global smooth solutions and exponential stability for a nonlinear beam in this paper we consider a dynamical system with boundary input and output describing the bending vibrations of a quasi linear beam where the nonlinearity comes from hooke s law first we derive an existence result for short time solutions of the system of equations then we show that the structure of the boundary input and output forces the system to admit global solutions at least when the initial data and the boundary input are small in a certain sense in particular we prove that the norm of the state of the system decays exponentially if the input becomes zero after a finite time the input being zero can be understood as a boundary feedback

feedback boundary control of linear hyperbolic systems with relaxation we consider boundary stabilization for one dimensional systems of linear hyperbolic partial differential equations with relaxation structure such equations appear in many applications by combining weighted lyapunov functions the structure is used to derive new stabilization results the result is illustrated with an application to boundary stabilization of water flows in open canals

distributed finite element kalman filter for field estimation the paper deals with decentralized state estimation for spatially distributed systems described by linear partial differential equations from discrete in space and time noisy measurements provided by sensors deployed over the spatial domain of interest a fully scalable approach is pursued by decomposing the domain into possibly overlapping subdomains assigned to different processing nodes interconnected to form a network each node runs a local finite dimensional discrete time kalman filter which exploits the finite element approach for spatial discretization a backward euler method for time discretization and the parallel schwarz method to iteratively enforce continuity of the field predictions over the boundaries of adjacent subdomains numerical stability of the adopted approximation scheme and stability of the proposed distributed finite element kalman filter are mathematically proved the effectiveness of the proposed approach is then demonstrated via simulation experiments concerning the estimation of a bi dimensional temperature field

brief paper boundary control of linearized saint venant equations oscillating modes the paper investigates the control of oscillating modes occurring in open channels due to the reflection of propagating waves on the boundaries these modes are well represented by linearized saint venant equations a set of hyperbolic partial differential equations which describe the dynamics of one dimensional open channel flow around a given stationary regime we use a distributed transfer function approach to compute a dynamic boundary controller that cancels the oscillating modes over all the canal pool this result is recovered with a riemann invariants approach in the case of a frictionless horizontal canal pool the effect of a proportional boundary control on the poles of the transfer matrix is then characterized by a root locus and we derive an asymptotic result for high frequencies closed loop poles

finite time boundary stabilization of general linear hyperbolic balance laws via fredholm backstepping transformation this paper is devoted to a simple and new proof on the optimal finite control time for general linear coupled hyperbolic system by using boundary feedback on one side the feedback control law is designed by first using a volterra transformation of the second kind and then using an invertible fredholm transformation both existence and invertibility of the transformations are easily obtained

model reference adaptive control of 2 times 2 coupled linear hyperbolic pdes we solve a model reference adaptive control problem for a class of linear 2 times 2 hyperbolic partial differential equations pdes with uncertain system parameters subject to harmonic disturbances from a single boundary measurement anticollocated with the actuation this is done by transforming the system into a canonical form from which filters are designed so that the states can be expressed as linear combinations of the filters and uncertain parameters a representation facilitating for the design of adaptive laws a stabilizing controller is then combined with the adaptive laws to make the measured signal asymptotically track the output of a reference model the reference model is taken as a simple transport pde moreover pointwise boundedness of all variables in the closed loop is proved provided the reference signal is bounded the theory is demonstrated in a simulation

global solutions and blow up solutions for a nonlinear string with boundary input and output abstract in this paper we investigate the global solutions and blow up solutions for a nonlinear string with boundary input and output and prove that under some conditions the system has global solutions and blow up solutions respectively

the second main theorem for holomorphic curves into semi abelian varieties we establish the second main theorem with the best truncation level one tf r l d n1 r f d tf r for an entire holomorphic curve f c a into a semi abelian variety a and an arbitrary effective reduced divisor d on a the low truncation level is important for applications we will actually prove this for the jet lifts of f finally we give some applications including the solution of a problem posed by mark green

dissipative boundary conditions for 2 2 hyperbolic systems of conservation laws for entropy solutions in bv in this article we investigate the bv stability of 2 2 hyperbolic systems of conservation laws with strictly positive velocities under dissipative boundary conditions more precisely we derive sufficient conditions guaranteeing the exponential stability of the system under consideration for entropy solutions in bv our proof is based on a front tracking algorithm used to construct approximate piecewise constants solutions whose bv norms are controlled through a lyapunov functional this lyapunov functional is inspired by the one proposed in j glimm s seminal work j glimm comm pure appl math 18 697 715 1965 modified with some suitable weights in the spirit of the previous works j m coron g bastin and b d andrea novel siam j control optim 47 3 1460 1498 2008 and j m coron b d andrea novel and g bastin ieee trans automat control 52 1 2 11 2007

an adaptive observer design for n 1 coupled linear hyperbolic pdes based on swapping in this paper we use swapping design filters to bring systems of n 1 partial differential equations of the hyperbolic type to static form standard parameter identification laws can then be applied to estimate unknown parameters in the boundary conditions proof of boundedness of the adaptive laws are offered and the results are demonstrated in simulations

the mixed initial boundary value problem for reducible quasilinear hyperbolic systems with linearly degenerate characteristics in this paper we prove that the c0 boundedness of solution implies the global existence and uniqueness of c1 solution to the mixed initial boundary value problem for linearly degenerate reducible quasilinear hyperbolic systems with nonlinear boundary conditions and we show by an example that the c0 norm of solution may blow up in a finite time this gives the mechanism of the formation of singularities caused by the interaction of boundary conditions with nonlinear hyperbolic waves the same result is still valid for the quasilinear hyperbolic system of rich type

the mixed initial boundary value problem for diagonalizable quasilinear hyperbolic systems in the first quadrant abstract in this paper we investigate the mixed initial boundary value problem for diagonalizable quasilinear hyperbolic systems with nonlinear boundary conditions on a half unbounded domain t x t 0 x 0 under the assumptions that system is strictly hyperbolic and linearly degenerate we obtain the global existence and uniqueness of c 1 solutions with the bounded l 1 l norm of the initial data as well as their derivatives and appropriate boundary condition based on the existence results of global classical solutions we also prove that when t tends to infinity the solutions approach a combination of c 1 travelling wave solutions under the appropriate assumptions of initial and boundary data the results can be applied to the equation of time like extremal surface in minkowski space r 1 1 n

an energy preserving nonlinear system modeling a string with input and output on the boundary abstract we study an energy conserving distributed parameter system described by a nonlinear string equation with the input and output at the boundary we prove the existence of global smooth solutions to this quasilinear hyperbolic system if the initial data and the boundary input are small if moreover the input function becomes zero after some finite time then the state trajectories decay exponentially

feedback boundary control of linear hyperbolic equations with stiff source term abstractwe consider feedback boundary control of hyperbolic systems with stiff source terms by combining weighted lyapunov functions the structure is used to derive stabilisation results in our analysis we give stabilising feedback laws that allow a robust uniform exponential stabilisation for a whole range of values of an uncertain parameter with a decay rate that is independent of the parameter in particular we are interested in the limit case when the relaxation parameter approaches zero the result is illustrated with the numerical analysis on the decay rate of the lyapunov function in terms of the stiff parameter and an application to boundary stabilisation of gas dynamics in pipes

on some nonlinear hyperbolic systems with damping boundary conditions in recent years the effect of boundary damping on the solutions to interior initial boundary value problems for hyperbolic equations has been extensively studied in 111 quinn and russell considered the wave equation with damping boundary condition on a part of the boundary it was proved by control theoretic method that if the boundary where the boundary condition was posed was star shaped then the solution of the wave equation had polynomial decay rate later on chen in l got the exponential decay of solutions to the wave equation under the more restrictive assumptions on the boundary and also lagnese in 6 got the exponential decay of solutions to the three dimensional elastic wave equations with damping boundary condition on a part of the boundary provided the geometry of the boundary is suitably restricted as for the nonlinear problem greenberg and li in 131 considered the nonlinear conservation system in one space dimension with damping boundary condition and proved the global existence of smooth solution when the initial data are small shen and zheng 14 and nagasawa 9 proved the global existence of classical solutions to the system of one dimensional thermoelasticity and to the system of one dimensional nonlinear problem in higher space dimension respectively for the nonlinear problem in higher space dimension to our knowledge the works were done by qin lo and zajaczkowski 26 only for the nonlinear wave equation with damping boundary conditions based on chen s results in l in this paper our aim is to derive the decay rate of solutions to the linear hyperbolic system of second order with damping boundary condition without any assumptions on the geometry of the boundary by using the spectral analysis method cf section 2 below and then to prove the global existence of smooth solutions to the corresponding nonlinear system with applications to nonlinear elastodynamics and to nonlinear acoustic wave equations our approach to obtaining the rate of decay of solutions to linear problem is completely different from l 6 and lo

brief paper boundary control with integral action for hyperbolic systems of conservation laws stability and experiments a strict lyapunov function for boundary control with integral actions of hyperbolic systems of conservation laws that can be diagonalised with riemann invariants is presented the time derivative of this lyapunov function can be made strictly negative definite by an appropriate choice of the boundary conditions and the integral control gains previous stability results are extended to guarantee the local convergence of the state towards a desired set point furthermore the control can be implemented as a feedback of the state measured only at the boundaries the control design method is illustrated with a hydraulic application namely the level and flow regulation in a reach of the sambre river and in the micro channel of valence respectively through simulations and experimentations

contraction analysis of nonlinear distributed systems contraction theory is a comparatively recent dynamic analysis and non linear control system design tool based on an exact differential analysis of convergence this paper extends contraction theory to local and global stability analysis of important classes of non linear distributed dynamics such as convection diffusion reaction processes lagrangian and hamilton jacobi dynamics and optimal controllers and observers by contrast with stability proofs based on energy dissipation stability and convergence can be determined for energy based systems excited by time varying inputs the hamilton jacobi bellman controller and a similar optimal non linear observer design are studied based on explicitly computable conditions on the convexity of the cost function these stability conditions extend the well known conditions on controllability and observability grammians for linear time varying systems without requiring the unknown transition matrix of the underlying differential dynamics

exact boundary controllability for 1 d quasilinear hyperbolic systems with a vanishing characteristic speed the general theory on exact boundary controllability for general first order quasilinear hyperbolic systems requires that the characteristic speeds of the system do not vanish this paper deals with exact boundary controllability when this is not the case some important models are also shown as applications of the main result the strategy uses the return method which allows one in certain situations to recover nonzero characteristic speeds

controllability and stabilization of a conservation law modeling a highly re entrant manufacturing system abstract in this paper we study the controllability and stabilization for a scalar conservation law modeling a highly re entrant manufacturing system with local and nonlocal velocity we prove a local state controllability result i e there exists a control that drives the solution from any given initial condition to any desired final condition in a certain time period provided that the initial and final data are both close to the origin a local result on nodal profile controllability is also given i e for any initial data and any given out flux in a neighborhood of the origin there exists a control under which the solution starts from any initial data reaches exactly any desired out flux over a fixed time period besides using a lyapunov function approach we can stabilize the system to the origin exponentially by output feedback control

boundary feedback control in networks of open channels this article deals with the regulation of water flow in open channels modelled by saint venant equations by means of a riemann invariants approach we deduce stabilizing control laws for a single horizontal reach without friction the stability condition is extended to a general class of hyperbolic systems which can describe canal networks with more general topologies a control law design based on this condition is illustrated with a simple case study two reaches in cascade the proof of the main stability theorem is based on a previous result from li ta tsien concerning the existence and decay of classical solutions of hyperbolic systems

control of 2 2 linear hyperbolic systems backstepping based trajectory generation and pi based tracking abstract we consider the problems of trajectory generation and tracking for general 2 2 systems of first order linear hyperbolic pdes with anti collocated boundary input and output we solve the trajectory generation problem via backstepping the reference input which generates the desired output incorporates integral operators acting on advanced and delayed versions of the reference output with kernels which were derived by vazquez krstic and coron for the backstepping stabilization of 2 2 linear hyperbolic systems we apply our approach to a wave pde with indefinite in domain and boundary damping for tracking the desired trajectory we employ a pi control law on the tracking error of the output we prove exponential stability of the closed loop system under the proposed pi control law when the parameters of the plant and the controller satisfy certain conditions by constructing a novel non diagonal lyapunov functional we demonstrate that the proposed pi control law compensates in the output the effect of in domain and boundary disturbances we illustrate our results with numerical examples

finite time stabilization of 2 times2 hyperbolic systems on tree shaped networks we investigate the finite time boundary stabilization of a one dimensional first order quasilinear hyperbolic system of diagonal form on 0 1 the dynamics of both boundary controls are governed by a finite time stable ode the solutions of the closed loop system issuing from small initial data in lip 0 1 are shown to exist for all times and to reach the null equilibrium state in finite time when only one boundary feedback law is available a finite time stabilization is shown to occur roughly in a twice longer time the above feedback strategy is then applied to the saint venant system for the regulation of water flows in a network of canals

brief paper lyapunov exponential stability of 1 d linear hyperbolic systems of balance laws explicit boundary dissipative conditions are given for the exponential stability in l 2 norm of one dimensional linear hyperbolic systems of balance laws email protected email protected email protected x 0 over a finite interval when the matrix m is marginally diagonally stable the result is illustrated with an application to boundary feedback stabilisation of open channels represented by linearised saint venant exner equations

estimation of boundary parameters in general heterodirectional linear hyperbolic systems abstract in this paper we extend recent results on state and boundary parameter estimation in coupled systems of linear partial differential equations pdes of the hyperbolic type consisting of n rightward and one leftward convecting equations to the general case which involves an arbitrary number of pdes convecting in both directions two adaptive observers are derived based on swapping design where one introduces a set of filters that can be used to express the system states as linear static combinations of the filter states and the unknown parameters standard parameter identification laws can then be applied to estimate the unknown parameters one observer which requires sensing at both boundaries generates estimates of the boundary parameters and system states while the second observer estimates the parameters from sensing limited to the boundary anti collocated with the uncertain parameters proof of boundedness of the adaptive laws is offered and sufficient conditions ensuring exponential convergence are derived the theory is verified in simulations

control of continuum models of production systems a production system which produces a large number of items in many steps can be modelled as a continuous flow problem the resulting hyperbolic partial differential equation pde typically is nonlinear and nonlocal modeling a factory whose cycle time depends nonlinearly on the work in progress one of the few ways to influence the output of such a factory is by adjusting the start rate in a time dependent manner we study two prototypical control problems for this case i demand tracking where we determine the start rate that generates an output rate which optimally tracks a given time dependent demand rate and ii backlog tracking which optimally tracks the cumulative demand the method is based on the formal adjoint method for constrained optimization incorporating the hyperbolic pde as a constraint of a nonlinear optimization problem we show numerical results on optimal start rate profiles for steps in the demand rate and for periodically varying demand rates and discuss the influence of the nonlinearity of the cycle time on the limits of the reactivity of the production system differences between perishable and non perishable demand demand versus backlog tracking are highlighted

switching rules for stabilization of linear systems of conservation laws in this paper the exponential convergence in l 2 norm is analyzed for a class of switched linear systems of conservation laws the boundary conditions are subject to switches we investigate the problem of synthesizing stabilizing switching controllers by means of lyapunov techniques three control strategies are developed based on steepest descent selection possibly combined with a hysteresis and a low pass filter for the first strategy we show the global exponential stabilizability but no result for the existence and uniqueness of trajectories can be stated for the other ones the problem is shown to be well posed and global exponential convergence can be obtained moreover we consider the robustness issues for these switching rules in presence of measurement noise some numerical examples illustrate our approach and show the merits of the proposed strategies particularly we have developed a model for a network of open channels with switching controllers in the gate operations

adaptive stabilization of 2 times 2 linear hyperbolic systems with an unknown boundary parameter from collocated sensing and control in this paper we solve an adaptive control problem for a class of 2 times 2 linear hyperbolic partial differential equations where sensing and actuation are restricted to the boundary anticollocated with an uncertain parameter this is done by combining a recently derived adaptive observer for the system states and the uncertain parameter with an adaptive control law proof of l 2 boundedness for all signals in the closed loop is given and the system states are proved to converge to zero pointwise in space the theory is demonstrated in a simulation

stable two dimensional solitary pulses in linearly coupled dissipative kadomtsev petviashvili equations we present a two dimensional 2d generalization of the stabilized kuramoto sivashinsky system based on the kadomtsev petviashvili kp equation including dissipation of the generic newell whitehead segel nws type and gain the system directly applies to the description of gravity capillary waves on the surface of a liquid layer flowing down an inclined plane with a surfactant diffusing along the layer s surface actually the model is quite general offering a simple way to stabilize nonlinear media combining the weakly 2d dispersion of the kp type with gain and nws dissipation other applications are internal waves in multilayer fluids flowing down an inclined plane double front flames in gaseous mixtures etc parallel to this weakly 2d model we also introduce and study a semiphenomenological one whose dissipative terms are isotropic rather than of the nws type in order to check if qualitative results are sensitive to the exact form of the lossy terms the models include an additional linear equation of the advection diffusion type linearly coupled to the main kp nws equation the extra equation provides for stability of the zero background in the system thus opening a way for the existence of stable localized pulses we focus on the most interesting case when the dispersive part of the system is of the kp i type which corresponds e g to capillary waves and makes the existence of completely localized 2d pulses possible treating the losses and gain as small perturbations and making use of the balance equation for the field momentum we find that the equilibrium between the gain and losses may select two steady state solitons from their continuous family existing in the absence of the dissipative terms the latter family is found in an exact analytical form and is numerically demonstrated to be stable the selected soliton with the larger amplitude is expected to be stable direct simulations completely corroborate the analytical predictions for both the physical and phenomenological models

existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in eulerian and lagrangian coordinates we prove the existence and compactness stability of entropy solutions for the hyperbolic systems of conservation laws corresponding to the isentropic gas dynamics where the pressure and density are related by a law for any 1 our results considerably extend and simplify the program initiated by diperna and provide a complete existence proof our methods are based on the compensated compactness and the kinetic formulation of systems of conservation laws 1996 john wiley sons inc

boundary control of hyperbolic conservation laws using a frequency domain approach the paper uses a frequency domain method for the boundary control of hyperbolic conservation laws we show that the transfer function of the hyperbolic system belongs to the callier desoer algebra which opens the way of sound results and in particular to the existence of a necessary and sufficient condition for the closed loop stability and the use of nyquist type test we examine the link between input output stability and exponential stability of the state specific results are then derived for the case of proportional diagonal boundary controllers the results are illustrated in the case of diagonal boundary control of open channel flow

a generalized bloch s theorem and the hyperbolicity of the complement of an ample divisor in an abelian variety bloch s theorem on the hyperbolicity of nonlinear subvarieties of abelian varieties states that the zariski closure of the image of a holomorphic map from c to an abelian variety is precisely the translate of an abelian subvariety b26 ks0 o77 w80 gg79 for a survey on the proofs of bloch s theorem see e g si95 chapter 7 the purpose of this note is to prove the following generalization of bloch s theorem if the image of a holomorphic map f from c to an abelian variety is zariski dense then the zariski closure of the image of the differential of any order of the map f is invariant under any translation of the abelian variety theorem 2 2 below this generalization has as a corollary the conjecture of lang la91 that the complement of an ample divisor in an abelian variety is hyperbolic the following slightly more general statement which together with bloch s theorem implies lang s conjecture see corollary 3 2 is a corollary of our generalized bloch s theorem the image from c to an abelian variety with zariski dense image must intersect any complex hypersurface of the abelian variety theorem 3

local stability theory of solitary pulses in an active medium abstract if the spatio temporal chaos of an extended active medium is driven by repulsive or attractive interaction among localized coherent structures such as kinks and pulses its dynamics can often be described by a low dimensional dynamical system this welcomed reduction in the degrees of freedom only occurs when the localized structures remain stable during their mutual interaction it is shown that the structures can be destroyed by two types of local disturbance an expanding radiation packet and a localized disturbance which travels with the structure these two instabilities and the bifurcations they induce select the proper coherent structure and specify the dynamics of the generalized kuramoto sivashinsky gks equation in an extended domain the possibility of describing pulse radiation interaction by resonance poles also suggests that a more general coherent structure theory can be developed

nonlinear waves in two fluid flows abstract typically the interaction of two miscible fluids leads to the formation of segregation patterns in the form of plane concentration waves and two or three dimensional voids or clusters the development of small amplitude plane travelling waves and the subsequent onset of higher dimensional instabilities can be elucidated analytically in addition numerical methods allow the computation of higher amplitude solutions as both direct approaches become costly although at different amplitude levels one is tempted to look for a simpler model problem therefore a nonlinear wave equation has been derived for two dimensional dispersed two phase flow in fluidized beds near the stability boundary using the froude number as small parameter this equation is a two dimensional perturbation of the korteweg de vries equation containing all low order terms it allows to identify the transition point between voids and clusters and to investigate these two regions for solitary and periodic waves a ginzburg landau equation is derived to describe transverse perturbations of one dimensional wavetrains bifurcating at the stability boundary

global smooth solutions for the quasilinear wave equation with boundary dissipation abstract we consider the existence of global solutions of the quasilinear wave equation with a boundary dissipation structure of an input output in high dimensions when initial data and boundary inputs are near a given equilibrium of the system our main tool is the geometrical analysis the main interest is to study the effect of the boundary dissipation structure on solutions of the quasilinear system we show that the existence of global solutions depends not only on this dissipation structure but also on a riemannian metric given by the coefficients and the equilibrium of the system some geometrical conditions on this riemannian metric are presented to guarantee the existence of global solutions in particular we prove that the norm of the state of the system decays exponentially if the input stops after a finite time which implies the exponential stabilization of the system by boundary feedback

disturbance rejection in general heterodirectional 1 d linear hyperbolic systems using collocated sensing and control abstract we solve a disturbance rejection problem for a general class of heterodirectional 1 d linear hyperbolic partial differential equations pdes the disturbance enters at one boundary while sensing and actuation is limited to the opposite boundary we do this by combining a separately designed controller and observer into an output feedback controller and proving stability of the overall closed loop system and exponential regulation of the control objective to zero the theory is demonstrated in a simulation

design of integral controllers for nonlinear systems governed by scalar hyperbolic partial differential equations the paper deals with the control and regulation by integral controllers for the nonlinear systems governed by scalar quasi linear hyperbolic partial differential equations both the control input and the measured output are located on the boundary the closed loop stabilization of the linearized model with the designed integral controller is proved first by using the method of spectral analysis and then by the lyapunov direct method based on the elaborated lyapunov function we prove local exponential stability of the nonlinear closed loop system with the same controller the output regulation to the set point with zero static error by the integral controller is shown upon the nonlinear system numerical simulations by the preissmann scheme are carried out to validate the robustness performance of the designed controller to face to unknown constant disturbances

finite dimensional regulators for a class of regular hyperbolic pde systems abstractin this paper the output regulation problem is addressed for a class of linear hyperbolic infinite dimensional systems with spatially varying coefficients modelling a large class of convection dominated transport reaction systems in particular distributed parameter systems with bounded input and unbounded output operators are considered first we demonstrate a general conclusion about the exponential stability of the considered system by relating the stability to the solution of an associated differential equation based on the assumption that the hyperbolic system satisfies the exponential stability conditions the main manuscript contribution is the development of two novel finite dimensional regulators output and error feedback regulators such that the controlled output of the plant tracks a reference signal generated by a known signal process exosystem in order to guarantee the feasibility of the proposed regulators the solvability of the corresponding sylvester equations is discusse

feedback stabilization for the mass balance equations of an extrusion process in this article we study the stabilization problem for an extrusion process in the isothermal case the model expresses the mass conservation in the extruder chamber and consists of a hyperbolic partial differential equation pde and a nonlinear ordinary differential equation ode whose dynamics describes the evolution of a moving interface by using a lyapunov approach we obtain the exponential stabilization for the closed loop system under natural feedback controls through indirect measurements numerical simulations are also provided with a comparison between the proposed approach and linear pi feedback controller

exact internal controllability of nodal profile for first order quasilinear hyperbolic systems abstract for 1 d first order quasilinear hyperbolic systems without zero eigenvalues based on the theory of semi global c 1 solution by means of the results on both the exact boundary controllability and the exact boundary controllability of nodal profile using an extension method the exact controllability of nodal profile can be obtained purely with the help of internal controls moreover using a perturbation method by adding suitable internal controls to the part of equations corresponding to zero eigenvalues the corresponding exact internal controllability of nodal profile for 1 d first order quasilinear hyperbolic systems with zero eigenvalues can be also obtained

a quadratic lyapunov function for saint venant equations with arbitrary friction and space varying slope the exponential stability problem of the nonlinear saint venant equations is addressed in this paper we consider the general case where an arbitrary friction and space varying slope are both included in the system which lead to non uniform steady states an explicit quadratic lyapunov function as a weighted function of a small perturbation of the steady states is constructed then we show that by a suitable choice of boundary feedback controls that we give explicitly the local exponential stability of the nonlinear saint venant equations for the h 2 norm is guaranteed

control of homodirectional and general heterodirectional linear coupled hyperbolic pdes research on stabilization of coupled hyperbolic pdes has been dominated by the focus on pairs of counter convecting heterodirectional transport pdes with distributed local coupling and with controls at one or both boundaries a recent extension allows stabilization using only one control for a system containing an arbitrary number of coupled transport pdes that convect at different speeds against the direction of the pde whose boundary is actuated in this paper we present a solution to the fully general case in which the number of pdes in either direction is arbitrary and where actuation is applied on only one boundary to all the pdes that convect downstream from that boundary to solve this general problem we solve as a special case the problem of control of coupled homodirectional hyperbolic linear pdes where multiple transport pdes convect in the same direction with arbitrary local coupling our approach is based on pde backstepping and yields solutions to stabilization by both full state and observer based output feedback and trajectory tracking problems

boundary feedback stabilization of a class of coupled hyperbolic equations with nonlocal terms this paper solves the problem of boundary feedback stabilization of a class of coupled ordinary differential equations hyperbolic equations with boundary trace and integral nonlocal terms using the backstepping approach the controller is designed by formulating an integral operator whose kernel is required to satisfy a coupled hyperbolic partial integral differential equation by applying the method of successive approximations the kernel s well posedness is given we prove the exponential stability of the origin of the system in a suitable hilbert space moreover a wave system with nonlocal terms is stabilized by applying the above result

stabilization of a system of n 1 coupled first order hyperbolic linear pdes with a single boundary input we solve the problem of stabilization of a class of linear first order hyperbolic systems featuring n rightward convecting transport pdes and one leftward convecting transport pde we design a controller which requires a single control input applied on the leftward convecting pde s right boundary and an observer which employs a single sensor on the same pde s left boundary we prove exponential stability of the origin of the resulting plant observer controller system in the spatial l2 sense

fast and accurate simulations of shallow water equations in large networks abstract we present a fast and accurate method for solving shallow water flows in large networks the governing equations consist of the one dimensional shallow water equations including friction forces a class of first order coupling conditions are implemented at the water intersections two dimensional counterparts for simple water networks are also considered for comparison as a numerical solver for shallow water equations we employed the discontinuous galerkin method in one and two space dimensions the method is simple highly accurate and exhibits enhanced stability properties in the vicinity of sharp gradients which are often present in the solution of shallow water flow problems the method belongs to a class of locally conservative finite element methods whose approximate solutions are discontinuous across inter element boundaries the performance of the proposed techniques is assessed on several applications for shallow water flows in networks the aim of such a method compared to the conventional two dimensional simulations is to solve shallow water flows in large networks efficiently and with an appropriate level of accuracy

stabilized kuramoto sivashinsky system a model consisting of a mixed kuramoto sivashinsky korteweg de vries equation linearly coupled to an extra linear dissipative equation is proposed the model applies to a description of surface waves on multilayered liquid films the extra equation makes it possible to stabilize the zero solution in the model thus opening the way to the existence of stable solitary pulses by means of perturbation theory treating the dissipation and the instability generating gain in the model but not the linear coupling between the two waves as small perturbations and making use of the balance equation for the net momentum we demonstrate that the perturbations may select two steady state solitons from their continuous family existing in the absence of the dissipation and gain in this case the selected pulse with the larger value of the amplitude is expected to be stable provided that the zero solution is stable the prediction is completely confirmed by direct simulations if the integration domain is not very large some pulses are stable even when the zero background is unstable an explanation for the latter finding is proposed furthermore stable bound states of two and three pulses are found numerically

null controllability of the stabilized kuramoto sivashinsky system with one distributed control this paper presents a control problem for a one dimensional nonlinear parabolic system which consists of a kuramoto sivashinsky korteweg de vries equation coupled to a heat equation we address the problem of controllability by means of a control supported in an interior open subset of the domain and acting on one equation only the local null controllability of the system is proved the proof is based on a carleman estimate for the linearized system around the origin a local inversion theorem is applied to get the result for the nonlinear system

on the modelling and stabilization of flows in networks of open canals in this paper we present a model for the controlled flow of a fluid through a network of canals using a coupled system of st venant equations we then generalize in a variety of ways recent results of coron d andrea novel and bastin concerning the stabilizability around equilibrium of the flow through a single channel this work is based on the theory of quasilinear hyperbolic systems and in particular on a delicate result of li ta tsien concerning the existence and decay of global classical solutions

adaptive stabilization of n 1 coupled linear hyperbolic systems with uncertain boundary parameters using boundary sensing abstract we design an adaptive control law stabilizing a class of systems of the form of n 1 linear partial differential equations pdes of the hyperbolic type from boundary sensing only we do this by combining a recently derived observer based on swapping design with a backstepping based adaptive control law with time varying gains boundedness of all signals in the closed loop system and asymptotic convergence to zero pointwise in space for the system states are proved the theory is demonstrated in a simulation

dissipative boundary conditions for one dimensional quasi linear hyperbolic systems lyapunov stability for the c 1 norm this paper is concerned with boundary dissipative conditions that guarantee the exponential stability of classical solutions of one dimensional quasi linear hyperbolic systems we present a comprehensive review of the results that are available in the literature the main result of the paper is then to supplement these previous results by showing how a new lyapunov stability approach can be used for the analysis of boundary conditions that are known to be dissipative for the c 1 norm

analysis and stabilization for networked linear hyperbolic systems of rationally dependent conservation laws in this paper for a networked linear hyperbolic partial differential equations pdes system of conservation laws the propagation periods of which are rationally dependent with coupled boundary conditions we propose a novel approach to analyze its controllability and observability in addition we propose a design method of a stabilizing controller where a boundary input with boundary valued feedback is considered first we characterize the control properties such as controllability of such a pde system in terms of the corresponding ones of a finite dimensional discrete time system defined on the boundaries of the pde system which is derived by fully exploiting the method of characteristics since the obtained discrete time system is low dimensional its analysis is relatively easier next we propose a design method of a stabilizing controller based on this discrete time system finally numerical simulations are presented to show that the proposed method is effective

adaptive control of linear 2 2 hyperbolic systems abstract we design two closely related state feedback adaptive control laws for stabilization of a class of 2 2 linear hyperbolic system of partial differential equations pdes with constant but uncertain in domain and boundary parameters one control law uses an identifier while the other is based on swapping design we establish boundedness of all signals in the closed loop system pointwise in space and time and convergence of the system states to zero pointwise in space the theory is demonstrated in simulations

minimum time control of heterodirectional linear coupled hyperbolic pdes we solve the problem of stabilizing a general class of linear first order hyperbolic systems considered systems feature an arbitrary number of coupled transport pdes convecting in either direction using the backstepping approach we derive a full state feedback law and a boundary observer enabling stabilization by output feedback unlike previous results finite time convergence to zero is achieved in the theoretical lower bound for control time

two sided boundary stabilization of heterodirectional linear coupled hyperbolic pdes we solve the problem of stabilizing a general class of linear first order hyperbolic systems using actuation at both boundaries of the spatial domain we design a novel fredholm transformation similarly to backstepping approaches to derive a boundary controller and a boundary observer enabling stabilization by output feedback this yields an explicit full state feedback law that achieves the theoretical lower bound for convergence to zero

numerical boundary feedback stabilisation of non uniform hyperbolic systems of balance laws in this paper numerical boundary stabilisation of a non uniform hyperbolic system of balance laws is studied for the numerical discretisation of the balance laws a first order explicit upwind sc

global classical solutions of mixed initial boundary value problem for quasilinear hyperbolic systems abstract in this paper we consider the mixed initial boundary value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant t x t 0 x 0 under the assumptions that the system is strictly hyperbolic and linearly degenerate or weakly linearly degenerate the global existence and uniqueness of c 1 solutions are obtained for small initial and boundary data we also present two applications for physical models

cylindrical solitary pulses in a two dimensional stabilized kuramoto sivashinsky system abstract by linearly coupling a generalized zakharov kuznetsov equation alias the two dimensional 2d benney equation to an extra linear dissipative equation a 2d extension of a recently proposed stabilized kuramoto sivashinsky system is developed the model applies to the description of surface waves on 2d liquid layers in various physical settings the extra equation provides for the stability of the zero state in the system thus paving a way to the existence of stable 2d localized solitary pulses sps a perturbation theory based on a family of cylindrical solitons existing in the conservative counterpart of the system is developed by treating dissipation and gain in the model as small perturbations it is shown that the system may select two steady state solitons from the continuous family provided by the conservative counterpart of which the one with larger amplitude is expected to be stable numerical simulations support the analytical predictions quite well additionally it is found that a shallow quasi one dimensional 1d trough is attached to the stable sp if the integration domain is not very large and an explanation to this feature is proposed stable double humped bound states of two pulses are found too

exponential stability of switched linear hyperbolic initial boundary value problems we consider the initial boundary value problem governed by systems of linear hyperbolic partial differential equations in the canonical diagonal form and study conditions for exponential stability when the system discontinuously switches between a finite set of modes the switching system is fairly general in that the system matrix functions as well as the boundary conditions may switch in time we show how the stability mechanism developed for classical solutions of hyperbolic initial boundary value problems can be generalized to the case in which weaker solutions become necessary due to arbitrary switching we also provide an explicit dwell time bound for guaranteeing exponential stability of the switching system when for each mode the system is exponentially stable our stability conditions only depend on the system parameters and boundary data these conditions easily generalize to switching systems in the nondiagonal form under a simple commutativity assumption we present tutorial examples to illustrate the instabilities that can result from switching

marcum q functions and explicit kernels for stabilization of 2 2 linear hyperbolic systems with constant coefficients abstract we find the exact analytical solution to a goursat pde system governing the kernels of a backstepping based boundary control law that stabilizes a constant coefficient 2 2 system of first order hyperbolic linear pdes the solution to the goursat system is related to the solution of a simpler explicitly solvable goursat system through a suitable infinite series of powers of partial derivatives which is summed explicitly in terms of special functions including bessel functions and the generalized marcum q functions of the first order the marcum functions are common in certain applications in communications but have not appeared previously in control design problems the dependence of the explicit solutions with respect to system parameters is analyzed through several examples including the stabilization of a constant equilibrium for a quasi linear plant

on the relaxation approximation of boundary control of the isothermal euler equations we consider the isothermal euler equations without friction that simulate gas flow through a pipe we consider the problem of boundary stabilisation of this system locally around a given stationary state we present a feedback law that is linear in the physical variables and yields exponential decay of the system state for the numerical solution of hyperbolic systems of conservation laws the jin xin relaxation scheme can be used therefore we also consider the boundary stabilisation of the relaxation system by the linear riemann feedback and present numerical examples that show the rapid exponential decay of the stabilised system

output feedback boundary control of 2 2 semilinear hyperbolic systems abstract we consider the control and state estimation of a class of 2 2 semilinear hyperbolic systems with actuation and sensing collocated at one boundary our approach exploits the dynamics on the characteristic lines of the hyperbolic system the control method using full state feedback can be used for both stabilization of an equilibrium and tracking at an arbitrary point in the domain the control objective is achieved globally in minimum time a lyapunov function is constructed to prove exponential convergence in the spatial supremum norm for linear systems the control input can be written explicitly as the inner product of kernels with the state and turns out to be equivalent to the control input obtained from previously known backstepping methods the observer achieves exact state estimation also in minimum time and combined with the state feedback controller solves the output feedback control problem the performance is demonstrated in a numerical example

model reference adaptive control of n 1 coupled linear hyperbolic pdes abstract we solve a model reference adaptive control mrac problem for a class of systems consisting of n 1 coupled linear hyperbolic partial differential equations the goal is achieved from a single boundary sensing anti collocated with the actuation with the only knowledge of the system being the system s transport delays and the sign of the product of the actuation and measurement scaling constants boundedness in l 2 is shown for all signals in the closed loop system moreover the adaptive output feedback stabilization problem is a subproblem of the mrac problem and it is shown that if the reference signal is set identically zero the system s l 2 norms are bounded and square integrable the theory is demonstrated in simulations

a note on jets of entire curves in semi abelian varieties we prove a product decomposition of the zariski closure of every jet lift of an entire curve f c a into a semi abelian variety a provided that f is of finite order on the other hand by giving an example of f into a three dimensional abelian variety we show that this product decomposition does not hold in general there was a gap in the proofs of 2 proposition 1 8 ii and of 6 theorem 2 2

regularity and lyapunov stabilization of weak entropy solutions to scalar conservation laws we consider the problem of lyapunov boundary stabilization of the weak entropy solution to a scalar conservation law with strictly convex flux in one dimension of space around a uniform equilibrium we show that for a specific class of boundary conditions the solution to the initial boundary value problem for an initial condition with bounded variations can be approximated arbitrarily closely in the l 1 norm by a piecewise smooth solution with finitely many discontinuities the constructive method we present designs explicit boundary conditions in this class which guarantee lyapunov stability of the weak entropy solution to the initial boundary value problem we show how the greedy control obtained by maximizing the decrease of the natural lyapunov function may fail to asymptotically stabilize and a brute force control generates unbounded variation of traces we then design a stabilizing control which avoid oscillations and propose a nonlocal technique depending on time and the whole initial datum which optimizes the convergence time controllers performance is illustrated on numerical benchmarks using the godunov scheme