energy levels of a two dimensional anharmonic oscillator characteristic function approach the eigenlevels of a two dimensional quartic anharmonic oscillator are identified as the zeros of a characteristic function derived from the resolvent numerical values are obtained by means of pade approximants

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

one dimensional transport equations with discontinuous coefficients we consider one dimensional linear transport equations with bounded but possibly discontinuous coefficient a the cauchy problem is studied from two different points of view in the first case we assume that a is piecewise continuous we give an existence result and a precise description of the solutions on the lines of discontinuity in the second case we assume that a satisfies a one sided lipschitz condition we give existence uniqueness and general stability results for backward lipschitz solutions and forward measure solutions by using a duality method we prove that the flux associated to these measure solutions is a product by some canonical representative of a key words linear transport equations discontinuous coefficients weak stability duality product of a measure by a discontinuous function nonnegative solutions 1991 mathematics subject classification primary 35f10 35b35 34a12 to appear in nonlinear analysis tma departement de mathematiques et applications umr cnrs 8553 ecole normale superieure et cnrs 45 rue d ulm 75230 paris cedex 05 france francois bouchut ens fr mathematiques applications et physique mathematique d orleans umr cnrs 6628 universite d orleans 45067 orleans cedex 2 france james math cnrs fr

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

the stability of two station multitype fluid networks this paper studies the fluid models of two station multiclass queueing networks with deterministic routing a fluid model is globally stable if the fluid network eventually empties under each nonidling dispatch policy we explicitly characterize the global stability region in terms of the arrival and service rates we show that the global stability region is defined by the nominal workload conditions and the virtual workload conditions and we introduce two intuitively appealing phenomena virtual stations and push starts that explain the virtual workload conditions when any of the workload conditions is violated we construct a fluid solution that cycles to infinity showing that the fluid network is unstable when all the workload conditions are satisfied we solve a network flow problem to find the coefficients of a piecewise linear lyapunov function the lyapunov function decreases to zero proving that the fluid level eventually reaches zero under any nonidling dispatch policy under certain assumptions on the inter arrival and service time distributions a queueing network is stable or positive harris recurrent if the corresponding fluid network is stable thus the workload conditions are sufficient to ensure the global stability of two station multiclass queueing networks with deterministic routing

adjoint and defect error bounding and correction for functional estimates we present two error estimation approaches for bounding or correcting the error in functional estimates such as lift or drag adjoint methods quantify the error in a particular output functional that results from residual errors in approximating the solution to the partial differential equation defect methods can be used to bound or reduce the error in the entire solution with corresponding improvements to functional estimates both approaches rely on smooth solution reconstructions and may be used separately or in combination to obtain highly accurate solutions with asymptotically sharp error bounds the adjoint theory is presented for both smooth and shocked problems numerical experiments confirm fourth order error estimates for a pressure integral of shocked quasi id euler flow by employing defect and adjoint methods together and accounting for errors in approximating the geometry it is possible to obtain functional estimates that exceed the order of accuracy of the discretization process and the reconstruction approach superconvergent drag estimates are obtained for subsonic euler flow over a lifting airfoil

finite difference order doubling in two dimensions an order doubling process previously used to obtain eighth order eigenvalues from the fourth order numerov method is applied to the perturbed oscillator in two dimensions a simple method of obtaining high order finite difference operators is reported and an odd parity boundary condition is found to be effective in facilitating the smooth operation of the order doubling process

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

a quick perturbative method for schrodinger equations within the framework of perturbation theory we propose firstly an iterative method which may serve as a source of optimal unperturbed solutions in both one and more dimensions it combines the runge kutta and newton algorithm and its efficiency is illustrated on a few quartic oscillators secondly admitting also an arbitrary perturbation of potentials we generalize the existing runge kutta one dimensional rayleigh schrodinger constructions of energies and wavefunctions to more dimensions

reaching a consensus in a dynamically changing environment convergence rates measurement delays and asynchronous events this paper uses recently established properties of compositions of directed graphs together with results from the theory of nonhomogeneous markov chains to derive worst case convergence rates for the headings of a group of mobile autonomous agents which arise in connection with the widely studied vicsek consensus problem the paper also uses graph theoretic constructions to solve modified versions of the vicsek problem in which there are measurement delays asynchronous events or a group leader in all three cases the conditions under which consensus is achieved prove to be almost the same as the conditions under which consensus is achieved in the synchronous delay free leaderless case

boundary observer design for a class of semi linear hyperbolic pde systems with recycle loop abstractthe present manuscript considers the observer design problem for a class of scalar semi linear hyperbolic partial differential equation pde systems with a recycle loop through the boundar

local error estimates for discontinuous solutions of nonlinear hyperbolic equations let u x t be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data suppose u varepsilon x t is the solution of an approximate viscosity regularization where varepsilon 0 is the small viscosity amplitude it is shown that by post processing the small viscosity approximation u varepsilon pointwise values of u and its derivatives with an error as close to e as desired can be recovered the analysis relies on the adjoint problem of the forward error equation which in this case amounts to a backward linear transport equation with discontinuous coefficients the novelty of our approach is to use a generalized e condition of the forward problem in order to deduce a w 1 infty energy estimate for the discontinuous backward transport equation this in turn leads to e uniform estimate on moments of the error u varepsilon u the approach presented does not follow the characteristics and therefore applies mutatis mutandis to o

existence of solution to supply chain models based on partial differential equation with discontinuous flux function abstract we consider a recently proposed model by armbruster et al 2011 2 for supply chains with finite buffers the continuous model is based on a scalar conservation law and discontinuous flux function a suitable reformulation of the model is introduced and studied analytically the latter include waves with infinite negative wave speed using wave front tracking existence of solutions is obtained for supply chains and sequences of supply chains numerical results are presented

a variational approach for continuous supply chain networks we consider a continuous supply chain network consisting of buffering queues and processors first proposed by d armbruster p degond and c ringhofer siam j appl math 6 6

analysis and control of a scalar conservation law modeling a highly re entrant manufacturing system abstract in this paper we study a scalar conservation law that models a highly re entrant manufacturing system as encountered in semi conductor production as a generalization of coron et al 2010 14 the velocity function possesses both the local and nonlocal character we prove the existence and uniqueness of the weak solution to the cauchy problem with initial and boundary data in l we also obtain the stability continuous dependence of both the solution and the out flux with respect to the initial and boundary data finally we prove the existence of an optimal control that minimizes in the l p sense with 1 p the difference between the actual out flux and a forecast demand over a fixed time period

a spectral algorithm for seriation and the consecutive ones problem in applications ranging from dna sequencing through archeological dating to sparse matrix reordering a recurrent problem is the sequencing of elements in such a way that highly correlated pairs of elements are near each other that is given a correlation function f reflecting the desire for each pair of elements to be near each other find all permutations pi with the property that if pi i pi j pi k then f i j ge f i k and f j k ge f i k this seriation problem is a generalization of the well studied consecutive ones problem we present a spectral algorithm for this problem that has a number of interesting features whereas most previous applications of spectral techniques provide only bounds or heuristics our result is an algorithm that correctly solves a nontrivial combinatorial problem in addition spectral methods are being successfully applied as heuristics to a variety of sequencing problems and our result helps explain and justify these applications

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

energy levels of two dimensional anharmonic oscillators with sextic and octic perturbations two renormalized versions of inner product and hypervirial techniques are presented together with the hill determinant approach for performing accurate calculations of the energy levels of two dimensional oscillators characterized by the potentials vn 6 x y x2 y2 axxx6 ayyy6 3axyx4 y2 3ayxx2y4 and vn 8 x y x2 y2 axxx8 ayyy8 6b xyx4y4 4axyx6y2 4ayxx2y6 the methods have been used effectively to calculate the energy eigenvalues for many eigenstates over a wide range values of perturbation parameter and various values of coefficients aij the results produced by various techniques are compared with each other

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

the sharp form of oleĭnik s entropy condition in several space variables we investigate the conditions under which the volpert kruzkov solution of a single conservation law in several space variables with flux f will satisfy the simplified entropy condition div f u s l t and when this condition guarantees uniqueness for given lx cauchy data we show that when f is c1 our condition guarantees uniqueness iff f is isotropic and that for such f the volpert kruzkov solution always satisfies our condition

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

nonlinear resonance for quasilinear hyperbolic equation the purpose of this paper is to study the wave behavior of hyperbolic conservation laws with a moving source resonance occurs when the speed of the source is too close to one of the characteristic speeds of the system for the nonlinear system characteristic speeds depend on the basic dependence variables and resonance gives rise to nonlinear interactions which lead to rich wave phenomena motivated by physical examples a scalar model is proposed and analyzed to describe the qualitative behavior of waves for a general system in resonance with the source analytical understanding is used to design a numerical scheme based on the random choice method an important physical example is transonic gas flow through a nozzle this analysis provides a transparent and revealing qualitative understanding of wave behavior of gas flow including such phenomena as nonlinear stability instability and changing types of waves

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

the confined system approximation for solving non separable potentials in three dimensions the hilbert space to which the wavefunction of the three dimensional schrodinger equation belongs has been replaced by where is a bounded region the energy spectrum of the usual unbounded system is then determined by showing that the dirichlet and neumann problems in generate upper and lower bounds respectively to the eigenvalues required highly accurate numerical results for the quartic and sextic oscillators are presented for a wide range of the coupling constants

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

brief paper synchronization in networks of identical linear systems the paper investigates the synchronization of a network of identical linear state space models under a possibly time varying and directed interconnection structure the main result is the construction of a dynamic output feedback coupling that achieves synchronization if the decoupled systems have no exponentially unstable mode and if the communication graph is uniformly connected the result can be interpreted as a generalization of classical consensus algorithms stronger conditions are shown to be sufficient but to some extent also necessary to ensure synchronization with the diffusive static output coupling often considered in the literature

on the smallest eigenvalue of grounded laplacian matrices we provide bounds on the smallest eigenvalue of grounded laplacian matrices which are obtained by removing certain rows and columns of the laplacian matrix of a given graph the gap between our upper and lower bounds depends on the ratio of the smallest and largest components of the eigenvector corresponding to the smallest eigenvalue of the grounded laplacian we provide a graph theoretic bound on this ratio and subsequently obtain a tight characterization of the smallest eigenvalue for certain classes of graphs specifically for weighted erdos renyi random graphs we show that when a sufficiently small set s of rows and columns is removed from the laplacian and the probability p of adding an edge is sufficiently large the smallest eigenvalue of the grounded laplacian asymptotically almost surely approaches mu w vert s vert p where mu w is the mean edge weight we also show that for weighted random d regular graphs with a single row and column removed the smallest eigenvalue is theta 1 n where n is the number of nodes in the network our bounds have applications to the study of the convergence rate in consensus dynamics with stubborn or leader nodes

a comparison of optimization based approaches for a model computational aerodynamics design problem abstract the objective of this paper is to compare three optimization based methods for solving aerodynamic design problems we use the euler equations for one dimensional duct flow as a model problem the optimization methods are i the black box method with finite difference gradients ii a modification where gradients are found by an algorithm based on the implicit function theorem and iii an all atonce method where the flow and design variables are simultaneously altered the three methods are applied to the model problem and compared for efficiency robustness and implementation difficulty we also show that the black box implicit gradient method is equivalent to applying the variational or optimal control approach to design optimization directly to the discretized analysis problem rather than to the continuous problem as is usually done the black box method with implicit gradients seems to provide a good compromise of features and can be retrofitted to most existing analysis codes to turn them into design codes although the all at once method was found to be less robust than the black box methods when it succeeded it was considerably more efficient

a simple iterative solution of the schrodinger equation in matrix representation form a simple procedure for solving the schrodinger equation is presented it is based upon an iterative solution of the secular equation a large enough convergence rate is obtained by using a basis set of properly scaled functions the effect of the scaling parameter on the convergence rate is studied in order to improve the calculation the method is applied to simple though non trivial quantum mechanical models such as the quartic sextic and octic anharmonic oscillators a double well potential and the linear confining potential highly accurate eigenvalues for all values of the coupling parameter are obtained

global convergence properties of conjugate gradient methods for optimization this paper explores the convergence of nonlinear conjugate gradient methods without restarts and with practical line searches the analysis covers two classes of methods that are globally convergent on smooth nonconvex functions some properties of the fletcher reeves method play an important role in the first family whereas the second family shares an important property with the polak ribiere method numerical experiments are presented

leader following consensus of multi agent systems under fixed and switching topologies the leader following consensus problem of higher order multi agent systems is considered in the system the dynamics of each agent and the leader is a linear system the control of each agent using local information is designed and detailed analysis of the leader following consensus is presented for both fixed and switching interaction topologies which describe the information exchange between the multi agent systems the design technique is based on algebraic graph theory riccati inequality and lyapunov inequality simulations indicate the capabilities of the algorithms

two sided eigenvalue bounds for the spherically symmetric states of the schro dinger equation the eigenvalues of the radial schr6dinger equation are calculated very accurately by obtaining exact upper and lower bounds by truncating the usual unbounded domain 0 cx of the system to a finite interval of the form 0 g two auxiliary eigenvalue problems are defined it is then proved that the eigenvalues of the resulting confined systems provide upper and lower bounds converging monotonically to the true eigenvalues required moreover each auxiliary eigenvalue problem gives rise to an orthonormal set involving bessel functions the matrix representation of the hamiltonian is therefore derived by expanding the wave function into a fourier bessel series numerical results for single and doublewell polynomial oscillators as well as gaussian type non polynomial potentials illustrate that the eigenvalues can be calculated to an arbitrary accuracy whenever the boundary parameter g is in the neighborhood of some critical value denoted by ecr 1998 elsevier science b v all rights reserved

optimal boundary control of nonlinear hyperbolic conservation laws with switched boundary data we consider the optimal control of initial boundary value problems for entropy solutions of scalar hyperbolic conservation laws in particular we consider initial boundary value problems where the initial and boundary data switch between different c 1 functions at certain switching points and both the functions and the switching points are controlled we show that the control to state mapping is differentiable in a certain generalized sense which implies frechet differentiability with respect to the control functions and the switching points for the composition with a tracking type functional even in the presence of shocks we also present an adjoint based formula for the gradient of the reduced objective functional

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

operator method for coupled anharmonic oscillators two coupled anharmonic oscillators are considered as a model for a nonperturbative description of the correlation and nonadiabatic effects which are typical for many dimensional quantum systems the eigenvalues and eigenfunctions for this model are found by means of the operator method modified for the case of degenerate solutions of the schr dinger equation it is shown that the zeroth approximation of the method allows one to find the analytical and uniformly suitable approximation for the energy levels and their splitting in the entire range of hamiltonian parameters and quantum numbers numerical calculations demonstrate the convergence of the successive approximations even for quasistationary states of the system the results are of interest for applied problems of spectroscopy and solid state physics

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

controllability for a scalar conservation law with nonlocal velocity abstract in this paper we study the state controllability and nodal profile controllability for a scalar conservation law with a nonlocal velocity that models a highly re entrant manufacturing system as encountered in semi conductor production we first prove a local state controllability result i e there exists a control that drives the solution from any given initial data to any desired final data in a certain time period provided that the initial and final data are both close to a given equilibrium 0 we also obtain a global state controllability result for the same system where there is no limitation on the distance between the initial and final data finally we prove a nodal profile controllability result i e there exists a control under which the solution starts from any initial data reaches exactly any given out flux over a fixed time period

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

convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws part 1 linearized approximations and linearized output functionals this paper analyzes the convergence of discrete approximations to the linearized equations arising from an unsteady one dimensional hyperbolic equation with a convex flux function a simple modified lax friedrichs discretization is used on a uniform grid and a key point is that the numerical smoothing increases the number of points across the nonlinear discontinuity as the grid is refined it is proved that this gives pointwise convergence almost everywhere for the solution to the linearized discrete equations with smooth initial data and also convergence in the discrete approximation of linearized output functionals in part 2 m giles and s ulbrich siam j numer anal 48 2010 pp 905 921 we extend the results to dirac initial data for the linearized equation and will prove the pointwise convergence almost everywhere for the solution of the adjoint discrete equations in addition we present numerical results illustrating the asymptotic behavior which is analyzed

opinion fluctuations and disagreement in social networks we study a tractable opinion dynamics model that generates long run disagreements and persistent opinion fluctuations our model involves an inhomogeneous stochastic gossip process of continuous opinion dynamics in a society consisting of two types of agents 1 regular agents who update their beliefs according to information that they receive from their social neighbors and 2 stubborn agents who never update their opinions and might represent leaders political parties or media sources attempting to influence the beliefs in the rest of the society when the society contains stubborn agents with different opinions the belief dynamics never lead to a consensus among the regular agents instead beliefs in the society fail to converge almost surely the belief profile keeps on fluctuating in an ergodic fashion and it converges in law to a nondegenerate random vector the structure of the graph describing the social network and the location of the stubborn agents within it shape the opinion dynamics the expected belief vector is proved to evolve according to an ordinary differential equation coinciding with the kolmogorov backward equation of a continuous time markov chain on the graph with absorbing states corresponding to the stubborn agents and hence to converge to a harmonic vector with every regular agent s value being the weighted average of its neighbors values and boundary conditions corresponding to the stubborn agents beliefs expected cross products of the agents beliefs allow for a similar characterization in terms of coupled markov chains on the graph describing the social network we prove that in large scale societies which are highly fluid meaning that the product of the mixing time of the markov chain on the graph describing the social network and the relative size of the linkages to stubborn agents vanishes as the population size grows large a condition of homogeneous influence emerges whereby the stationary beliefs marginal distributions of most of the regular agents have approximately equal first and second moments

state and observer based feedback control of normal flow equations abstract we investigate wellposedness and stability of normal flow equations with two different control schemes the first one considers the velocity field as a control action and relies on a proportional regulator that is proved to be stable in the second one control acts on the source term and is based on an observer providing an estimate of the gradient of the normal flow solution convergence properties are demonstrated in this case numerical results are presented to show the effectiveness of the proposed approaches

method for optimally controlling unsteady shock strength in one dimension this paper presents a new formulation and computational solution of an optimal control problem concerning unsteady shock wave attenuation the adjoint system of equations for the unsteady euler system in one dimension is derived and used in an adjoint based solution procedure for the optimal control a novel algorithm is used to satisfy all necessary first order optimality conditions while locally minimizing an appropriate cost functional distributed control solutions with certain physical constraints are calculated for attenuating blast waves similar to those generated by ignition overpressure from the shuttle s solid rocket booster during launch results are presented for attenuating shocks traveling at mach 1 5 and 3 5 down to 85 80 and 75 of the uncontrolled wave s driving pressure the control solutions give insight into the magnitude and location of energy dissipation necessary to decrease a given blast wave s overpressure to a set target level over a given spatial domain while using only as m

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

adjoint based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws we propose a rigorous procedure to obtain the adjoint based gradient representation of cost functionals for the optimal control of discontinuous solutions of conservation laws hereby it is not necessary to introduce adjoint variables for the shock positions our approach is based on stability properties of the adjoint equation we give a complete analysis for the case of convex scalar conservation laws the adjoint equation is a transport equation with discontinuous coefficients and special reversible solutions must be considered to obtain the correct adjoint based gradient formula reversible solutions of the adjoint transport equation and the required stability properties are analyzed in detail

consensus seeking in multiagent systems under dynamically changing interaction topologies this note considers the problem of information consensus among multiple agents in the presence of limited and unreliable information exchange with dynamically changing interaction topologies both discrete and continuous update schemes are proposed for information consensus this note shows that information consensus under dynamically changing interaction topologies can be achieved asymptotically if the union of the directed interaction graphs have a spanning tree frequently enough as the system evolves

inner product perturbation theory for a perturbed two dimensional oscillator with mixed parity potential ground and excited state energy levels of the schrodinger equation for various model potentials in two dimensional space are calculated using inner product and renormalized series techniques over wide ranges of relevant perturbation parameters mixed parity potentials are treated whereas earlier works have treated only even parity perturbations

the perturbed two dimensional oscillator eigenvalues and infinite field limits via continued fractions renormalised perturbation theory and moment methods the author employs two independent methods to obtain consistent and accurate values of eigenvalues e lambda of the perturbed two dimensional oscillator hamiltonian h h 0 lambda x2y2 i continued fraction representations of the rayleigh schrodinger perturbation series to large order as well as a renormalised version of perturbation theory and ii a method of moments based on the positivity properties of the factorised wavefunction the latter generates converging upper and lower bounds to e lambda the two methods are also used to obtain estimates for the eigenvalues f 0 of the infinite field limit hamiltonian hinfinity px2 py2 x2y2

the eigenvalues of the schrodinger equation for spherically symmetric states for various types of potentials in two three and n dimensions by using perturbative and non perturbative methods the energy levels of the schrodinger equation involving various model potentials in two three and n dimensional space are calculated by using perturbative and non perturbative methods the methods have been used effectively to calculate the energy eigenvalues even for high state numbers and large values of the perturbation parameters

regularity theory and adjoint based optimality conditions for a nonlinear transport equation with nonlocal velocity in this contribution the optimal boundary control problem for a first order nonlinear nonlocal hyperbolic pde is studied motivated by various applications ranging from re entrant manufacturing systems to particle synthesis processes we establish the regularity of solutions for w 1 p data based on a general l 2 tracking type cost functional the existence uniqueness and regularity of the adjoint system in w 1 p is derived using the special structure induced from the nonlocal flux function of the state equation the assumption of w 1 p and not l p regularity comes thereby due to the fact that the adjoint equation asks for more regularity to be well defined this problem is discussed in detail and we give a solution by defining a special type of cost functional such that the corresponding optimality system is well defined

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

analysis and applications of spectral properties of grounded laplacian matrices for directed networks abstract in depth understanding of the spectral properties of grounded laplacian matrices is critical for the analysis of convergence speeds of dynamical processes over complex networks such as opinion dynamics in social networks with stubborn agents we focus on grounded laplacian matrices for directed graphs and show that their eigenvalues with the smallest real part must be real lower and upper bounds for such eigenvalues are provided utilizing tools from nonnegative matrix theory for those eigenvectors corresponding to such eigenvalues we discuss two cases when we can identify the vertex that corresponds to the smallest eigenvector component we then discuss an application in leader follower social networks where the grounded laplacian matrices arise naturally with the knowledge of the vertex corresponding to the smallest eigenvector component for the smallest eigenvalue we prove that by removing or weakening specific directed couplings pointing to the vertex having the smallest eigenvector component all the states of the other vertices converge faster to that of the leading vertex this result is in sharp contrast to the well known fact that when the vertices are connected together through undirected links removing or weakening links does not accelerate and in general decelerates the converging process

converging bounds for the eigenvalues of multiminima potentials in two dimensional space the eigenvalue problem in of schrodinger operators with a polynomial perturbation has been replaced by one corresponding to the system confined in a box with impenetrable walls it is shown that the dirichlet and the von neumann problems in generate upper and lower bounds respectively to the eigenvalues of the unbounded system to illustrate the method rapidly converging two sided bounds for the energy levels of two and three well oscillators are presented using simple trigonometric basis functions

schur reduction of trees and extremal entries of the fiedler vector abstract we study the eigenvectors of laplacian matrices of trees the laplacian matrix is reduced to a tridiagonal matrix using the schur complement this preserves the eigenvectors and allows us to provide formulas for the ratio of eigenvector entries we also obtain bounds on the ratio of eigenvector entries along a path in terms of the eigenvalue and perron values the results are then applied to the fiedler vector here we locate the extremal entries of the fiedler vector and study classes of graphs such that the extremal entries can be found at the end points of the longest path

a sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms we present a sensitivity and adjoint calculus for the control of entropy solutions of scalar conservation laws with controlled initial data and source term the sensitivity analysis is based on shift variations which are the sum of a standard variation and suitable corrections by weighted indicator functions approximating the movement of the shock locations based on a first order approximation by shift variations in l1 we introduce the concept of shift differentiability which is applicable to operators having functions with moving discontinuities as images and implies differentiability for a large class of tracking type functionals in the main part of the paper we show that entropy solutions are generically shift differentiable at almost all times t 0 with respect to the control hereby we admit shift variations for the initial data which allows us to use the shift differentiability result repeatedly over time slabs this is useful for the design of optimization methods with time domain decomposition our analysis especially of the shock sensitivity combines structural results by using generalized characteristics and an adjoint argument our adjoint based shock sensitivity analysis does not require us to restrict the richness of the shock structure a priori and admits shock generation points the analysis is based on stability results for the adjoint transport equation with discontinuous coefficients satisfying a one sided lipschitz condition as a further main result we derive and justify an adjoint representation for the derivative of a large class of tracking type functionals

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

optimal control of flow with discontinuities optimal control of the 1 d riemann problem of euler equations is studied with the initial values for pressure and density as control parameters the least squares type cost functional employs either distributed observations in time or observations calculated at the end of the assimilation window existence of solutions for the optimal control problem is proven smooth and nonsmooth optimization methods employ the numerical gradient respectively a subgradient of the cost functional obtained from the adjoint of the discrete forward model the numerical flow obtained with the optimal initial conditions obtained from the nonsmooth minimization matches very well with the observations the algorithm for smooth minimization converges for the shorter time horizon but fails to perform satisfactorily for the longer time horizon except when the observations corresponding to shocks are detected and removed

a continuum model for a re entrant factory high volume multistage continuous production flow through a re entrant factory is modeled through a conservation law for a continuous density variable on a continuous production line augmented by a state equation for the speed of the production along the production line the resulting nonlinear nonlocal hyperbolic conservation law allows fast and accurate simulations little s law is built into the model it is argued that the state equation for a re entrant factory should be nonlinear comparisons of simulations of the partial differential equation pde model and discrete event simulations are presented a general analysis of the model shows that for any nonlinear state equation there exist two steady states of production below a critical start rate a high volume high throughput time state and a low volume low throughput time state the stability of the low volume state is proved output is controlled by adjusting the start rate to a changed demand rate two linear factories and a re entrant factory each one modeled by a hyperbolic conservation law are linked to provide proof of concept for efficient supply chain simulations instantaneous density and flux through the supply chain as well as work in progress wip and output as a function of time are presented extensions to include multiple product flows and preference rules for products and dispatch rules for re entrant choices are discussed

compensation of input delay that depends on delayed input abstract for nonlinear systems we develop a pde based predictor feedback control design which compensates actuator dynamics governed by a transport pde with outlet boundary value dependent propagation velocity global asymptotic stability under the predictor feedback control law is established assuming spatially uniform strictly positive transport velocity the stability proof is based on a lyapunov like argument and employs an infinite dimensional backstepping transformation that is introduced an equivalent representation of the transport pde nonlinear ode cascade via a nonlinear system with an input delay that is defined implicitly through an integral of the past input is also provided and the equivalent predictor feedback control design for the delay system is presented the validity of the proposed controller is illustrated applying a predictor feedback bang bang boundary control law to a pde model of a production system with a queue consistent simulation results are provided that support the theoretical developments

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

stability results for the continuity equation abstract we provide a thorough study of stability of the 1 d continuity equation which models many physical conservation laws in our system theoretic perspective the velocity is considered to be an input an additional input appears in the boundary condition boundary disturbance stability estimates are provided in all l p state norms with p 1 including the case p however in our input to state stability estimates the gain and overshoot coefficients depend on the velocity moreover the logarithmic norm of the state appears instead of the usual norm the obtained results can be used in the stability analysis of larger models that contain the continuity equation in particular it is shown that the obtained results can be used in a straightforward way for the stability analysis of non local nonlinear manufacturing models under feedback control

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