adaptive stabilization of a kirchhoff s non linear beam with output disturbances abstract this paper is concerned with adaptive stabilization of an undamped non linear beam with disturbed outputs by boundary feedback control the adaptive controller is constructed by the concept of high gain adaptive feedback and the estimation mechanism for the unknown parameters of the measurement noise the global existence and uniqueness of the solution of the closed loop system is justified the stability of the closed loop system is proved such that the convergence of the system state to zero and the convergence of the estimated parameter to the unknown parameter are guaranteed for the smooth and small initial data

viscous scalar conservation law with nonlinear flux feedback and global attractors abstract in this paper we study the forced viscous scalar conservation law on 0 1 with the nonlinear flux feedback at the boundary global existence and uniqueness are established for l bounded initial conditions and forcing functions under an appropriate growth condition on the flux function and nonlinear dissipation at the boundary we show the existence of an absorbing set that absorbs the whole space l 0 1 and the existence of a compact global attractor in the l topology

abstract functional differential equations and reaction diffusion systems functional differential equations and reaction diffusion systems r h martin jr and h l smith abstract several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays suppose q is a bounded domain in rn with aq smooth and a is the laplacian operator on q also let m be a positive integer z a positive number and f fi m a continuous bounded function from 0 xc x q x c z 0 m into rm where c t 0 is the space of continuous functions from z 0 into r the purpose of this paper is to apply abstract results for semilinear functional differential equations in banach spaces to reactiondiffusion systems with time delays having the form a tu x t diau x t i t x ut x t a xeq i 1 m rdd ai x ui x t au i x t 81 x t t a x eaq u x a 0 x x 0 5 t 0 di 0 and ca q 0 o is c and i q x 0 oo r is c2 here an is the outward normal derivative on aq and if di 0 it is assumed that no boundary conditions are specified for this i also tu x t denotes the partial with respect to t whereas ut x denotes the member of c z 0 defined by 0 u x t 0 u x t 0 m our techniques provide basic existence criteria but the main point is that they can also be effectively applied to obtain estimates for solutions especially received by the editors october 7 1987 and in revised form october 7 1988 1980 mathematics subject classification 1985 revision primary 35r10 34k30

global stability of an hiv 1 infection model with saturation infection and intracellular delay in this paper an hiv 1 infection model with a saturation infection rate and an intracellular delay accounting for the time between viral entry into a target cell and the production of new virus particles is investigated by analyzing the characteristic equations the local stability of an infection free equilibrium and a chronic infection equilibrium of the model is established by using suitable lyapunov functionals and the lasalle invariant principle it is proved that if the basic reproduction ratio is less than unity the infection free equilibrium is globally asymptotically stable if the basic reproduction ratio is greater than unity the chronic infection equilibrium is globally asymptotically stable

a numerical solution for non classical parabolic problem based on chebyshev spectral collocation method a numerical method based on chebyshev polynomials and local interpolating functions is proposed for solving the one dimensional parabolic partial differential equation subject to non classical conditions to assess its validity and accuracy the method is applied to solve several test problem

an all frequency weakly singular surface integral equation for electromagnetism in dielectric media reformulation and well posedness analysis abstract we develop and analyze a surface integral equation sie whose solution pertains to numerical simulations of propagating time harmonic electromagnetic waves in three dimensional dielectric media the formulae to evaluate the far field pattern and propagation of the electric and magnetic fields in the interior and exterior of a dielectric body through surface integrals require the solution of a 2 2 system of weakly singular sies for the two unknown electric and magnetic fields at the interface surface of the dielectric body the sie is governed by an operator that is of the classical identity plus compact form the tangential surface currents and normal surface charges of the dielectric model can be easily computed from the surface electric and magnetic fields a novel feature of that second kind sie is that when augmented with two stabilization scalar equations the corresponding reformulation does not suffer from spurious resonances or low frequency breakdown we analyze this novel feature of the sie in both the sobolev and continuous function space settings without any tangential constraints by proving the uniqueness of the stabilized solution of the sie for all positive incident wave frequencies and physically appropriate real complex dielectric media constants using numerical examples we also demonstrate that without our stabilization constraints the 2 2 second kind sie suffers from some spurious resonances and incorporating the two additional scalar equations removes these spurious resonances at the zero frequency limit the two stabilization scalar equations are automatically satisfied and the 2 2 formulation nicely decouples to a well posed system at the zero frequency limit our analysis to establish the equivalence of the classical dielectric model governed by the maxwell equations and the all frequency sie based representation provides a natural decomposition of the surface unknowns into their tangential currents and normal charges the salient features of the representation and well posedness analysis provide a tool to develop a new class of efficient and high order algorithms with numerical analysis for simulation of the three dimensional dielectric media model from low to high frequencies

pointwise control of the boussinesq system abstract we study a pointwise control problem for the boussinesq system in two dimensions the control is a source term in the heat equation and the cost functional takes into account the distance between the solution to the boussinesq system and a given profile we have recently studied similar problems for the linearized boussinesq system nguyen and raymond j optim theory appl 141 2009 147 165 studying these problems in the nonlinear case is more difficult there is some literature on the pointwise control of linear and nonlinear partial differential equations however to the authors knowledge nothing has been done for the boussinesq system in this study we prove the existence of optimal solutions and derive optimality conditions

on the existence of solutions for a class of fourth order differential equations abstract we study the existence of the homoclinic solution of a class of nonlinear fourth order ordinary differential equations which involve particular cases of the extended fisher kolmogorov and swift hohenberg equations we address the case when the source term possesses a nonlinear dependence on the first order derivative of the solution the main result follows from an iterative method using the sequence of solutions obtained through the mountain pass technique in order to illustrate this method some numerical examples are shown

a priori error estimates for three dimensional parabolic optimal control problems with pointwise control in this paper we provide an a priori error analysis for parabolic optimal control problems with a pointwise dirac type control in space on three dimensional domains the two dimensional case was treated in d leykekhman and b vexler siam j numer anal 51 2013 pp 2797 2821 however the three dimensional case is technically much more involved to approximate the problem we use standard continuous piecewise linear elements in space and the piecewise constant discontinuous galerkin method in time despite low regularity of the state equation we establish mathcal o sqrt k h order of convergence rate for the control in the l 2 norm this result improves almost twice the previously known estimate in w gong m hinze and z zhou siam j control optim 52 2014 pp 97 119 and in addition does not require any relationship between the time step k and the mesh size h the main technical tools are discrete maximal parabolic regularity results and the best approximation type estim

flow control of fluids through porous media this paper discusses flow control of fluids through porous media specifically related to its application in petroleum science the flow of fluids is described by the boussinesq equation with mixed boundary conditions a neumann boundary condition describes no flow at the outer boundary of the porous media and a nonlinear boundary condition describes the interaction between the porous media and the source potential in many ways the source potential can be controlled this leads to a boundary control problem of the boussinesq equation the problem can be analyzed by the lyapunov method numerical examples are performed to provide a clear understanding of the concept of boundary control for these fluid flow systems

wave simulation in frozen porous media abstract we propose a numerical algorithm for simulation of wave propagation in frozen porous media where the pore space is filled with ice and water the model based on a biot type three phase theory predicts three compressional waves and two shear waves and models the attenuation level observed in rocks attenuation is modeled with exponential relaxation functions which allow a differential formulation based on memory variables the wavefield is obtained using a grid method based on the fourier differential operator and a runge kutta time integration algorithm since the presence of slow quasistatic modes makes the differential equations stiff a time splitting integration algorithm is used to solve the stiff part analytically the modeling is second order accurate in the time discretization and has spectral accuracy in the calculation of the spatial derivatives

optimal control of damped klein gordon equations with state constraints abstract in this paper we study the maximum principles for optimal control problems governed by the damped klein gordon equations with state constraints and we prove the existence of the optimal parameter and deduce the necessary conditions on the optimal parameter

a numerical method for delayed partial differential equations describing infectious diseases abstract in this paper we propose a numerical method for delayed partial differential equations that describe the dynamics of viral infections such as the human immunodeficiency virus hiv and the hepatitis b virus hbv we first prove that the proposed numerical method preserves the positivity and boundedness of solutions in order to ensure the well posedness of the problem by constructing appropriate discrete lyapunov functionals we show that the proposed method also preserves the global stability of equilibria of the corresponding continuous system with no restriction on the space and time step sizes moreover the discrete model and main results presented in qin et al 2014 are extended and generalized

boundary control for a class of pseudo parabolic differential equations abstract the boundary stabilization problem for a class of linear and nonlinear pseudo parabolic differential equations is considered the proposed control laws are used to achieve global exponential stability for the linear system and semi global exponential stability for the nonlinear system in the h 1 sense an h 2 bound of the solution for the nonlinear system is also derived a numerical example is included to illustrate the application of the proposed control laws

boundary control of viscosity degasperis procesi equation abstract this paper investigates the boundary control problem of the viscosity degasperis procesi equation in 0 1 results of the existence of a solution to the viscosity degasperis procesi equation in a short time interval under the boundary condition are given in the paper it proves the global exponential stability under the given boundary condition in l 2 h 1 h 2 h 3 space

on single integral equations for the transmission problem of acoustics the transmission problem namely scattering of time harmonic waves in a compressible fluid by a fluid inclusion with different material properties is usually formulated as a pair of coupled boundary integral equations over the interface s between the inclusion and the exterior fluid in this paper however we consider methods for solving the transmission problem using a single integral equation over s for a single unknown function in fact we derive four different integral equations using a hybrid of the direct green s theorem and indirect layer ansatz methods and give conditions for the unique solvability of each and for the subsequent construction of the solution to the transmission problem some of our single integral equations are fredholm integral equations of the second kind with weakly singular kernels thus these equations all of which appear to be new are attractive computationally

a generalized hbv model with diffusion and two delays an hepatitis b virus hbv model with spatial diffusion general incidence rate and time delays subject to homogeneous neumann boundary conditions is formulated the stability of the disease free equilibrium and the chronic infection equilibrium is analyzed by using the linearization method and by constructing appropriate lyapunov functionals furthermore several diffusive hbv models existing in the literature are generalized and extended

global stability for an sei model of infectious disease with immigration we study an sei model of disease transmission with immigration into all three classes for incidence we allow for a nonlinear response to the number of infectives including mass action and saturating incidence as special cases there is no disease free equilibrium and therefore no basic reproduction number for all parameter values the only equilibrium is an endemic equilibrium using a lyapunov function we show that this equilibrium is globally asymptotically stable

optimal control of mobile sources for heat conductivity processes abstractwe consider a problem of optimal control of the mobile sources for heat conductivity processes described by parabolic equation and systems of ordinary differential equations sufficient conditions for frechet differentiability of the performance criterion and an expression for its gradient were determined the necessary optimality condition was established in the form of the integral principles of maximum the theoretical conclusions are illustrated by the solution of a numerical example

survey paper well posed systems the lti case and beyond this survey is an introduction to well posed linear time invariant lti systems for non specialists we recall the more general concept of a system node classical and generalized solutions of system equations criteria for well posedness the subclass of regular linear systems some of the available linear feedback theory motivated by physical examples we recall the concepts of impedance passive and scattering passive systems conservative systems and systems with a special structure that belong to these classes we illustrate this theory by examples of systems governed by heat and wave equations we develop local and global well posedness results for lti systems with nonlinear in particular bilinear feedback by extracting the abstract idea behind various proofs in the literature we apply these abstract results to derive well posedness results for the burgers and navier stokes equations

on global stabilization of burgers equation by boundary control abstract although often referred to as a one dimensional cartoon of navier stokes equation because it does not exhibit turbulence the burgers equation is a natural first step towards developing methods for control of flows recent references include burns and kang nonlinear dynamics 2 1991 235 262 choi et al j fluid mech 253 1993 509 543 ito and kang siam j control optim 32 1994 831 854 ito and yan j math anal appl 227 1998 271 299 byrnes et al j dynam control systems 4 1998 457 519 and van ly et al numer funct anal optim 18 1997 143 188 while these papers have achieved tremendous progress in local stabilization and global analysis of attractors the problem of global asymptotic stabilization has remained open this problem is non trivial because for large initial conditions the quadratic convective term which is negligible in a linear local analysis dominates the dynamics we derive nonlinear boundary control laws that achieve global asymptotic stability we consider both the viscous and the inviscid burgers equation using both neumann and dirichlet boundary control we also study the case where the viscosity parameter is uncertain as well as the case of stochastic burgers equation for some of the control laws that would require the measurement in the interior of the domain we develop the observer based versions

strongly elliptic boundary integral equations for electromagnetic transmission problems we study a boundary integral equation method for transmission problems for strongly elliptic differential operators which yields a strongly elliptic system of pseudodifferential operators and which therefore can be used for numerical computations with galerkin s procedure the method is shown to work for the vector helmholtz equation in 3 with electromagnetic transmission conditions we propose a slightly modified system of boundary values in order for the corresponding bilinear form to be coercive over h 1 we analyse the boundary integral equations using the calculus of pseudodifferential operators here the concept of the principal symbol is used to derive existence and regularity results for the solution

nearly cloaking the electromagnetic fields the approximate cloaking is investigated for time harmonic maxwell s equations via the approach of transformation optics the problem is reduced to certain boundary effect estimates due to an inhomogeneous electromagnetic inclusion with an asymptotically small support but an arbitrary content enclosed by a thin high conducting layer sharp estimates are established in terms of the asymptotic parameter which are independent of the material tensors of the small electromagnetic inclusion the result implies that the blow up a small region construction via the transformation optics approach yields a near cloak for the electromagnetic waves a novelty lies in the fact that the geometry of the cloaking construction of this work can be very general moreover by incorporating the conducting layer developed in the present paper right between the cloaked region and the cloaking region arbitrary electromagnetic contents can be nearly cloaked our mathematical technique extends the general one developed in h y

global stabilization of a class of nonlinear reaction diffusion partial differential equations by boundary feedback this paper provides global exponential stabilization results by means of boundary feedback control for one dimensional nonlinear unstable reaction diffusion partial differential equations pdes wi

optimal controls of 3 dimensional navier stokes equations with state constraints this work is concerned with the maximum principles for optimal control problems governed by 3 dimensional navier stokes equations some types of state constraints time variables are considered

on non standard finite difference models of reaction diffusion equations reaction diffusion equations arise in many fields of science and engineering often their solutions enjoy a number of physical properties we design in a systematic way new non standard finite difference schemes which replicate three of these properties the first property is the stability instability of the fixed points of the associated space independent equation this property is preserved by non standard one and two stage theta methods presented in the general setting of stiff or non stiff systems of differential equations schemes which preserve the principle of conservation of energy for the corresponding stationary equation second property are constructed by nonlocal approximation of nonlinear reactions assembling of theta methods in the time variable with energy preserving schemes in the space variable yields non standard schemes which under suitable functional relation between step sizes display the boundedness and positivity of the solution third property a spectral method in the space variable coupled with a suitable non standard scheme in the time variable is also presented numerical experiments are provided

a variational approach to video segmentation for botanical data in order to engage in photosynthesis plant leaves absorb co 2 via the opening of pores in their surfaces called stomata water evaporates through open stomata however which is a detriment to plant function thus a plant will seek a stomatal aperture that balances its need for co 2 with its aversion to h 2 o loss in order to visualize a particular leaf s stomatal aperture an experimentalist injects the leaf with dye so that it fluoresces regions with a higher relative intensity that do not correspond to veins in the leaf then correspond to areas in which the stomata are closed and the darker nonvein regions to areas in which the stomata are open a camera is used to collect the emitted light and a fluorescence pattern is measured images are continually recorded as these patterns change with time resulting in a video sequence the goal of this paper is to segment one such video sequence into fluorescing and nonfluorescing regions after preprocessing the video we take a variational approach to the segmentation problem and solve the associated three dimensional evolution equation using a semi implicit numerical scheme a fixed point formulation of the method is presented as are results of the segmentation for actual leaf data

adaptive optimal control of highly dissipative nonlinear spatially distributed processes with neuro dynamic programming highly dissipative nonlinear partial differential equations pdes are widely employed to describe the system dynamics of industrial spatially distributed processes sdps in this paper we consider the optimal control problem of the general highly dissipative sdps and propose an adaptive optimal control approach based on neuro dynamic programming ndp initially karhunen loeve decomposition is employed to compute empirical eigenfunctions eefs of the sdp based on the method of snapshots these eefs together with singular perturbation technique are then used to obtain a finite dimensional slow subsystem of ordinary differential equations that accurately describes the dominant dynamics of the pde system subsequently the optimal control problem is reformulated on the basis of the slow subsystem which is further converted to solve a hamilton jacobi bellman hjb equation hjb equation is a nonlinear pde that has proven to be impossible to solve analytically thus an adaptive optimal control method is developed via ndp that solves the hjb equation online using neural network nn for approximating the value function and an online nn weight tuning law is proposed without requiring an initial stabilizing control policy moreover by involving the nn estimation error we prove that the original closed loop pde system with the adaptive optimal control policy is semiglobally uniformly ultimately bounded finally the developed method is tested on a nonlinear diffusion convection reaction process and applied to a temperature cooling fin of high speed aerospace vehicle and the achieved results show its effectiveness

boundary control of burgers equation a numerical approach we describe a methodology for solving boundary control problems for the viscous burgers equation the aim is to identify boundary forcing in order to ensure the best fit between data and model results by minimizing a functional which measures model and data discrepancies a continuous variational formulation involving the adjoint technique is used and its counterpart discretized version is obtained with a matrix approach as a guideline a particular discretization of the nonlinear term of the equation is performed in order to insure for the gradient of the functional to be minimized a discretized expression which can be directly deduced from its continuous counterpart numerical experiments validate the proposed optimization algorithm

numerical solution of a non classical parabolic problem an integro differential approach a numerical method based on an integro differential formulation and approximation by local interpolating functions is proposed for solving a one dimensional parabolic partial differential equation subject to non classical conditions some specific test problems are solved using the proposed method numerical results obtained indicate that it can give accurate solutions and that it is an interesting and viable alternative to existing numerical methods for solving the class of problems under consideration

fully implicit finite differences methods for two dimensional diffusion with a non local boundary condition three new fully implicit methods which are based on the 5 5 crank nicolson method the 5 5 n h noye hayman implicit method and the 9 9 n h implicit method are developed for solving the heat equation in two dimensional space with non local boundary conditions the latter is fourth order while the others are second order while the implicit methods developed here like the scheme based on the standard implicit backward time centered space btcs method use a large amount of central processor cpu time the high accuracy of the new fourth order fully implicit scheme is significant like the btcs method the new methods are also unconditionally stable

pointwise control of burgers equation a numerical approach abstract we describe a methodology for numerically solving pointwise control problems for the viscous burger s equation in one space dimension the solution methods are based on a combination of finite difference and finite element techniques for the time and space discretizations with iterative algorithms such as quasi newton methods and gmres numerical experiments validate the methods

on a singular two dimensional nonlinear evolution equation with nonlocal conditions abstract this paper deals with a nonclassical initial boundary value problem for a two dimensional parabolic equation with bessel operator we prove the existence and uniqueness of the weak solution of the given nonlinear problem we start by solving the associated linear problem after writing this latter in its operator form we establish an a priori bound from which we deduce the uniqueness of the strong solution for the solvability of the associated linear problem we prove that the range of the operator generated by the considered problem is dense on the basis of the obtained results of the linear problem we apply an iterative process to establish the existence and uniqueness of the nonlinear problem

a meshless approach for solution of burgers equation a new meshless method called gradient reproducing kernel particle method grkpm is proposed for numerical solutions of one dimensional burgers equation with various values of viscosity and different initial and boundary conditions discretization is first done in the space via grkpm and subsequently the reduced system of nonlinear ordinary differential equations is discretized in time by the gear s method comparison with the exact solutions which are only available for restricted initial conditions and values of viscosity approves the efficacy of the proposed method for challenging cases involving small viscosities comparison with the results obtained using other numerical schemes in the literature further attests the desirable features of the presented methodology

boundary and distributed control of the viscous burgers equation in this paper the dynamics of the forced burgers equation ut vuxx uux f x subject to both neumann boundary conditions and periodic boundary conditions using boundary and distributed control is analyzed for the boundary control problem we show that the controlled unforced burgers equation i e the closed loop system is exponentially stable when the viscosity v is known and globally asymptotically stable when v is unknown as for the distributed control problem we apply karhunen loeve decomposition on the dynamics of the forced burgers equation to generate a low dimensional dynamical system whose dynamics is similar to that of burgers equation then a feedback linearization control is used on the reduced system to exponentially stabilize the dynamics of the equation numerical simulations for the boundary and distributed controls are presented to support the analytical results

a priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control we consider finite element approximations of parabolic control problems with pointwise control the state equation exhibits low regularity due to the control imposed pointwisely this introduces some difficulties for both theoretical and numerical analysis to discretize the optimal control problem we use variational discretization together with piecewise linear and continuous finite elements for the space discretization of the state and we use the backward euler scheme for time discretization we prove a priori error estimates for the control state and adjoint state numerical experiments are provided which support the theoretical results

a numerical approach to an optimal boundary control of the viscous burgers equation the dynamics of the forced burgers equation subject to dirichlet boundary conditions using boundary control is analyzed with the objective of minimizing the distance between the final state function and target profile along with the energy of the control an efficient method is suggested to solve the optimal boundary control of the burgers equation the solution method involves the transformation of the original problem into one with homogeneous boundary conditions this modifies the problem from one in which there are boundary controls to one in which there are distributed controls the modal space technique is applied on the distributed controls of the forced burgers equation to generate a low dimensional dynamical systems the time variant controls are approximated by a finite term of the fourier series whose coefficients and frequencies giving optimal solutions are to be determined thereby converting the optimal control problem into mathematical programming problem the approximate solution space based on the control parameterization is obtained by using the runge kutta method numerical simulations for the boundary controls are presented for various target functions to assess the efficiency of the proposed method

a dissipative pseudospectral method for the two dimensional navier stokes equations a pseudospectral method for solving the two dimensional navier stokes equations with periodic boundary conditions is developed the numerical method preserves the zero dissipation property of the nonlinear convective term using weak variational forms the convergence properties of the authors scheme are analyzed

fourth order finite difference method for solving burgers equation abstract in this paper we present fourth order finite difference method for solving nonlinear one dimensional burgers equation this method is unconditionally stable the convergence analysis of the present method is studied and an upper bound for the error is derived numerical comparisons are made with most of the existing numerical methods for solving this equation

reaction diffusion systems with time delays monotonicity invariance comparison and convergence in a recent paper we have obtained differential inequality results developed comparison theorems and established invariance criteria for mild solutions of abstract functional differential equations in a banach space the primary application for these results is to reaction diffusion systems in which the reaction nonlinearity contains time delays in the present paper we apply these results to reaction diffusion systems with time delays of the form u t d u f x u t x x u u n 0 x where is a bounded open set in r n with smooth boundary u r n d diag d 1 d n with d i 0

stability enhancement by boundary control in 2 d channel flow in this paper we stabilize the parabolic equilibrium profile in a two dimensional 2 d channel flow using actuators and sensors only at the wall the control of channel flow was previously considered by speyer and coworkers and bewley and coworkers who derived feedback laws based on linear optimal control and implemented by wall normal actuation with an objective to achieve global lyapunov stabilization we arrive at a feedback law using tangential actuation using teamed pairs of synthetic jets or rotating disks and only local measurements of wall shear stress allowing to embed the feedback in microelectromechanical systems mems hardware without need for wiring this feedback is shown to guarantee global stability in at least h sup 2 norm which by sobolev s embedding theorem implies continuity in space and time of both the flow field and the control as well as their convergence to the desired steady state the theoretical results are limited to low values of reynolds number however we present simulations that demonstrate the effectiveness of the proposed feedback for values five order of magnitude higher

gegenbauer approximation and its applications to differential equations on the whole line abstract a gegenbauer approximation is discussed several imbedding inequalities and inverse inequalities are obtained some approximation results are given by variable transformation differential equations on the whole line are changed to certain equations on a finite interval gegenbauer polynomials are used for their numerical solutions the stabilities and convergences of proposed schemes are proved the main idea and techniques used in this paper are also applicable to other multiple dimensional problems in unbounded domains

the utilization of total mass to determine the switching points in the symmetric boundary control problem with a linear reaction term abstract the authors study the problem u t u x x u 0 x 1 t 0 u x 0 0 and u 0 t u 1 t t where t u 0 for t 2 k t t 2 k 1 and t 0 for t 2 k 1 t t 2 k 2 k 0 1 2 with t 0 0 and the sequence t k is determined by the equations 0 1 u x t k d x m for k 1 3 5 and 0 1 u x t k d x m for k 2 4 6 and where 0 m m note that the switching points t k k 1 2 3 are unknown existence and uniqueness are demonstrated theoretical estimates of the t k and t k 1 t k are obtained and numerical verifications of the estimates are presented the case of u x 0 t u x 1 t t is also considered and analyzed

static output feedback control design for linear mimo systems with actuator dynamics governed by diffusion pdes this paper deals with the problem of static output feedback sof control design for a class of diffusion partial differential equation pde and ordinary differential equation ode cascades where the ode model is used to describe the dynamics of the multi input and multi output mimo plant and the diffusion pde model is employed to represent the dynamics of actuators the objective of this paper is to develop a simple as well as effective sof controller via the lyapunov s direct method such that the resulting closed loop system is globally exponentially stable by constructing a quadratic lyapunov function the sufficient condition on the globally exponential stability of the closed loop cascaded system is presented in terms of linear matrix inequality lmi then an lmi based design method of the sof controller is developed on the basis of the obtained stability analysis result finally two numerical examples are provided to illustrate the effectiveness of the proposed design method

explicit integral operator feedback for local stabilization of nonlinear thermal convection loop pdes a state feedback boundary control law that stabilizes fluid flow in a 2d thermal convection loop is presented the fluid is enclosed between two cylinders heated from above and cooled from below which makes its motion unstable for a large enough rayleigh number the actuation is at the boundary through rotation direct velocity actuation and heat flux heating or cooling of the outer boundary the design is a new approach for this kind of a coupled pde problem based on a combination of singular perturbation theory and the backstepping method for infinite dimensional linear systems stability is proved by lyapunov method though only a linearized version of the plant is considered in the design an extensive closed loop simulation study of the nonlinear pde model shows that the result holds for reasonably large initial conditions a highly accurate approximation to the control law is found in closed form

backstepping boundary control of burgers equation with actuator dynamics in this paper we propose a backstepping boundary control law for burgers equation with actuator dynamics while the control law without actuator dynamics depends only on the signals u 0 t and u 1 t the backstepping control also depends on ux 0 t ux 1 t uxx 0 t and uxx 1 t making the regularity of the control inputs the key technical issue of the paper with elaborate lyapunov analysis we prove that all these signals are suciently regular and the closed loop system including the boundary dynamics is globally h 3 stable and well posed c 2000 elsevier science b v all rights reserved

control input separation by actuation mode expansion for flow control problems a control input separation method is proposed for reduced order modelling in boundary control problems the dynamics of flow systems are typically described by partial differential equations where the input affects the system through boundary conditions from a control design perspective it is most desirable and natural to employ finite dimensional representations in which the input enters the dynamics directly the method proposed here to resolve the input from the boundary conditions is based on obtaining a proper orthogonal decomposition of the unforced flow of the system and then augmenting this decomposition by optimally computed actuation modes built using snapshots of the actuated flow a reduced order galerkin model is then derived for this expansion in which the input appears as an explicit term in the system dynamics the model reduces exactly to the original baseline case under zero input conditions the proposed method is then compared to an existing input separation technique namely the s

a de giorgi iteration based approach for the establishment of iss properties for burgers equation with boundary and in domain disturbances this note addresses input to state stability iss properties with respect to w r t boundary and in domain disturbances for burgers equation the developed approach is a combination of the method of de giorgi iteration and the technique of lyapunov functionals by adequately splitting the original problem into two subsystems the iss properties in l 2 norm for burgers equation have been established using this method moreover as an application of de giorgi iteration iss in l infty norm w r t in domain disturbances and actuation errors in boundary feedback control for a one dimensional linear unstable reaction diffusion equation have also been established it is the first time that the method of de giorgi iteration is introduced in the iss theory for infinite dimensional systems and the developed method can be generalized for tackling some problems on multidimensional spatial domains and to a wider class of nonlinear partial differential equations

dynamics and control of the 2 d navier stokes equations abstract this paper deals with the dynamics and control of the two dimensional 2 d navier stokes n s equations with a spatially periodic and temporally steady forcing term first we construct a dynamical system of nine nonlinear differential equations by fourier expansion and truncation of the 2 d n s equations then we study the dynamics of the obtained reduced order system by analyzing the system s attractors for different values of the reynolds number r e by applying the symmetry of the equations on one of the system s attractors a symmetric limit trajectory that is part of the dynamics is obtained moreover a lyapunov based control strategy to control the dynamics of the system for a given r e is designed finally numerical simulations are undertaken to validate the theoretical developments

low dimensional modelling and dirichlet boundary controller design for burgers equation modelling and boundary control for the burgers equation is studied in this paper modelling has been done via processing of numerical observations through proper orthogonal decomposition pod with galerkin projection this results in a set of spatial basis functions together with a set of ordinary differential equations odes describing the temporal evolution since the dynamics described by the burgers equation are non linear the corresponding reduced order dynamics turn out to be non linear the presented analysis explains how the free boundary condition appears as a control input in the odes and how controller design can be accomplished the issues of control system synthesis are discussed from the point of practicality performance and robustness the numerical results obtained are in good compliance with the theoretical claims a comparison of various different approaches is presented

numerical solution of one dimesional burgers equation explicit and exact explicit finite difference methods this paper presents finite difference solution and analytical solution of the finite difference approximations based on the standard explicit method to the one dimensional burgers equation which arises frequently in the mathematical modelling used to solve problems in fluid dynamics results obtained by these ways for some modest values of viscosity have been compared with the exact fourier one it is shown that they are in good agreement with each other

nonstandard finite difference scheme for a diffusive within host virus dynamics model with both virus to cell and cell to cell transmissions abstract in this paper we first propose a diffusive within host virus dynamics model with both virus to cell and cell to cell transmissions then we consider the discretization of the model by using nonstandard finite difference scheme it is then followed by the investigation of the global stability of the equilibria of the discrete model our study shows that if the basic reproduction number r 0 1 then the infection free equilibrium is globally asymptotically stable however if r 0 1 then the infection equilibrium is globally asymptotically stable numerical simulations are presented to illustrate our theoretical results

on the kleinman martin integral equation method for electromagnetic scattering by a dielectric body the interface problem describing the scattering of time harmonic electromagnetic waves by a dielectric body is often formulated as a pair of coupled boundary integral equations for the electric and magnetic current densities on the interface gamma in this paper following an idea developed by kleinman and martin siam j appl math 48 1988 pp 307 325 for acoustic scattering problems we consider methods for solving the dielectric scattering problem using a single integral equation over gamma for a single unknown density one knows that such boundary integral formulations of the maxwell equations are not uniquely solvable when the exterior wave number is an eigenvalue of an associated interior maxwell boundary value problem we obtain four different families of integral equations for which we can show that by choosing some parameters in an appropriate way they become uniquely solvable for all real frequencies we analyze the well posedness of the integral equations in the space of finite ener

brief paper sliding mode boundary control of a parabolic pde system with parameter variations and boundary uncertainties this paper considers the stabilization problem of a one dimensional unstable heat conduction system rod modeled by a parabolic partial differential equation pde powered with a dirichlet type actuator from one of the boundaries by applying the volterra integral transformation a stabilizing boundary control law is obtained to achieve exponential stability in the ideal situation when there are no system uncertainties the associated lyapunov function is used for designing an infinite dimensional sliding manifold on which the system exhibits the same type of stability and robustness against certain types of parameter variations and boundary disturbances it is observed that the relative degree of the chosen sliding function with respect to the boundary control input is zero a continuous control law satisfying the reaching condition is obtained by passing a discontinuous signum signal through an integrator

a dynamic compensator for in domain control of burgers equation this paper addresses the problem of output regulation control of a burgers equation under pointwise in domain actuation a dynamic compensator that generates the in domain control is developed for

optimal control of fluid equations with buoyancy we consider the optimal control of a set of fluid equations made up of the momentum equations coupled with an equation modeling thermal effects control is exerted by means of heat sources within the fluid with the goal of controlling the temperature or velocity distribution of the fluid existence as well as the regularity properties of optimal controls are analyzed stability with respect to observations is discussed finally compactness of the solution set and uniqueness properties are determined

numerical analysis of pattern formation on the surface of transpiring leaves abstract the process of photosynthesis is facilitated by pores on the leaf surface called stomata when a particular stoma is open co 2 is absorbed through its aperture but h 2 o is also lost due to evaporation thus a plant will seek a stomatal aperture that balances its need for co 2 with its aversion to h 2 o loss in order to visualize a particular leaf s stomatal aperture distribution and how it changes with time fluorescence data is collected at regular intervals as digital images resulting in a video sequence it has been observed that stomatal apertures are often synchronized into spatially extended patches in order objectively to analyze this phenomenon we have developed a technique to isolate patches via a three dimensional pde based segmentation method the resulting segmented data is then collapsed to a vector valued time series of much smaller dimension with a hybrid pca archetypal analysis approach this allows for a unique interpretation of the data in terms of statistical measures of the motions of representative patches the technique is illustrated with a data set from a particularly complicated regime collected by the complexity and stomatal behavior research lab at utah state university

constrained control of 1 d parabolic pdes using sampled in space sensing and actuation abstract we consider distributed control of a class of 1 d parabolic pdes under distributed in domain point actuation and measurements in the presence of control constraints this class includes unstable diffusion reaction equations as well as stable burgers equations where we aim to locally improve the convergence we suggest an observer based control law that employs the averaged values of the observer state this allows to regionally stabilize the system we derive linear matrix inequalities lmis conditions that provide an estimate on the set of initial conditions starting from which the state trajectories of the system are exponentially converging to zero a numerical example validates the efficiency of the method

nonlinear stabilization of shock like unstable equilibria in the viscous burgers pde we stabilize the unstable ldquoshock likerdquo equilibrium profiles of the viscous burgers equation using control at the boundaries these equilibria are not stabilizable even locally using the standard ldquoradiation feedback boundary conditions rdquo using a nonlinear spatially scaled transformation that employs three ingredients of which one is the hopf cole nonlinear integral transformation and linear backstepping we design an explicit nonlinear full state control law that achieves exponential stability with a region of attraction for which we give an estimate the region of attraction is not the entire state space since the burgers pde is known not to be globally controllable

a dissipative feedback control synthesis for systems arising in fluid dynamics a dissipative feedback control synthesis is constructed to regulate the systems arising in fluid dynamics the feedback law is obtained by utilizing nonlinear dynamic programming techniques the control law is designed for driving the system to a prescribed equilibrium state and enhancing the energy dissipation effects of the dynamical system two dimensional navier stokes equations and burgers equation are used for numerical experiments to illustrate the effects of the feedback synthesis and the theoretical results

numerical solution of a parabolic equation subject to specification of energy in this paper special attention is devoted to one particular problem the diffusion equation subject to the specification of mass in a portion of the domain which has been studied quite extensively both analytically and numerically in recent years parabolic partial differential equations with a non local constraint in place of one of the standard boundary specifications feature in the mathematical modeling of many phenomena in this paper the application of the method of lines mol to such problems is considered the mol semi discretization approach will be used to transform the model partial differential equation into a system of first order linear ordinary differential equations odes the mol is a method of solving pdes by discretizing the equation with respect to all but one variable usually time the spatial partial derivative is approximated by a finite difference method the solution of the resulting system of first order odes satisfies a recurrence relation which involves a matrix exponential function numerical techniques are developed by approximating the exponential matrix function in this recurrence relation the matrix exponential function is approximated by a rational approximation consisting of four parameters new parallel algorithms are developed using the resulting approximation numerical experiments on two challenging examples are presented to illustrate the performance of the algorithms

infinite dimensional and reduced order observers for burgers equation obtaining a representative model in feedback control system design problems is a key step and is generally a challenge for spatially continuous systems it becomes more difficult as the dynamics is infinite dimensional and the well known techniques of systems and control engineering are difficult to apply directly in this paper observer design is reported for one dimensional burgers equation which is a non linear partial differential equation an infinite dimensional form of the observer is demonstrated to converge asymptotically to the target dynamics and proper orthogonal decomposition is used to obtain the reduced order observer when this is done the corresponding observer is shown to be successful under certain circumstances the paper unfolds the connections between target dynamics observer and their finite dimensional counterparts a set of simulation results has been presented to justify the theoretical claims of the paper

using lyapunov s direct method for wave suppression in reactive systems this paper proposes a new approach for stabilizing a homogeneous solution in reaction convection diffusion system with oscillatory kinetics in which moving or stationary patterns emerge in the absence of control specifically we aim to suppress patterns by using a spatially weighted finite dimensional feedback control that assures stability of the solution according to lyapunov s direct method a practical design procedure based on spectral representation of the system and the dissipative nature of parabolic pdes is presented 2006 elsevier b v all rights reserved

compensation of actuator dynamics governed by quasilinear hyperbolic pdes we present a methodology for stabilization of general nonlinear systems with actuator dynamics governed by a certain class of quasilinear first order hyperbolic pdes since for such pde ode cascades the speed of propagation depends on the pde state itself which implies that the prediction horizon cannot be a priori known analytically the key design challenge is the determination of the predictor state we resolve this challenge and introduce a pde predictor feedback control law that compensates the transport actuator dynamics due to the potential formation of shock waves in the solutions of quasilinear first order hyperbolic pdes which is related to the fundamental restriction for systems with time varying delays that the delay rate is bounded by unity we limit ourselves to a certain feasibility region around the origin and we show that the pde predictor feedback law achieves asymptotic stability of the closed loop system providing an estimate of its region of attraction our analysis combines lyapunov like arguments and iss estimates since it may be intriguing as to what is the exact relation of the cascade to a system with input delay we highlight the fact that the considered pde ode cascade gives rise to a system with input delay with a delay that depends on past input values defined implicitly via a nonlinear equation the developed control design methodology is applied to the control of vehicular traffic flow at distant bottlenecks

some implicit methods for the numerical solution of burgers equation in this article we develop multisymplectic box methods on burgers equation burgers equation has some discontinuous solutions because of effects of viscosity term these discontinuities raise phenomenon of shock waves some explicit and implicit numerical methods have been experimented on burgers equation but these schemes have not been seen proper in this case because of their conditional stability and existence of viscosity term to avoid these problems multisymplectic box scheme which has been constructed for kdv equation is purposed for burgers equation multisymplectic box scheme is a very effective box scheme in diminishing artificial wiggles which appear in approximation solutions we consider two types of box schemes and implement on burgers equation to get better results with no artificial wiggles

optimal a priori error estimates of parabolic optimal control problems with pointwise control in this paper we consider a parabolic optimal control problem with a pointwise dirac type control in space but variable in time in two space dimensions to approximate the problem we use the standard continuous piecewise linear approximation in space and the piecewise constant discontinuous galerkin method in time despite low regularity of the state equation we show almost optimal h 2 k convergence rate for the control in l 2 norm this result improves almost twice the previously known estimate in w gong m hinze and z zhou a priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control tech report 2011 07 hamburger beitrage zur angewandten mathematik hamburg germany 2011

numerical solution of a flow control problem vorticity reduction by dynamic boundary action in order to laminarize an unsteady internal flow the vorticity field is minimized in a least squares sense using an optimal control approach the flow model is the navier stokes equation for a viscous incompressible fluid and the flow is controlled by suction and blowing on a part of the boundary a quasi newton method is used for the minimization of a quadratic objective function involving a measure of the vorticity and a regularization term the navier stokes equations are approximated using a finite difference scheme in time and finite element approximations in space accurate expressions for the gradient of the discrete objective function are needed to obtain a satisfactory convergence rate of the minimization algorithm therefore first order necessary conditions for a minimizer of the objective function are derived in the fully discrete case a memory saving device is discussed without which problems of any realistic size especially in three space dimensions would remain computationally intractable the feasibility of the optimal control approach for flow control problems is demonstrated by numerical experiments for a two dimensional flow in a rectangular cavity at a reynolds number high enough for nonlinear effects to be important

a non standard finite difference scheme for a delayed and diffusive viral infection model with general nonlinear incidence rate abstract a non standard finite difference scheme is proposed to solve a delayed and diffusive viral infection model with general nonlinear incidence rate the results show that the discrete model preserves the positivity and boundedness of solutions in order to ensure the well posedness of the problem moreover this method preserves all equilibria of the original continuous model by constructing lyapunov functionals we show that the global stability of equilibria is completely determined by the basic reproduction number 0 which implies that the proposed discrete model can efficiently blue preserve the global stability of equilibria of the corresponding continuous model numerical experiments are carried out to support the theoretical results

boundary integral method for multi layered electromagnetic scattering problems abstract in this paper we consider the scattering of time harmonic electromagnetic plane waves by an impenetrable obstacle which is embedded in a piecewise homogeneous medium applying potential theory the problem can be reformulated as a boundary integral system we obtain the existence and uniqueness of the solution to the system by using the generalized garding inequality and the fredholm theory

global dynamics of a cell mediated immunity in viral infection models with distributed delays abstract in this paper we investigate global dynamics for a system of delay differential equations which describes a virus immune interaction in vivo the model has two distributed time delays describing time needed for infection of cell and virus replication our model admits three possible equilibria an uninfected equilibrium and infected equilibrium with or without immune response depending on the basic reproduction number for viral infection r 0 and for ctl response r 1 such that r 1 r 0 it is shown that there always exists one equilibrium which is globally asymptotically stable by employing the method of lyapunov functional more specifically the uninfected equilibrium is globally asymptotically stable if r 0 1 an infected equilibrium without immune response is globally asymptotically stable if r 1 1 r 0 and an infected equilibrium with immune response is globally asymptotically stable if r 1 1 the immune activation has a positive role in the reduction of the infection cells and the increasing of the uninfected cells if r 1 1

the boundary penalty method for the diffusion equation subject to the specification of mass in this paper we concern with a new boundary penalty finite element method for the one dimensional time dependent diffusion equation subject to the specification of mass instead of solving the diffusion equation directly we will use the finite element method to solve a boundary penalty problem which is derived from the diffusion equation and analyse the error estimate of the semi discrete and fully discrete approximations when the penalty parameter e is sufficiently small the numerical experiments show that our method is very effective and supports our analysis

regularity and lyapunov stabilization of weak entropy solutions to scalar conservation laws we consider the problem of lyapunov boundary stabilization of the weak entropy solution to a scalar conservation law with strictly convex flux in one dimension of space around a uniform equilibrium we show that for a specific class of boundary conditions the solution to the initial boundary value problem for an initial condition with bounded variations can be approximated arbitrarily closely in the l 1 norm by a piecewise smooth solution with finitely many discontinuities the constructive method we present designs explicit boundary conditions in this class which guarantee lyapunov stability of the weak entropy solution to the initial boundary value problem we show how the greedy control obtained by maximizing the decrease of the natural lyapunov function may fail to asymptotically stabilize and a brute force control generates unbounded variation of traces we then design a stabilizing control which avoid oscillations and propose a nonlocal technique depending on time and the whole initial datum which optimizes the convergence time controllers performance is illustrated on numerical benchmarks using the godunov scheme