multiple criterion control a convex programming approach the performance of linear dynamic systems with respect to a non linear increasing value function that aggregates convex functionals is investigated within a convex programming framework the methodology developed in this paper combines fundamental properties of convex sets in order to decompose the multicriterion control problem into a two level structure the upper level of this structure comprises only decision making aspects of the problem and involves the solution of a relaxed and static multicriterion problem the lower level of the structure comprises in turn only optimal control aspects of the problem the overall procedure is easily implemented the solution of the aggregated problem is obtained as the limit of a sequence of optimal control problems that preserve their original solution properties relationships between the approach adopted here and existing solution techniques are discussed the solutions of the classical minimax and salukvadze problems are obtained as special cases of the proposed approach

1 2 laplacian problems with exponential nonlinearity abstract by exploiting a suitable trudinger moser inequality for fractional sobolev spaces we obtain existence and multiplicity of solutions for a class of one dimensional nonlocal equations with fractional diffusion and nonlinearity at exponential growth

rearrangements of functions let g be measurable non negative functions on r and let tf be their equimeasurable symmetric decreasing rearrangements let f r r r be continuous and suppose that the associated rectangle function defined by f r f a c f b d f a d f b c for r x y r2 a x b c y d is non negative then f g d f tf g d where is lebesgue measure the concept of equimeasurable rearrangement is also defined for functions on a more general class of measure spaces and the inequality holds in the general case if f x y x y where is convex and 0 0 then we obtain tf g d g d in particular if x x p 1 p then we find that the operator s tf is a contraction on lp for 1 p

covariance controllers for linear continuous time systems abstract necessary and sufficient conditions are given for the existence of a controller of specified order n c which will assign a specified matrix value x to the covariance of the closed loop system if a solution exists the entire set of stabilizing controllers that assign x are parameterized by a certain skew symmetric matrix

error estimates for regularization methods in hilbert scales in this paper we study regularization methods to reconstruct the solution x of the linear ill posed problem ax y a x to y from noisy data y delta in y the regularization methods are of the general form x alpha delta bar x g alpha b s a a b s a y delta a bar x where b denotes an unbounded self adjoint strictly positive definite operator in the hilbert space x assuming ax sim x a and x bar x p le e for some bar x in x a ge 0 and p ge 0 x r b r 2 x is the norm in a hilbert scale x r r in rnus we derive error estimates which show that the accuracy of the above regularization methods is order optimal provided that the function g alpha lambda the regularization parameter alpha and the parameter s are chosen properly

lqg control with an h sup infinity performance bound a riccati equation approach an lqg linear quadratic gaussian control design problem involving a constraint on h sup infinity disturbance attenuation is considered the h sup infinity performance constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on l sub 2 performance in contrast to the pair of separated riccati equations of standard lqg theory the h sup infinity constrained gains are given by a coupled system of three modified riccati equations the coupling illustrates the breakdown of the separation principle for the h sup infinity constrained problem both full and reduced order design problems are considered with an h sup infinity attenuation constraint involving both state and control variables an algorithm is developed for the full order design problem and illustrative numerical results are given

h 2 optimal control with an h constraint the state feedback case in this paper we consider a mixed hz optimal control problem it is assumed that the plant as well as the feedback controller are finite dimensional and linear time invariant and that the plant state is available for feedback more specifically among all the state feedback controllers that minimize the he norm of a closed loop transfer matrix we give necessary and sufficient conditions for the existence of a controller that also satisfies a prescribed h norm bound on some other closed loop transfer matrix when these conditions are met the solution to the above problem is also a global solution to the contrained optimization problem of minimizing an he norm performance measure subject to an h norm constraint we also give state space formulae for computing the solutions some easily checkable sufficient conditions for the existence of solutions are given finally we give an example in which all solutions to the constrained optimization problem are necessarily dynamic i e there is no static gain solution even though plant state is available for feedback

singular trudinger moser inequality and fractional p laplace equations in rn abstract in this paper we establish a singular trudinger moser inequality in r n applying that result we study the existence of solution to n s fractional laplacian equation l n s s u x v x u n s 2 u f x u x where 0 n 0 s 1 l n s s is a nonlocal fractional operator with singular kernel k r n 0 r f is a continuous function on r n r which does not satisfy the ambrosetti rabinowitz condition by using mountain pass theorem we obtain the existence of solution to above equation furthermore when f satisfies the ambrosetti rabinowitz condition we obtain the existence to solution of that equation with negative energy in our best knowledge it is the first time the existence to solution of above equation is studied

mathcal y scale regularization many inverse problems are ill posed in the sense that their solution is unstable with respect to data perturbations and hence regularization methods have to be used for their stable solution two possible drawbacks of standard regularization methods are a saturation i e only suboptimal approximations can be found for smooth solutions this is the case e g for tikhonov regularization b numerical effort e g the large number of iterations of methods like landweber iteration a framework that allows us to overcome both effects at least for certain classes of inverse problems is regularization in hilbert scales where a solution is searched in a scale of spaces hilbert scale but convergence is achieved in the original space regularization methods in hilbert scales can be viewed as modified preconditioned versions of standard methods in order to make the advantages of the hilbert scale approach applicable to a new class of problems we propose to use a scale of spaces over the image space instead this results in a new family of methods which we call mathcal y scale regularization and whose optimal convergence properties are analyzed one of the key steps in the analysis is the formulation of an adequate a posteriori stopping rule and the proof of optimal convergence rates the theoretical results are illustrated in several numerical examples

further convergence results on the general iteratively regularized gauss newton methods under the discrepancy principle we consider the general iteratively regularized gauss newton methods xk 1 x0 g k f xk f xk f xk f xk y f xk xk x0 for solving nonlinear inverse problems f x y using the only available noise y of y satisfying y y with a given small noise level 0 in order to produce reasonable approximation to the sought solution we terminate the iteration by the discrepancy principle under much weaker conditions we derive some further convergence results which improve the existing ones and thus expand the applied range

an iterative multi level algorithm for solving nonlinear ill posed problems the convergence analysis of landweber s iteration for the solution of nonlinear ill posed problem has been developed recently by hanke neubauer and scherzer in concrete applications sufficient conditions for convergence of the landweber iterates developed there although quite natural turned out to be complicated to verify analytically however in numerical realizations when discretizations are considered sufficient conditions for local convergence can usually easily be stated this paper is motivated by these observations initially a discretization is fixed and a discrete landweber iteration is implemented until an appropriate stopping criterion becomes active the output is used as an initial guess for a finer discretization an advantage of this method is that the convergence analysis can be considered in a family of finite dimensional spaces the numerical performance of this multi level algorithm is compared with landweber s iteration

state space solutions to hadamard weighted ℋ and ℋ 2 control problems variations of the standard h and h 2 control problems are presented state space solutions are given for the problems of weighting the individual elements of a closed loop transfer function while i satisfying an h frobenius norm bound or ii minimizing the h 2 norm of the closed loop the optimal h frobenius case is analysed and it is shown that the optimal controller is often unique in contrast to the standard h control problem the solutions of an algebraic riccati equation are examined in the h 2 case to reveal the structure of the h 2 optimal controller

stationary kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity this paper deals with the existence and the asymptotic behavior of non negative solutions for a class of stationary kirchhoff problems driven by a fractional integro differential operator lk and involving a critical nonlinearity in particular we consider the problem m u 2 lku f x u u 2s 2uin u 0in rn where rn is a bounded domain 2s is the critical exponent of the fractional sobolev space hs rn the function f is a subcritical term and is a positive parameter the main feature as well as the main difficulty of the analysis is the fact that the kirchhoff function m could be zero at zero that is the problem is degenerate the adopted techniques are variational and the main theorems extend in several directions previous results recently appeared in the literature

adaptive differential dynamic programming for multiobjective optimal control an efficient numerical solution scheme entitled adaptive differential dynamic programming is developed in this paper for multiobjective optimal control problems with a general separable structure for a multiobjective control problem with a general separable structure the optimal weighting coefficients for various performance indices are time varying as the system evolves along any noninferior trajectory recognizing this prominent feature in multiobjective control the proposed adaptive differential dynamic programming methodology combines a search process to identify an optimal time varying weighting sequence with the solution concept in the conventional differential dynamic programming convergence of the proposed adaptive differential dynamic programming methodology is addressed

the generalized h 2 control problem abstract in this paper we consider the problem of finding an internally stabilizing controller such that the controlled or regulated signals have a guaranteed maximum peak value in response to arbitrary but bounded energy exogenous inputs more specifically we give a complete solution to the problem of finding a stabilizing controller such that the closed loop gain from l 2 0 to l 0 is below a specified level we consider both state feedback and output feedback problems in the state feedback case it is shown that if this synthesis problem is solvable then a solution can be chosen to be a constant state feedback gain necessary and sufficient conditions for the existence of solutions as well as a formula for a state feedback gain that solves this control problem are obtained in terms of a finite dimensional convex feasibility program after showing the separation properties of this synthesis problem the output feedback case is reduced to a state feedback problem it is shown that in the output feedback case generalized h 2 controllers can be chosen to be observer based controllers the theory is demonstrated with a numerical example

a multi grid method with a priori and a posteriori level choice for the regularization of nonlinear ill posed problems in this paper we study a multi grid method for the numerical solution of nonlinear systems of equations arising from the discretization of ill posed problems where the special eigensystem structure of the underlying operator equation makes it necessary to use special smoothers we provide uniform contraction factor estimates and show that a nested multigrid iteration together with an a priori or a posteriori chosen stopping index defines a regularization method for the ill posed problem i e a stable solution method that converges to an exact solution of the underlying infinite dimensional problem as the data noise level goes to zero with optimal rates under additional regularity conditions

alternating convex projection methods for covariance control design the problem of designing a static state feedback or full order dynamic controller is formulated as a problem of designing an appropriate plant state covariance matrix we show that closed loop stability and multiple output norm constraints imply that the plant state covariance matrix lies at the intersection of some specified closed convex sets in the space of symmetric matrices we address the covariance feasibility problem to determine the existence and compute a covariance matrix to satisfy assignability and output norm performance constraints we address the covariance optimization problem to construct an assignable covariance matrix which satisfies output performance constraints and is as close as possible to a given desired covariance we also treat inconsistent constraints where we look for an assignable covariance matrix which best approximates desired but non achievable output performance objectives we call this the infeasible covariance optimization problem all these problems are of a conve

towards a general convergence theory for inexact newton regularizations we develop a general convergence analysis for a class of inexact newton type regularizations for stably solving nonlinear ill posed problems each of the methods under consideration consists of two components the outer newton iteration and an inner regularization scheme which applied to the linearized system provides the update in this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including landweber and steepest decent iterations iterated tikhonov method and method of conjugate gradients

a dynamical system method for solving nonlinear ill posed problems this paper investigates the dynamical systems for solving nonlinear ill posed problems and uses lyapunov theory which is constructed in this paper to prove the stability of the solution of the dynamical system problems finally numerical experiments are presented with pc matlab

on the discrepancy principle for some newton type methods for solving nonlinear inverse problems we consider the computation of stable approximations to the exact solution x dagger of nonlinear ill posed inverse problems f x y with nonlinear operators f x y between two hilbert spaces x and y by the newton type methods x k 1 delta x 0 g alpha k left f x k delta f x k delta right f x k delta left f x k delta y delta f x k delta x k delta x 0 right of y satisfying y delta y le delta with a given small noise level delta 0 we terminate the iteration by the discrepancy principle in which the stopping index k delta is determined as the first integer such that f x k delta delta y delta le tau delta f x k delta y delta quad 0 le k k delta converges to x dagger as delta rightarrow 0 and establish various order optimal convergence rate results it is remarkable that we even can show the order optimality under merely the lipschitz condition on the frechet derivative f of f if x 0 x dagger is smooth enough

accelerated landweber iterations for the solution of ill posed equations in this paper the potentials of so calledlinear semiiterative methods are considered for the approximate solution of linear ill posed problems and ill conditioned matrix equations several efficient two step methods are presented most of which have been introduced earlier in the literature stipulating certain conditions concerning the smoothness of the solution a notion of optimal speed of convergence may be formulated various direct and converse results are derived to illustrate the properties of this concept if the problem s right hand side data are contaminated by noise semiiterative methods may be used asregularization methods assuming optimal rate of convergence of the iteration for the unperturbed problem the regularized approximations will be of order optimal accuracy to derive these results specific properties of polynomials are used in connection with the basic theory of solving ill posed problems rather recent results onfast decreasing polynomials are applied to answer an open question of brakhage numerical examples are given including a comparison to the method of conjugate gradients

on landweber iteration for nonlinear ill posed problems in hilbert scales in this paper we derive convergence rates results for landweber iteration in hilbert scales in terms of the iteration index k for exact data and in terms of the noise level delta for perturbed data these results improve the one obtained recently for landweber iteration for nonlinear ill posed problems in hilbert spaces for numerical computations we have to approximate the nonlinear operator and the infinite dimensional spaces by finite dimensional ones we also give a convergence analysis for this finite dimensional approximation the conditions needed to obtain the rates are illustrated for a nonlinear hammerstein integral equation numerical results are presented confirming the theoretical ones

the multiple linear quadratic gaussian problem in this paper we define and solve the multiple linear quadratic gaussian lqg problem for discrete time systems using salukvadze s ideal point method which treats dynamic but deterministic multicriteria optimization problems the solution consists in designing a scalar regulator where the weighting matrices are replaced by the sums of their counterparts in the individual criteria the resulting control is optimal in the sense that it minimizes the difference between the cost of the implemented policy from the cost that would occur if we optimized each criterion separately and added the outcomes all the nice properties of the scalar lqg regulator are enjoyed by the multicriteria regulator as well extension to the continuous time case is straightforward yielding identical results

inexact newton regularization methods in hilbert scales we consider a class of inexact newton regularization methods for solving nonlinear inverse problems in hilbert scales under certain conditions we obtain the order optimal convergence rate result

local identification of nonparametric and semiparametric models in parametric nonlinear structural models a classical sufficient condition for local identification like fisher 1966 and rothenberg 1971 is that the vector of moment conditions is differentiable at the true parameter with full rank derivative matrix we derive an analogous result for the nonparametric nonlinear structural models establishing conditions under which an infinite dimensional analog of the full rank condition is sufficient for local identification importantly we show that additional conditions are often needed in nonlinear nonparametric models to avoid nonlinearities overwhelming linear effects we give restrictions on a neighborhood of the true value that are sufficient for local identification we apply these results to obtain new primitive identification conditions in several important models including nonseparable quantile instrumental variable iv models and semiparametric consumption based asset pricing models

computation of generalized h 2 optimal controllers the generalized h 2 optimal control problem for a linear time invariant system is one in which the conventional h 2 norm is replaced by an operator norm the closed loop system is described in terms of a mapping between the space of time domain input disturbances in l 2 and the space of time domain regulated outputs in l a minimum of this norm is then sought over all stabilizing controllers it is shown that optimal controllers for such problems have the structure of a kalman filter with estimated state feedback where the feedback gains are obtained from the solution to a weighted lqr problem a computational algorithm is presented to determine the weights in this lqr problem and examples are given which demonstrate various problems which may arise in obtaining the optimal weights in particular it is shown that the generalized h 2 problem may involve the solution to a singular lqr problem

multiobjective h sub 2 h sub spl infin control for a linear time invariant system with several disturbance inputs and controlled outputs we show how to minimize the nominal h sub 2 norm performance in one channel while keeping bounds on the h sub 2 norm or h sub spl infin norm performance implying robust stability in the other channels this multiobjective h sub 2 h sub spl infin problem in an infinite dimensional space is reduced to sequences of finite dimensional convex optimization problems we show how to compute the optimal value and how to numerically detect the existence of a rational optimal controller if it exists we reveal how the novel trick of optimizing the trace norm of the youla parameter over certain convex constraints allows one to design a nearly optimal controller whose youla parameter is of the same order as the optimal one

on the order optimality of the regularization via inexact newton iterations inexact newton regularization methods have been proposed by hanke and rieder for solving nonlinear ill posed inverse problems every such a method consists of two components an outer newton iteration and an inner scheme providing increments by regularizing local linearized equations the method is terminated by a discrepancy principle in this paper we consider the inexact newton regularization methods with the inner scheme defined by landweber iteration the implicit iteration the asymptotic regularization and tikhonov regularization under certain conditions we obtain the order optimal convergence rate result which improves the suboptimal one of rieder we in fact obtain a more general order optimality result by considering these inexact newton methods in hilbert scales

a general theory for the wiener hopf design of multivariable control systems multivariable control system design in a quadratic cost setting is treated for a completely general configuration the class of all controllers is determined for which the general configuration is internally asymptotically stable and a quadratic cost functional is finite this controller class is parameterized in terms of an arbitrary real rational strictly proper matrix z s which is analytic in the closed right hand side of the s plane the choice z identical to 0 gives the controller which minimizes the quadratic cost functional

a theory of state covariance assignment for discrete systems in many regulation control problems it is desired to design the controller so that various system states have acceptable root mean squared values this is a multiple objective problem one way to satisfy these objectives is to assign a specified state covariance to the system this paper introduces and solves the following problem 1 characterize the entire set of state covariances which may be assigned to a linear discrete time system by state feedback and 2 find the set of all state feedback gains which will assign an admissible state covariance to the system extensions are also presented for state estimate feedback

iterative estimation of solutions to noisy nonlinear operator equations in nonparametric instrumental regression this paper discusses the solution of nonlinear integral equations with noisy integral kernels as they appear in nonparametric instrumental regression we propose a regularized newton type iteration and establish convergence and convergence rate results a particular emphasis is on instrumental regression models where the usual conditional mean assumption is replaced by a stronger independence assumption we demonstrate for the case of a binary instrument that our approach allows the correct estimation of regression functions which are not identifiable with the standard model this is illustrated in computed examples with simulated data

a convergence analysis of the landweber iteration for nonlinear ill posed problems in this paper we prove that the landweber iteration is a stable method for solving nonlinear ill posed problems for perturbed data with noise level delta we propose a stopping rule that yields the convergence rate o delta 1 2 under appropriate conditions we illustrate these conditions for a few examples

parameter selection rules for path following with singular embedding singular embedding methods require appropriately adjusted parameters to guarantee the contraction of locally quadratically convergent iterative methods firstly some general rule for the parameter selection is proposed and its rate of convergence is analyzed secondly some modifications in the case of polynomial error and contraction bounds are studied finally these results are applied to the embedding of an elliptic boundary value problem with discontinuous nonlinearities into a family of smooth problems here the regularization is done in such a way that the solutions of the resulting auxiliary problems require only one step of newton s method

p n equations with critical exponential nonlinearities in rn abstract the paper deals with the existence of nontrivial solutions for p n equations in r n with critical exponential growth the main features and novelty of the paper are the p n growth of the elliptic operator combined with the double lack of compactness in order to state a multiplicity result we introduce a tricky step analysis based on the application of a completely new brezis and lieb type lemma for exponential nonlinearities

a convergence analysis of regularization by discretization in preimage space in this paper we investigate the regularizing properties of discretization in preimage space for linear and nonlinear ill posed operator equations with noisy data we propose to choose the discretization level that acts as a regularization parameter in this context by a discrepancy principle while general convergence has been shown not to hold see 17 we provide convergence results under appropriate conditions on the exact solution

inexact newton regularization using conjugate gradients as inner iteration in our papers inverse problems 15 1999 pp 309 327 and numer math 88 2001 pp 347 365 we proposed algorithm tt reginn an inexact newton iteration for the stable solution of nonlinear ill posed problems tt reginn consists of two components the outer iteration which is a newton iteration stopped by the discrepancy principle and an inner iteration which computes the newton correction by solving the linearized system the convergence analysis presented in both papers covers virtually any linear regularization method as inner iteration especially landweber iteration nu methods and tikhonov phillips regularization in the present paper we prove convergence rates for tt reginn when the conjugate gradient method which is nonlinear serves as inner iteration thereby we add to a convergence analysis of hanke who had previously investigated tt reginn furnished with the conjugate gradient method numer funct anal optim 18 1997 pp 971 993 by numerical experiments we illustrate that the conjugate gradient method outperforms the nu method as inner iteration

pareto suboptimal controllers in multi objective disturbance attenuation problems abstract a multi objective disturbance attenuation problem is considered as a novel framework for control and filtering problems under multiple exogenous disturbances there are n potentially possible disturbance inputs of a system on each of which may act a disturbance from a certain class a disturbance attenuation level is defined for each channel as an induced norm of the operator mapping signals of the corresponding class to the objective output of the system necessary conditions of the pareto optimality are derived it is established that the optimal solutions with respect to a multi objective cost parameterized by weights from an n dimensional simplex are pareto suboptimal solutions and their relative losses compared to the pareto optimal ones do not exceed 1 n n these results are extended to the case when the disturbances acting on different inputs are combined into coalitions the approach is applied to multiple classes of l 2 bounded and impulsive disturbances for which the h 0 optimal controllers as the pareto suboptimal solutions are synthesized in terms of linear matrix inequalities lmis illustrative examples demonstrate the effectiveness of the approach proposed

successive method for general multiple linear quadratic control problem in discrete time a successive method is proposed for the general multiple linear quadratic control problem in discrete time a family of auxiliary parametric linear quadratic control problems is constructed such that its solution sequence converges to the optimal solution of the original problem theoretical analysis for the computational procedure and global convergence of the successive method is provided the successive method utilizes the special structure of the problem thus being computationally efficient and being able to furnish a closed loop optimal control law

optimal a posteriori parameter choice for tikhonov regularization for solving nonlinear ill posed problems the authors propose an a posteriors strategy for choosing the regularization parameter in tikhonov regularization for solving nonlinear ill posed problems and show that under certain conditions the convergence rate obtained with this strategy is optimal as a by product a new stability estimate for the regularized solutions is given which applies to a class of parameter identification problems the authors compare the parameter choice strategy with morozov s discrepancy principle finally numerical results are presented

multilevel dynamic programming for general multiple linear quadratic control in discrete time systems abstract this paper extends the reach of dynamic programming to general multiple linear quadratic control problems in discrete time systems where the overall objective is a nonlinear function of multiple quadratic performance indices a multilevel dynamic programming solution procedure is proposed to obtain an analytical linear control law for this nonseparable control problem at the lower level the formulated auxiliary lagrangian problem is of a parametric linear quadratic structure and is solved by dynamic programming the weighting vector in the auxiliary lagrangian problem is adjusted by the upper level iteratively this two level process repeats until an optimal condition is satisfied

multiple objective optimal control of linear systems the quadratic norm case control of a linear system to meet requirements embodied in multiple performance criteria is recognized as the typical design problem the authors study the design of control when every criterion of interest takes the form of an h sub 2 norm defined over certain exogenous inputs and controlled outputs of the system the objective is to find all stabilizing controllers that are noninferior with respect to the defined objectives that is to find the set of pareto optimal controllers the novel feature is the use of the youla parametrization to represent the admissible controllers this representation enables the authors to show that each criterion and hence their weighted sum is convex over the class of admissible controllers since the set of admissible controllers is itself convex the pareto optimal controllers can be determined by the standard scalarization technique it is concluded that in general the controller order is higher than the order of the plant

persistent inputs and the standard h 2 multivariable control problem the h 2 control problem is formulated with exogenous inputs having unstable shape deterministic components such inputs are called persistent some issues in connection with past solutions of the h 2 control problem in this case are carefully described and addressed in particular it is established that two and three degree of freedom 2dof and 3dof systems with persistent inputs cannot be treated within the framework of the standard configuration in addition past treatments of persistent inputs within the framework of the generalised 2dof configuration focused on solutions which yielded stable error transforms without explicitly requiring the same for controller outputs in this article a more general configuration is treated and physical considerations are invoked to justify imposition of the requirement that both the regulated variable z s and the controller output u s be stable a persistent input model for which there exists a stabilising controller that makes both z s and u s stable is cal

on general multiple linear quadratic control problems general multiple linear quadratic control problems are studied where the overall objective is a nonlinear function of multiple quadratic performance indices a multilevel solution procedure is proposed to obtain an analytical linear control law for this nonseparable control problem the auxiliary lagrangian problem formulated at the lower level is of a parametric linear quadratic structure in which the weighting vector is adjusted by the upper level until a derived optimal condition is met

on critical cases of sobolev s inequalities abstract we present a new form of the trudinger type inequality which shows an explicit dependence moreover we give an alternative proof of the brezis gallouet wainger inequality

on a regularized levenberg marquardt method for solving nonlinear inverse problems we consider a regularized levenberg marquardt method for solving nonlinear ill posed inverse problems we use the discrepancy principle to terminate the iteration under certain conditions we prove the convergence of the method and obtain the order optimal convergence rates when the exact solution satisfies suitable source wise representations

on steepest descent kaczmarz methods for regularizing systems of nonlinear ill posed equations we investigate modified steepest descent methods coupled with a loping kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill posed operator equations we show that the proposed method is a convergent regularization method numerical tests are presented for a linear problem related to photoacoustic tomography and a nonlinear problem related to the testing of semiconductor devices

bouligand landweber iteration for a non smooth ill posed problem this work is concerned with the iterative regularization of a non smooth nonlinear ill posed problem where the forward mapping is merely directionally but not g teaux differentiable using a bouligand subderivative of the forward mapping a modified landweber method can be applied however the standard analysis is not applicable since the bouligand subderivative mapping is not continuous unless the forward mapping is g teaux differentiable we therefore provide a novel convergence analysis of the modified landweber method that is based on the concept of asymptotic stability and merely requires a generalized tangential cone condition these conditions are verified for an inverse source problem for an elliptic pde with a non smooth lipschitz continuous nonlinearity showing that the corresponding bouligand landweber iteration converges strongly for exact data as well as in the limit of vanishing data if the iteration is stopped according to the discrepancy principle this is illustrated with a numerical example

a convergence analysis of nonlinear implicit iterative method for nonlinear ill posed problems abstract implicit iterative method acquires good effect in solving linear ill posed problems we have ever applied the idea of implicit iterative method to solve nonlinear ill posed problems under the restriction that is appropriate large we proved the monotonicity of iterative error and obtained the convergence and stability of iterative sequence numerical results show that the implicit iterative method for nonlinear ill posed problems is efficient in this paper we analyze the convergence and stability of the corresponding nonlinear implicit iterative method when k are determined by hanke criterion

control of level crossings in stationary gaussian random processes a new optimal stochastic control problem that minimizes the intensity for a signal to upcross a level is solved by rewriting it as a one parametric optimization problem over a set of lqg problem solutions the solution can sometimes be thought of as finding optimal weightings in an lqg problem

a homotopy method for nonlinear inverse problems we develop a homotopy method for nonlinear inverse problems where the forward problems are governed by some forms of differential equations a tikhonov style regularization approach yields an optimization problem ordinary iterative methods may fail to solve this problem due to their locally convergent properties then the fixed point homotopy method is introduced to solving the normal equation of the optimization problem and a new and globally convergent algorithm is constructed which is highly effective in the aspects of speed of computation ability of noise suppression and wide region of convergence as a practical application the method is used to solve the inverse problem of 2 d acoustic wave equation we demonstrate the merits and effectiveness of our algorithm on two realistic model problems

fractional adams moser trudinger type inequalities abstract extending several works we prove a general adams moser trudinger type inequality for the embedding of bessel potential spaces h n p p into orlicz spaces for an arbitrary domain with finite measure in particular we prove sup u h n p p n 2 p u l p 1 e n p u p p 1 d x c n p for a positive constant n p whose sharpness we also prove we further extend this result to the case of lorentz spaces i e n 2 p u l p q the proofs are simple as they use green functions for fractional laplace operators and suitable cut off procedures to reduce the fractional results to the sharp estimate on the riesz potential proven by adams and its generalization proven by xiao and zhai we also discuss an application to the problem of prescribing the q curvature and some open problems

a critical kirchhoff type problem involving a nonlocal operator abstract in this paper we show the existence of non negative solutions for a kirchhoff type problem driven by a nonlocal integrodifferential operator that is m u z 2 l k u f x u u 2 2 u in u 0 in r n where l k is an integrodifferential operator with kernel k is a bounded subset of r n m and f are continuous functions z is a functional norm and 2 is a fractional sobolev exponent

on convergence rates of inexact newton regularizations reginn is an algorithm of inexact newton type for the regularization of nonlinear ill posed problems inverse problems 15 1999 pp 309 327 in the present article convergence is shown under weak smoothness assumptions source conditions moreover convergence rates are established some computational illustrations support the theoretical results

the structured h2 optimization problem in this note the structured h 2 optimization problem is defined and two methods are outlined for its solution the first method involves a vectorization kronecker product technique and reduces the optimization problem to two nested projections to circumvent the dimension inflation implicit in this method an iterative algorithm is proposed as an alternative this is an application of von neumann s alternating projection algorithm in hilbert space and can be shown to be applicable to the present problem under two standard rank assumptions a short version of this paper has appeared in halikias 1993

on multiple criteria stationary linear quadratic control multiple control criteria and design specifications are in general present in control systems design in this study all pareto optimal solutions are characterized for a large class of multiple criteria stationary linear quadratic control problems in terms of associated scalar criterion linear quadratic control problem solutions based on a supporting hyperplane argument and thus the results are fully constructive this argument which was used to establish these results is believed to be new

existence of solutions for kirchhoff type problem involving the non local fractional p laplacian abstract the purpose of this paper is to investigate the existence of weak solutions for a kirchhoff type problem driven by a non local integro differential operator of elliptic type with homogeneous dirichlet boundary conditions as follows m r 2 n u x u y p k x y d x d y l k p u f x u in u 0 in r n where l k p is a non local operator with singular kernel k is an open bounded subset of r n with lipshcitz boundary m is a continuous function and f is a caratheodory function satisfying the ambrosetti rabinowitz type condition we discuss the above mentioned problem in two cases when f satisfies sublinear growth condition the existence of nontrivial weak solutions is obtained by applying the direct method in variational methods when f satisfies suplinear growth condition the existence of two nontrivial weak solutions is obtained by using the mountain pass theorem

multiplicity of solutions for p x polyharmonic elliptic kirchhoff equations abstract in this paper we establish the existence of an unbounded sequence of solutions for a class of quasilinear elliptic p x polyharmonic kirchhoff equations including the new delicate degenerate case not yet covered in the literature the main tool is the symmetric mountain pass theorem of ambrosetti and rabinowitz

h sub spl infin multiple objective robust controllers for infinite horizon single measurement single control input problems a graphical method is introduced that solves the robust infinite horizon h sub spl infin multiple objective control problem for single measurement single control input systems the solution is obtained by describing boundaries on the nichols chart each boundary defines the set of all admissible gain and phase values for the loop transmission at a given frequency these boundaries are obtained by using the well known parameterization of all the solutions for a single objective h sub spl infin control problem the new method links between the theories of h sub spl infin and quantitative feedback theory qft it can be used to design robust h sub spl infin controllers with almost no overdesign and it provides a convenient solution of h sub spl infin multiple objective problems that are difficult to solve by the standard four block setting it also extends the methods of siso qft to deal with a vector of disturbances the latter may affect the controlled plant through any input coupling matrix and not necessarily through the controller input or the measurement output

control of mean time between failures a new optimal stochastic control problem is posed the criterion is to maximize the mean time between failures criterion given a certain reference value it is shown that this control problem is closely related both to the problem of minimizing the variance of the signal minimum variance control and to the problem of minimizing the so called upcrossing probability minimum upcrossing control it is made plausible that the upcrossing probability is a better approximating criterion to minimize than the variance criterion the problem of minimizing the upcrossing probability can be thought of as finding optimal weighting matrices in an lqg problem the new controller is compared with the minimum variance controller for a first order process it is seen that the new controller causes a lower upcrossing probability and a larger mean time between failures the improvement in the example is up to about 25 this makes it possible to choose the reference value closer to the critical level without causing smaller

existence and multiplicity of solutions to equations of n laplacian type with critical exponential growth in rn in this paper we deal with the existence of solutions to the nonuniformly elliptic equation of the form 0 1 div a x u v x u n 2u f x u x eh x in rn when f rn r r behaves like exp u n n 1 when u and satisfies the ambrosetti rabinowitz condition in particular in the case of n laplacian i e a x u u n 2 u we obtain multiplicity of weak solutions of 0 1 moreover we can get the nontriviality of the solution in this case when e 0 finally we show that the main results remain true if one replaces the ambrosetti rabinowitz condition on the nonlinearity by weaker assumptions and thus we establish the existence and multiplicity results for a wider class of nonlinearity see section 7 for more details

mimo nonlinear ultrasonic tomography by propagation and backpropagation method this paper develops a fast ultrasonic tomographic imaging method in a multiple input multiple output mimo configuration using the propagation and backpropagation pbp method by this method ultrasonic excitation signals from multiple sources are transmitted simultaneously to probe the objects immersed in the medium the scattering signals are recorded by multiple receivers utilizing the nonlinear ultrasonic wave propagation equation and the received time domain scattered signals the objects are to be reconstructed iteratively in three steps first the propagation step calculates the predicted acoustic potential data at the receivers using an initial guess second the difference signal between the predicted value and the measured data is calculated third the backpropagation step computes updated acoustical potential data by backpropagating the difference signal to the same medium computationally unlike the conventional pbp method for tomographic imaging where each source takes turns to excite the acoustical field until all the sources are used the developed mimo pbp method achieves faster image reconstruction by utilizing multiple source simultaneous excitation furthermore we develop an orthogonal waveform signaling method using a waveform delay scheme to reduce the impact of speckle patterns in the reconstructed images by numerical experiments we demonstrate that the proposed mimo pbp tomographic imaging method results in faster convergence and achieves superior imaging quality

multiple solutions for n kirchhoff type problems with critical exponential growth in rn abstract in this paper we consider a class of nonlocal n kirchhoff type problems involving a nonlinearity term having critical exponential growth by applying variational methods together with trudinger moser inequality in whole r n we obtain the existence of at least two positive solutions

multiple objective optimization approach to adaptive and learning control this paper formulates a new approach to the classical learning adaptive control problem our approach is based on two key observations 1 the inherent conflict between control and identification as they compete for the only available resource namely the input to the plant 2 when designing and optimizing the performance of a control system the current task as well as the repertoire of other typical future tasks which the system may encounter during its life time should be considered our approach is formulated for a general nonlinear time varying plant thus unlike existing adaptive control theory the theory for a linear time invariant system evolves as a special case of the general case the design for the full lifetime of the system creates a methodology that specifies what current actions should be taken in addition to the tracking of the current reference trajectory at the expense of some performance degradation in the current task so as to improve the performance of future tasks this is

performance first control for discrete time lqg problems performance first control for discrete time lqg is considered in this paper to minimize the probability that the performance index exceeds a preselected threshold via constructing a closed loop feedback control law this problem can be converted into a mean variance control problem which can be solved by developing a nested form of the variance and using polynomial optimization as a solution scheme

a convergent algorithm for the output covariance constraint control problem this paper considers the optimal control problem of minimizing control effort subject to multiple performance constraints on output covariance matrices y i of the form y i leq overline y i where overline y i is given the contributions of this paper are a set of conditions that characterize global optimality and an iterative algorithm for finding a solution to the optimality conditions this iterative algorithm is completely described up to a user specified parameter we show that under suitable assumptions on problem data the iterative algorithm converges to a solution of the optimality conditions provided that this parameter is properly chosen both discrete and continuous time problems are considered

regularization in model updating this paper presents the theory of sensitivity based model updating with a special focus on the properties of the solution that result from the combination of optimization of the response prediction with a priori information about the uncertain parameters model updating together with the additional regularization criterion is an optimization with two objective functions and must be linearized to obtain the solution structured solutions are obtained based on the generalized singular value decomposition gsvd and specific features of the parameter and response paths as the regularization parameter varies are explored the four different types of spaces that arise in the solution are discussed together with the characteristics of the regularized solution families these concepts are demonstrated on a simulated discrete example and on an experimental case study copyright 2007 john wiley sons ltd

newton lavrentiev regularization for ill posed operator equations in hilbert scales in this paper we consider the two step method for approximately solving the ill posed operator equation f x f where f d f x x is a nonlinear monotone operator defined on a real hilbert space x in the setting of hilbert scales we derive the error estimates by selecting the regularization parameter a according to the adaptive method considered by pereverzev and schock in 2005 when the available data is f d with f f email protected x is of optimal order

an analysis of lavrentiev regularization method and newton type process for nonlinear ill posed problems abstract in this paper we consider the lavrentiev regularization method and a modified newton method for obtaining stable approximate solution to nonlinear ill posed operator equations f x y where f d f x x is a nonlinear monotone operator or f x 0 is nonnegative selfadjoint operator defined on a real hilbert space x we assume that only a noisy data y x with y y are available further we assume that frechet derivative f of f satisfies center type lipschitz condition a priori choice of regularization parameter is presented we proved that under a general source condition on x 0 x the error x x n between the regularized approximation x n x 0 x 0 and the solution x is of optimal order in the concluding section the algorithm is applied to numerical solution of the inverse gravimetry problem

optimal multivariable controller design using an itse performance index this article deals with the design of optimal controllers and as an embedded problem with the computation of achievable performance bounds in the control of non minimum phase unstable multi input multi output systems the cost function comprises a time weighted measure of the tracking error itse and an incremental quadratic penalisation of the control effort the proposed methodology relies on the properties of a frequency domain generalisation of the itse index the solution to the optimisation problem can be computed using orthonormal basis function expansions and a closed form expression is found for the optimal coefficients of the expansion numerical examples are presented to illustrate the results

existence and multiplicity of solutions for fractional schrodinger kirchhoff equations with trudinger moser nonlinearity abstract we study the existence and multiplicity of solutions for a class of fractional schrodinger kirchhoff type equations with the trudinger moser nonlinearity more precisely we consider m u n s n s s u v x u n s 1 u f x u h x u p 2 u in r n u r 2 n u x u y n s x y 2 n d x d y r n v x u n s d x s n where m 0 0 is a continuous function s 0 1 n 2 0 is a parameter 1 p n s s is the fractional n s laplacian v r n 0 is a continuous function f r n r r is a continuous function and h r n 0 is a measurable function first using the mountain pass theorem a nonnegative solution is obtained when f satisfies exponential growth conditions and is large enough and we prove that the solution converges to zero in w v s n s r n as then using the ekeland variational principle a nonnegative nontrivial solution is obtained when is small enough and we show that the solution converges to zero in w v s n s r n as 0 furthermore using the genus theory infinitely many solutions are obtained when m is a special function and is small enough we note that our paper covers a novel feature of kirchhoff problems that is the kirchhoff function m 0 0