internal stabilization of semilinear parabolic systems this work is concerned with the internal stabilization of the steady state solutions to semilinear parabolic systems via finite dimensional feedback controllers the internal controller is active on a nonempty open subset and in one equation only

higher order implicit strong numerical schemes for stochastic differential equations higher order implicit numerical methods which are suitable for stiff stochastic differential equations are proposed these are based on a stochastic taylor expansion and converge strongly to the corresponding solution of the stochastic differential equation as the time step size converges to zero the regions of absolute stability of these implicit and related explicit methods are also examined

an iterative method for linear discrete ill posed problems with box constraints many questions in science and engineering give rise to linear discrete ill posed problems often it is desirable that the computed approximate solution satisfies certain constraints e g that some or all elements of the computed solution be nonnegative this paper describes an iterative method of active set type for the solution of large scale problems of this kind the method employs conjugate gradient iteration with a stopping criterion based on the discrepancy principle and allows updates of the active set by more than one index at a time

a divide and conquer method for unitary and orthogonal eigenproblems leth n xn be a unitary upper hessenberg matrix whose eigenvalues and possibly also eigenvectors are to be determined we describe how this eigenproblem can be solved by a divide and conquer method in which the matrixh is split into two smaller unitary upper hessenberg matricesh 1 andh 2 by a rank one modification ofh the eigenproblems forh 1 andh 2 can be solved independently and the solutions of these smaller eigenproblems define a rational function whose zeros on the unit circle are the eigenvalues ofh the eigenvector ofh can be determined from the eigenvalues ofh and the eigenvectors ofh 1 andh 2 the outlined splitting of unitary upper hessenberg matrices into smaller such matrices is carried out recursively this gives rise to a divide and conquer method that is suitable for implementation on a parallel computer whenh n xn is orthogonal the divide and conquer scheme simplifies and is described separately our interest in the orthogonal eigenproblem stems from applications in signal processing numerical examples for the orthogonal eigenproblem conclude the paper

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

a fourth order algorithm for solving the multi dimensional kramers equation in langevin form we propose a new fourth order numerical method for solving the multi dimensional kramers equation in standard and overdamped regimes the algorithm is developed for modelling the trajectories of airborne particles propagating through porous media such as three dimensional fibre networks it disentangles the deterministic and stochastic parts of the solution so that standard fourth order numerical methods can be used to calculate the deterministic component the stochastic correction to the solution is obtained using a perturbative expansion of the stochastic force in the overdamped regime our method avoids the need for extremely small time steps required by conventional methods we compare the convergence rate of our algorithm to that of other methods by solving the one dimensional duffing equation the proposed algorithm is finally used to evaluate the performance of various filters made from fibres of two different sizes

necessary optimality conditions for infinite dimensional state constrained control problems abstract this paper is concerned with first order necessary optimality conditions for state constrained control problems in separable banach spaces assuming inward pointing conditions on the constraint we give a simple proof of pontryagin maximum principle relying on infinite dimensional neighboring feasible trajectories theorems proved in 20 further we provide sufficient conditions guaranteeing normality of the maximum principle we work in the abstract semigroup setting but nevertheless we apply our results to several concrete models involving controlled pdes pointwise state constraints as positivity of the solutions are allowed

a non stationary multi scale oscillating free boundary for the laplace and heat equations the paper is concerned with the homogenization problem for the poisson equation in a domain a part of whose boundary varies in time and is rapidly oscillating in space at each time such a problem arises in growing semiconductor film taking into account the fact that the surface of the film is changing in time by chemical vapor deposition or by etching we prove existence and uniqueness of the solution and establish its homogenizated approximation we obtain an l estimate on the error between the free boundary and the homogenized boundary

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

optimal control problems with state constraint governed by navier stokes equations abstract this work deals with the existence of optimal solution and the maximum principle for optimal control problem governed by navier stokes equations with state constraint of pointwise type in 3d strong results in 2d are also given

modulus based iterative methods for constrained tikhonov regularization tikhonov regularization is one of the most popular methods for the solution of linear discrete ill posed problems in many applications the desired solution is known to lie in the nonnegative cone it is then natural to require that the approximate solution determined by tikhonov regularization also lies in this cone the present paper describes two iterative methods that employ modulus based iterative methods to compute approximate solutions in the nonnegative cone of large scale tikhonov regularization problems the first method considered consists of two steps first the given linear discrete ill posed problem is reduced to a small problem by a krylov subspace method and then the reduced tikhonov regularization problems so obtained is solved the second method described explores the structure of certain image restoration problems computed examples illustrate the performances of these methods

stabilization by sparse controls for a class of semilinear parabolic equations stabilization problems for parabolic equations with polynomial nonlinearities are investigated in the context of an optimal control formulation with a sparsity enhancing cost functional this formulation allows that the optimal control completely shuts down once the trajectory is sufficiently close to a stable steady state such a property is not present for commonly chosen control mechanisms to establish these results it is necessary to develop a function space framework for a class of optimal control problems posed on infinite time horizons which is otherwise not available

fourth order algorithms for solving the multivariable langevin equation and the kramers equation we develop a fourth order simulation algorithm for solving the stochastic langevin equation the method consists of identifying solvable operators in the fokker planck equation factorizing the evolution operator for small time steps to fourth order and implementing the factorization process numerically a key contribution of this paper is to show how certain double commutators in the factorization process can be simulated in practice the method is general applicable to the multivariable case and systematic with known procedures for doing fourth order factorizations the fourth order convergence of the resulting algorithm allowed very large time steps to be used in simulating the brownian dynamics of 121 yukawa particles in two dimensions the converged result of a first order algorithm can be obtained by using time steps 50 times as large to further demonstrate the versatility of our method we derive two new classes of fourth order algorithms for solving the simpler kramers equation without requiring the derivative of the force the convergence of many fourth order algorithms for solving this equation are compared

quasi newton approach to nonnegative image restorations abstract image restoration or deblurring is the process of attempting to correct for degradation in a recorded image typically the blurring system is assumed to be linear and spatially invariant and fast fourier transform fft based schemes result in efficient computational image restoration methods however real images have properties that cannot always be handled by linear methods in particular an image consists of positive light intensities and thus a nonnegativity constraint should be enforced this constraint and other ways of incorporating a priori information have been suggested in various applications and can lead to substantial improvements in the reconstructions nevertheless such constraints are rarely implemented because they lead to nonlinear problems which require demanding computations we suggest efficient implementations for three nonnegatively constrained restorations schemes constrained least squares maximum likelihood and maximum entropy we show that with a certain parameterization and using a quasi newton scheme these methods are very similar in addition our formulation reveals a connection between our approach for maximum likelihood and the expectation maximization em method used extensively by astronomers numerical experiments illustrate that our approach is superior to em both in terms of accuracy and efficiency

maximum principle for a hybrid system via singular perturbations we prove a maximum principle of pontryagin type for the time optimal control of a hybrid system the system is nonlinear and consists of a controlled coupled ode pde the control input acts via the boundary and the interior of the domain systems of this type frequently arise in modeling population dynamics in a contaminated environment for the regularization we use ekeland s variational principle along with a singular perturbation technique for this purpose we introduce a new trajectory on an additional exterior domain whose size may be considered as a singular perturbation parameter it turns out that the maximum principle is stable with respect to the size of the additional exterior domain the technique allows us to obtain necessary optimality conditions without involving measure boundary data

optimal control of semilinear parabolic systems with state constraint abstract the aim of this work is to obtain the existence of optimal solution and maximum principle for optimal control problem with pointwise type state constraint governed by semilinear parabolic systems with certain polynomial like nonlinearity application to optimal control problems of the phase transition system is given

an interior point method for large constrained discrete ill posed problems ill posed problems are numerically underdetermined it is therefore often beneficial to impose known properties of the desired solution such as nonnegativity during the solution process this paper proposes the use of an interior point method in conjunction with truncated iteration for the solution of large scale linear discrete ill posed problems with box constraints an estimate of the error in the data is assumed to be available numerical examples demonstrate the competitiveness of this approach

pontryagin maximum principle for semilinear and quasilinearparabolic equations with pointwise state constraints this paper studies the first order necessary conditions for the optimal controls of semilinear and quasilinear parabolic partial differential equations with pointwise state constraints a pontryagin type maximum principle is obtained

conditional gradient tikhonov method for a convex optimization problem in image restoration in this paper we consider the problem of image restoration with tikhonov regularization as a convex constrained minimization problem using a kronecker decomposition of the blurring matrix and the tikhonov regularization matrix we reduce the size of the image restoration problem therefore we apply the conditional gradient method combined with the tikhonov regularization technique and derive a new method we demonstrate the convergence of this method and perform some numerical examples to illustrate the effectiveness of the proposed method as compared to other existing methods

tikhonov regularization with a solution constraint many numerical methods for the solution of linear ill posed problems apply tikhonov regularization this paper presents a modification of a numerical method proposed by golub and von matt for quadratically constrained least squares problems and applies it to tikhonov regularization of large scale linear discrete ill posed problems the method is based on partial lanczos bidiagonalization and gauss quadrature computed examples illustrating its performance are presented

minimax control for evolutionary variational bilateral problem this paper deals with a minimax control problem for semilinear evolutionary variational inequalities associated with bilateral constraints the control domain is not necessarily convex the cost functional which is to be minimized is the sup norm of some function of the state and the control the major novelty of such a problem lies in the simultaneous presence of the nonsmooth state equation variational inequality and the nonsmooth cost functional the sup norm in this paper the existence and the pontryagin type necessary condition for optimal controls are established

inertial effects on the escape rate of a particle driven by colored noise an instanton approach a recent calculation in the weak noise limit of the rate of escape of a particle over a one dimensional potential barrier is extended by including an inertial term in the langevin equation specifically we consider a system described by the langevin equation m ddot x dot x v prime x xi where is a gaussian colored noise with mean zero and correlator t t d exp t t a pathintegral formulation is augmented by a steepest descent calculation valid in the weak noise d 0 limit this yields an escape rate exp s d where the action s is the minimum over paths characterizing escape over the barrier of a generalized onsager machlup functional the extremal path being an instanton of the theory the extremal actions is calculated analytically for smallm and for general potentials and numerical results fors are displayed for various ranges ofm and for the typical case of the quartic potentialv x x2 2 x4 4

nonexistence and existence results of an optimal control problem governed by a class of multisolution semilinear elliptic equations abstract an optimal control problem governed by a class of multisolution semilinear elliptic equations is studied in this paper it is pointed out that the control domain and cost functional are nonconvex by analyzing the maximum condition for an optimal relaxed control one nonexistence and some existence results of an optimal pair are obtained the idea of using relaxed controls is mainly derived from the paper lou 2007 by lou where a linear and well posed controlled system was considered the present paper is mainly devoted to discussing the case not contained in lou 2007

the qr algorithm for unitary hessenberg matrices abstract let h be an n n unitary right hessenberg matrix with positive subdiagonal elements using what we call the schur parameterization of h we show how one step of the shifted qr algorithm for h can be carried out in o n arithmetic operations coupled with the shift strategy of eberlein and huang 3 this will permit computation of the spectrum of h to machine precision in o n 2 operations one potential application is the computation of gauss szego quadrature formulas 12 given the associated schur parameters 7 the weights can also be computed by direct analogy with 6

pontryagin s principle for state constrained boundary control problems of semilinear parabolic equations this paper deals with state constrained optimal control problems governed by semilinear parabolic equations we establish a minimum principle of pontryagin s type to deal with the state constraints we introduce a penalty problem by using ekeland s principle the key tool for the proof is the use of a special kind of spike perturbations distributed in the domain where the controls are defined conditions for normality of optimality conditions are given

state constrained optimal control problems governed by coupled nonlinear wave equations with memory this work considers the optimal control problems governed by coupled nonlinear wave equations with memory where the state constraint is a type of integral first we investigate the existence of the optimal solution for the optimal control problem and then we deduce the maximum principle for the optimal solution with state constraint moreover we give a concrete example which is covered by the main result

stationary solution for the color driven duffing oscillator adiabatic methods may be used to find approximate reduced stationary distributions for the linearly damped duffing oscillator driven by correlated noise of finite bandwidth here we show that a stationary distribution may be obtained which explains the insensitivity of the system to variations in damping coefficient and noise color as observed in numerical simulations exact analytic results are obtained for parabolic potentials for comparison

solving the trust region subproblem using the lanczos method the approximate minimization of a quadratic function within an ellipsoidal trust region is an important subproblem for many nonlinear programming methods when the number of variables is large the most widely used strategy is to trace the path of conjugate gradient iterates either to convergence or until it reaches the trust region boundary in this paper we investigate ways of continuing the process once the boundary has been encountered the key is to observe that the trust region problem within the currently generated krylov subspace has a very special structure which enables it to be solved very efficiently we compare the new strategy with existing methods the resulting software package is available as hsl vf05 within the harwell subroutine library

riccati based feedback control of the monodomain equations with the fitzhugh nagumo model feedback control for the monodomain equations is studied the dynamics of interest are governed by a coupled pde ode reaction diffusion system with nonmonotone nonlinearity of fitzhugh nagumo type a localized distributed control is used to locally stabilize the nonlinear system this is achieved by a riccati based feedback law determined by the linearized system it is shown that the riccati equation corresponding to the pde variable suffices for exponential stabilization of the linearized pde ode system the theoretical findings are underlined by several numerical examples

computer simulations of multiplicative stochastic differential equations a class of robust algorithms for the computer simulation of stochastic differential equations with multiplicative noise is investigated excellent agreement is obtained with the known analytic behaviour of the kubo oscillator in the white noise limit the algorithms include a known first order one dimensional explicit method as well as implicit methods of increased stability a distinction is drawn between classes of stochastic differential equations depending on the type of spatial variation or curvature defined by the diffusion tensor this allows greatly simplified numerical implementations of the new algorithms in certain cases the results of different techniques are compared for the case of the kubo oscillator where a semi implicit technique gives the greatest accuracy

regularity of solutions of evolution integrodifferential equations with deviating argument abstract in this paper we prove the existence of mild and classical solutions of delay integrodifferential equation and delay integrodifferential evolution equations with nonlocal condition in banach spaces the regularity solutions of integrodifferential evolution equations interconnected with viscoelastic material is derived to guarantee the stabilization the results are established by using the resolvent operators and the fixed point principles finally an example is given to show the potential of the proposed techniques

numerical simulations of generalized langevin equations with deeply asymptotic parameters a unified algorithm for solving langevin equations with deeply asymptotic parameters is proposed and tested the method consists of identifying solvable linear friction and implementing the force evaluations by use of the runge kutta method we apply the present scheme to the periodic motion of an overdamped particle subjected to a multiplicative white noise the accurate calculations for the temporal velocity of the particle and its correlation function can be realized by introducing an inertial term it is shown that the fluctuation around the steady quantity increases with decreasing time step in the overdamped white noise algorithm however a massive white noise technique greatly reduces this spurious drift and the result can converge to the correct value if the added inertia approaches zero the other application is the simulation of generalized langevin equation with an exponential memory friction this allows us to treat a weak non markovian process

necessary conditions for optimal control of distributed parameter systems the ekeland variational principle is used to prove a maximum principle for the optimal controls of a general semilinear evolutionary control system in a banach space with strictly convex dual the boundary constraint is of general type which includes the optimal periodic control problem as a special case

finite dimensional controller design for semilinear parabolic systems this paper considers the stabilization of steady state solutions of a semilinear parabolic system using finite dimensional feedback controllers with support in an arbitrary open subset and which are active in one equation only it is shown that such a controller with dimension given by the largest algebraic multiplicity of the unstable eigenvalues of the linearized system exponentially stabilizes the steady state solution an optimal design methodology for these types of controllers which is based on the finite element approximation of the semilinear parabolic system is introduced and illustrated by numerical simulation examples

feedback stabilization of the cahn hilliard type system for phase separation abstract this article is concerned with the internal feedback stabilization of the phase field system of cahn hilliard type modeling the phase separation in a binary mixture under suitable assumptions on an arbitrarily fixed stationary solution we construct via spectral separation arguments a feedback controller having its support in an arbitrary open subset of the space domain such that the closed loop nonlinear system exponentially reaches the prescribed stationary solution this feedback controller has a finite dimensional structure in the state space of solutions in particular every constant stationary solution is admissible

optimal control for semilinear evolutionary variational bilateral problem this paper is concerned with an optimal control problem for some semilinear evolutionary variational inequalities associated with bilateral constraints the control domain is a general separable metric space and has no algebraic structure in particular it is not necessarily convex existence and optimality conditions of optimal pairs are established

pontryagin s principle for local solutions of control problems with mixed control state constraints this paper deals with optimal control problems of semilinear parabolic equations with pointwise state constraints and coupled integral state control constraints we obtain necessary optimality conditions in the form of a pontryagin s minimum principle for local solutions in the sense of l p p leq infty

null controllability of some reaction diffusion systems with one control force this work is concerned with the null controllability of semilinear parabolic systems by a single control force acting on a subdomain