optimal controllers and output feedback stabilization abstract in this paper the output feedback stabilizability problem is explored in terms of control lyapunov functions sufficient conditions for stabilization are provided for a certain class of systems by means of output feedback stabilizers that can be obtained from an optimization problem our main results extends those developed in 31 and generalize a theorem due to sontag 23

a local stabilization theorem for interconnected systems abstract we prove that if a nonlinear control system is locally asymptotically stabilizable at its equilibrium by means of a continuous feedback then adding an integrator the resulting system is also locally asymptotically stabilizable by means of a feedback law which is smooth except possibly at the equilibrium our approach is based on previous works of arstein sontag and the author

penetration through slots in conducting cylinders part 2 tm case three methods for determining the penetration through small apertures in closed conducting surfaces are outlined and their salient features discussed these methods are designated 1 the scatterer method 2 the short circuit current method and 3 the equivalent current method they are implemented by integral equation techniques but are amenable to differential equation or hybrid methods procedures for applying each method are outlined as are schemes for repairing singular equations rendered invalid by the presence of false resonances reasons for inaccuracies in the three methods are also delineated data determined for a given structure by all three methods are presented and numerical examples that illustrate important features of the methods and their relative accuracies are described it is pointed out that some methods yield incorrect penetration data an outline is given of the integral equation formulations and numerical schemes needed to accurately determine the field that penetrates through a slot in a conducting cylinder excited by an axially independent te source

solutions to the kaup broer system and its 2 1 dimensional integrable generalization via the dressing method abstract in this paper we formulate the nonlocal dbar problem dressing method of manakov and zakharov zakharov and manakov 1984 1985 zakharov 1989 for the 4 scaling classes of the 1 1 dimensional kaup broer system broer 1975 kaup 1975 the applications of the method for the 1 1 dimensional kaup broer systems are reductions of a method for a complex valued 2 1 dimensional completely integrable partial differential equation first introduced in rogers and pashaev 2011 this method allows computation of solutions to all scaling classes of the kaup broer system we then consider the case of non capillary waves with gravitational forcing and use the dressing method to compute n soliton solutions and more general solutions in the closure of the n soliton solutions in the topology of uniform convergence in compact sets called primitive solutions these more general solutions are analogous to the solutions derived in dyachenko and zakharov 2016 zakharov and dyachenko 2016 zakharov et al 2016 for the kdv equation we derive dressing functions for finite gap solutions and compute counter propagating dispersive shockwave type solutions numerically

lyapunov design of stabilizing controllers for cascaded systems the design of a state feedback law for an affine nonlinear system to render a as small as possible compact neighborhood of the equilibrium of interest globally attractive is discussed following z artstein s theorem 1983 the problem can be solved by designing a so called control lyapunov function for systems which are in a cascade form a lyapunov function meeting artstein s conditions is designed assuming the knowledge of a control law stabilizing the equilibrium of the head nonlinear subsystem in particular for planar systems this gives sufficient and necessary conditions for a compact neighborhood of the equilibrium to be stabilized

period preserving nonisospectral flows and the moduli space of periodic solutions of soliton equations flows on the moduli space of the algebraic riemann surfaces preserving the periods of the corresponding solutions of the soliton equations are studied we show that these flows are gradient with respect to some indefinite symmetric flat metric arising in the hamiltonian theory of the whitham equations the functions generating these flows are conserved quantities for all the equations simultaneously we show that for 1 1 systems these flows can be imbedded in a larger system of ordinary nonlinear differential equations with a rational right hand side finally these flows are used to give a complete description of the moduli space of algebraic riemann surfaces

stabilization of nonlinear systems in the plane abstract it is shown that every small time locally controllable system in the plane can locally be asymptotically stabilized by employing locally holder continuous feedback laws as essentially was conjectured by e sontag an explicit algorithm for the construction of such feedback laws is given

controllability and stabilization of one dimensional systems near bifurcation points abstract bifurcation theory deals with the change of qualitative behavior in a parameter dependent family of differential equations for one dimensional equations the possible bifurcation scenarios are well understood if the family of differential equations can be controlled by admissible controls with compact range the question arises whether the systems are controllable near a bifurcation point and whether stabilization around unstable bifurcation branches via bounded feedback is possible in this paper we show that controllability for parametrized families of one dimensional control systems can be characterized in terms of two parameters the original bifurcation parameter and the size of the control range these results are used to construct nonsmooth stabilizing feedbacks and to describe the set of initial values from which stabilization is possible furthermore robustness properties of the stabilizing feedback are discussed

feedback stabilization of real analytic systems in the plane abstract we investigate the local feedback stabilization of single input control affine analytic systems x tf x ug x in the plane new necessary and sufficient conditions for local stabilization with feedback laws of the form u v x 1 x 2 x 2 v x 1 0 0 2 v x 1 0 0 2 0 0 v 0 0 0 are obtained by using lyapunov s stability theorems on two dimensional analytic systems if the sufficient conditions are satisfied we also provide explicit feedback laws

asymptotic stabilization of a class of smooth two dimensional systems this paper studies the asymptotic stabilizability of two dimensional control systems the class under consideration includes c infty systems that satisfy a certain genericity assumption and all real analytic systems necessary and sufficient conditions for feedback stabilization using continuous feedback and a sufficient condition for c 1 feedback stabilization are given this latter condition is given in terms of an inequality involving two indices if the direction of the inequality is changed an obstruction to c infty feedback stabilizability is obtained a subclass of polynomial systems is also studied and given complete necessary and sufficient conditions for global asymptotic stabilization using c 1 feedback

em wave scattering by an axial slot on a circular pec cylinder with an eccentrically layered inner coating a dual series solution for te polarization a dual series approach is used to formulate scattering of a te polarized plane wave by a circular pec cylinder having a longitudinal slot and enclosing two eccentric cylindrical dielectric layers the dual series equations are analytically regularized to a matrix fredholm equation of the second kind by use of the abel integral transform method the resulting infinite set of linear algebraic equations i s l a e is truncated and solved numerically the aforementioned analytical numerical solution intrinsically takes into consideration the correct edge behavior of the field furthermore the addition theorem of cylindrical wave functions is used to match the field components on any cylindrical interface the numerical application manifests how the structure of the inner loading of this cylindrical cavity is imprinted on backscattering moreover it is shown how the location of the innermost dielectric cylinder may contribute to the suppression or enhancement of resonances

a universal construction of artstein s theorem on nonlinear stabilization abstract this note presents an explicit proof of the theorem due to artstein which states that the existence of a smooth control lyapunov function implies smooth stabilizability moreover the result is extended to the real analytic and rational cases as well the proof uses a universal formula given by an algebraic function of lie derivatives this formula originates in the solution of a simple riccati equation

a riemann hilbert problem for the finite genus solutions of the kdv equation and its numerical solution abstract we derive a riemann hilbert problem satisfied by the baker akhiezer function for the finite gap solutions of the korteweg de vries kdv equation as usual for riemann hilbert problems associated with solutions of integrable equations this formulation has the benefit that the space and time dependence appears in an explicit linear and computable way we make use of recent advances in the numerical solution of riemann hilbert problems to produce an efficient and uniformly accurate numerical method for computing all periodic and quasi periodic finite genus solutions of the kdv equation

dual series solution to em wave scattering by a circular slit pec cylinder enclosing multiple dielectric cylindrical rods a cylindrical structure composed of a circular pec cylinder having a longitudinal slot and enclosing an arbitrary number of eccentric dielectric cylinders embedded in a dielectric sheath is considered in this paper excitation is provided by a plane wave normally incident upon the structure both te and tm polarizations are considered for the incident wave the scattered wave as well as the electromagnetic em field within the scatterer are determined by using a dual series approach combined with the translational addition theorem for cylindrical wave functions the abel integral transform method is applied in order to regularize the dual series equations to a matrix fredholm equation of the second kind the resulting infinite set of linear algebraic equations i s l a e is truncated and solved numerically the solution inherently contains the behavior of the field near the aperture rim as required by meixner s edge condition the numerical application manifests how the internal structure of the scatterer and the polarization of the incident wave affect backscattering by the cylindrical cavity

output feedback stabilization the stabilization of control systems using output feedback is addressed necessary and sufficient lyapunov conditions are established generalizing the artstein s stabilization theorem for linear control systems a necessary and sufficient lyapunov condition is provided for stabilization by means of a linear output feedback law

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

asymptotic stabilization of two dimensional real analytic systems abstract we investigate the local asymptotic stabilizability of a real analytic control system around an equilibrium point and give necessary and sufficient conditions for local asymptotic stabilization using continuous feedback

primitive potentials and bounded solutions of the kdv equation abstract we construct a broad class of bounded potentials of the one dimensional schrodinger operator that have the same spectral structure as periodic finite gap potentials but that are neither periodic nor quasi periodic such potentials which we call primitive are non uniquely parametrized by a pair of positive holder continuous functions defined on the allowed bands primitive potentials are constructed as solutions of a system of singular integral equations which can be efficiently solved numerically simulations show that these potentials can have a disordered structure primitive potentials generate a broad class of bounded non vanishing solutions of the kdv hierarchy and we interpret them as an example of integrable turbulence in the framework of the kdv equation

algebro geometric finite gap solutions to the korteweg de vries equation as primitive solutions abstract in this paper we show that all algebro geometric finite gap solutions to the korteweg de vries equation can be realized as a limit of n soliton solutions as n diverges to infinity see remark 1 for the precise meaning of this statement this is done using the primitive solution framework initiated by dyachenko et al 2016 and zakharov et al 2016 25 26 one implication of this result is that the n soliton solutions can approximate any bounded periodic solution to the korteweg de vries equation arbitrarily well in the limit as n diverges to infinity we also study primitive solutions numerically that have the same spectral properties as the algebro geometric finite gap solutions but are not algebro geometric solutions

feedback stabilization of nonlinear systems by locally bounded controls abstract in this paper results are presented on output regulation and stabilization of nonlinear control systems using lyapunov based methods sufficient conditions for global output regulation and stabilization using locally bounded state feedback are obtained the approach used here is motivated by the work of artstein 1983 sontag 1989 and tsinias 1989 on the relationship between control lyapunov functions and feedback stabilization

accurate simulation of reflector antennas by the complex source dual series approach the radiation from circular cylindrical reflector antennas is treated in an accurate manner for both polarizations the problem is first formulated in terms of the dual series equations and then is regularized by the riemann hilbert problem technique the resulting matrix equation is solved numerically with a guaranteed accuracy and remarkably little cpu time is needed the feed directivity is included in the analysis by the complex source point method various characteristic patterns are obtained for the front and offset fed reflector antenna geometries with this analysis and some comparisons are made with the high frequency techniques the directivity and radiated power properties are also studied