boundary feedback stabilization for an unstable time fractional reaction diffusion equation in this paper we consider boundary feedback stabilization for unstable time fractional reaction diffusion equations new state feedback controls with actuation on one end are designed by the backstepping method for both dirichlet and neumann boundary controls by the riesz basis approach and the fractional lyapunov method we prove the existence and uniqueness and the mittag leffler stability for the closed loop systems for both cases the observers and the observer based output feedback are designed to stabilize the systems

relaxation in nonconvex optimal control problems described by fractional differential equations abstract we consider the minimization problem of an integral functional with integrand that is not convex in the control on solutions of a control system described by fractional differential equation with mixed nonconvex constraints on the control a relaxation problem is treated along with the original problem it is proved that under general assumptions the relaxation problem has an optimal solution and that for each optimal solution there is a minimizing sequence of the original problem that converges to the optimal solution with respect to the trajectory the control and the functional in appropriate topologies simultaneously

controllability of fractional integro differential evolution equations with nonlocal conditions this paper concerns the controllability for a class of fractional integro differential evolution equations with nonlocal initial conditions by using the fractional calculus measure of noncompactness and the monch fixed point theorem we obtain a controllability result for the nonlocal cauchy problem of the fractional integro differential evolution equations involving noncompact semigroups and the nonlocal functions without lipschitz continuity an example is given to illustrate the effectiveness of the abstract results

approximate controllability of fractional nonlocal delay semilinear systems in hilbert spaces we study the existence and approximate controllability of a class of fractional nonlocal delay semilinear differential systems in a hilbert space the results are obtained by using semigroup theory fractional calculus and schauder s fixed point theorem multi delay controls and a fractional nonlocal condition are introduced furthermore we present an appropriate set of sufficient conditions for the considered fractional nonlocal multi delay control system to be approximately controllable an example to illustrate the abstract results is given

approximate controllability of fractional evolution systems with riemann liouville fractional derivatives in this paper we deal with the control systems governed by fractional evolution differential equations involving riemann liouville fractional derivatives in banach spaces our main purpose in this article is to establish suitable assumptions to guarantee the existence and uniqueness results of mild solutions under these conditions the approximate controllability of the associated fractional evolution systems involving riemann liouville fractional derivatives is formulated and proved

asymptotic behavior of solutions of a nonlinear evolution equation the purpose of this paper is to study the asymptotic behavior of the solutions of a certain nonlinear evolution equation in banach spaces see 1 below this time dependent initial value problem involves an accretive operator a and a function g r f r it was first considered by browder in 11 and was the basis of the iterative procedures for constructing zeros of accretive operators presented in 2 and 5 these papers treat only the case when 0 e r a we begin with several results e g theorems 3 and 6 that are of interest when 0 r a these results are in some sense analogs of known results for nonlinear semigroups and resolvents there is however a difference see for example corollary 4 then we show that under certain mild restrictions 0 e r a if and only if the solutions to 1 are bounded theorem 7 when an additional condition is imposed on g we can shown that a is zero free if and only if lim u t co for each solution of 1 theorem 8 and proposition 9 in the last part of the paper we assume that 0 e r a and obtain several strong convergence results for solutions of 1 e g theorem 12 these results show that the asymptotic behavior of the solutions to 1 resembles the behavior of resolvents more than that of nonlinear semigroups theorem 12 yields improvements of the convergence results for the iterative procedure of s and leads to a geometric result concerning the fixed point set of a nonexpansive mapping corollary 13 we also mention that the sequence constructed from the iterative procedure of 5 imitates the behavior of the solutions of 1 not only when 0 e r a but also when a is zero free finally we remark that many of our results are new even in hilbert space let e be a real banach space and let i denote the identity operator recall that a subset a of e x e with domain d a and range r a is said to be accretive if lx x2 0 we denote the closure of a subset d of e by cl d its closed convex hull by clco d and its distance from a point x in e by d x 0 we 43 0022 247x 81 090043 11 02 00 0

a fractional differential equation for a mems viscometer used in the oil industry a mathematical model is developed for a micro electro mechanical system mems instrument that has been designed primarily to measure the viscosity of fluids that are encountered during oil well exploration it is shown that in one mode of operation the displacement of the device satisfies a fractional differential equation fde the theory of fdes is used to solve the governing equation in closed form and numerical solutions are also determined using a simple but efficient central difference scheme it is shown how knowledge of the exact and numerical solutions enables the design of the device to be optimised it is also shown that the numerical scheme may be extended to encompass the case of a nonlinear spring where the resulting fde is nonlinear

mild solutions to the time fractional navier stokes equations in r n abstract this paper addresses the existence and uniqueness of mild solutions to the navier stokes equations with time fractional differential operator of order 0 1 several interesting properties about the solution are also highlighted like regularity and decay rate in lebesgue spaces which will depend on the fractional exponent moreover it is shown that the l p exponent range which the solution belongs to is different from the range for the solution of the classical problem with 1

approximate controllability of fractional order semilinear systems with bounded delay abstract in this paper sufficient conditions are established for the approximate controllability of a class of semilinear delay control systems of fractional order the existence and uniqueness of mild solution of the system is also proved the results are obtained by using contraction principle and the schauder fixed point theorem some examples are given to illustrate the theory

existence of mild solutions for fractional neutral evolution equations abstract in this paper by using the fractional power of operators and some fixed point theorems we discuss a class of fractional neutral evolution equations with nonlocal conditions and obtain various criteria on the existence and uniqueness of mild solutions in the end we give an example to illustrate the applications of the abstract results

approximate controllability of second order distributed implicit functional systems abstract in this paper we study the approximate controllability of control systems with states and controls in hilbert spaces and described by a second order semilinear abstract functional differential equation with infinite delay initially we establish a characterization for the approximate controllability of a second order abstract linear system and in the last section we compare the approximate controllability of a semilinear abstract functional system with the approximate controllability of the associated linear system

approximate controllability of fractional nonlinear differential inclusions fractional differential equations have wide applications in both physical and social sciences this paper addresses the issue of approximate controllability for a class of fractional nonlinear differential inclusions in banach spaces a new set of sufficient conditions are formulated and proved for the approximate controllability of fractional nonlinear differential inclusions in particular the results are established with the assumption that the associated linear part of system is approximately controllable further the result is extended to obtain the conditions for the solvability of controllability results for fractional inclusions with nonlocal conditions finally an example is presented to illustrate the theory of the obtained result

controllability of a backward fractional semilinear differential equation abstract in this paper we study the approximate controllability of a fractional semilinear differential equation involving the right fractional caputo derivative more precisely we construct by means of tikhonov type regularization method the controllability operator then under certain condition on this operator we obtain that the associate backward fractional linear system can be steered to an arbitrary small neighborhood of the state at initial time this allows us to prove the approximate controllability of the semilinear system

on the approximate controllability of fractional evolution equations with compact analytic semigroup in this paper we discuss the approximate controllability of fractional evolution equations involving caputo fractional derivative the results are obtained with the help of the theory of fractional calculus semigroup theory and the schauder s fixed point theorem under the assumption that the corresponding linear system is approximately controllable finally an example is provided to illustrate the obtained results

on the gibbs phenomenon and its resolution the nonuniform convergence of the fourier series for discontinuous functions and in particular the oscillatory behavior of the finite sum was already analyzed by wilbraham in 1848 this was later named the gibbs phenomenon this article is a review of the gibbs phenomenon from a different perspective the gibbs phenomenon as we view it deals with the issue of recovering point values of a function from its expansion coefficients alternatively it can be viewed as the possibility of the recovery of local information from global information the main theme here is not the structure of the gibbs oscillations but the understanding and resolution of the phenomenon in a general setting the purpose of this article is to review the gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function with the same order of accuracy as in the smooth case this is done by using the finite expansion series to construct a different rapidly convergent approximation

degenerate type fractional evolution hemivariational inequalities and optimal controls via fractional resolvent operators in this paper we mainly investigate controlled fractional evolution hemivariational inequalities of degenerate type in caputo and riemann liouville fractional derivatives of order in 1 2 respect

fractional cauchy problems with almost sectorial operators this paper concerns the abstract cauchy problem of fractional evolution equations with almost sectorial operators the suitable mild solutions of evolution equations with riemann liouville derivative and caputo derivative are introduced respectively the existence theorems of mild solutions of cauchy problems are established the results obtained here improve and generalize some known results finally an example is given for demonstration

center stable manifold for planar fractional damped equations in this paper we discuss the existence of a center stable manifold for planar fractional damped equations by constructing a suitable lyapunovperron operator via giving asymptotic behavior of mittagleffler function we obtain an interesting center stable manifold theorem finally an example is provided to illustrate the result

approximate controllability of a delayed semilinear control system with growing nonlinear term abstract this paper deals with the approximate controllability of a semilinear control system with delay where the nonlinear term satisfies the linear growth condition sufficient conditions for the approximate controllability of a semilinear control system have been established by assuming that the corresponding linear control system with delay is approximately controllable to prove our results the schauder fixed point theorem is applied and instead of a co semigroup associated with the mild solution of the system we use the so called fundamental solution

controllability of fractional evolution nonlocal impulsive quasilinear delay integro differential systems in this work the controllability result of a class of fractional evolution nonlocal impulsive quasilinear delay integro differential systems in a banach space has been established by using the theory of fractional calculus fixed point technique and also we introduced a new concept called a u resolvent family as an application that illustrates the abstract results an example is given

approximate controllability for fractional semilinear parabolic equations abstract in this paper we deal with the control systems described by a large class of fractional semilinear parabolic equations firstly we reformulate the fractional parabolic equations into abstract fractional differential equations associated with a semigroup on an appropriate banach space secondly we introduce a suitable concept on a mild solution for this kind of fractional parabolic equations and present the existence and uniqueness of mild solution by utilizing the theory of semigroup of linear operator nonlinear analysis method and fixed point theorem then the approximate controllability of the fractional semilinear parabolic equations is formulated and proved at the end of the paper an example is given to illustrate our main results

families of quasi rational solutions of the nls equation and multi rogue waves we construct a multi parametric family of the solutions of the focusing nonlinear schr dinger equation nls from the known results describing the multi phase almost periodic elementary solutions given in terms of riemann theta functions we give a new representation of their solutions in terms of wronskians determinants of order 2n composed of elementary trigonometric functions when we perform a special passage to the limit when all the periods tend to infinity we obtain a family of quasi rational solutions this leads to efficient representations for the peregrine breathers of orders n 1 2 3 first constructed by akhmediev and his co workers and also allows us to obtain a simpler derivation of the generic formulas corresponding the three or six rogue wave formation in the frame of the nls model first explained by v b matveev in 2010 our formulation allows us to isolate easily the second or third order peregrine breathers from generic solutions and also to compute the peregrine breathers of orders 2 and 3 easily with respect to other approaches in the cases n 2 3 we obtain the comfortable formulas to study the deformation of a higher peregrine breather of order 2 to the three rogue wave or order 3 to the six rogue wave solutions via the variation of the free parameters of our construction

on the approximate controllability of semilinear fractional differential systems fractional differential equations have wide applications in science and engineering in this paper we consider a class of control systems governed by the semilinear fractional differential equations in hilbert spaces by using the semigroup theory the fractional power theory and fixed point strategy a new set of sufficient conditions are formulated which guarantees the approximate controllability of semilinear fractional differential systems the results are established under the assumption that the associated linear system is approximately controllable further we extend the result to study the approximate controllability of fractional systems with nonlocal conditions an example is provided to illustrate the application of the obtained theory

approximate controllability of a multi valued fractional impulsive stochastic partial integro differential equation with infinite delay in this paper the approximate controllability of a new class of multi valued fractional impulsive stochastic partial integro differential equations with infinite delay in hilbert spaces is studied firstly by using stochastic analysis analytic resolvent operator fractional powers of closed operators and a fixed point theorem for multi valued maps we prove an existence result of mild solutions for the control systems under the mixed lipschitz and caratheodory conditions secondly we discuss a new set of sufficient conditions for the approximate controllability of the systems the results are obtained under the assumption that the associated linear system is approximately controllable finally an example is provided to illustrate the proposed theory

approximate controllability of fractional delay dynamic inclusions with nonlocal control conditions we introduce a nonlocal control condition and the notion of approximate controllability for fractional order quasilinear control inclusions approximate controllability of a fractional control nonlocal delay quasilinear functional differential inclusion in a hilbert space is studied the results are obtained by using the fractional power of operators multi valued analysis and sadovskii s fixed point theorem main result gives an appropriate set of sufficient conditions for the considered system to be approximately controllable as an example a fractional partial nonlocal control functional differential inclusion is considered

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

numerical investigation of the stability of the rational solutions of the nonlinear schrdinger equation hightlightsa broad numerical investigation involving large ensembles of perturbed initial data indicates the peregrine and second order rational solutions of the nls equation are linearly unstable a highly accurate chebyshev pseudo spectral method is developed for solving the nls equation that uses the map x cot in combination with the fast fourier transform to approximate uxx on an infinite domain a modified fourier spectral method that can treat initial data with discontinuous derivatives over periodic domains is developed for solving the nls equation of which the resolution of the peregrine solution is a particular example the rational solutions of the nonlinear schrdinger nls equation have been proposed as models for rogue waves in this article we develop a highly accurate chebyshev pseudo spectral method cps4 to numerically study the stability of the rational solutions of the nls equation the scheme cps4 using the map x cot and the fft to approximate uxx correctly handles the infinite line problem a broad numerical investigation using cps4 and involving large ensembles of perturbed initial data indicates the peregrine and second order rational solutions are linearly unstable although standard fourier integrators are often used in current studies of the nls rational solutions they do not handle solutions with discontinuous derivatives correctly using standard fourier pseudo spectral method fps4 for peregrine initial data yields tiny gibbs oscillations in the first steps of the numerical solution these oscillations grow to o 1 providing further evidence of the instability of the peregrine solution to resolve the gibbs oscillations we modify fps4 using a spectral splitting technique which significantly improves the numerical solution

gauge theory of ba cklund transformations ii abstract a class of gauge transformations is constructed for hamiltonian hierarchies of completely integrable systems on semi simple lie algebras these transformations create or annihilate poles in the scattering data hence create or annihilate solitons in the potential q in that sense they generalize backlund transformations to which they reduce precisely for the 2 2 akns systems to each backlund gauge transformation is associated an automorphism in the weyl group of the lie algebra two proofs are given to show that these backlund gauge transformations preserve the nonlinear evolution equation one proof uses the transformation of the scattering data under the backlund transformation the second proof is purely gauge theoretic

exploring delayed mittag leffler type matrix functions to study finite time stability of fractional delay differential equations abstract in this paper we introduce a concept of delayed two parameters mittag leffler type matrix function which is an extension of the classical mittag leffler matrix function with the help of the delayed two parameters mittag leffler type matrix function we give an explicit formula of solutions to linear nonhomogeneous fractional delay differential equations via the variation of constants method in addition we prove the existence and uniqueness of solutions to nonlinear fractional delay differential equations thereafter we present finite time stability results of nonlinear fractional delay differential equations under mild conditions on nonlinear term finally an example is presented to illustrate the validity of the main theorems

nonlocal initial value problems for differential equations with hilfer fractional derivative in this paper we discuss the existence of solutions to nonlocal initial value problem for differential equations with hilfer fractional derivative to begin with we establish an equivalent mixed type integral equation for our problem then we utilize basic properties of hilfer fractional calculus and fixed point methods to derive three fundamental existence results in the weighted space of continuous functions some examples are given to illustrate our results

controllability of semilinear control systems dominated by the linear part while various equivalent conditions for controllability have been obtained in the case of linear control systems controllability problems of semilinear control systems usually require some complicated and limited assumptions in this paper we show the approximate controllability of an abstract semilinear control system under the assumption which has a simple form and can be easily checked in many examples