comparative study of methods of various orders for finding repeated roots of nonlinear equations abstract in this paper we are considering 20 families of methods for finding repeated roots of a nonlinear equation the methods are of order up to 8 we use the idea of basin of attraction to compare the methods we found that 4 methods performed best based on 3 quantitative criteria

constructing higher order methods for obtaining the multiple roots of nonlinear equations this paper concentrates on iterative methods for obtaining the multiple roots of nonlinear equations using the computer algebra system mathematica we construct an iterative scheme and discuss the conditions to obtain fourth order methods from it all the presented fourth order methods require one function and two derivative evaluation per iteration and are optimal higher order iterative methods for obtaining multiple roots we present some special methods from the iterative scheme including some known already numerical examples are also given to show their performance

energy decay estimates for the damped plate equation with a local degenerated dissipation we consider the euler bernoulli plate equation in a bounded open set of r2 with a degenerated local damping term this dissipation is effective in a subset of and the damping coefficient may vanish in some subset of dimension one of we show that the usual observability inequality for the undamped problem implies polynomial decay estimates for the damped problem our method can be applied for other pde s such as the wave equation or the schrodinger equation

two equivalence theorems of different kinds of optimal control problems for schrodinger equations this paper builds up two equivalence theorems for different kinds of optimal control problems of internally controlled schrodinger equations the first one is an equivalence of the minimal norm and the minimal time control problems where the target sets are the origin of the state space the minimal time control problems are also called the first type of optimal time control problems the second one is an equivalence of three optimal control problems optimal target control problems optimal norm control problems and the second type of optimal time control problems

modified jarratt method for computing multiple roots abstract in this paper we present a fourth order method for computing multiple roots of nonlinear equations the method is based on jarratt scheme for simple roots p jarratt some efficient fourth order multipoint methods for solving equations bit 9 1969 119 124 the method is optimal since it requires three evaluations per step namely one evaluation of function and two evaluations of first derivative the efficacy is tested on a number of relevant numerical problems it is observed that the present scheme is competitive with other similar robust methods

on the stabilization of persistently excited linear systems we consider control systems of the type dot x ax alpha t bu where u in mathbf r a b is a controllable pair and alpha is an unknown measurable time varying signal with values in 0 1 satisfying a persistent excitation condition of the type int t t t alpha s ds geq mu for every t geq0 with 0 mu leq t independent of t we prove that such a system is stabilizable with a linear feedback depending only on the pair t mu if the eigenvalues of a have a nonpositive real part we also show that stabilizability does not hold for an arbitrary matrix a moreover the question of whether the system can be stabilized or not with an arbitrarily large rate of convergence gives rise to a bifurcation phenomenon depending on the parameter mu t

regularity of a schrodinger equation with dirichlet control and colocated observation this paper studies the basic properties of the schrodinger equation defined on a bounded domain of r n n 2 with partial dirichlet control and colocated observation it is shown that the system is not only wellposed in the sense of d salamon but also regular in the sense of g weiss it is also shown that the corresponding feedthrough operator is zero 2005 elsevier b v all rights reserved

stabilization of persistently excited linear systems by delayed feedback laws this paper considers the stabilization to the origin of a persistently excited linear system by means of a linear state feedback where we suppose that the feedback law is not applied instantaneously but after a certain positive delay not necessarily constant the main result is that under certain spectral hypotheses on the linear system stabilization by means of a linear delayed feedback is indeed possible generalizing a previous result already known for non delayed feedback laws

ballistic deposition on deterministic fractals observation of discrete scale invariance the growth of ballistic aggregates on deterministic fractal substrates is studied by means of numerical simulations first we attempt the description of the evolving interface of the aggregates by applying the well established family vicsek dynamic scaling approach systematic deviations from that standard scaling law are observed suggesting that significant scaling corrections have to be introduced in order to achieve a more accurate understanding of the behavior of the interface subsequently we study the internal structure of the growing aggregates that can be rationalized in terms of the scaling behavior of frozen trees i e structures inhibited for further growth lying below the growing interface it is shown that the rms height h s and width w s of the trees of size s obey power laws of the form h s ensuremath propto s ensuremath nu ensuremath parallel and w s ensuremath propto s ensuremath nu ensuremath perp respectively also the tree size distribution n s behaves according to n s ensuremath sim s ensuremath ensuremath tau here ensuremath nu ensuremath parallel and ensuremath nu ensuremath perp are the correlation length exponents in the directions parallel and perpendicular to the interface respectively also ensuremath tau is a critical exponent however due to the interplay between the discrete scale invariance of the underlying fractal substrates and the dynamics of the growing process all these power laws are modulated by logarithmic periodic oscillations the fundamental scaling ratios characteristic of these oscillations can be linked to the spatial fundamental scaling ratio of the underlying fractal by means of relationships involving critical exponents we argue that the interplay between the spatial discrete scale invariance of the fractal substrate and the dynamics of the physical process occurring in those media is a quite general phenomenon that leads to the observation of logarithmic periodic modulations of physical observables

an observability estimate for the heat equation from a product of two measurable sets this paper presents an observability estimate for the heat equation in 0 t where has a real analytic boundary the observation region is e where and e are positive measurable subsets of and 0 t respectively the building of this estimate is based on a slightly modified version of the new estimate established in apraiz and escauriaza 2012 15 the technique developed in phung and wang 2012 11 and the shortcut of the lebeau robbiano strategy provided in miller 2010 3

equilibrium restricted solid on solid growth model on fractal substrates the equilibrium restricted solid on solid growth model on fractal substrates is studied by introducing a fractional langevin equation the growth exponent beta and the roughness exponent alpha defined respectively by the surface width via w approximately t beta and the saturated width via w sat approximately l alpha l being the system size were obtained by a power counting analysis and the scaling relation 2alpha d f z rw was found to hold the numerical simulation data on sierpinski gasket checkerboard fractal and critical percolation cluster were found to agree well with the analytical predictions of the fractional langevin equation

on the controllability of parabolic systems with a nonlinear term involving the state and the gradient we present some results concerning the controllability of a quasi linear parabolic equation with linear principal part in a bounded domain of mathbb r n with dirichlet boundary conditions we analyze the controllability problem with distributed controls supported on a small open subset and boundary controls supported on a small part of the boundary we prove that the system is null and approximately controllable at any time if the nonlinear term f y nabla y grows slower than y log 3 2 1 y nabla y nabla y log 1 2 1 y nabla y at infinity generally in this case in the absence of control blow up occurs the proofs use global carleman estimates parabolic regularity and the fixed point method

on a class of quadratic convergence iteration formulae without derivatives a class of iterative formulae without derivatives are proposed in this paper these methods are shown to be quadratic convergence some numerical tests are given

new blow up rates for fast controls of schrodinger and heat equations abstract we consider the null controllability problem for the schrodinger and heat equations with boundary control we concentrate on short time or fast controls we improve recent estimates see l miller geometric bounds on the growth rate of null controllability cost for the heat equation in small time j differential equations 204 2004 202 226 l miller how violent are fast controls for schrodinger and plate vibrations arch ration mech anal 172 2004 429 456 l miller controllability cost of conservative systems resolvent condition and transmutation j funct anal 218 2005 425 444 l miller the control transmutation method and the cost of fast controls siam j control optim 45 2006 762 772 on the norm of the operator associating to any initial state the minimal norm control driving the system to zero our main results concern the schrodinger and heat equations in one space dimension they yield new estimates concerning window problems for series of exponentials as described in t i seidman the coefficient map for certain exponential sums nederl akad wetensch indag math 48 1986 463 478 and in t i seidman s a avdonin s a ivanov the window problem for series of complex exponentials j fourier anal appl 6 2000 233 254 these results are used following l miller the control transmutation method and the cost of fast controls siam j control optim 45 2006 762 772 to deal with the case of several space dimensions

scaling properties of surface growth on rough substrates generated by standard models abstract in surface growth simulations usually the substrate is initially considered smooth with no roughness but as known in reality many substrates in nature are initially rough in this article we study the sensitivity of dynamic scaling of different models on rough substrates generated by standard surface growth models we implement the three growth procedures which are as follows for the first two procedures two separate substrates are grown with the random deposition rd and restricted solid on solid rsos for different time steps t 1 before growing them both with the ballistic deposition model bd but in the third growth procedure as an inverse of the second procedure particles deposit on the substrates by the bd model and become rough by the rsos model we conclude that the surfaces need a characteristic growth time t x to lose information about the substrate we show that the characteristic time depends on both the time needed for roughening the substrate and the special surface growth models in addition we show that the dependence of the two parameters t x and t 1 is a power law dependence only for adequate short times of t 1 which is also calculated

a new fourth order iterative method for finding multiple roots of nonlinear equations in this paper we present a new fourth order method for finding multiple roots of nonlinear equations it requires one evaluation of the function and two of its first derivative per iteration finally some numerical examples are given to show the performance of the presented method compared with some known third order methods

the time optimal control with constraints of the rectangular type for linear time varying odes in this paper we study a time optimal control problem of some linear time varying ordinary differential equations where the control constraint set is of the rectangular type we derive a necessary and sufficient condition and provide an algorithm for the optimal time as well as the optimal control for this problem the basis consists of an equivalence theorem between the time optimal control problem and an associated norm optimal control problem and the variational characterization of the latter problem

continuous dependence of optimal control to controlled domain of actuator for heat equation abstract in this paper we consider continuous dependence of the optimal control with respect to the actuator domain which is varying as open subset in the spatial domain for a multi dimensional heat equation both time optimal control and norm optimal control problems are considered the reason behind combining these two problems together is that these two problems are actually equivalent the energy to be used to drive the system to target set in minimal time interval is actually the minimal energy of driving the system to target set in this minimal time interval and visa versa it is shown that both optimal control and optimal cost are continuous with respect to open controlled actuator domain under the lebesgue measure

stabilization of two dimensional persistently excited linear control systems with arbitrary rate of convergence we consider the control system dot x a x alpha t b u where the pair a b is controllable x in mathbb r 2 u is a scalar control and the unknown signal alpha satisfies a persistent excitation condition we study the stabilization of this system and we prove that it is globally asymptotically stable with arbitrarily large exponential rate uniformly with respect to all signals satisfying a common persistent excitation condition and a common lipschitz continuity bound

time optimal boundary controls for the heat equation the fact that the time optimal controls for parabolic equations have the bang bang property has been recently proved for controls distributed inside the considered domain interior control the main result in this article asserts that the boundary controls for the heat equation have the same property at least in rectangular domains this result is proved by combining methods from traditionally distinct fields the lebeau robbiano strategy for null controllability and estimates of the controllability cost in small time for parabolic systems on one side and a remez type inequality for muntz spaces and a generalization of turan s inequality on the other side

asymptotic stability for intermittently controlled second order evolution equations motivated by several works on ordinary differential equations we are interested in the asymptotic stability of intermittently controlled partial differential equations we give a condition of asymptotic stability for second order evolution equations uniformly damped by an on off feedback this result extends to the case of partial differential equations a previous result of r a smith concerning ordinary differential equations

comparing the basins of attraction for kanwar bhatia kansal family to the best fourth order method there are relatively few optimal fourth order methods for solving nonlinear algebraic equations having roots of known multiplicity m in a previous paper we have compared 5 such methods two of which require the evaluation of the m 1 st root we have used the basin of attraction idea to recommend the best optimal fourth order method here we compare the family of methods developed by kanwar et al 2013 to the best known method we will also point out some mistake in deriving that family

on conditions for asymptotic stability of dissipative infinite dimensional systems with intermittent damping abstract we study the asymptotic stability of a dissipative evolution in a hilbert space subject to intermittent damping we observe that even if the intermittence satisfies a persistent excitation condition if the hilbert space is infinite dimensional then the system needs not being asymptotically stable not even in the weak sense exponential stability is recovered under a generalized observability inequality allowing for time domains that are not intervals weak asymptotic stability is obtained under a similarly generalized unique continuation principle finally strong asymptotic stability is proved for intermittences that do not necessarily satisfy some persistent excitation condition evaluating their total contribution to the decay of the trajectories of the damped system our results are discussed using the example of the wave equation schrodinger s equation and for strong stability also the special case of finite dimensional systems

stability of solutions for some classes of nonlinear damped wave equations we consider two classes of semilinear wave equations with nonnegative damping which may be of type on off or integrally positive in both cases we give a sufficient condition for the asymptotic stability of the solutions in the case of integrally positive damping we show that such a condition is also necessary

new optimal class of higher order methods for multiple roots permitting f xn 0 finding multiple zeros of nonlinear functions pose many difficulties for many of the iterative methods a major difficulty in the application of iterative methods is the selection of initial guess such that neither guess is far from zero nor the derivative is small in the vicinity of the required root otherwise the methods would fail miserably finding a criterion for choosing initial guess is quite cumbersome and therefore more effective globally convergent algorithms for multiple roots are still needed therefore the aim of this paper is to present an improved optimal class of higher order methods having quartic convergence permitting f x 0 in the vicinity of the required root the present approach of deriving this optimal class is based on weight function approach all the methods considered here are found to be more effective and comparable to the similar robust methods available in literature

perturbations of time optimal control problems for a class of abstract parabolic systems in this work we study the asymptotic behavior of the solutions of a class of abstract parabolic time optimal control problems when the generators converge in an appropriate sense to a given strictly negative operator our main application to pdes systems concerns the behavior of optimal time and of the associated optimal controls for parabolic equations with highly oscillating coefficients as we encounter in homogenization theory our main results assert that provided that the target is a closed ball centered at the origin and of positive radius the solutions of the time optimal control problems for the systems with oscillating coefficients converge in the usual norms to the solution of the corresponding problem for the homogenized system in order to prove our main theorem we provide several new results which could be of a broader interest on time and norm optimal control problems

controllability of impulse controlled systems of heat equations coupled by constant matrices abstract this paper studies the approximate and null controllability for impulse controlled systems of heat equations coupled by a pair a b of constant matrices we present a necessary and sufficient condition for the approximate controllability which is exactly kalman s controllability rank condition of a b we prove that when such a system is approximately controllable the approximate controllability over an interval 0 t can be realized by adding controls at arbitrary q a b different control instants 0 1 2 q a b t provided that q a b 1 d a where d a min im a and q a b n we also show that in general such systems are not null controllable

on the existence of time optimal controls with constraints of the rectangular type for heat equations this paper presents a time optimal control problem with control constraints of the rectangular type for internally controlled heat equations an existence result of time optimal controls for such a problem is established the rectangular type of control constraints originates from the study of time optimal control problems for ordinary differential equations in the finite dimensional case there is no difference between such problems with control constraints of the rectangular type and those of the ball type from the perspective of the study on the existence of optimal controls interestingly in the infinite dimensional case the problem with control constraints of the rectangular type differs essentially from that with control constraints of the ball type for infinite dimensional systems the existence for time optimal controls with constraints of the ball type has already been discussed in the literature while the study of the rectangular type has not been touched upon as far as we know

transfer functions of regular linear systems part i characterizations of regularity we recall the main facts about the representation of regular linear systems essentially that they can be described by equations of the form x t ax t bu t y t cx t du t like finite dimensional systems but now a b and c are in general unbounded operators regular linear systems are a subclass of abstract linear systems we define transfer functions of abstract linear systems via a generalization of a theorem of foures and segal we prove a formula for the transfer function of a regular linear system which is similar to the formula in finite dimensions the main result is a simple to state but hard to prove necessary and sufficient condition for an abstract linear system to be regular in terms of its transfer function other conditions equivalent to regularity are also obtained the main result is a consequence of a new tauberian theorem which is of independent interest

some fourth order nonlinear solvers with closed formulae for multiple roots in this paper we present six new fourth order methods with closed formulae for finding multiple roots of nonlinear equations the first four of them require one function and three derivative evaluation per iteration the last two require one function and two derivative evaluation per iteration several numerical examples are given to show the performance of the presented methods compared with some known methods

equivalence of three different kinds of optimal control problems for heat equations and its applications this paper presents an equivalence theorem for three different kinds of optimal control problems which are optimal target control problems optimal time control problems and optimal norm control problems controlled systems in this study are internally controlled heat equations with the aid of this theorem we establish an optimal norm feedback law and build up some explicit algorithms for solutions of optimal norm and optimal time control problems

time varying bang bang property of time optimal controls for heat equation and its application abstract in this paper the bang bang property of time optimal controls for heat equation is established compared with the existing results on these problems the bound of control variables is not a constant but a time varying function as an application of the bang bang property some kind of relation between the time optimal control problem and its corresponding target optimal control problem is considered

on some modified families of multipoint iterative methods for multiple roots of nonlinear equations in this paper we propose a new one parameter family of schroder s method for finding the multiple roots of nonlinear equations numerically further we derive many new cubically convergent families of schroder type methods proposed families are derived from the modified newton s method for multiple roots and one parameter family of schroder s method furthermore we introduce new families of third order multipoint iterative methods for multiple roots free from second order derivative by semi discrete modifications of the above proposed methods one of the families requires two evaluations of the function and one evaluation of its first order derivative and the other family requires one evaluation of the function and two evaluations of its first order derivative per iteration numerical examples are also presented to demonstrate the performance of proposed iterative methods

converse lyapunov theorems for switched systems in banach and hilbert spaces we consider switched systems on banach and hilbert spaces governed by strongly continuous one parameter semigroups of linear evolution operators we provide necessary and sufficient conditions for their global exponential stability uniform with respect to the switching signal in terms of the existence of a lyapunov function common to all modes

maximum principle and bang bang property of time optimal controls for schrodinger type systems we consider the time optimal control problem with a point target for a class of infinite dimensional systems with a dynamics governed by an abstract schrodinger type equation the main results establish a pontryagyn type maximum principle and give sufficient conditions for the bang bang property of optimal controls the results are then applied to some systems governed by partial differential equations the paper ends by a discussion of possible extensions and by stating some open problems

spectral inequalities for non selfadjoint elliptic operators and application to the null controllability of parabolic systems we consider elliptic operators a on a bounded domain that are compact perturbations of a selfadjoint operator we first recall some spectral properties of such operators localization of the spectrum and resolvent estimates we then derive a spectral inequality that measures the norm of finite sums of root vectors of a through an observation with an exponential cost following the strategy of g lebeau and l robbiano 1995 we deduce the construction of a control for the non selfadjoint parabolic problem partial t u a u b g in particular the l 2 norm of the control that achieves the extinction of the lower modes of a is estimated examples and applications are provided for systems of weakly coupled parabolic equations and for the measurement of the level sets of finite sums of root functions of a

an approach to the optimal time for a time optimal control problem of an internally controlled heat equation in this paper we study the optimal time for a time optimal control problem mathcal p where the state system is an internally controlled heat equation by projecting the original problem via the finite element method we obtain another time optimal control problem mathcal p hl governed by a linear system of ordinary differential equations here h and l are the mesh sizes of the finite element spaces of the state space and the control space respectively the purpose of this study is to approach the optimal time for the problem mathcal p through the optimal time for the problem mathcal p hl we prove that the optimal time for the problem mathcal p hl converges to the optimal time for the problem mathcal p when h and l are approaching zero more significantly we obtain error estimates between the optimal times in terms of h and l for certain cases

tauberian theorems for the laplace stieltjes transform let a 0 oo c be a function of locally bounded variation with a 0 0 whose laplace stieltjes transform g z fo e z da t is absolutely convergent for re z 0 let e be the singular set of g in ir and suppose that 0 e various estimates for lim supt la t g o l are obtained in particular a t g 0 as t oo if i e is null ii sup sup e aysda s oo yee t o o iii lim lim sup sup a s a t 0 610 t oo t 6 s t this result contains tauberian theorems for laplace transforms power series and dirichlet series

approximation of time optimal controls for heat equations with perturbations in the system potential in this paper we study a certain approximation property for a time optimal control problem of the heat equation with l infty potential we prove that the optimal time and the optimal control to the same time optimal control problem for the heat equation where the potential has a small perturbation are close to those for the original problem we also verify that for the heat equation with a small perturbation in the potential one can construct a new time optimal control problem which has the same target as that of the original problem but has a different control constraint bound from that of the original problem such that the new and the original problems share the same optimal time and meanwhile the optimal control of the new problem is close to that of the original one the main idea to approach such approximation is an appropriate use of an equivalence theorem of minimal norm and minimal time control problems for the heat equations under consideration this theorem was first established by wa

semidiscrete finite element approximation of time optimal control problems for semilinear heat equations with nonsmooth initial data abstract this paper is devoted to the study of semidiscrete finite element approximation of time optimal control problem governed by semilinear heat equation with nonsmooth initial data when the control is acted locally an error estimate for the optimal time is obtained by making use of the approximation results for the equations moreover with the help of pontryagin s maximum principle and the unique continuation property for the heat equation a better error estimate for the optimal time can be derived when the control is acted globally into the state equation

high order nonlinear solver for multiple roots a method of order four for finding multiple zeros of nonlinear functions is developed the method is based on jarratt s fifth order method for simple roots and it requires one evaluation of the function and three evaluations of the derivative the informational efficiency of the method is the same as previously developed schemes of lower order for the special case of double root we found a family of fourth order methods requiring one less derivative thus this family is more efficient than all others all these methods require the knowledge of the multiplicity

on the equivalence of minimal time and minimal norm controls for internally controlled heat equations we prove the equivalence of the minimal time and minimal norm control problems for heat equations on bounded smooth domains of the euclidean space with homogeneous dirichlet boundary conditions and controls distributed internally on an open subset of the domain where the equation evolves we consider the problem of null controllability whose aim is to drive solutions to rest in a finite final time as a consequence of this equivalence using the well known variational characterization of minimal norm controls we establish necessary and sufficient conditions for the minimal time and the corresponding control

a sixth order family of three point modified newton like multiple root finders and the dynamics behind their extraneous fixed points a class of three point sixth order multiple root finders and the dynamics behind their extraneous fixed points are investigated by extending modified newton like methods with the introduction of the multivariate weight functions in the intermediate steps the multivariate weight functions dependent on function to function ratios play a key role in constructing higher order iterative methods extensive investigation of extraneous fixed points of the proposed iterative methods is carried out for the study of the dynamics associated with corresponding basins of attraction numerical experiments applied to a number of test equations strongly support the underlying theory pursued in this paper relevant dynamics of the proposed methods is well presented with a variety of illustrative basins of attraction applied to various test polynomials

attainable subspaces and the bang bang property of time optimal controls for heat equations in this paper we study two subjects on internally controlled heat equations with time varying potentials the attainable subspaces and the bang bang property for some time optimal control problems we present some equivalent characterizations on the attainable subspaces and provide a sufficient condition to ensure the bang bang property both the above mentioned characterizations and the sufficient condition are closely related to some function spaces consisting of some solutions to the adjoint equations it seems for us that the existing ways to derive the bang bang property for heat equations with time invariant potentials see for instance h o fattorini infinite dimensional linear control systems the time optimal and norm optimal problems elsevier amsterdam 2005 f gozzi and p loreti siam j control optim 37 1999 pp 1195 1221 v mizel and t seidman siam j control optim 35 1997 pp 1204 1216 and g wang siam j control optim 47 2008 pp 1701 1720 do not work

controllability for distributed bilinear systems this paper studies controllability of systems of the form dw dt mathcal a w p t mathcal b w where mathcal a is the infinitesimal generator of a c 0 semigroup of bounded linear operators e mathcal a t on a banach space x mathcal b x to x is a c 1 map and p in l 1 0 t mathbb r is a control the paper i gives conditions for elements of x to be accessible from a given initial state w 0 and ii shows that controllability to a full neighborhood in x of w 0 is impossible for dim x infty examples of hyperbolic partial differential equations are provided

l infty null controllability for the heat equation and its consequences for the time optimal control problem in this paper we establish a certain l infty null controllability for the internally controlled heat equation in omega times 0 t with the control restricted to a product set of an open nonempty subset in omega and a subset of positive measure in the interval 0 t based on this we obtain a bang bang principle for the time optimal control of the heat equation with controls taken from the set mathcal u ad u cdot t 0 infty rightarrow l 2 omega measurable u cdot t in u a e in t where u is a closed and bounded subset of l 2 omega namely each optimal control u cdot t of the problem satisfies necessarily the bang bang property u cdot t in partial u for almost all t in 0 t where partial u denotes the boundary of the set u and t is the optimal time we also get the uniqueness of the optimal control when the target set s is convex and the control set u is a closed ball

persistently damped transport on a network of circles in this paper we address the exponential stability of a system of transport equations with intermittent damping on a network of n geq 2 circles intersecting at a single point o the n equations are coupled through a linear mixing of their values at o described by a matrix m the activity of the intermittent damping is determined by persistently exciting signals all belonging to a fixed class the main result is that under suitable hypotheses on m and on the rationality of the ratios between the lengths of the circles such a system is exponentially stable uniformly with respect to the persistently exciting signals the proof relies on an explicit formula for the solutions of this system which allows one to track down the effects of the intermittent damping

an abstract bang bang principle and time optimal boundary control of the heat equation a principal technical result of this paper is that the one dimensional heat equation with boundary control is exactly null controllable with control restricted to an arbitrary set e subset 0 t of positive measure a general abstract argument is presented to show that in contrast to previous results this implies the bang bang property for time optimal controls i e such a control can take only extreme values of the hull of the constraint set without imposing any condition regarding the target state

on a class of quadratically convergent iteration formulae in this paper a new class of iterative formulae having quadratic convergence is presented furthermore these algorithms are comparable to the well known method of newton and the computed results support this theory

a new asymptotic stability criterion for nonlinear time variant differential equations a new sufficient condition for asymptotic stability of ordinary differential equations is proposed unlike classical lyapunov theory the time derivative along solutions of the lyapunov function may have positive and negative values the classical lyapunov approach may be regarded as an infinitesimal version of the present theorem verification in practical problems is harder than in the classical case an example is included in order to indicate how the present theorem may be applied

on the dynamics of a triparametric family of optimal fourth order multiple zero finders with a weight function of the principal mth root of a function to function ratio abstract under the assumption of known root multiplicity m n a triparametric family of two point optimal quartic order methods locating multiple zeros are investigated in this paper by introducing a weight function dependent on a function to function ratio special cases of weight functions with selected free parameters are considered and studied through various test equations and numerical experiments to support the theory developed in this paper in addition we explore the relevant dynamics of proposed methods via mobius conjugacy map when applied to a prototype polynomial z a m z b m the results of such dynamics are visually illustrated through a variety of parameter spaces as well as dynamical planes

discrete growth models on deterministic fractal substrate abstract the growth of the modified family model and the etching model on the sierpinski carpet is studied by means of numerical simulations the evolving interface of the aggregates is described by the well established family vicsek dynamic scaling approach the results of the modified family model prove the universality of the fractional langevin equation introduced by lee and kim s b lee j m kim phys rev e 80 2009 021101 the etching model also shows good scaling behavior we conjecture that the systematic deviations of the data found in the ballistic deposition c m horowitz f roma e v albano phys rev e 78 2008 061118 may be due to the finite size effects of the ballistic deposition model

the surface statistics of a granular aggregate the problem of the surface fluctuations in a settled granular material is posed a simple model is given which describes the process by which a particle settles and comes to rest on the existing surface of the packing and from this a set of langevin equations for the fourier modes of the surface are derived these equations imply that the fourier amplitudes behave like the velocities of a set of independent brownian particles we show that this results in logarithmically divergent surface fluctuations if the flux of particles onto the surface is random the divergence being removed by a more accurate description of the settling material for example by having the granules fall through a sieve

some results on the controllability of forward stochastic heat equations with control on the drift abstract in this paper we establish the null approximate controllability for forward stochastic heat equations with control on the drift the null controllability is obtained by a time iteration method and an observability estimate on partial sums of eigenfunctions for elliptic operators as a consequence of the null controllability we obtain the observability estimate for backward stochastic heat equations which leads to a unique continuation property for backward stochastic heat equations and hence the desired approximate controllability for forward stochastic heat equations it deserves to point out that one needs to introduce a little stronger assumption on the controller for the approximate controllability of forward stochastic heat equations than that for the null controllability this is a new phenomenon arising in the study of the controllability problem for stochastic heat equations

constructing a family of optimal eighth order modified newton type multiple zero finders along with the dynamics behind their purely imaginary extraneous fixed points abstract an optimal family of eighth order multiple zero finders and the dynamics behind their basins of attraction are proposed by considering modified newton type methods with multivariate weight functions extensive investigation of purely imaginary extraneous fixed points of the proposed iterative methods is carried out for the study of the dynamics associated with corresponding basins of attraction numerical experiments strongly support the underlying theory pursued in this paper an exploration of the relevant dynamics of the proposed methods is presented along with illustrative basins of attraction for various polynomials