asymptotic behavior of the solution to the system for a viscous reactive gas abstract in this paper we investigate the asymptotic behavior of the generalized solution to the compressible navier stokes equations which describes a thermal explosion in a viscous reactive perfect gas confined between two infinite parallel plates in combustion theory

a novel multigrid based preconditioner for heterogeneous helmholtz problems an iterative solution method in the form of a preconditioner for a krylov subspace method is presented for the helmholtz equation the preconditioner is based on a helmholtz type differential operator with a complex term a multigrid iteration is used for approximately inverting the preconditioner the choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with fourier analysis multigrid analysis results are verified by numerical experiments high wavenumber helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner

on tracking control of a class of complementary slackness hybrid mechanical systems abstract in this note we consider the problem of feedback control of n dof degree of freedom rigid manipulators subject to a scalar frictionless unilateral constraint f x 0 x r n is the vector of generalized coordinates the stability analysis relies on a stability concept that incorporates the hybrid and nonsmooth dynamical feature of the overall system it is shown that stability of the closed loop system can be obtained this work generalizes the results in brogliato et al ieee trans automat control 42 2 1997 200 215 which were mainly restricted to the 1 dof case it also clarifies some concepts related to the hybrid nature of closed loop complementary slackness mechanical systems

robust trajectory tracking for a class of hybrid systems an internal model principle approach this paper deals with the asymptotic tracking of periodic trajectories for hybrid systems having linear dynamics in each operating mode and isolated discrete switching events switching systems parametric uncertainties are considered and the dimension of the state vector is allowed to vary among modes to deal with the hybrid nature of the system and the possible discontinuities of its solutions at switching times a properly amended tracking control problem is defined and a feedback control law based on a discontinuous version of the classical internal model principle is proposed the innovative design of the discrete time dynamics of the compensator guarantees the robust existence of a steady state response giving zero tracking error in the controlled output and local convergence to it

global existence of large solutions to initial boundary value problems for a viscous heat conducting one dimensional real gas abstract the existence of global classical solutions to initial boundary value problems in the dynamics of a one dimensional viscous heat conducting gas is established the nonlinear dissipative effects turn out to be sufficiently strong to prevent the development of singularities

the linear quadratic regulator for periodic hybrid systems abstract the main objective of this paper is to characterize feedback control laws that are optimal with respect to a quadratic cost functional in the framework of linear hybrid systems undergoing time driven periodic jumps namely the so called hybrid linear quadratic regulator lqr problem the optimal solution to the hybrid lqr problem is determined both in the case of finite horizon and infinite horizon optimal control problems by introducing a hybrid periodic extension of the classic differential and difference riccati equations thus leading to the notion of monodromy riccati equation interestingly due to the periodic nature of the discrete time events the computation of the optimal feedback hinges upon the solution of a differential rather than algebraic riccati equation also in the infinite horizon case hence yielding a time varying periodic control law necessary and sufficient conditions that ensure asymptotic stability of the closed loop system are provided and discussed in detail in the case of infinite horizon optimal control problems

arbitrary high order c0 tensor product galerkin finite element methods for the electromagnetic scattering from a large cavity the paper is concerned with the electromagnetic scattering from a large cavity embedded in an infinite ground plane the electromagnetic cavity problem is described by the helmholtz equation with a nonlocal boundary condition on the aperture of the cavity and dirichlet or neumann boundary conditions on the walls of the cavity a tensor product galerkin finite element method fem is proposed in which spaces of c 0 piecewise polynomials of degree k 1 are employed by the fast fourier transform and the toeplitz type structure of the approximation to the nonlocal operator in the nonlocal boundary condition a fast algorithm is designed for solving the linear system arising from the cavity problem with vertically layered media which requires o n 2logn operations on an nxn uniform partition numerical results for model problems illustrate the efficiency of the fast algorithm and exhibit the expected optimal global convergence rates of the finite element methods moreover our numerical results also show that the high order approximations are especially effective for problems with large wave numbers

mathcal l _2 gain for hybrid linear systems with periodic jumps a game theoretic approach for analysis and design in this paper the disturbance attenuation problem is formulated and solved for a class of linear hybrid systems in the presence of periodic jumps the results are achieved both in the finite and infinite horizon cases by borrowing ideas from the theory of dynamic games in the considered formulation independent disturbances affecting the continuous time and the discrete time components of the hybrid system are allowed moreover the analysis is carried out by introducing easily verifiable conditions involving the definition of a monodromy riccati equation i e a classical riccati equation defined for the one period discrete time equivalent model interestingly as a by product the main statements essentially characterize the solution of zero sum noncooperative dynamic games for periodic linear hybrid systems which is of interest per se

efficient iterative solution of the three dimensional helmholtz equation we examine preconditioners for the discrete indefinite helmholtz equation on a three dimensional box shaped domain with sommerfeld like boundary conditions the preconditioners are of two types the first is derived by discretization of a related continuous operator that differs from the original only in its boundary conditions the second is derived by a block toeplitz approximation to the descretized problem the resulting preconditioning matrices allow the use of fast transform methods and differ from the discrete helmholtz operator by an operator of low rank we present experimental results demonstrating that when these methods are combined with krylov subspace iteration convergence rates depend only mildly on both the wave number and discretization mesh size in addition the methods display high efficiencies in an implementation on an ibm sp 2 parallel computer

asymptotic controllability implies feedback stabilization it is shown that every asymptotically controllable system can be globally stabilized by means of some discontinuous feedback law the stabilizing strategy is based on pointwise optimization of a smoothed version of a control lyapunov function iteratively sending trajectories into smaller and smaller neighborhoods of a desired equilibrium a major technical problem and one of the contributions of the present paper concerns the precise meaning of solution when using a discontinuous controller

synchronization of impacting mechanical systems with a single constraint abstract this paper addresses the synchronization problem of mechanical systems subjected to a single geometric unilateral constraint the impacts of the individual systems induced by the unilateral constraint generally do not coincide even if the solutions are arbitrarily close to each other the mismatch in the impact time instants demands a careful choice of the distance function to allow for an intuitively correct comparison of the discontinuous solutions resulting from the impacts we propose a distance function induced by the quotient metric which is based on an equivalence relation using the impact map the distance function obtained in this way is continuous in time when evaluated along jumping solutions the property of maximal monotonicity which is fulfilled by most commonly used impact laws is used to significantly reduce the complexity of the distance function based on the simplified distance function a lyapunov function is constructed to investigate the synchronization problem for two identical one dimensional mechanical systems sufficient conditions for the uncoupled individual systems are provided under which local synchronization is guaranteed furthermore we present an interaction law which ensures global synchronization also in the presence of grazing trajectories and accumulation points zeno behavior the results are illustrated using numerical examples of a 1 dof mechanical impact oscillator which serves as stepping stone in the direction of more general systems

solutions to hybrid inclusions via set and graphical convergence with stability theory applications motivated by questions in stability theory for hybrid dynamical systems we establish some fundamental properties of the set of solutions to such systems using the notion of a hybrid time domain and general results on set and graphical convergence we establish under weak regularity and local boundedness assumptions that the set of solutions is sequentially compact and upper semicontinuous with respect to initial conditions and system perturbations the general facts are then used to establish several results for the behavior of hybrid systems that have asymptotically stable compact sets these results parallel what is already known for differential inclusions and difference inclusions for example the basin of attraction for a compact attractor is relatively open the attractivity is uniform from compact subsets of the basin of attraction and asymptotic stability is robust with respect to small perturbations

an iterative method for the helmholtz equation an iterative algorithm for the solution of the helmholtz equation is developed the algorithm is based on a preconditioned conjugate gradient iteration for the normal equations the preconditioning is based on an ssor sweep for the discrete laplacian numerical results are presented for a wide variety of problems of physical interest and demonstrate the effectiveness of the algorithm

stability criteria for switched and hybrid systems the study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems the objective of this paper is to outline some of these problems to review progress made in solving them in a number of diverse communities and to review some problems that remain open an important contribution of our work is to bring together material from several areas of research and to present results in a unified manner we begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions closely related to the concept of stability are the notions of exponential growth rates and converse lyapunov theorems both of which are discussed in detail in particular results on common quadratic lyapunov functions and piecewise linear lyapunov functions are presented as they represent constructive methods for proving stability and also represent problems in which significant progress has been made we also comment on the inherent difficulty in determining stability of switched systems in general which is exemplified by np hardness and undecidability results we then proceed by considering the stability of switched systems in which there are constraints on the switching rules through both dwell time requirements and state dependent switching laws also in this case the theory of lyapunov functions and the existence of converse theorems are reviewed we briefly comment on the classical lur e problem and on the theory of stability radii both of which contain many of the features of switched systems and are rich sources of practical results on the topic finally we present a list of questions and open problems which provide motivation for continued research in this area

sweeping preconditioner for the helmholtz equation hierarchical matrix representation the paper introduces the sweeping preconditioner which is highly efficient for iterative solutions of the variable coefficient helmholtz equation including very high frequency problems the first central idea of this novel approach is to construct an approximate factorization of the discretized helmholtz equation by sweeping the domain layer by layer starting from an absorbing layer or boundary condition given this specific order of factorization the second central idea is to represent the intermediate matrices in the hierarchical matrix framework in two dimensions both the construction and the application of the preconditioners are of linear complexity the generalized minimal residual method gmres solver with the resulting preconditioner converges in an amazingly small number of iterations which is essentially independent of the number of unknowns this approach is also extended to the three dimensional case with some success numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach

spectral analysis of the discrete helmholtz operator preconditioned with a shifted laplacian shifted laplace preconditioners have attracted considerable attention as a technique to speed up convergence of iterative solution methods for the helmholtz equation in this paper we present a comprehensive spectral analysis of the helmholtz operator preconditioned with a shifted laplacian our analysis is valid under general conditions the propagating medium can be heterogeneous and the analysis also holds for different types of damping including a radiation condition for the boundary of the computational domain by combining the results of the spectral analysis of the preconditioned helmholtz operator with an upper bound on the gmres residual norm we are able to provide an optimal value for the shift and to explain the mesh dependency of the convergence of gmres preconditioned with a shifted laplacian we illustrate our results with a seismic test problem

invariance principles for hybrid systems with connections to detectability and asymptotic stability this paper shows several versions of the lasalle s invariance principle for general hybrid systems the broad framework allows for nonuniqueness of solutions zeno behaviors and does not insist on continuous dependence of solutions on initial conditions instead only a mild structural property involving graphical convergence of solutions is posed the general invariance results are then specified to hybrid systems given by set valued data further results involving invariance as well as observability detectability and asymptotic stability are given

a numerical study on the stability of a class of helmholtz problems this paper concerns the stability of a class of helmholtz problems in rectangular domains a well known application is the electromagnetic scattering from a rectangular cavity embedded in an infinite ground plane error analysis of numerical methods for cavity problems relies heavily on the stability estimates however it is extremely difficult to derive an optimal stability bound with the explicit dependency on wave numbers in this paper a high order finite element approximation is proposed for calculating the stability bound numerical experiments show that the stability depends strongly on wave numbers in extreme case and it is almost independent on the wave numbers in an average sense our numerical results also help to understand the stability of the multi frequency inverse problems

frequency independent approach to calculate physical optics radiations with the quadratic concave phase variations in this work we develop the numerical steepest descent path nsdp method to calculate the physical optics po radiations with the quadratic concave phase variations with the surface integral equation method the physical optics po scattered fields are formulated and further reduced to the surface integrals the high frequency physical critical points contributions including the stationary phase points the boundary resonance points and the vertex points are comprehensively studied via the proposed nsdp method the key contributions of this work are twofold one is that together with the po integrals taking the quadratic parabolic and hyperbolic phase terms this work makes the nsdp theory be complete for treating the po integrals with quadratic phase variations another is that in order to illustrate the transition effect of the high frequency physical critical points in this work we consider and further extend the nsdp method to calculate the po integrals with the coalescence of the high frequency critical points numerical results for the highly oscillatory po integral with the coalescence of the critical points are given to verify the efficiency of the proposed nsdp method the nsdp method could achieve the frequency independent computational workload and error controllable accuracy in all the numerical experiments especially for the case of the coalescence of the high frequency critical points

tracking control for hybrid systems with state triggered jumps this paper addresses the tracking problem in which the controller should stabilize time varying reference trajectories of hybrid systems despite the fact that discrete events or jumps in hybrid systems can often not be controlled directly as e g is the case in impacting mechanical systems the controller should still stabilize the desired trajectory a major complication in the analysis of this hybrid tracking problem is that in general the jump times of the plant do not coincide with those of the reference trajectory consequently the conventional euclidean tracking error does not converge to zero even if trajectories converge to the reference trajectory in between jumps and the jump times converge to those of the reference trajectory hence standard control approaches can not be applied we propose a novel definition of the tracking error that overcomes this problem and formulate lyapunov based conditions for the global asymptotic stability of the hybrid reference trajectory using these conditions we design hysteresis based controllers that solve the hybrid tracking problem for two exemplary systems including the well known bouncing ball problem

eigenanalysis of some preconditioned helmholtz problems in this work we calculate the eigenvalues obtained by preconditioning the discrete helmholtz operator with sommerfeld like boundary conditions on a rectilinear domain by a related operator with boundary conditions that permit the use of fast solvers the main innovation is that the eigenvalues for two and three dimensional domains can be calculated exactly by solving a set of one dimensional eigenvalue problems this permits analysis of quite large problems for grids fine enough to resolve the solution for a given wave number preconditioning using neumann boundary conditions yields eigenvalues that are uniformly bounded located in the first quadrant and outside the unit circle in contrast dirichlet boundary conditions yield eigenvalues that approach zero as the product of wave number with the mesh size is decreased these eigenvalue properties yield the first insight into the behavior of iterative methods such as gmres applied to these preconditioned problems

two transparent boundary conditions for the electromagnetic scattering from two dimensional overfilled cavities we consider electromagnetic scattering from two dimensional 2d overfilled cavities embedded in an infinite ground plane the unbounded computational domain is truncated to a bounded one by using a transparent boundary condition tbc proposed on a semi ellipse for overfilled rectangular cavities with homogeneous media another tbc is introduced on the cavity apertures which produces a smaller computational domain the existence and uniqueness of the solutions of the variational formulations for the transverse magnetic and transverse electric polarizations are established in the exterior domain the 2d scattering problem is solved in the elliptic coordinate system using the mathieu functions in the interior domain the problem is solved by a finite element method numerical experiments show the efficiency and accuracy of the new boundary conditions

fast directional multilevel algorithms for oscillatory kernels this paper introduces a new directional multilevel algorithm for solving n body or n point problems with highly oscillatory kernels these systems often result from the boundary integral formulations of scattering problems and are difficult due to the oscillatory nature of the kernel and the non uniformity of the particle distribution we address the problem by first proving that the interaction between a ball of radius r and a well separated region has an approximate low rank representation as long as the well separated region belongs to a cone with a spanning angle of o 1 r and is at a distance which is at least o r 2 away from from the ball we then propose an efficient and accurate procedure which utilizes random sampling to generate such a separated low rank representation based on the resulting representations our new algorithm organizes the high frequency far field computation by a multidirectional and multiscale strategy to achieve maximum efficiency the algorithm performs well on a large group of highly oscillatory kernels our algorithm is proved to have o n log n computational complexity for any given accuracy when the points are sampled from a two dimensional surface we also provide numerical results to demonstrate these properties

a two dimensional helmhotlz equation solution for the multiple cavity scattering problem here considered is the mathematical analysis and numerical computation of the electromagnetic wave scattering by multiple cavities embedded in an infinite ground plane above the ground plane the space is filled with a homogeneous medium while the interiors of the cavities are filled with inhomogeneous media characterized by variable permittivities by introducing a new transparent boundary condition on the cavity apertures the multiple cavity scattering problem is reduced to a boundary value problem of the two dimensional helmholtz equation imposed in the separated interior domains of the cavities the existence and uniqueness of the weak solution for the model problem is achieved via a variational approach a block gauss seidel iterative method is introduced to solve the coupled system of the multiple cavity scattering problem where only a single cavity scattering problem is required to be solved at each iteration numerical examples demonstrate the efficiency and accuracy of the proposed method

phase reduction theory for hybrid nonlinear oscillators hybrid dynamical systems characterized by discrete switching of smooth dynamics have been used to model various rhythmic phenomena however the phase reduction theory a fundamental framework for analyzing the synchronization of limit cycle oscillations in rhythmic systems has mostly been restricted to smooth dynamical systems here we develop a general phase reduction theory for weakly perturbed limit cycles in hybrid dynamical systems that facilitates analysis control and optimization of nonlinear oscillators whose smooth models are unavailable or intractable on the basis of the generalized theory we analyze injection locking of hybrid limit cycle oscillators by periodic forcing and reveal their characteristic synchronization properties such as ultrafast and robust entrainment to the periodic forcing and logarithmic scaling at the synchronization transition we also illustrate the theory by analyzing the synchronization dynamics of a simple physical model of biped locomotion

follow the bouncing ball global results on tracking and state estimation with impacts in this paper we formulate tracking and state estimation problems of a translating mass in a polyhedral billiard as a stabilization problem for a suitable set due to the discontinuous trajectories arising from the impacts we use hybrid systems stability analysis tools to establish the results using a novel concept of mirrored images of the target mass we prove that 1 a tracking control algorithm and 2 an observer algorithm guarantee global exponential stability results for specific classes of polyhedral billiards including rectangles moreover we combine these two algorithms within dynamic controllers that guarantee global output feedback tracking the results are illustrated via simulations

dynamical properties of hybrid automata hybrid automata provide a language for modeling and analyzing digital and analogue computations in real time systems hybrid automata are studied here from a dynamical systems perspective necessary and sufficient conditions for existence and uniqueness of solutions are derived and a class of hybrid automata whose solutions depend continuously on the initial state is characterized the results on existence uniqueness and continuity serve as a starting point for stability analysis lyapunov s theorem on stability via linearization and lasalle s invariance principle are generalized to hybrid automata

disturbance decoupling in hybrid linear systems with state jumps this work investigates the problem of making the output of a hybrid linear system totally insensitive to a disturbance input while attaining global asymptotic stability of the compensated hybrid dynamics in the presence of possibly nonuniformly spaced state jumps meeting the only condition that the minimum time between two consecutive occurrences is no smaller than a given positive real constant both existential and constructive necessary and sufficient conditions for solvability of the problem addressed are proven

a domain decomposition solver for acoustic scattering by elastic objects in layered media a finite element solution procedure is presented for accurately computing time harmonic acoustic scattering by elastic targets buried in sediment an improved finite element discretization based on trilinear basis functions leading to fourth order phase accuracy is described for sufficiently accurate discretizations 100 million to 1 billion unknowns are required the resulting systems of linear equations are solved iteratively using the gmres method with a domain decomposition preconditioner employing a fast direct solver due to the construction of the discretization and preconditioner iterations can be reduced onto a sparse subspace associated with the interfaces numerical experiments demonstrate capability to evaluate the scattered field with hundreds of wavelengths

a finite element capacitance matrix method for exterior helmholtz problems we introduce an algorithm for the efficient numerical solution of exterior boundary value problems for the helmholtz equation the problem is reformulated as an equivalent one on a bounded domain using an exact non local boundary condition on a circular artificial boundary an fft based fast helmholtz solver is then derived for a finite element discretization on an annular domain the exterior problem for domains of general shape are treated using an imbedding or capacitance matrix method the imbedding is achieved in such a way that the resulting capacitance matrix has a favorable spectral distribution leading to mesh independent convergence rates when krylov subspace methods are used to solve the capacitance matrix equation

a fast method for the solution of the helmholtz equation in this paper we consider the numerical solution of the helmholtz equation arising from the study of the wave equation in the frequency domain the approach proposed here differs from those recently considered in the literature in that it is based on a decomposition that is exact when considered analytically so the only degradation in computational performance is due to discretization and roundoff errors in particular we make use of a multiplicative decomposition of the solution of the helmholtz equation into an analytical plane wave and a multiplier which is the solution of a complex valued advection diffusion reaction equation the use of fast multigrid methods for the solution of this equation is investigated numerical results show that this is an efficient solution algorithm for a reasonable range of frequencies

trajectory tracking for a particle in elliptical billiards an infinitely rigid unitary mass particle is considered moving on a planar region delimited by a rigid elliptical barrier elliptical billiards under the action of proper control forces a class of periodic trajectories involving an infinite sequence of non smooth impacts between the mass and the barrier at fixed times is found by using an lmis based procedure the jumps in the velocities at the impact times render difficult if not impossible to obtain the classical stability and attractivity properties for the dynamic system describing the tracking error behaviour hence the tracking control problem is properly stated using notions similar to the quasi stability concept in v lakshmikantham d d bainov and p s simeonov theory of impulsive differential equations 6 world scientific 1989 a controller whose state is subject to discontinuities based on the internal model principle is shown to solve the proposed tracking problem giving rise to control forces that are piecewise continuous fun

a parallel multigrid based preconditioner for the 3d heterogeneous high frequency helmholtz equation we investigate the parallel performance of an iterative solver for 3d heterogeneous helmholtz problems related to applications in seismic wave propagation for large 3d problems the computation is no longer feasible on a single processor and the memory requirements increase rapidly therefore parallelization of the solver is needed we employ a complex shifted laplace preconditioner combined with the bi cgstab iterative method and use a multigrid method to approximate the inverse of the resulting preconditioning operator a 3d multigrid method with 2d semi coarsening is employed we show numerical results for large problems arising in geophysical applications

hybrid output regulation for linear systems with periodic jumps solvability conditions structural implications and semi classical solutions the problem of output regulation for a class of hybrid linear systems characterized by periodic jumps is considered in this paper necessary and sufficient conditions for its solution are provided both in the full information and error feedback case by a detailed analysis of such conditions several interesting properties are derived including the fact that the regulator must contain a suitably defined copy of the flow zero dynamics of the plant in addition to the usual copy of the exosystem dynamics and the fact that the output regulation problem is generically solvable for fat plants having more inputs than outputs and generically not solvable for square plants in addition semi classical solutions to the hybrid output regulation problem are introduced and discussed such solutions are characterized by the property that the required steady state input can be generated as a simple linear function of the state of the exosystem as in the non hybrid case although the associated steady state response has a genuinely hybrid structure

a fast algorithm for the electromagnetic scattering from a large cavity a fast algorithm is presented for solving electromagnetic scattering from a rectangular open cavity embedded in an infinite ground plane the medium inside the cavity is assumed to be vertically layered by introducing a transparent artificial boundary condition the problem in the open cavity is reduced to a bounded domain problem a simple finite difference method is then applied to solve the model helmholtz equation the fast algorithm is designed for solving the resulting discrete system in terms of the discrete fourier transform in the horizontal direction a gaussian elimination in the vertical direction and a preconditioning conjugate gradient method with a complex diagonal preconditioner for the indefinite interface system the existence and uniqueness of the finite difference solution are established for arbitrary wave numbers our numerical experiments for large numbers of mesh points up to 16 million unknowns and for large wave numbers e g between 100 and 200 wavelengths show that the algorithm is extremely efficient the cost for calculating the radar cross section which is of significant interest in practice is o m2 for an m times m mesh the proposed algorithm may be extended easily to solve discrete systems from other discretization methods of the model problem

a composite preconditioner for the electromagnetic scattering from a large cavity in this paper we propose a composite preconditioner for the electromagnetic scattering from a large cavity the electromagnetic cavity problem is described by the helmholtz equation with a nonlocal boundary condition on the aperture of the cavity and dirichlet or neumann boundary conditions on the walls of the cavity the preconditioner proposed here combines the optimal sine transform based approximation with a layered medium model using fast fourier transforms the computational cost of every iteration is o n 2logn on an nxn uniform partition of the unit square numerical results for a model problem show that the new preconditioner is more efficient than those recently considered in the literature for the cavity with a small portion of non layered media we propose a sparse preconditioned conjugate orthogonal conjugate gradient solver combined with the new preconditioner numerical results for a model problem are reported to demonstrate the efficiency of the sparse solver

a domain decomposition method for the helmholtz equation in a multilayer domain the two dimensional helmholtz equation for problems where the physical domain consists of layers with different material properties is studied an efficient preconditioner for iterative solution of the problem is constructed the problem is discretized with fourth order accurate finite difference operators for the construction of the radiation boundary conditions a fourth order finite element method also is used the large sparse complex indefinite and ill conditioned system of equations that arises is solved with preconditioned restarted gmres a domain decomposition method is used in which the preconditioning is based on the schur complement algorithm with fast poisson type preconditioners for the subdomains the memory requirements for the preconditioner are nearly linear in the number of unknowns the arithmetic complexity for each iteration is low whereas the construction of the preconditioner is a bit more expensive electromagnetic wave propagation in a three layered waveguide is used as a model problem numerical experiments show that convergence is achieved in a few iterations compared with banded gaussian elimination which is a standard solution method the iterative method shows significant gain in both memory requirements and arithmetic complexity furthermore the relative gain grows when the problem size increases

on the control of finite dimensional mechanical systems with unilateral constraints this paper focuses on the problem of the control of a class of mechanical systems with a finite number of degrees of freedom subject to unilateral constraints on the position roughly speaking those systems are described by a set of ordinary differential equations that represent smooth dynamics together with an algebraic inequality condition f q spl ges 0 where q is the vector of generalized coordinates and an impact rule relating the interaction impulse and the velocity nonsmooth dynamics is at the core of the study of such systems this implies that one can suitably define solutions and stability concepts that fit with the considered model we then discuss the closed loop control problem and analyze various switching control strategies

towards a stability theory of general hybrid dynamical systems in recent work we proposed a general model for hybrid dynamical systems whose states are defined on arbitrary metric space and evolve along some notion of generalized abstract time for such systems we introduced the usual concepts of lyapunov and lagrange stability we showed that it is always possible to transform this class of hybrid dynamical systems into another class of dynamical systems with equivalent qualitative properties but defined on real time r 0 the motions of this class of systems are in general discontinuous this class of systems may be finite or infinite dimensional for the above discontinuous dynamical systems and hence for the above hybrid dynamical systems we established the principal lyapunov stability theorems as well as lagrange stability theorems for some of these we also established converse theorems we demonstrated the applicability of these results by means of specific classes of hybrid dynamical systems in the present paper we continue the work described above in doing so we first develop a general comparison theory for the class of hybrid dynamical systems resp discontinuous dynamical systems considered herein making use of stability preserving mappings we then show how these results can be applied to establish some of the principal lyaponov stability theorems for the latter we also state and prove a converse theorem not considered previously finally to demonstrate the applicability of our results we consider specific examples throughout the paper

sine transform based preconditioners for symmetric toeplitz systems abstract the optimal circulant preconditioner for a given matrix a is defined to be the minimizer of c a f over the set of all circulant matrices c here f is the frobenius norm optimal circulant preconditioners have been proved to be good preconditioners in solving toeplitz systems with the preconditioned conjugate gradient method in this paper we construct an optimal sine transform based preconditioner which is defined to be the minimizer of b a f over the set of matrices b that can be diagonalized by sine transforms we will prove that for general n by n matrices a these optimal preconditioners can be constructed in o n 2 real operations and in o n real operations if a is toeplitz we will also show that the convergence properties of these optimal sine transform preconditioners are the same as that of the optimal circulant ones when they are employed to solve toeplitz systems numerical examples are given to support our convergence analysis

output regulation by error dynamic feedback in hybrid systems with periodic state jumps abstract this work deals with output regulation in multivariable hybrid systems featuring a continuous time linear dynamics periodically affected by instantaneous changes of the state more precisely given a hybrid linear plant and a hybrid linear exogenous system with periodic state jumps the problem consists in finding a hybrid feedback regulator with the same characteristics achieving global asymptotic stability of the closed loop dynamics and asymptotic tracking of the reference generated by the exogenous system for all the initial states starting from a general necessary and sufficient condition for the existence of a solution the discussion leads to a more specific sufficient condition which outlines the computational framework for a straightforward synthesis of the compensator the internal model principle is shown to hold in a more general formulation than the original one adapted to the hybrid systems considered a numerical example is worked out with the aim of illustrating how to implement the devised technique the geometric approach is the key methodology in attaining these results

trajectory tracking control of multiconstraint complementarity lagrangian systems in this study one considers the tracking control problem of a class of nonsmooth fully actuated lagrangian systems subject to frictionless unilateral constraints the task under consideration contains both free motion and constraint motion phases a switching controller that guarantees an approximate tracking is designed particular attention is paid to transition impacting and detachment between different phases of motion the exogenous signals that assure the stabilization on take off from some constraints are explicitly defined this paper extends previous works on the topic as it considers more than one constraint for n degree of freedom systems numerical examples illustrate the main results

internal model principle for linear systems with periodic state jumps this paper deals with the problem of output regulation for a class of hybrid linear siso systems and exosystems whose dynamics have jumps according to the value of a clock variable the problem of designing controllers embedding an hybrid internal model of the switching exosystem in order to achieve the regulation objective is addressed key concepts and tools well known in the field of output regulation for continuous time systems such as the concept of steady state response of regulator equations and the internal model property are thus generalized to the hybrid setting emphasis is placed on internal models and stabilizers that are robust to uncertainties entering both in the flow and jump dynamics of the hybrid controlled system time varying and time independent design principles are presented simulation results show the effectiveness of the proposed solutions in meaningful cases

a domain decomposition method for the helmholtz equation and related optimal control problems we present an iterative domain decomposition method to solve the helmholtz equation and related optimal control problems the proof of convergence of this method relies on energy techniques this method leads to efficient algorithms for the numerical resolution of harmonic wave propagation problems in homogeneous and heterogeneous media

distance function design and lyapunov techniques for the stability of hybrid trajectories the comparison between time varying hybrid trajectories is crucial for tracking observer design and synchronisation problems for hybrid systems with state triggered jumps in this paper a generic distance function is designed that can be used for this purpose the so called peaking phenomenon which occurs when using the euclidean distance to compare two hybrid trajectories is circumvented by taking the hybrid nature of the system explicitly into account based on the proposed distance function we define the stability of a trajectory and present sufficient lyapunov type conditions for hybrid system with state triggered jumps a constructive lyapunov function design is presented for hybrid systems with affine flow and jump maps and a jump set that is a hyperplane the stability conditions can then be verified using linear matrix conditions finally for this class of systems we present a tracking controller that asymptotically stabilises a given hybrid reference trajectory and we illustrate our results with an example

smooth lyapunov functions for hybrid systems part i existence is equivalent to robustness hybrid systems are dynamical systems where the state is allowed to either evolve continuously flow on certain subsets of the state space or evolve discontinuously jump from other subsets of the state space for a broad class of such systems characterized by mild regularity conditions on the data we establish the equivalence between the robustness of stability with respect to two measures and a characterization of such stability in terms of a smooth lyapunov function this result unifies and generalizes previous results for differential and difference inclusions with outer semicontinuous and locally bounded right hand sides furthermore we give a description of forward completeness of a hybrid system in terms of a smooth lyapunov like function

compact optimal quadratic spline collocation methods for the helmholtz equation quadratic spline collocation methods are formulated for the numerical solution of the helmholtz equation in the unit square subject to non homogeneous dirichlet neumann and mixed boundary conditions and also periodic boundary conditions the methods are constructed so that they are a of optimal accuracy and b compact that is the collocation equations can be solved using a matrix decomposition algorithm involving only tridiagonal linear systems using fast fourier transforms the computational cost of such an algorithm is o n 2logn on an nxn uniform partition of the unit square the results of numerical experiments demonstrate the optimal global accuracy of the methods as well as superconvergence phenomena in particular it is shown that the methods are fourth order accurate at the nodes of the partition

robust hybrid output regulation for linear systems with periodic jumps semiclassical internal model design a complete procedure for the design of a robust output feedback regulator is proposed for a class of uncertain linear hybrid systems with periodic jumps using a hybrid extension of the classical internal model principle simple conditions testable on the plant nominal data for the problem solvability are given the plant is not restricted to be minimum phase square or single input single output the proposed regulator has a key feature of containing an internal model composed by two main units a flow internal model in charge of providing the correct input to achieve regulation during flows and a jump internal model in charge of suitably resetting the state of the regulator at each period the proposed procedure is illustrated by its application to a physically motivated example for which the output regulation problem is not solvable by methods appeared thus far in the literature

structural properties of a class of linear hybrid systems and output feedback stabilization in this paper we deal with the problem of output feedback stabilization for a class of linear hybrid systems this problem is addressed by characterizing the structural properties of such a class of systems namely reachability controllability stabilizability observability constructibility and detectability are framed in terms of algebraic and geometric conditions on the data of the system two canonical forms recalling the classical kalman decompositions with respect to reachability and observability are given by taking advantage of this characterization duality between control and observation structural properties is established and necessary and sufficient conditions for the existence of a linear time invariant output feedback compensator are stated compared with previous results no assumption is needed on the plant about minimum phaseness relative degree or squareness

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

on input allocation based regulation for linear over actuated systems results concerning the output regulation problem for over actuated linear systems are presented in this paper the focus is on the characterization of the solution of the full information regulator problem for systems which are right invertible but not left invertible and the input operator is injective the intrinsic redundancy in the plant model is exploited by parameterizing all solutions of the regulator equations and performing a static or dynamic optimization on the space of solutions this approach effectively shapes the non unique steady state of the system so that the long term behavior optimizes a given performance index in particular nonlinear cost functions that account for constraints on the inputs are considered within the general form of a hybrid system assumed for the allocation mechanism an example is given to illustrate the proposed methodology

frequency domain analysis of linear systems with periodic jumps definition of hybrid transfer function pole and zero abstract a frequency domain analysis is carried out for a class of linear systems in the presence of periodic time driven jumps the result is achieved by introducing the notion of transfer function in this context and by relying on a suitably defined transform for hybrid arcs whose properties are comprehensively discussed it is firstly shown that knowledge of the transfer function permits the computation in closed form of the complete state and output response of the hybrid plant which may be otherwise intractable from the computational point of view moreover the hybrid transfer function resulting from series interconnection of two plants is derived and exploited to introduce and discuss in the single input single output case the concepts of zero and pole of the hybrid transfer function differently from the classical purely continuous or discrete time cases only a subset of the latter is related to the stability properties of the hybrid system

a multigrid method enhanced by krylov subspace iteration for discrete helmholtz equations standard multigrid algorithms have proven ineffective for the solution of discretizations of helmholtz equations in this work we modify the standard algorithm by adding gmres iterations at coarse levels and as an outer iteration we demonstrate the algorithm s effectiveness through theoretical analysis of a model problem and experimental results in particular we show that the combined use of gmres as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as 200 for fixed wave numbers it displays grid independent convergence rates and has costs proportional to the number of unknowns

asymptotic tracking of periodic trajectories for a simple mechanical system subject to nonsmooth impacts an infinitely rigid mass is considered which moves on a plane under the action of proper control forces within a closed region delimited by an infinitely massive and rigid circular barrier a suitably amended tracking problem for a class of periodic trajectories involving an infinite number of nonsmooth impacts between the mass and the barrier is formally stated and solved

velocity observers for non linear mechanical systems subject to non smooth impacts this paper is concerned with the estimation of the velocity variables when the position variables are the measured outputs for non linear mechanical systems subject to non smooth impacts both elastic and inelastic i e with coefficient of restitution e 1 and e 0 1 respectively a reduced order observer is proposed which guarantees that the corresponding error system despite the possible presence of an infinite sequence of non smooth impacts is locally exponentially stable an estimate of the basin of attraction is also given

on the solvability of a boundary value problem for model equations of linearly viscous materials this paper deals with model equations for linearly viscous materials viscosity is allowed to vary with temperature the global solvability of an initial boundary value problem for nonlinear equations is proved by continuation of a local solution with the help of a priori estimates the main attention is paid to the global estimates

on the graphical stability of hybrid solutions with non matching jump times we investigate stability of a solution of a hybrid system in the sense that the graphs of solutions from nearby initial conditions remain close and tend towards the graph of the given solution in this manner a small continuous time mismatch is allowed between the jump times of neighbouring solutions and the peaking phenomenon is avoided we provide conditions such that this stability notion is implied by stability with respect to a specifically designed distance like function hence stability of solutions in the graphical sense can be analysed with existing lyapunov techniques

observer design for unilaterally constrained lagrangian systems a passivity based approach this paper addresses the problem of state estimation in nonlinear lagrangian dynamical systems with frictionless unilateral constraints using the position measurement as output the discontinuous velocity variable in such systems is modeled as a function of bounded variation so that zeno phenomenon is not ruled out since the derivative of such functions is represented with the lebesgue stieltjes measure the framework of measure differential inclusions mdis is used to describe the dynamics a class of estimators is proposed which also uses the framework of mdis and is shown to generate asymptotically converging state estimates the existence and uniqueness of solutions for the proposed estimators is rigorously proven the global stability of error dynamics is analyzed using the generalized lyapunov methods for functions of bounded variation as particular cases of our estimators we provide an explicit construction of a full order observer and a reduced order observer

lyapunov conditions for input to state stability of impulsive systems this paper introduces appropriate concepts of input to state stability iss and integral iss for impulsive systems i e dynamical systems that evolve according to ordinary differential equations most of the time but occasionally exhibit discontinuities or impulses we provide a set of lyapunov based sufficient conditions for establishing these iss properties when the continuous dynamics are iss but the discrete dynamics that govern the impulses are not the impulses should not occur too frequently which is formalized in terms of an average dwell time adt condition conversely when the impulse dynamics are iss but the continuous dynamics are not there must not be overly long intervals between impulses which is formalized in terms of a novel reverse adt condition we also investigate the cases where i both the continuous and discrete dynamics are iss and ii one of these is iss and the other only marginally stable for the zero input while sharing a common lyapunov function in the former case we obtain a stronger notion of iss for which a necessary and sufficient lyapunov characterization is available the use of the tools developed herein is illustrated through examples from a micro electro mechanical system mems oscillator and a problem of remote estimation over a communication network

numerical solution of electromagnetic scattering from a large partly covered cavity the paper focuses on the numerical study of electromagnetic scattering from two dimensional 2d large partly covered cavities which is described by the helmholtz equation with a nonlocal boundary condition on the aperture the classical five point finite difference method is applied for the discretization of the helmholtz equation and a linear approximation is used for the nonlocal boundary condition we prove the existence and uniqueness of the numerical solution when the medium in the cavity is y direction layered or the number of the mesh points on the aperture is large enough the fast algorithm proposed in bao and sun 2005 2 for open cavity models is extended to solving the partly covered cavity problem with vertically layered media a preconditioned krylov subspace method is proposed to solve the partly covered cavity problem with a general medium in which a layered medium model is used as a preconditioner of the general model numerical results for several types of partly covered cavities with different wave numbers are reported and compared with those by ilu type preconditioning algorithms our numerical experiments show that the proposed preconditioning algorithm is more efficient for partly covered cavity problems particularly with large wave numbers

lyapunov stability theory of nonsmooth systems this paper develops nonsmooth lyapunov stability theory and lasalle s invariance principle for a class of nonsmooth lipschitz continuous lyapunov functions and absolutely continuous state trajectories computable tests based on filipov s differential inclusion and clarke s generalized gradient are derived the primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right hand side such as in nonsmooth dynamic systems or variable structure control