a low gain controller and here our results are a partial extension of results by hamalainen and pohjolainen in their papers the plant is required to have an exponentially stable transfer function in the callier desoer algebra while in this paper we only require the plant to be well posed and exponentially stable these conditions are sufficiently unrestrictive to be verifiable for many partial differential equations in more than one space variable our second approach concerns the case when the second component of the plant transfer function from control input to tracking error is positive in this case we identify a very simple stabilizing controller which is again an internal model but which does not require low gain we apply our results to two problems involving systems modeled by partial differential equations the problem of rejecting external noise in a model for structure acoustics interactions and a similar problem for two coupled beams

disturbance rejection in 2 x 2 linear hyperbolic systems many interesting problems in the oil and gas industry face the challenge of responding to disturbances from afar typically the disturbance occurs at the inlet of a pipeline or the bottom of an oil well while sensing and actuation equipment is installed at the outlet only kilometers away from the disturbance the present paper develops an output feedback control law for such cases based on modelling the transport phenomenon as a 2 times 2 linear partial differential equation pde of hyperbolic type and the disturbance as a finite dimensional linear system affecting the left boundary of the pde sensing and actuation are co located at the right boundary of the pde the design provides a separation principle allowing a disturbance attenuating full state feedback control law to be combined with an observer the results are applied to a relevant problem from the oil and gas industry and demonstrated in simulations

output and error feedback regulator designs for linear infinite dimensional systems abstract this manuscript addresses the output regulation problem for linear distributed parameter systems dpss with bounded input and unbounded output operators we introduce novel methods for the design of the output feedback and error feedback regulators in the output feedback regulator design the measurements available for the regulator do not belong to the set of controlled outputs the proposed output feedback regulator with the injection of the measurement y m t and reference y r t can realize both the plant and the exosystem states estimation disturbance rejection and reference signal tracking simultaneously moreover new design approach provides an alternative choice for seeking the output injection gain in a traditional error feedback regulator design the regulator parameters are easily configured to solve the output regulation problems and to ensure the stability of the closed loop systems the results are demonstrated via computer simulation in two types of representative systems the parabolic partial differential equation pde system and the first order hyperbolic pde system

backstepping observers for a class of parabolic pdes in this paper we design exponentially convergent observers for a class of parabolic partial integro differential equations p i des with only boundary sensing available the problem is posed as a problem of designing an invertible coordinate transformation of the observer error system into an exponentially stable target system observer gain output injection function is shown to satisfy a well posed hyperbolic pde that is closely related to the hyperbolic pde governing backstepping control gain for the state feedback problem for several physically relevant problems the observer gains are obtained in closed form the observer gains are then used for an output feedback design in both collocated and anti collocated setting of sensor and actuator the order of the resulting compensator can be substantially lowered without affecting stability explicit solutions of a closed loop system are found in particular cases

blow up for solutions of some linear wave equations with mixed nonlinear boundary conditions this paper deals with the blow up in solutions of some linear wave equations with mixed nonlinear boundary conditions the finite time blow up in solutions are caused from the nonlinear boundary conditions even if the initial data are smooth and well defined for all times the pointwise blow up in exact solution and blow up in energy solution are investigated

backstepping design of robust output feedback regulators for boundary controlled parabolic pdes in this article a backstepping based solution of the robust output regulation problem for boundary controlled parabolic pdes with spatially varying parameters is presented this is achieved by stabilizing the plant extended with a finite dimensional internal model of the exogenous signals for the resulting pde ode system with spatially varying parameters a systematic backstepping based compensator design method is developed the robustness of the achieved output regulation is verified for non destabilizing model uncertainties an uncertain parabolic system illustrates the results of the article