quadratic stabilizability of uncertain linear systems existence of a nonlinear stabilizing control does not imply existence of a linear stabilizing control this note is concerned with the problem of stabilizing an uncertain linear system via state feedback control an uncertain system which admits a stabilizing state feedback control and some associated quadratic lyapunov function is said to be quadratically stabilizable in a number of recent papers conditions are given under which quadratic stabilizability via nonlinear control implies quadratic stabilizability via linear control these papers restrict the manner in which the uncertain parameters are permitted to enter structurally into the state equation in order to establish this result this note presents an example which shows that this implication is not true for more general uncertain linear systems to this end we describe an uncertain linear system which is quadratically stabilizable via nonlinear control but not quadratically stabilizable via linear control

using nu ssga to reduce the searching time in inverse problem of a buried metallic object we describe an inverse scattering problem with the aim of reducing the computation time for recovering the details of a perfectly conducting cylindrical object buried in a half space first we use fourier series or cubic spline methods to describe the shape and reformulate the inverse problem into an optimization one then we solved it by the improved steady state genetic algorithm ssga and simple genetic algorithm sga respectively and compare the cost time in finding out the global extreme solution of the objective function it is found the searching ability of ssga is much powerful than that of the sga even when the initial guess is far away from the exact one the cost time for converging to a global extreme solution using by ssga is much less than that by sga numerical results are given to show that the inverse problem by using ssga is much better than sga in time costing

a novel method for solving the inverse scattering problem for time harmonic acoustic waves in the resonance region ii in a previous paper siam j appl math 1985 pp 1039 1053 we presented a new method for determining the shape of an acoustically soft obstacle from a knowledge of the time harmonic incident wave and the far field pattern of the scattered wave the method given there was based on knowing the far field pattern for an interval of values of the square of the wave number k such that this interval contained the first eigenvalue lambda 1 of the interior dirichlet problem the purpose of this paper is to extend the methods of our earlier paper to treat the case when the far field pattern is only known for a single value of the wave number such that k 2 is not equal to lambda 1 in addition we show how to combine the results of this paper with the first to obtain a method that is valid for any fixed value of k independent of whether or not k 2 is an eigenvalue we also extend our analysis to include the case of the impedance boundary value problem numerical examples are given showing the pra

properties of the far field operator for a multilayered scatterer in acoustic scattering theory in this paper we consider the inverse scattering problem which is to determine the shape of a multilayered scatterer the adjoint of the far field operator is found in scattering theory the boundary data are given by the traces of plane waves the closure of the span of the images of these plane waves under the far field operator is characterized furthermore taking the affine hull instead of the span of the plane waves on the boundary some results are obtained

an inversion algorithm in two dimensional elasticity abstract in this paper scattering problems for the rigid body and the cavity in two dimensional linear elasticity are considered in each case the corresponding far field scattering amplitudes are presented and the herglotz condition and herglotz wavefunctions are introduced a pair of integral equations are constructed in the far field region the properties of the herglotz functions are used to derive solvability conditions and to built approximate far field equations a method for solving inverse scattering problems is proposed and the support of the scattering obstacle is found by noting the unboundedness of the l2 norm of the herglotz densities as an interior point approaches the boundary of the scattering object from inside the scatterer illustration of the unboundedness property on the boundary is carried out for rigid circular cylinders and cavities numerical results for rigid bodies are also given showing the applicability of this method

the inverse scattering problem for time harmonic acoustic waves the inverse scattering problem we are considering in this paper is to determine the shape of a sound soft bounded connected obstacle from a knowledge of the time harmonic incident wave with frequency in the resonance region and the far field pattern of the scattered wave after some introductory remarks we begin by describing the method of integral equations and the null field method for solving the direct scattering problem this enables us to define an operator t mapping the boundary of the scattering obstacle and the incident field onto the far field pattern the inverse scattering problem is to invert this operator in order to do this we first must examine the range of t i e to characterize the class of far field patterns and to establish the existence of t on the range of t i e to show the uniqueness of the solution to the inverse scattering problem this analysis shows that for a given measured far field pattern in general no solution exists to the inverse scattering problem and if a solution does exist it does not depend continuously on the measured data i e the problem is improperly posed this motivates us to consider a linearized model and to examine various methods for studying linearized improperly posed problems based on the ideas of a priori assumptions and compactness arguments we then consider a simple model problem that focuses on the nonlinear character of the inverse scattering problem and motivated by our study of the linearized model reformulate the inverse scattering problem as a problem in constrained optimization we conclude by considering the numerical solution of this nonlinear optimization problem approach your problems from the right end and begin with the answers then one day perhaps you will find the final question from the hermit clad in crane feathers in the chinese maze murders by r van gulik

the inverse scattering problem for time harmonic acoustic waves in an inhomogeneous medium with complex refraction index abstract the inverse scattering problem of determining the complex refraction index of a compact inhomogeneous medium from a knowledge of the far field patterns of the scattered fields corresponding to many incident time harmonic plane acoustic waves is solved by using the colton monk method again by using the same method the analogue of karp s theorem for the scattering of acoustic waves through an inhomogeneous medium with compact support is proven

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

the three dimensional inverse scattering problem for acoustic waves abstract we consider the inverse scattering problem for an acoustically soft obstacle in r 3 by assuming a priori that the unknown scattering obstacle is starlike and has its boundary lying in a compact family of holder continuously differentiable surfaces it is shown that an optimal solution can be constructed which depends continuously on the measured far field data remarks are made on the numerical approximation of the optimal solution

effective permittivity of log normal isotropic random media a method of deriving the effective permittivity epsilon eff of log normal d dimensional isotropic media is developed the approach is based on applying the regularized green function technique for the laplace operator the feasibility of exact determining of the effective permittivity is analysed by deriving epsilon eff at third order in the log permittivity variance it is shown that the 3d effective permittivity is a functional of the spectrum of inhomogeneities and therefore cannot be expressed by a closed formula in particular the formula for the effective permittivity discussed previously by landau and lifshitz 1960 and matheron 1967 deviates from the exact one in the third order term

on the numerical solution of the three dimensional inverse obstacle scattering problem abstract the inverse problem under consideration is to determine the shape of an impenetrable sound soft obstacle from the knowledge of a time harmonic incident plane acoustic wave and far field or near field measurements of the scattered wave we present a method for the approximate solution which avoids the solution of the corresponding direct problem and stabilizes the ill posed inverse problem by reformulating it as a nonlinear optimization problem the numerical implementation of the method is described and some three dimensional examples of reconstructions are given

a new non iterative singular sources method for the reconstruction of piecewise constant media we describe a novel non iterative method for the reconstruction of a piecewise constant inhomogeneous medium in acoustic scattering which we call the singular sources method the basic idea of the method is to use the behaviour of the scattered field for singular incident fields multipoles to calculate the size of the refractive index n at some point z0 on the boundary of the support of the scatterer and then eliminate this value from the data by subtracting a known piecewise constant background medium the paper includes the theory for the singular sources method to locate the unknown support of an inhomogeneous medium for a known inhomogeneous background medium also we give a new uniqueness proof for the reconstruction of a piecewise constant medium in two or three dimensions using techniques that differ from those used to prove previous well known results

image reconstruction of perfectly conducting objects by a cascaded method in this paper we propose a method which combines a particle swarm optimization pso algorithm with a newton kantorovitch algorithm for image reconstruction of perfectly conducting objects first the inverse problem is recast as a global nonlinear optimization problem which is solved by a pso then the solution obtained by the pso is taken as an initial guess for the newton kantorovitch algorithm to obtain the more accuracy solution in a few iterations the inversion algorithm which is based on the rigorous mathematics makes use of the received scattered field and appropriate boundary condition to derive a set of nonlinear integral equations the moment method is used to transform the nonlinear integral equations into matrix form then the pseudoinverse transformation is employed to overcome the ill posedness to obtain a convergent and stable solution the particle swarm optimization algorithm is employed to find out the global extreme solution of the object function numerical results show that the performance of this cascaded method is better than the individual pso and the individual newton kantorovitch algorithm satisfactory reconstruction has been obtained by using this cascaded method

a fast method for acoustic imaging of multiple three dimensional objects this paper is concerned with the inverse problem for imaging multiple three dimensional objects using the information of the far field pattern of the scattered wave a spatially dependent function which has noticeably different values inside and outside the obstacle is derived a numerical method based on the characterization is developed to obtain a visualization of the obstacle the most remarkable advantage of this method is that it does not need any prior knowledge about the geometry and physical properties of the scatterer and requires only the information of the far field measurements for a finite number of directions of incidence and observation distributed over a limited range furthermore the scheme is very simple and fast since it avoids the use of the iterative procedure and requires only the solution of a linear system some numerical examples with synthetic far field data are given showing the practicality and efficiency of this scheme

the limited aperture problem of inverse acoustic scattering dirichlet boundary conditions let d be a bounded simply connected domain in the plane and let f theta k underset raise0 3em hbox smash scriptscriptstyle thicksim alpha l be the far field pattern arising from the scattering of a time harmonic acoustic plane wave u l underset raise0 3em hbox smash scriptscriptstyle thicksim x exp ik underset raise0 3em hbox smash scriptscriptstyle thicksim x cdot underset raise0 3em hbox smash scriptscriptstyle thicksim alpha l where underset raise0 3em hbox smash scriptscriptstyle thicksim alpha l is a unit vector and k is the wave number and the time harmonic factor e i omega t has been factored out it is assumed in addition that the total field satisfies homogeneous dirichlet boundary conditions on partial d in this paper a method is presented for recovering partial d given the far field patterns f theta k underset raise0 3em hbox smash scriptscriptstyle thicksim alpha l

elastic herglotz functions solutions of the spectral navier equation in the linearized theory of elasticity that satisfy the herglotz boundness condition at infinity are introduced the leading asymptotic terms in a neighborhood of infinity provide the far field patterns and the herglotz norm is expressed as the sum of the l 2 norms of these patterns over the unit sphere basic integral representations that connect the solid spherical navier eigenvectors to the vector spherical harmonics over the unit sphere are utilized to prove the fundamental representation theorem for the elastic herglotz solutions in full space it is shown that the longitudinal and the transverse herglotz kernels are exactly the corresponding far field patterns of the irrotational and the solenoidal parts of the displacement field particular methods to obtain the displacement field from the far field patterns and vice versa are also described

paper linear ultimate boundedness control of uncertain dynamical systems given a state equation whose description includes uncertain parameters it is often desirable to design a feedback controller which renders all solutions uniformly and ultimately bounded when facing such a problem one often has available or assumes an upper bound for the excursions of the uncertain parameters within this framework the objective is to guarantee no matter how nature selects the uncertainty that the state will eventually end up and remain within some pre specified region when this region is a small neighborhood about the origin the concept of uniform ultimate boundedness is tantamount to practical asymptotic stability the objectives of this paper are twofold first and foremost our goal is to show that a class of ultimate boundedness problems which have been solved to date via nonlinear control can in point of fact be solved via linear control this is the main result and is given in theorem 4 1 secondly we show that some of the so called matching assumptions which have been required in the earlier literature can be weakened somewhat this enables one to design controllers for a larger class of dynamical systems than previously considered

properties of min max controllers in uncertain dynamical systems we discuss properties of min max controllers previously obtained for uncertain dynamical systems in particular we present a new model for switching action and re examine asymptotic stability we demonstrate the attractiveness of switching surfaces and comment on the insensitivity of solutions in those surfaces to the uncertainty

the no response test a sampling method for inverse scattering problems we describe a novel technique which we call the no response test to locate the support of a scatterer from knowledge of a far field pattern of a scattered acoustic wave the method uses a set of sampling surfaces and a special test response to detect the support of a scatterer without a priori knowledge of the physical properties of the scatterer specifically the method does not depend on information about whether the scatterer is penetrable or impenetrable nor does it depend on any knowledge of the nature of the scatterer absorbing reflecting etc in contrast to previous sampling algorithms the techniques described here enable one to locate obstacles or inhomogeneities from the far field pattern of only one incident field the no response test is a one wave method we investigate the theoretical basis for the no response test and derive a one wave uniqueness proof for a region containing the scatterer we show how to find the object within this region we demonstrate the applicability of the method by reconstructing sound soft sound hard impedance and inhomogeneous medium scatterers in two dimensions from one wave with full and limited aperture far field data

a parallel algorithm for solving the inverse scattering moment problem abstract a plane incident wave impinges upon an unknown acoustically soft scatterer it is assumed that the scatterer has a polynomial boundary it is star shaped and its characteristic dimension is much less than the wavelength of the incident field an analytic procedure is proposed to reduce the shape reconstruction process to the evaluation of certain expansion coefficients and the solution of a linear algebraic system then a parallel numerical algorithm is constructed which recovers the exact shape of the scatterer from the knowledge of a finite number of generalized low frequency moments generated by the leading low frequency approximation the efficiency of the proposed algorithm as well as the detailed parametric influence on the performance of the algorithm are extensively investigated

inverse scattering for conductors by the equivalent source method the inverse scattering for two dimensional conductors is investigated assume that an incident wave is incident upon conducting cylinders of unknown shape and the scattered field is measured outside by properly processing the scattering data the position and shape of the conducting scatterers can then be reconstructed the proposed theory of inverse scattering is a combination of the diffraction tomography and the equivalent source method to overcome the ill posedness the condition of the measured scattered field and the boundary condition on the conductor surfaces are combined and recast into an optimization problem using the least squares concept by comparing the reconstructed and the prescribed results the inversion algorithms are numerically examined note that if a priori information of the shape is sufficient and the equivalent sources are properly placed the algorithms can be flexibly applied to the cases of multiple conductors also the effects of noise contamination and multiple scattering on the reconstruction result are numerically investigated

inverse scattering via low frequency moments an acoustically soft scatterer defined by a closed and star shape polynomial surface of any degree disturbs the propagation of a time harmonic plane incident wave it is demonstrated that under the hypotheses of schiffer s uniqueness theorem all the generalized low frequency moments corresponding to the capacity potential can be obtained from the scattering amplitude an analytic algorithm is proposed that recovers the geometry of the body whenever a finite number of generalized moments generated by the leading low frequency approximation are given what is striking here is the fact that a surface measure generated by a potential problem is enough to recover the geometry of the scatterer the idea here is to relate the given moments to a set of particular combined spherical moments that appear as coefficients of an algebraic linear system whose solution provides the coefficients of the scattering surface in spherical harmonics this is done with the help of an inner product defined over the surface of

a new class of stabilizing controllers for uncertain dynamical systems this paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems the vector of uncertain parameters q cdot is time varying and its values q t lie within a prespecified bounding set q in r p furthermore no statistical description of q cdot is assumed and the controller is shown to render the closed loop system practically stable in a so called guaranteed sense that is the desired stability properties are assured no matter what admissible uncertainty q cdot is realized within the perspective of previous research in this area this paper contains one salient feature the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time invariant in contrast in much of the previous literature see for example 1 2 7 and 9 a linear system is stabilized via nonlinear control another feature of this paper is the fact that the

a riccati equation approach to the stabilization of uncertain linear systems this paper presents a method for designing a feedback control law to stabilize a class of uncertain linear systems the systems under consideration contain uncertain parameters whose values are known only to within a given compact bounding set furthermore these uncertain parameters may be time varying the method used to establish asymptotic stability of the closed loop system obtained when the feedback control is applied involves the use of a quadratic lyapunov function the main contribution of this paper involves the development of a computationally feasible algorithm for the construction of a suitable quadratic lyapunov function once the lyapunov function has been obtained it is used to construct the stabilizing feedback control law the fundamental idea behind the algorithm presented involves constructing an upper bound for the lyapunov derivative corresponding to the closed loop system this upper bound is a quadratic form by using this upper bounding procedure a suitable lyapunov function can be found by solving a certain matrix riccati equation another major contribution of this paper is the identification of classes of systems for which the success of the algorithm is both necessary and sufficient for the existence of a suitable quadratic lyapunov function

a two step method in inverse scattering problem for a crack in this paper we present a method for solving a time harmonic acoustic inverse scattering problem for a sound soft crack in r2 based on the integral equation method our method splits the nonlinear severely ill posed inverse problem into a linear well posed direct problem and a nonlinear ill posed problem at this setting not only the regularized newton s method can still be used to solve the inverse problem numerically but also with the advantage of keeping the structure of the algorithm simple and efficient

structural stabilization of uncertain systems necessity of the matching condition this paper investigates one aspect of the problem of stabilizing an uncertain linear system that is the systems under consideration contain uncertain parameters which are unknown but bounded the question arises as to whether such a system can be stabilized via feedback control in some of the previous papers in this area the system is assumed to satisfy a so called matching condition this type of assumption is used to assure that the uncertain system can be stabilized it is known however that the matching condition is not a necessary condition for stabilizability this paper introduces a strengthened notion of stabilizability referred to as structural stabilizability via a nominally determined quadratic lyapunov function it is then shown that if a system is to have this stronger property the matching condition must necessarily be satisfied

on ultimate boundedness control of uncertain systems in the absence of matching assumptions this paper describes an approach to ultimate boundedness control under hypotheses less stringent than heretofore required instead of imposing the so called matching conditions of previous authors a certain decomposition of the dynamics is performed the resulting decomposed system has two parts a matched portion and a mismatched portion a certain measure of mismatch m is defined and it is shown that effective control is possible as long as m does not exceed some critical mismatch threshold m it is seen that this threshold depends on the choice of feedback hence it becomes possible to investigate the tradeoffs between the size of the mismatch threshold and the sizes of the feedback gains which achieve the desired ultimate boundedness of solutions