on existence of limit occupational measures set of a controlled stochastic differential equation we establish that under certain conditions the set of occupational measures as well as the set of mathematical expectations of occupational measures generated by the admissible controls and the corresponding solutions of a controlled stochastic differential equation csde converge with the time horizon tending to infinity to a set called limit occupational measures set loms and we show that this limit set coincides with the set of stationary marginal distributions of the csde we also demonstrate the applicability of our results for averaging of singularly perturbed csde

tracking fast trajectories along a slow dynamics a singular perturbations approach controlled coupled slow and fast motions are examined in a singular perturbations setting the objective is to minimize a cost functional that takes into account both the fast motion supposing say tracking a fast target and the slow dynamics a method is offered to cope with the possibility that the fast flow has nonstationary limits invariant measures of the fast motion are then the controlled objects on the infinitesimal scale optimal amalgamation of them on the slow scale induces the variational limit whose solutions are near optimal solutions of the perturbed system

singularly perturbed control systems with one dimensional fast dynamics the order reduction approach to singularly perturbed control systems suggests employing as a variational limit the differential algebraic system obtained when the small parameter is set to be zero it is known that the method is valid only under restrictive convergence conditions on the fast dynamics we verify in this paper that when the fast state variable is one dimensional the order reduction method is valid in general this is true however when appropriate relaxation is allowed in the reduced order system we also indicate how to extract near optimal solutions to the original system from optimal solutions of the order reduction one along the traditional reasoning of separating time scales examples are displayed showing that without allowing the relaxation the order reduction may not provide the correct limit

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

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

on a representation of the limit occupational measures set of a control system with applications to singularly perturbed control systems a representation of the limit occupational measures set of a control system in terms of the vector function defining the system s dynamics is established applications in averaging of singularly perturbed control systems are demonstrated

multiscale singularly perturbed control systems limit occupational measures sets and averaging an averaging technique for nonlinear multiscale singularly perturbed control systems is developed issues concerning the existence and structure of limit occupational measures sets generated by such systems are discussed general results are illustrated with special cases

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

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

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

singularly perturbed ordinary differential equations with dynamic limits coupled slow and fast motions generated by ordinary differential equations are examined the qualitative limit behaviour of the trajectories as the small parameter tends to zero is sought after invariant measures of the parametrised fast flow are employed to describe the limit behaviour rather than algebraic equations which are used in the standard reduced order approach in the case of a unique invariant measure for each parameter the limit of the slow motion is governed by a chattering type equation without the uniqueness the limit of the slow motion solves a differential inclusion the fast flow in turn converges in a statistical sense to the direct integral respectively the set valued direct integral of the invariant measures

convergence to convex compact sets in infinite dimensions we provide a simple test for the convergence of a sequence of subsets in an infinite dimensional space to a compact convex one the test reduces roughly the convergence problem to finite dimensions applications are portrayed demonstrating how convexification phenomena established in finite dimensions are easily generalized to infinite dimensions

compensation of wave actuator dynamics for nonlinear systems the problem of stabilization of pde ode cascades has been solved in the linear case for several pde classes whereas in the nonlinear case the problem has been solved only for the transport delay pde namely for compensation of an arbitrary delay at the input of a nonlinear plant motivated by a specific engineering application in off shore drilling we solve the problem of stabilization of the cascade of a wave pde with a general nonlinear ode due to the presence of nonlinearities of arbitrary growth and the time reversibility of the wave pde and due to the possibility of using arguments based on lyapunov functionals or explicit solutions several stability analysis approaches are possible we present stability results in the h 2 times h 1 and c 1 times c 0 norms for general nonlinear odes as well as in the h 1 times l 2 norm for linear odes we specialize our general design for wave pde ode cascades to the case of a wave pde whose uncontrolled end does not drive an ode but is instead governed by a nonlinear robin boundary condition a nonlinear spring as in the friction law in drilling this is the first global stabilization result for wave equations that incorporate non collocated destabilizing nonlinearities of superlinear growth we present two numerical examples one with a nonlinear ode and one with a nonlinear spring at the uncontrolled boundary of the wave pde

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

on the construction of asymptoticallly optimal controls for singularly perturbed systems we construct asymptotically optimal controls for singularly perturbed nonlinear systems where both the slow and the fast motions are controlled for this purpose we give necessary optimality conditions for an averaged system for the slow motion

pi regulation of a reaction diffusion equation with delayed boundary control the general context of this work is the feedback control of an infinite dimensional system so that the closed loop system satisfies a fading memory property and achieves the setpoint tracking of a given reference signal more specifically this paper is concerned with the proportional integral pi regulation control of the left neumann trace of a one dimensional reaction diffusion equation with a delayed right dirichlet boundary control in this setting the studied reaction diffusion equation might be either open loop stable or unstable the proposed control strategy goes as follows first a finite dimensional truncated model that captures the unstable dynamics of the original infinite dimensional system is obtained via spectral decomposition the truncated model is then augmented by an integral component on the tracking error of the left neumann trace after resorting to the artstein transformation to handle the control input delay the pi controller is designed by pole shifting stability of the resulting closed loop infinite dimensional system consisting of the original reaction diffusion equation with the pi controller is then established thanks to an adequate lyapunov function in the case of a time varying reference input and a time varying distributed disturbance our stability result takes the form of an exponential input to state stability iss estimate with fading memory finally another exponential iss estimate with fading memory is established for the tracking performance of the reference signal by the system output in particular these results assess the setpoint regulation of the left neumann trace in the presence of distributed perturbations that converge to a steady state value and with a time derivative that converges to zero numerical simulations are carried out to illustrate the efficiency of our control strategy

performance output tracking for multidimensional heat equation subject to unmatched disturbance and noncollocated control this paper investigates performance output tracking for a boundary controlled multidimensional heat equation it is assumed that the so called total disturbance which is composed of internal possibly nonlinear uncertainty and external disturbance the equation is subject to is on one part of the boundary and that the control is applied on the rest of the boundary using only partial boundary measurement we first propose an extended state observer to estimate both system state and the total disturbance this allows us to design a servomechanism and then an output feedback controller under the condition that both the reference signal and the disturbance vanish or belong to spaces h 1 0 infty l 1 gamma 1 and l 2 0 infty l 2 gamma 0 respectively we show that the over all control strategy achieves three objectives on the system performance first exponentially output tracking for arbitrary given reference signal second uniformly boundedness of all internal signals and third the internal asymptotic stability of the closed loop system in addition the control strategy turns out to be robust to the measurement noise we provide numerical experiments to illustrate the effectiveness of the proposed control strategy

on boundary feedback stabilization of non uniform linear 2 2 hyperbolic systems over a bounded interval conditions for boundary feedback stabilizability of non uniform linear 2 2 hyperbolic systems over a bounded interval are investigated the main result is to show that the existence of a basic quadratic control lyapunov function requires that the solution of an associated ode is defined on the considered interval this result is used to give explicit conditions for the existence of stabilizing linear boundary feedback control laws the analysis is illustrated with an application to the boundary feedback stabilization of open channels represented by linearized saint venant equations with non uniform steady states

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

viscosity solutions methods for singular perturbations in deterministic and stochastic control viscosity solutions methods are used to pass to the limit in some penalization problems for first order and second order degenerate parabolic hamilton jacobi bellman equations this characterizes the limit of the value functions of singularly perturbed optimal control problems for deterministic systems and for controlled degenerate diffusions the results apply to cases where the usual order reduction method does not give the correct limit and to systems with fast state variables depending nonlinearly on the control some connections with ergodic control and periodic homogenization are discussed

characterization of the optimal trajectories for the averaged dynamics associated to singularly perturbed control systems abstract the aim of this paper is to study singularly perturbed control systems firstly we provide linearized formulation version for the calculus of the value function associated with the averaged dynamics secondly we obtain necessary and sufficient conditions in order to identify the optimal trajectory of the averaged system

a reduction procedure for generalized riemann problems with application to nonlinear transmission lines generalized simple wave solutions to quasilinear hyperbolic nonhomogeneous systems of pdes are obtained through the differential constraint method these solutions prove to be flexible enough to solve generalized riemann problems where discontinuous initial data are involved within such a theoretical framework the governing model of nonlinear transmission lines is investigated throughout

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

koopman operator based model reduction for switched system control of pdes we present a new framework for optimal and feedback control of pdes using koopman operator based reduced order models k roms the koopman operator is a linear but infinite dimensional operator which describes the dynamics of observables a numerical approximation of the koopman operator therefore yields a linear system for the observation of an autonomous dynamical system in our approach by introducing a finite number of constant controls the dynamic control system is transformed into a set of autonomous systems and the corresponding optimal control problem into a switching time optimization problem this allows us to replace each of these systems by a k rom which can be solved orders of magnitude faster by this approach a nonlinear infinite dimensional control problem is transformed into a low dimensional linear problem using a recent convergence result for the numerical approximation via extended dynamic mode decomposition edmd we show that the value of the k rom based objective function converges in measure to the value of the full objective function to illustrate the results we consider the 1d burgers equation and the 2d navier stokes equations the numerical experiments show remarkable performance concerning both solution times and accuracy

flatness of heavy chain systems in this paper the flatness m fliess j levine p martin and p rouchon internat j control 61 1995 pp 1327 1361 m fliess j levine p martin and p rouchon ieee trans automat control 44 1999 pp 922 937 of heavy chain systems i e trolleys carrying a fixed length heavy chain that may carry a load is addressed in the partial derivatives equations framework we parameterize the system trajectories by the trajectories of its free end and solve the motion planning problem namely steering from one state to another state when considered as a finite set of small pendulums these systems were shown to be flat r m murray in proceedings of the ifac world congress san francisco ca 1996 pp 395 400 our study is an extension to the infinite dimensional case under small angle approximations these heavy chain systems are described by a one dimensional 1d partial differential wave equation dealing with this infinite dimensional description we show how to get the explicit parameterization of the chain trajectory using distributed and punctual advances and delays of its free end this parameterization results from symbolic computations replacing the time derivative by the laplace variable s yields a second order differential equation in the spatial variable where s is a parameter its fundamental solution is for each point considered along the chain an entire function of s of exponential type moreover for each we show that thanks to the liouville transformation this solution satisfies modulo explicitly computable exponentials of s the assumptions of the paley wiener theorem this solution is in fact the transfer function from the flat output the position of the free end of the system to the whole state of the system using an inverse laplace transform we end up with an explicit motion planning formula involving both distributed and punctual advances and delays operators

discontinuous control problems with state constraints linear formulations and dynamic programming principles abstract this paper aims at studying a class of discontinuous deterministic control problems under state constraints using a linear programming approach as for classical control problems gaitsgory and quincampoix 2009 16 goreac and serea 2011 19 the primal linear problem is stated on some appropriate space of probability measures naturally the support of these measures is contained in the set of constraints this linearized value function and its dual can alternatively be seen as the limit of standard penalized problems second we provide a semigroup property for this set of probability measures leading to dynamic programming principles for control problems under state constraints an abstract principle is provided for general bounded costs linearized versions are obtained under further semi continuity assumptions

gas flow in fan shaped networks classical solutions and feedback stabilization we analyze the subcritical gas flow through fan shaped networks of pipes that is through tree shaped networks with exactly one node where more than two pipes meet the gas flow in pipe networks is modeled by the isothermal euler equations a hyperbolic pde system of balance laws for this system we analyze stationary states and classical nonstationary solutions locally around a stationary state on a finite time interval furthermore we present a lyapunov function and boundary feedback laws to stabilize a fan shaped network around a given stationary state

duality in linear programming problems related to deterministic long run average problems of optimal control it has been established recently that under mild conditions deterministic long run average problems of optimal control are asymptotically equivalent to infinite dimensional linear programming problems lpps and that these lpps can be approximated by finite dimensional lpps in this paper we introduce the corresponding infinite and finite dimensional dual problems and study duality relationships we also investigate the possibility of using solutions of finite dimensional lpps and their duals for numerical construction of the optimal controls in periodic optimization problems the construction is illustrated with a numerical example

spectral theory of some degenerate elliptic operators with local singularities abstract this paper is based on our previous results haroske and skrzypczak 2008 23 haroske and skrzypczak in press 25 on compact embeddings of muckenhoupt weighted function spaces of besov and triebel lizorkin type with example weights of polynomial growth near infinity and near some local singularity our approach also extends haroske and triebel 1994 21 in various ways we obtain eigenvalue estimates of degenerate pseudodifferential operators of type b 2 p x d b 1 where b i l r i r n w i w i a i 1 2 and p x d 1 0 0 finally we deal with the negative spectrum of some operator h a v for where the potential v may have singularities in terms of muckenhoupt weights and a is a positive elliptic pseudodifferential operator of order 0 self adjoint in l 2 r n this part essentially relies on the birman schwinger principle we conclude this paper with a number of examples also comparing our results with preceding ones

local exponential h2 stabilization of 2x2 quasilinear hyperbolic systems using backstepping in this work we consider the problem of boundary stabilization for a quasilinear 2 times2 system of first order hyperbolic pdes we design a new full state feedback control law with actuation on only one end of the domain which achieves h 2 exponential stability of the closed loop system our proof uses a backstepping transformation to find new variables for which a strict lyapunov function can be constructed the kernels of the transformation are found to verify a goursat type 4 times4 system of first order hyperbolic pdes whose well posedness is shown using the method of characteristics and successive approximations once the kernels are computed the stabilizing feedback law can be explicitly constructed from them

limits of singularly perturbed control problems with statistical dynamics of fast motions we describe the limit behavior of admissible trajectories in a singularly perturbed control system as the small parameter tends to zero a general case is considered where in the limit the fast motion may infinitely rapidly oscillate in time invariant measures of the parameterized fast flow are employed to describe the limit behavior and construct the limit control problem the notion of relatively slow controls is introduced approximating properties of the limit problem within the families of relatively slow controls are verified the results are illustrated by examples

suboptimization of singularly perturbed control systems a technique different from the boundary layer method is developed to deal with singularly perturbed optimal control problems the technique is applicable in particular in the case when the optima

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

disturbance rejection in 2 x 2 linear hyperbolic systems many interesting problems in the oil and gas industry face the challenge of responding to disturbances from afar typically the disturbance occurs at the inlet of a pipeline or the bottom of an oil well while sensing and actuation equipment is installed at the outlet only kilometers away from the disturbance the present paper develops an output feedback control law for such cases based on modelling the transport phenomenon as a 2 times 2 linear partial differential equation pde of hyperbolic type and the disturbance as a finite dimensional linear system affecting the left boundary of the pde sensing and actuation are co located at the right boundary of the pde the design provides a separation principle allowing a disturbance attenuating full state feedback control law to be combined with an observer the results are applied to a relevant problem from the oil and gas industry and demonstrated in simulations

robust state feedback regulator design for general linear heterodirectional hyperbolic systems the topic of this paper is the backstepping based solution of the robust state feedback regulation problem for general linear heterodirectional hyperbolic systems with spatially varying coefficients thereby a large class of outputs is considered i e mixed distributed and pointwise in domain or boundary outputs the state feedback regulator design is based on the boldsymbol p copy internal model principle this results in a stabilization problem for a cascade of boldsymbol m p boundary controlled transport partial differential equations pdes with boldsymbol p ordinary differential equations odes for this a systematic backstepping approach is presented in which all design equations are explicitly solvable in backstepping coordinates furthermore the solvability conditions for the considered output regulation problem can be evaluated in closed form utilizing the plant transfer behavior robust output regulation is verified for nondestabilizing model uncertainties and unknown disturbance input locations a boldsymbol 4 times 4 hyperbolic system with uncertain transport velocities and couplings in the transport pdes illustrates the results of this paper

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

the problem of optimal control with reflection studied through a linear optimization problem stated on occupational measures abstract we obtain a linear programming characterization for the minimum cost associated with finite dimensional reflected optimal control problems in order to describe the value functions we employ an infinite dimensional dual formulation instead of using the characterization via hamilton jacobi partial differential equations in this paper we consider control problems with both infinite and finite horizons the reflection is given by the normal cone to a proximal retract set

limit occupational measures set for a control system and averaging of singularly perturbed control systems abstract we discuss conditions for the existence of the limit occupational measures set for a control system we approximate slow components of the trajectories of a singularly perturbed control system by the solutions of a differential inclusion the differential inclusion is obtained via averaging the slow subsystem over measures from the limit occupational measures set constructed for the associated system describing the fast motions with frozen slow ones