nonlinear stability analysis for a class of differential integral systems arising from nuclear reactor dynamics a nonlinear stability analysis has been performed for mixed differential integral equations particularly applicable to nuclear reactors new general theorems of stability in bounded domains of initial perturbations have been introduced for a class of reactor models with an arbitrary reactivity feedback also effective stability criteria have been established for reactors with linear reactivity feedbacks for many kinds of models of phenomena typically encountered in nuclear reactors these new criteria give sharper stability bounds than criteria previously published the theoretical results obtained have been illustrated with examples of selected reactor models

almost invariant subspaces an approach to high gain feedback design part ii almost conditionally invariant subspaces in part i of this paper almost controlled invariant subspaces were studied in this part we will consider their duals the almost conditionally invariant subspaces these concepts give immediately by dualization of the almost disturbance decoupling control by state feedback the solution of the almost disturbance decoupled estimation problem finally we consider the problem of approximate disturbance decoupling by measurement feedback and it is shown that this problem is solvable to any arbitrary degree of accuracy if and only if 1 almost disturbance decoupling by state feedback and 2 almost disturbance decoupled estimation of the to be controlled output are both possible

stability of discrete bilinear systems with time delayed feedback functions the stability of discrete time varying bilinear systems is studied the control u t in the bilinear systems is considered as output feedback functions with time delay i e u t f y t y t 1 y t r 1 which is an important case and is absent in the literature also the authors assumed that the feedback function f is of larger classes than the classes given by current literature the sufficient conditions derived in theorems in the paper depend only on the coefficient matrices of the bilinear systems so that these results are convenient to check and to apply in engineering problems

on global solutions of the maxwell dirac equations we prove for the maxwell dirac equations in 1 3 dimensions that modified wave operators exist on a domain of small entire test functions of exponential type and that the cauchy problem inr r3 has a unique solution for each initial condition att 0 which is in the image of the wave operator the modification of the wave operator which eliminates infrared divergences is given by approximate solutions of the hamilton jacobi equation for a relativistic electron in an electromagnetic potential the modified wave operator linearizes the maxwell dirac equations to their linear part

interaction equations for short and long dispersive waves abstract we show the time local well posedness for a system of nonlinear dispersive equations for the water wave interaction formula it is shown that for any initial data u0 v0 hs r hs 1 2 r s 0 the solution for the above equation uniquely exists in a subset ofc t t hs c t t hs 1 2 and depends continuously on the data by virtue of a special structure of the nonlinear coupling the solution is stable under a singular limiting process

on the derivative nonlinear schro dinger equation abstract in this paper we discuss the cauchy problem for the derivative nonlinear schrodinger equation i t 2i x 2 0 0 x x where 0 under an explicit smallness condition of the initial data we prove the unique global existence of solutions to this problem in the usual sobolev spaces in the weighted sobolev spaces and in the schwartz class we describe the smoothing effect in detail furthermore for the data decaying exponentially at infinity we prove that the above equation has unique local solutions which are analytic in the space direction

stabilizing controllers for bilinear systems the problem of finding stabilizing feedback controllers for continuous bilinear systems where the controls act additively and multiplicatively simultaneously is treated the applicability of the quadratic control law of jacobson 5 and others is extended to the case when the a matrix has arbitary eigenvalues under certain conditions a class of bilinear systems common among biochemical flow systems is defined the dyadic bilinear systems a control scheme the so called division controller for dyadic bilinear systems is suggested the practicality of the control schemes is demonstrated on the problem of controlling the neutron level in a fission reactor

novel all pass factorization all solutions to rational matrix equation and control application for an arbitrary rational matrix rm a new descriptor realization of its all pass factorization is presented the coefficients of the factors are expressed explicitly in the solution of a linear matrix inequality which is well known in control theory using the all pass factorization necessary and sufficient conditions for existence of a solution of an equation with rms where the unknown rm has its poles in the left closed complex half plane and it can be improper are presented as well as all solutions as an application of the latter result the notion of almost disturbance decoupling i e when the infimum of the mathscr h infty norm of the transfer matrix from the disturbance input to the controlled output over the stabilizing controllers is zero is generalized the generalization is such that instead of the whole frequency axis the frequency axis with excluded intervals of small lengths is considered the controller consists of output feedback plus disturbance feedforward designed independently a necessary and sufficient solvability condition is presented which is expressed in terms of the given data the three main results are illustrated by an example

nonexistence of asymptotically free solutions for a nonlinear schrodinger equation let u be a nontrivial smooth solution to iut u u p 1u if n 1 and 2 p 3 then there does not exist any finite energy free solution v such that u t v t 2 0 as t this extends a theorem of strauss in which the same result was proved for 1 p 2

on a stability theorem for nonlinear systems with slowly varying inputs recently m kelemen ieee trans automat contr vol ac 31 766 768 aug 1986 presented a stability result that deals with the response of a nonlinear system to slowly varying input signals in this work a proof is given under a weaker hypothesis on the input signal by constructing a lyapunov function that is standard in the control literature this proof may be more accessible than the other approaches for completeness brief proofs of some intermediate results that are regarded as well known are included a simple example is given to show that this approach can yield explicit bounds in certain situations of interest

adaptive suboptimal filtering of bilinear systems the optimal filter minimum mean square error of discrete bilinear stochastic systems with output feedback is studied here the sequential filter of bilinear systems is derived for suboptimal adaptive estimation of the unknown a priori state and observation noise statistics simultaneously with the bilinear system state the unbiased estimations of state noise variance q and observation noise variance r are obtained under some usual conditions for on line operation this paper gives the recursive form of this adaptive suboptimal filter asf computer simulations show that asf approximates the minimum mean square error mse filter very well and asf provides improved state estimates at little computing expense

global existence of solutions for a subcritical nonlinear schrodinger equation abstract we consider the one dimensional nonlinear schrodinger equation i u t 1 2 u x x n u u x r t 1 u 1 x u 0 x x r where the nonlinearity n u u u 2 u 3 u 3 2 u 2 the exponent 0 is sufficiently small our purpose in this paper is to prove a global existence in time of small solutions

stability of discrete time bilinear systems with constant inputs in this paper discrete time bilinear systems are considered necessary and sufficient conditions are developed which ensure that for any constant input in an unbounded control domain the trajectories of the system tend to the origin as time increases

almost disturbance decoupling for single input single output nonlinear systems the almost disturbance decoupling problem for nonlinear single input single output systems is addressed by using singular perturbation methods and high gain feedback sufficient conditions and the explicit high gain nonlinear state feedback in solvable cases are given they generalize both previous almost results for linear systems and exact ones for nonlinear systems the necessity of the conditions is discussed in particular an example is given where the main structural condition is not satisfied and the high gain control designed on the basis of linear approximations fails to achieve almost disturbance decoupling for the original system

the extended j spectral factorization for descriptor systems in this paper we study an extended j spectral factorization problem i e j spectral factorization problem for descriptor systems which may not have full column normal rank and may exhibit poles and zeros on the extended imaginary axis we present necessary and sufficient solvability conditions and develop a numerically reliable method for the underlying problem our method is implemented by using only orthogonal transformations and has an acceptable computational complexity

new results and application of singular control we reconsider the singular control problem with marginal stability of the closed loop system when the transfer matrix from the input to the output can have linearly dependent columns and zeros on the extended imaginary axis we present a new theoretical existence result and a new numerical algorithm based on finding an orthogonal transformation of the matrix pencil associated to the euler lagrange differential equations into a block triangular form we present an application in linear quadratic control of descriptor systems under the constraints of physical realizability of the control and impulse free and marginally stable closed loop system

impulse controllability and observability of rectangular descriptor systems the presence of impulsive responses in descriptor systems and how it relates to impulse controllability and impulse observability is considered it is shown that the equivalence between impulse controllability observability and the existence of an impulse eliminating semistate feedback output injection gain although true for square descriptor systems does not hold for rectangular descriptor systems in general hence necessary and sufficient conditions for the existence of an impulse eliminating semistate feedback output injection gain are presented

global existence and asymptotic behavior of solutions for the maxwell schrodinger equations in three space dimensions in this paper we study the global existence and asymptotic behavior of solutions for the maxwell schrodinger equations under the coulomb gauge condition in three space dimensions with the final states given att this leads to the construction of the modified wave operator for certain scattered data it is also shown that for the initial data in the range of the modified wave operator the initial value problem of the maxwell schrodinger equations has the global solutions in time

a stability property in this note we obtain a stability property not for a point of equilibrium but for a certain family of asymptotically stable equilibria of a dynamical system with an input we show that this property depends not only on the initial condition but also on the time derivative of the input function we suggest possible applications of this result to some control problems

optimal small data scattering for the generalized derivative nonlinear schrodinger equations abstract in this work we consider the following generalized derivative nonlinear schrodinger equation i t u x x u i u 2 x u 0 t x r r we prove that when 2 the solution is global and scattering when the initial data is small in h s r 1 2 s 1 moreover we show that when 0 2 there exist a class of solitary wave solutions c satisfying c h 1 r 0 when c tends to some endpoint which is against the small data scattering statement therefore in this model the exponent 2 is optimal for small data scattering we remark that this exponent is larger than the short range exponent and the strauss exponent

constructive solutions to spectral and inner outer factorizations with respect to the disk we use state space realizations to solve the spectral and inner outer factorization problems with respect to the unit disk formulated for a completely general rational matrix the algorithm is based on orthogonal transformations and standard reliable procedures for solving stein and riccati equations the formulas apply in particular to general descriptor systems or polynomial matrices the main novelty is that we allow for arbitrary rank poles and zeros on the unit circle or at infinity

robust stabilization of uncertain systems in this paper we consider the systems described by dx axdt budt sum i sigma i f i xd beta i qquad text or qquad dot x ax bu sum i b i f i x t c i x and we will derive conditions under which there exists a feedback control law u kx such that the closed loop system is stable for all sigma i or smooth nonlinearities f i the nonlinearities f i and the noisy gains sigma i d beta i are unknown uncertainties in the system and the problem considered is to obtain a control law which is robust against these uncertainties as far as stability is concerned

stabilization of a class of singularly perturbed bilinear systems the lyapunoy stability of a form of singularly perturbed bilinear system is investigated a theorem is proposed to show how to calculate the stabilizing controller for the whole system from a knowledge of the stability of the reduced order model a numerical example is given

stability of discrete deterministic and stochastic nonlinear systems abstract in this paper we derive some sufficient conditions for local asymptotic stability and instability for discrete nonlinear deterministic and stochastic systems these theorems are easy to apply to specific discrete nonlinear systems some applications of our theorems to bilinear systems illustrate the above

algorithms for optimal control with invariant zeros on the extended imaginary axis abstract we transform an control problem with invariant zeros on the extended imaginary axis into a regular problem then using the nevanlinna pick theorem we find necessary and sufficient existence conditions for the original control problem an algorithm and examples with invariant zeros on the extended imaginary axis are given by the same approach we also solve the singular 2 control problem

a quadratic nonlinear schrodinger equation in one space dimension abstract in this paper we study the cauchy problem for the quadratic derivative nonlinear schrodinger equation iu t u xx u x 2 t x r 2 u 0 x u 0 x r we suppose that the initial data are small in the weighted sobolev space h 3 1 r then we prove that there exist global in time small solutions to the cauchy problem we study the large time behavior of solutions and construct the modified asymptotics for large time values

computation of general inner outer and spectral factorizations in this paper we solve two problems in linear systems theory the computation of the inner outer and spectral factorizations of a continuous time system considered in the most general setting we show that these factorization problems rely essentially on solving for the stabilizing solution a standard algebraic riccati equation of order usually much smaller than the mcmillan degree of the transfer function matrix of the system the proposed procedures are completely general being applicable for a polynomial proper improper system whose transfer function matrix could be rank deficient and could have poles zeros on the imaginary axis or at infinity as an application we discuss the extension to the case of rational matrices of the complete orthogonal decomposition of a constant matrix numerical refinements and examples illustrating the proposed approach are discussed in detail

long range scattering for nonlinear schrodinger equations in one space dimension we consider the scattering problem for the nonlinear schrodinger equation in 1 1 dimensions open image in new window where x r 0 r p 3 we show that modified wave operators for exist on a dense set of a neighborhood of zero in the lebesgue spacel2 r or in the sobolev spaceh1 r the modified wave operators are introduced in order to control the long range nonlinearity u 2u

stabilization analysis of discrete nonlinear systems in this paper we set a new framework for the analysis of nonlinear discrete systems to carry out the stabilization theorem and to design stable controllers of the systems within that framework this framework includes many practically important systems described for example by lipschitz functions functions with arbitrary order bounded derivatives bilinear functions and all polynomial functions an important existence theorem for stabilizing has been proven for this framework the theorem basically states that under definite rather broad conditions the zero solution of the plants can be made stable by a nonlinear output feedback in addition this nonlinear feedback is from a larger class of functions than these classes given by current literature following these stabilization theorems a new design procedure for the output feedback control law is presented all results presented are easy to use and convenient to apply the application is illustrated by several examples including the design of stabilizing control for a bilinear control system

final state problem for korteweg de vries type equations we study the final state problem for the korteweg de vries type equations ut 1 x 1ux u2ux t x r r u t fs t l2 0 as t where r the function fs t we call a final state defined by the final data u we show that there does not exist a nontrivial solution of this equation in the case of fs t u t u where u t is the free evolution group of this equation we construct the modified wave operator for the korteweg de vries type equations under the conditions that the final data u arc real valued functions and the fourier transform u vanishes at the origin

the output stabilization of siso bilinear systems a quite general approach to the problem of stabilizing a siso single input single output bilinear system by output feedback is presented following a lyapunov like method a family of admissible controls is defined and an easy way for deriving its general features is proposed previous results can be obtained as special cases of this approach numerical examples are included to show the feasibility of this algorithm in determining simple control laws

brief paper almost disturbance decoupling and pole placement we revisit here the almost disturbance decoupling problem addp willems 1981 by state feedback with the objective to solve addp and simultaneously place the maximal number of poles in the closed loop solution indeed when addp is solvable we show that whatever be the choice of a particular feedback solution the obtained closed loop system always has a set of fixed poles we characterize these fixed poles of addp the other non fixed poles can be placed freely and we characterize the optimal solutions in terms of ad hoc subspaces and feedbacks which allow us to solve addp with maximal pole placement from our contribution which treats the most general case for studying addp with maximal usually partial pole placement directly follow the solutions of addp with complete pole placement when there are no addp fixed poles and addp with internal stability when all the fixed poles of addp are stable without requiring the use of stabilizability subspaces as in willems 1981 we extend the concept of self bounded controlled invariant subspaces basile marro 1992 to almost ones an example is proposed that illustrates our contributions

stability analysis of bilinear systems the stability of the differential bilinear systems is studied the authors consider time varying bilinear systems with output feedback the input or control u t is not only a signal but also an input with output feedback i e u t f y t it is assumed that the feedback function f is in a function space which contains the class of linear functions functions satisfying the lipschitz condition and quadratic functions

robustness of feedback stabilized systems in the presence of non linear and random perturbations the general robustness problem is discussed for a controlled discrete time system subject to a lur e type feedback non linearity together with random multiplicative noises effective on state input and the non linearity a sufficient condition is given utilizing the discrete stochastic version of the second method of lyapunov to guarantee that any stabilizing controller designed for the deterministic linear part of the system renders the closed loop system with perturbations asymptotically stable with probability one then specializations to systems given in controllability and controller canonical forms are made independent noise signals are assumed to be effective on individual parameters of these canonical forms whose inputs are designed to assign the closed loop poles to zero in these cases stronger results are obtained in terms of system parameters and known characteristics of perturbations

final state problem for the cubic nonlinear klein gordon equation we study the final state problem for the nonlinear klein gordon equation utt u uxx u3 t r x r where r we prove the existence of solutions in the neighborhood of the approximate solutions 2 re u t w t where u t is the free evolution group defined by u t f 1e it f x 1 x2 f and f 1 are the direct and inverse fourier transformations respectively and w t x f 1 u e 3 2 i 2 u 2log t with a given final data u is a real valued function and 3 i u l is small

ott sudan ostrovskiy equation on a half line abstract we consider the initial boundary value problem for the ott sudan ostrovskiy equation on a half line we study traditionally important problems of the theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time

large time asymptotics of solutions to the generalized benjamin ono equation we study the asymptotic behavior for large time of solutions to the cauchy problem for the generalized benjamin ono bo equation ut u 1u x huxx 0 where h is the hilbert transform x t r when the initial data are small enough if the power of the nonlinearity is greater than 3 then the solution of the cauchy problem has a quasilinear asymptotic behavior for large time in the case 3 critical for the asymptotic behavior i e for the modified benjamin ono equation we prove that the solutions have the same l time decay as in the corresponding linear bo equation also we find the asymptotics for large time of the solutions of the cauchy problem for the bo equation in the critical and noncritical cases for the critical case we prove the existence of modified scattering states if the initial function is sufficiently small in certain weighted sobolev spaces these modified scattering states differ from the free scattering states by a rapidly oscillating factor

kalman yakubovic popov lemma for descriptor systems abstract for a given descriptor realization of para hermitian rational matrix s we present a generalization of kalman yakubovic popov lemma i e necessary and sufficient conditions for 0 on the imaginary axis in terms of an inequality with constant matrices the result is quite general since can have poles and zeros on the extended imaginary axis hence the nonstrict inequality j 0 r can hold instead of the strict inequality can be singular the descriptor realization is required to be only impulse controllable and controllable or stabilizable and det 0 a spectral factorization of is given by the above mentioned constant matrices three consequences of the generalized kyp lemma and an illustrative numerical example are given

asymptotics for large time of solutions to the nonlinear schrodinger and hartree equations we study the asymptotic behavior in time of solutions to the cauchy problems for the nonlinear schrodinger equation with a critical power nonlinearity and the hartree equation we prove the existence of modified scattering states and the sharp time decay estimate in the uniform norm of solutions to the cauchy problem with small initial data this estimate is very important for the proof of the existence of modified scattering states to the nonlinear schrodinger equations with a critical nonlinearity and the hartree equation in order to derive the desired estimates we introduce a certain phase function since the previous methods based solely on a priori estimates of the operator x it acting on the solution without specifying any phase function do not work for the critical case under consideration the well known nonexistence of the usual l 2 scattering states shows that our result is sharp

h optimization without assumptions on finite or infinite zeros explicit algebraic conditions are presented for the suboptimality of some parameter in the h infty optimization problem by output measurement control apart from two strict properness conditions no artificial assumptions restrict the underlying system in particular the plant may have zeros on the imaginary axis or at infinity these suboptimality characterizations are applied to show how to compute the optimal value by quadratically convergent algorithms and to solve the almost disturbance decoupling problem with internal stability

existence of solutions for schrodinger evolution equations we study the existence uniqueness and regularity of the solution of the initial value problem for the time dependent schrodinger equationi u t 1 2 u v t x u u 0 u0 we provide sufficient conditions onv t x such that the equation generates a unique unitary propagatoru t s and such thatu t s u0ec1 l2 c0 h2 n foru0eh2 n the conditions are general enough to accommodate moving singularities of type x 2 n 4 or x n 2 n 3

inhomogeneous neumann initial boundary value problem for the nonlinear schrodinger equation abstract we consider the inhomogeneous neumann initial boundary value problem for the nonlinear schrodinger equation formulated on a half line we study traditionally important problems of the theory of nonlinear partial differential equations such as global in time existence of solutions to the initial boundary value problem and the asymptotic behavior of solutions for large time

stability of discrete bilinear systems with output feedback abstract time varying discrete bilinear systems with output feedback are considered sufficient conditions ensuring the stability and asymptotic stability of discrete bilinear systems with output feedback are derived these sufficient conditions depend only on the coefficient matrices of the bilinear systems thus they are convenient to check and easy to apply in engineering practice

on asymptotically stabilizing feedback control of bilinear systems asymptotically stabilizing feedback controls for a class of bilinear systems are discussed a simple modification which strengthens a result of slemrod 1 is noted and illustrated by example