equations of motion for variational electrodynamics abstract we extend the variational problem of wheeler feynman electrodynamics by generalizing the electromagnetic functional to a local space of absolutely continuous trajectories possessing a derivative velocities of bounded variation we show here that the gateaux derivative of the generalized functional defines two partial lagrangians for variations in our generalized local space one for each particle we prove that the critical point conditions of the generalized variational problem are i the euler lagrange equations must hold lebesgue almost everywhere and ii the momentum of each partial lagrangian and the legendre transform of each partial lagrangian must be absolutely continuous functions generalizing the weierstrass erdmann conditions

the optimal sampling pattern for linear control systems in digital control systems the state is sampled at given sampling instants and the input is kept constant between two consecutive instants with the optimal sampling problem we mean the selection of sampling instants and control inputs such that a given function of the state and input is minimized in this paper we formulate the optimal sampling problem and we derive a necessary condition for the optimality of a set of sampling instants in the linear quadratic regulator problem since the numerical solution of the optimal sampling problem is very time consuming we also propose a new quantization based sampling strategy that is computationally tractable and capable of achieving near optimal cost finally and probably most interesting of all we prove that the quantization based sampling is optimal in first order systems for a large number of samples experiments demonstrate that quantization based sampling has near optimal performance even when the system has a higher order however it is still an open question whether quantization based sampling is asymptotically optimal in any case

geometric formulation of mechanical systems with one sided constraints a geometric formulation of analytical mechanics of systems with holonomic and holonomic one sided constraints is proposed this is an extension of work by tulczyjew 1986 in which non holonomic constraints were also considered but not the one sided case the introduction of lagrange multipliers to give explicit equations of motion was not considered by tulczyjew either this allows the authors to study problems of motion of particles hitting moving walls they verify that the conditions of collision are satisfied

fredholm transform and local rapid stabilization for a kuramoto sivashinsky equation this paper is devoted to the study of the local rapid exponential stabilization problem for a controlled kuramoto sivashinsky equation on a bounded interval we build a feedback control law to force the solution of the closed loop system to decay exponentially to zero with arbitrarily prescribed decay rates provided that the initial datum is small enough our approach uses a method we introduced for the rapid stabilization of a korteweg de vries equation it relies on the construction of a suitable integral transform and can be applied to many other equations

superstability and rigorous asymptotics in singularly perturbed state dependent delay differential equations abstract we study the singularly perturbed state dependent delay differential equation e x t x t k x t r r r x t 1 x t which is a special case of the equation e x t g x t x t r r r x t one knows that for every sufficiently small e 0 eq possesses at least one so called slowly oscillating periodic solution and moreover the graph of every such solution approaches a specific sawtooth like shape as e 0 in this paper we obtain higher order asymptotics of the sawtooth including the location of the minimum and maximum of the solution with the form of the solution near these turning points and as well an asymptotic formula for the period using these and other asymptotic formulas we further show that the solution enjoys the property of superstability namely the nontrivial characteristic multipliers are of size o e for small e this stability property implies that this solution is unique among all slowly oscillating periodic solutions again for small e

global stabilization of the generalized korteweg de vries equation posed on a finite domain this paper is concerned with the internal stabilization of the generalized korteweg de vries equation on a bounded domain the global well posedness and the exponential stability are investigated when the exponent in the nonlinear term ranges over the interval 1 4 the global exponential stability is obtained whatever the location where the damping is active confirming positively a conjecture of perla menzala vasconcellos and zuazua quart appl math 60 2002 pp 111 129

boundary constrained control of delayed nonlinear schrodinger equation this paper studies regional boundary stabilization of nonlinear schrodinger equation with state delay and bounded internal disturbance the boundary constrained control law is designed by using the backstepping method regional input to state stability of the perturbed system with time delay is established by a lyapunov function and a generalized halanay s inequality estimates on the set of initial conditions are found starting from which the solutions are exponentially attracted to a bounded set a numerical example demonstrates the efficiency of the results

impulse output rapid stabilization for heat equations abstract the main aim of this paper is to provide a new feedback law for the heat equations in a bounded domain with dirichlet boundary condition two constraints will be compulsory first the controls are active in a subdomain of and at discrete time points second the observations are made in another subdomain and at different discrete time points our strategy consists in linking an observation estimate at one time minimal norm impulse control approximate inverse source problem and rapid output stabilization

input to state stabilization of linear systems with positive outputs this paper considers the problem of stabilization of linear systems for which only the magnitudes of outputs are measured it is shown that if a system is controllable and observable then one can find a stabilizing controller which is robust with respect to observation noise in the iss sense

mechanical systems subjected to generalized non holonomic constraints we study mechanical systems subject to constraint functions that can be dependent at some points and independent at the rest such systems are modelled by means of generalized codistributions we discuss how the constraint force can transmit an impulse to the motion at the points of dependence and derive an explicit formula to obtain the post impact momentum in terms of the pre impact momentum

a periodic solution of a differential equation with state dependent delay we study a differential equation for delayed negative feedback which models a situation where the delay depends on the present state and becomes effective in the future the main result is existence of a periodic solution in case the equilibrium is linearly unstable the proof employs the ejective fixed point principle on a compact convex set k0 c h 0 r of lipschitz continuous functions and uses that the equation generates a smooth semiflow on an infinite dimensional submanifold of the space c1 h 0 r

output feedback stabilization of the korteweg de vries equation this paper presents an output feedback control law for the korteweg de vries equation the control design is based on the backstepping method and the introduction of an appropriate observer the local exponential stability of the closed loop system is proven some numerical simulations are shown to illustrate this theoretical result

ideality criterion for unilateral constraints in time dependent impulsive mechanics we construct a new geometric framework based on the concepts of left and right jet bundles of a classical space time v in order to analyze the impulsive behavior of a unilateral constraint s the setup allows deep insights into how one can choose an ideality criterion for the constraint s when the hypothesis of conservation of kinetic energy is assumed we show that the conservation of kinetic energy alone univocally determines the impulsive reaction when the codimension of s is 1 and that it leaves the impulsive reaction partially undetermined when the codimension of s is greater than 1 if the codimension of s is greater than 1 we prove that an additional minimality requirement determines a physically meaningful constitutive characterization of s we show that both the newton like and the poisson like approaches to the description of the reactive impulse are equivalent in the sense that both give the same results about the ideality criterion moreover we prove that the same results hold using the cl

merging homoclinic solutions due to state dependent delay abstract for 2 5 2 the linear differential equation x t x t 1 with constant time lag is unstable and hyperbolic for sufficiently close to 5 2 we construct a state dependent delay d u 0 2 on an open set in the space c 1 2 0 with d u 1 for close to 0 so that the nonlinear equation x t x t d u x t has a one parameter family of entire solutions c r r r r r u which are homoclinic to zero and merge in finite time at some t y 0 c r t c 0 t for r r u and t t y this complements earlier results on complicated solution behaviour caused by state dependent delay and shows that there exist delay functionals for which certain transversality and regularity conditions along homoclinic solutions are violated

note on the cost of the approximate controllability for the heat equation with potential abstract we prove the approximate controllability for the heat equation with potential with a cost of order ec e when the target is in h 0 1 with a precision in l 2 norm also a quantification estimate of the unique continuation for initial data in l 2 of the heat equation with potential is established

closed form boundary state feedbacks for a class of 1 d partial integro differential equations in this paper a problem of boundary stabilization of a class of linear parabolic partial integro differential equations p i des in one dimension is considered using the method of backstepping avoiding spatial discretization required in previous efforts the problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic p i de the kernel p i de is then converted into an equivalent integral equation and by applying the method of successive approximations the equation s well posedness and the kernel s smoothness are established it is shown how to extend this approach to design optimally stabilizing controllers an adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller and the performance bounds are derived for a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed loop solutions are found in closed form a numerical scheme for the kernel p i de is proposed its numerical effort compares favorably with that associated with operator riccati equations

partial differential equations with discrete and distributed state dependent delays this work is an attempt to treat partial differential equations with discrete concentrated state dependent delay the main idea is to approximate the discrete delay term by a sequence of distributed delay terms all with state dependent delays we study local existence and long time asymptotic behavior of solutions and prove that the model with distributed delay has a global attractor while the one with discrete delay possesses the trajectory attractor

rapid boundary stabilization of linear distributed systems we prove that under rather general assumptions an exactly controllable problem is uniformly stabilizable with arbitrarily prescribed decay rates our approach is direct and constructive and avoids many of the technical difficulties associated with the usual methods based on riccati equations we give several applications for the wave equation and for petrovsky systems

robustness with respect to sampling for stabilization of riesz spectral systems we suppose that a continuous time feedback is input output stabilizing for an infinite dimensional system we address the question of whether the sampled data controller obtained by applying idealized sample and hold to this continuous time feedback is also input output stabilizing if the sampling time is small enough this question has been previously addressed for fairly general systems under various conditions in this note we restrict our attention to riesz spectral systems for which we generalize the existing results specifically we give two relatively simple conditions which combined are sufficient for the sampled data controller to be stabilizing the first condition is a spectrum decomposition for the open loop system generator which by itself is necessary but not sufficient for the system to be stabilizable by sampled data control the second is a summability condition relating the real part of the spectrum of the generator and the expansion coefficients for the input and feedback operators

rapid exponential feedback stabilization with unbounded control operators we study the wellposedness and the main features of a class of feedback control systems the involved control system is composed of the generator of a strongly continuous group for the free part and of an unbounded control operator so that the results can be applied to boundary or point control problems for partial differential equations of hyperbolic or petrowski type the feedback operator is explicit and one can achieve an arbitrary large decay rate for the closed loop system these results are proved under a controllability assumption and the proofs rely on general results about the algebraic riccati equation associated with the linear quadratic regulator problem

sampled data boundary feedback control of 1 d parabolic pdes abstract the paper provides results for the application of boundary feedback control with zero order hold zoh to 1 d linear parabolic systems on bounded domains it is shown that the continuous time boundary feedback applied in a sample and hold fashion guarantees closed loop exponential stability provided that the sampling period is sufficiently small two different continuous time feedback designs are considered the reduced model design and the backstepping design the obtained results provide stability estimates for weighted 2 norms of the state and robustness with respect to perturbations of the sampling schedule is guaranteed

singular hopf bifurcation in a differential equation with large state dependent delay we study the onset of sustained oscillations in a classical state dependent delay sdd differential equation inspired by control theory owing to the large delays considered the hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold using asymptotic techniques we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations the dependence of the delay on the solution can be either linear or nonlinear with at least quadratic dependence in the former case an asymptotic connection is made with the rayleigh oscillator in the latter van der pol s equation is derived for the small amplitude oscillations sdd differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant delay differential equation but explicit analytical construction of solutions are rare this paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for sdd problems with large delays

boundary control of coupled reaction advection diffusion systems with spatially varying coefficients recently the problem of boundary stabilization for unstable linear constant coefficient coupled reaction diffusion systems was solved by means of the backstepping method the extension of this result to systems with advection terms and spatially varying coefficients is challenging due to complex boundary conditions that appear in the equations verified by the control kernels in this technical note we address this issue by showing that these equations are essentially equivalent to those verified by the control kernels for first order hyperbolic coupled systems which were recently found to be well posed the result therefore applies in this case allowing us to prove h 1 stability for the closed loop system it also unveils a previously unknown connection between backstepping kernels for coupled parabolic and hyperbolic problems

boundary stabilization of a 1 d wave equation with in domain antidamping we consider the problem of boundary stabilization of a 1 d one dimensional wave equation with an internal spatially varying antidamping term this term puts all the eigenvalues of the open loop system in the right half of the complex plane we design a feedback law based on the backstepping method and prove exponential stability of the closed loop system with a desired decay rate for plants with constant parameters the control gains are found in closed form our design also produces a new lyapunov function for the classical wave equation with passive boundary damping

non linear partial differential equations with discrete state dependent delays in a metric space we investigate a class of non linear partial differential equations with discrete state dependent delays the existence and uniqueness of strong solutions for initial functions from a banach space are proved to get the well posed initial value problem we restrict our study to a smaller metric space construct the dynamical system and prove the existence of a compact global attractor

output feedback stabilization of the linearized korteweg de vries equation with right endpoint controllers abstract in this paper we prove the output feedback stabilization for the linearized korteweg de vries kdv equation posed on a finite domain in the case the full state of the system cannot be measured we assume that there is a sensor at the left end point of the domain capable of measuring the first and second order boundary traces of the solution this allows us to design a suitable observer system whose states can be used for constructing boundary feedbacks acting at the right endpoint so that both the observer and the original plant become exponentially stable stabilization of the original system is proved in the l 2 sense while the convergence of the observer system to the original plant is also proved in higher order sobolev norms the standard backstepping approach used to construct a left endpoint controller fails and presents mathematical challenges when building right endpoint controllers due to the overdetermined nature of the related kernel models in order to deal with this difficulty we use the method of ozsari and batal 2019 which is based on using modified target systems involving extra trace terms in addition we show that the number of controllers and boundary measurements can be reduced to one with the cost of a slightly lower exponential rate of decay we provide numerical simulations illustrating the efficacy of our controllers

global hopf bifurcation for differential equations with state dependent delay abstract we develop a global hopf bifurcation theory for a system of functional differential equations with state dependent delay the theory is based on an application of the homotopy invariance of s 1 equivariant degree using the formal linearization of the system at a stationary state our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state dependent delay

local exponential stabilization of fisher s equation using the backstepping technique abstract this paper deals with the boundary stabilization problem for the 1d fisher s partial differential equation pde defined on a bounded interval which is a nonlinear unstable distributed parameter system the stabilizing controller derived using the backstepping technique for linear parabolic pdes associated with fisher s pde can be extended to the nonlinear case in the work we can prove that the boundary controller based on the linear backstepping synthesis approach can make the closed loop nonlinear fisher s system exponentially stable with any desired decay rate locally in both l 2 0 1 and h 1 0 1 respectively the effectiveness of the control synthesis approach is illustrated by a numerical simulation

on the geometry of non holonomic lagrangian systems we present a geometric framework for non holonomic lagrangian systems in terms of distributions on the configuration manifold if the constrained system is regular an almost product structure on the phase space of velocities is constructed such that the constrained dynamics is obtained by projecting the free dynamics if the constrained system is singular we develop a constraint algorithm which is very similar to that developed by dirac and bergmann and later globalized by gotay and nester special attention to the case of constrained systems given by connections is paid in particular we extend the results of koiller for caplygin systems an application to the so called non holonomic geometry is given

output feedback stabilization of an unstable wave equation we consider the problem of stabilization of a one dimensional wave equation that contains instability at its free end and control on the opposite end in contrast to classical collocated boundary damper feedbacks for the neutrally stable wave equations with one end satisfying a homogeneous boundary condition the controllers and the associated observers designed in the paper are more complex due to the open loop instability of the plant the controller and observer gains are designed using the method of backstepping which results in explicit formulae for the gain functions we prove exponential stability and the existence and uniqueness of classical solutions for the closed loop system we also derive the explicit compensators in frequency domain the results are illustrated with simulations

backstepping observers for a class of parabolic pdes in this paper we design exponentially convergent observers for a class of parabolic partial integro differential equations p i des with only boundary sensing available the problem is posed as a problem of designing an invertible coordinate transformation of the observer error system into an exponentially stable target system observer gain output injection function is shown to satisfy a well posed hyperbolic pde that is closely related to the hyperbolic pde governing backstepping control gain for the state feedback problem for several physically relevant problems the observer gains are obtained in closed form the observer gains are then used for an output feedback design in both collocated and anti collocated setting of sensor and actuator the order of the resulting compensator can be substantially lowered without affecting stability explicit solutions of a closed loop system are found in particular cases

periodic output feedback stabilization of neutral systems a new feedback control technique called periodic output feedback is investigated in the context of infinite dimensional linear systems modeled by neutral functional differential equations in this method discrete output samples are multiplied by a periodic gain function to generate a continuous feedback control this work focuses on stabilization of neutral systems with delayed control modeled in the state space w sub 2 sup 1 r 0 r sup n spl times l sup 2 r 0 r sup p where w sub 2 sup 1 r 0 r sup n denotes the sobolev space of r sup n valued absolutely continuous functions with square integrable derivatives on r 0 we show that a class of these systems can be stabilized by periodic output feedback even though their input operators are unbounded we overcome this difficulty by representing the system state using an abstract integral variation of constants formula an algorithm is presented at the end of this paper to construct a periodic output feedback gain function an example is provided to illustrate the construction of the gain function

sample period effects in optimally designed digital control systems a systematic procedure for obtaining an optimal sampled data control system which treats the sample period as a free design parameter is developed the sensitivity of the closed loop performance with respect to the sample period is examined for various choices of the penalty weighting factors and the compensator structure an optimal nonzero sample period may exist depending on the penalty weighting assigned to the system states even when a boundary condition is imposed that requires the optimal sampled data control system to converge to an optimal continuous control system as the sample period approaches zero

exact controllability of a nonlinear korteweg de vries equation on a critical spatial domain we consider the boundary controllability problem for a nonlinear korteweg de vries equation with the dirichlet boundary condition we study this problem for a spatial domain with a critical length for which the linearized control system is not controllable in order to deal with the nonlinearity we use a power series expansion of second order we prove that the nonlinear term gives the local exact controllability around the origin provided that the time of control is large enough

boundary value problems for higher order linear impulsive differential equations in this paper higher order linear impulsive differential equations with fixed moments of impulses subject to linear boundary conditions are studied green s formula is defined for piecewise differentiable functions properties of green s functions for higher order impulsive boundary value problems are introduced an appropriate example of the green s function for a boundary value problem is provided furthermore eigenvalue problems and basic properties of eigensolutions are considered

the solution manifold and c1 smoothness for differential equations with state dependent delay abstract let h 0 u c 1 h 0 r n open and f u r n continuously differentiable if f satisfies two mild additional smoothness conditions then the set x u 0 f is a c1 submanifold of codimension n in c 1 h 0 r n the maximal solutions x of the initial value problems x t f x t x 0 x define a continuous semiflow f on x and all operators f t are continuously differentiable their derivatives d2f t are given in the usual way by solutions v to the variational equation along x with segments vt in the tangent spaces tf t x the additional conditions on f are motivated by properties of differential equations with state dependent delay and are verified for an example

impulse and sampled data optimal control of heat equations and error estimates we consider the optimal control problem of minimizing some quadratic functional over all possible solutions of an internally controlled multidimensional heat equation with a periodic terminal state constraint this problem has a unique optimal solution which can be characterized by an optimality system derived from the pontryagin maximum principle we define two approximations of this optimal control problem the first one is an impulse approximation and consists of considering a system of linear heat equations with impulse control the second one is obtained by the sample and hold procedure applied to the control resulting in a sampled data approximation of the controlled heat equation we prove that both problems have a unique optimal solution and we establish precise error estimates for the optimal controls and optimal states of the initial problem with respect to its impulse and sampled data approximations

boundary sampled data feedback stabilization for parabolic equations abstract the aim of this work is to design an explicit finite dimensional boundary feedback controller of sampled data form for locally exponentially stabilizing the equilibrium solutions to semilinear parabolic equations the feedback controller is expressed in terms of the eigenfunctions corresponding to unstable eigenvalues of the linearized equation this stabilizing procedure is applicable for any sampling rate and when the sampling period tends to zero the feedback converges to certain feedback designed for stabilizing the parabolic equations with continuous time boundary feedback control

metastable periodic patterns in singularly perturbed state dependent delayed equations abstract we consider the scalar delayed differential equation x t x t f x t r where 0 r r x 1 r x and f represents either a monotone positive feedback d f d x 0 or a monotone negative feedback d f d x 0 plus some other hypotheses when the delay is a constant i e r x 1 this equation can support metastable rapidly oscillating solutions that are transients whose duration is of order exp c for some c 0 in this paper combining analytic and numerical techniques we investigate whether this metastable behavior persists when the delay r x depends non trivially on the state variable x more precisely we proceed in three steps first we explore theoretically and numerically the existence of branches of rapid periodic solutions for these equations then to predict conditions for metastability we introduce transition layer equations that depict the asymptotic shape of transient oscillations when tends to zero and finally these predictions are validated by numerical simulations numerical explorations are also performed for other scalings of with our conclusions are i when 0 0 there are no metastable transients ii when 0 0 and 0 is finite for monotone negative feedback the metastable oscillations exist irrespective of choices of f and r iii for monotone positive feedback they require that the feedback f and the delay r satisfy extra conditions such as f being an odd function and the delay r x an even function one novel result is that state dependent delays may lead to metastable dynamics in equations that cannot support such regimes when the delay is constant

on the simultaneous presence of unilateral and kinetic constraints in time dependent impulsive mechanics the simultaneous presence of unilateral and kinetic constraints acting on a mechanical system with a finite number of degrees of freedom is framed in the geometric context of left and right jet bundles of the classical space time bundle of the system the survey gives three main cases and several subcases some of which are mathematically correct but physically meaningless the existence of at least one frame of reference for which the whole set of constraints can be at rest is the criterion selecting the physically relevant systems for these systems the conservation of kinetic energy possibly together with a standard gauss s requirement on the impulsive reaction is shown to give a well posed criterion of ideality of the constraints the application of the criterion to several examples is presented and the corresponding results are critically analyzed

rapid stabilization for a korteweg de vries equation from the left dirichlet boundary condition this paper deals with the stabilization problem for the korteweg de vries equation posed on a bounded interval the control acts on the left dirichlet boundary condition at the right end point dirichlet and neumann homogeneous boundary conditions are considered the proposed feedback law forces the exponential decay of the system under a smallness condition on the initial data moreover the decay rate can be tuned to be as large as desired the feedback control law is designed by using the backstepping method

mechanical systems subjected to impulsive constraints a geometrical description of mechanical systems subjected to impulsive non holonomic constraints is given their motions are determined by means of suitable projection operators which allow us to evaluate the instantaneous jumps of momenta due to the action of impulsive reaction forces several paradigmatic examples are investigated from this viewpoint a disk falling into a plane a rolling ball hitting a wall and the collision of two balls

differential equations with discrete state dependent delay uniqueness and well posedness in the space of continuous functions abstract partial differential equations with discrete concentrated state dependent delays in the space of continuous functions are investigated in general the corresponding initial value problem is not well posed so we find an additional assumption on the state dependent delay function to guarantee the well posedness for the constructed dynamical system we study the long time asymptotic behavior and prove the existence of a compact global attractor

chemical principle and pde of variational electrodynamics abstract the two body problem of variational electrodynamics possesses differential delay equations of motion with state dependent delays of neutral type and solutions that can have velocity discontinuities on countable sets from a periodic orbit possessing some mild properties at breaking points we define a synchronization function in r r 3 which is further used to construct two bounded oscillatory functions vanishing at breaking points and whose first derivatives are continuous and defined everywhere but at breaking points the oscillatory functions are associated with a natural pde identity in w 2 2 r 3 we postulate an asymptotic ordering for a generalized pde identity which defines a fredholm schroedinger operator in w 2 2 r 3 with a forcing term belonging to l 2 r 3 o 1 r 2 as an application we introduce the chemical principle criterion to select orbits with asymptotically vanishing far fields and estimate the bohr radius parameter of the fredholm schroedinger pde using the boundary layers of orbits chosen according to the chemical principle criterion last working backward we derive an orbital property of the chemical principle type from the ordering conditions

a non local pde model for population dynamics with state selective delay local theory and global attractors we propose a non local pde model for the evolution of a single species population that involves delayed feedback where the delay such as the maturation time in the delayed birth rate is selective and the selection depends on the status of the system this delay selection in contrast with the usual state dependent delay widely used in ordinary delay differential equation ensures the lipschitz continuity of the nonlinear functional in the classical phase space we also develop the local theory and the existence and upper semi continuity of the global attractor with respect to parameters

backstepping boundary control of burgers equation with actuator dynamics in this paper we propose a backstepping boundary control law for burgers equation with actuator dynamics while the control law without actuator dynamics depends only on the signals u 0 t and u 1 t the backstepping control also depends on ux 0 t ux 1 t uxx 0 t and uxx 1 t making the regularity of the control inputs the key technical issue of the paper with elaborate lyapunov analysis we prove that all these signals are suciently regular and the closed loop system including the boundary dynamics is globally h 3 stable and well posed c 2000 elsevier science b v all rights reserved

exact controllability stabilization and perturbations for distributed systems exact controllability is studied for distributed systems of hyperbolic type or for petrowsky systems like plate equations the control is a boundary control or a local distributed control exact controllability consists in trying to drive the system to rest in a given finite time the solution of the problems depends on the function spaces where the initial data are taken and also depends on the function space where the control can be chosen a systematic method named hum for hilbert uniqueness method is introduced as the terminology indicates it is based on uniqueness results classical or new and on hilbert spaces constructed in infinitely many ways by using uniqueness a number of applications are indicated having a general method for exact controllability implies having a general method for stabilization this leads to new and complicated nonlinear riccati s type pdes to be compared with direct methods when available in the last part of the paper we consider how all this behaves fo

a condition on delay for differential equations with discrete state dependent delay abstract parabolic differential equations with discrete state dependent delay are studied the approach based on an additional condition on the delay function introduced in a v rezounenko differential equations with discrete state dependent delay uniqueness and well posedness in the space of continuous functions nonlinear anal 70 11 2009 3978 3986 is developed we propose and study an analogue of the condition which is sufficient for the well posedness of the corresponding initial value problem on the whole space of continuous functions c the dynamical system is constructed in c and the existence of a compact global attractor is proved

on the existence of time optimal controls with constraints of the rectangular type for heat equations this paper presents a time optimal control problem with control constraints of the rectangular type for internally controlled heat equations an existence result of time optimal controls for such a problem is established the rectangular type of control constraints originates from the study of time optimal control problems for ordinary differential equations in the finite dimensional case there is no difference between such problems with control constraints of the rectangular type and those of the ball type from the perspective of the study on the existence of optimal controls interestingly in the infinite dimensional case the problem with control constraints of the rectangular type differs essentially from that with control constraints of the ball type for infinite dimensional systems the existence for time optimal controls with constraints of the ball type has already been discussed in the literature while the study of the rectangular type has not been touched upon as far as we know

null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach using the backstepping approach we recover the null controllability for the heat equations with variable coefficients in space in one dimension and prove that these equations can be stabilized in finite time by means of periodic time varying feedback laws to this end on the one hand we provide a new proof of the well posedness and the optimal bound with respect to damping constants for the solutions of the kernel equations this allows one to deal with variable coefficients even with a weak regularity of these coefficients on the other hand we establish the well posedness and estimates for the heat equations with a nonlocal boundary condition at one side

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

output feedback stabilization for heat equations with sampled data controls abstract in this paper we build up an output feedback law to stabilize a sampled data controlled heat equation with a potential in a bounded domain the feedback law abides the following rules first we divide equally the time interval 0 into infinitely many disjoint time periods and divide each time period into three disjoint subintervals second for each time period we observe a solution over an open subset of in the first subinterval take sample from outputs at one time point of the first subinterval add a time invariant output feedback control over another open subset of in the second subinterval let the equation evolve free in the last subinterval thus the corresponding feedback control is of sampled data our feedback law has the following advantages the sampling period which is the length of the above time period can be arbitrarily taken the feedback law has an explicit expression in terms of the sampling period the behaviors of the norm of the feedback law when the sampling period goes to zero or infinity are clear the construction of the feedback law is based on two kinds of approximate null controllability for heat equations one has time invariant controls while another has impulse controls the studies of the aforementioned controllability with time invariant controls need a new observability inequality for heat equations built up in the current work

nonideal unilateral constraints in impulsive mechanics a geometric approach we use the differential geometric framework of the first jet bundle of the classical space time bundle to study the impulsive behavior of a mechanical system with a finite number of degrees of freedom subject to nonideal unilateral constraints we show that this framework allows deeper insights on the concepts of nonideal constitutive characterization and of coefficient of restitution of the constraints we study the relations among newton poisson and stronge definitions of coefficient of restitution we reveal the inconsistency of the criticisms based on the energy balance of the impact for the newton definition we show the equivalence of the three definitions in the nonideal smooth case we prove the equivalence of newton and poisson ones and the insufficiency of the stronge one in the nonideal rough case we analyze the relation between coefficient of restitution and coulomb s friction coefficient in the rough case we present also several physically meaningful examples

boosting the decay of solutions of the linearised korteweg de vries burgers equation to a predetermined rate from the boundary abstractthe aim of this article is to extend recent results on the boundary feedback controllability of the korteweg de vries equation to the korteweg de vries burgers equation which is posed on a bounded domain in the first part of the paper it is proven that all the sufficiently small solutions can be steered to zero at any desired exponential rate by means of a suitably constructed boundary feedback controller in the second part an observer is proposed when a type of boundary measurement is available while there is no full access to the medium

geometric formulation of carnot s theorem a geometrical approach for one sided constraints is given the riemannian metric is used in order to define convenient projectors which give the post impulses in terms of the pre impulses a formulation of carnot s theorem within this geometric framework is exhibited

two approaches for the stabilization of nonlinear kdv equation with boundary time delay feedback this paper concerns the nonlinear korteweg de vries equation with boundary time delay feedback under appropriate assumption on the coefficients of the feedbacks delayed or not we first prove that this nonlinear infinite dimensional system is well posed for small initial data the main results of our study are two theorems stating the exponential stability of the nonlinear time delay system two different methods are employed a lyapunov function approach allowing to have an estimation on the decay rate but with a restrictive assumption on the length of the spatial domain of the kdv equation and an observability inequality approach with a contradiction argument for any noncritical lengths but without estimation on the decay rate some numerical simulations are given to illustrate the results

backstepping forwarding control and observation for hyperbolic pdes with fredholm integrals an integral transform is introduced which allows the construction of boundary controllers and observers for a class of first order hyperbolic pides with fredholm integrals these systems do not have a strict feedback structure and thus the standard backstepping approach cannot be applied sufficient conditions for the existence of the backstepping forwarding transform are given in terms of spectral properties of some integral operators and more conservatively but easily verifiable in terms of the norms of the coefficients in the equations an explicit transform is given for particular coefficient structures in the case of strict feedback systems the procedure detailed in this paper reduces to the well known backstepping design the results are illustrated with numerical simulations

generalized sampled data feedback control of distributed parameter systems abstract in the sampled data control literature there are necessary conditions and sufficient conditions for stabilizability of distributed parameter systems by generalized sampled data control for finite dimensional systems the necessary conditions are also known to be sufficient we show that this equivalence extends to the infinite dimensional case if the underlying semigroup is analytic however for general systems the necessary conditions are not sufficient nor are the sufficient conditions necessary we prove this by a single example with a free parameter one choice of parameter shows that the necessary conditions are too weak and another choice shows that the sufficient conditions are too strong

small time local stabilization for a korteweg de vries equation abstract this paper focuses on the local small time stabilization of a korteweg de vries equation on bounded interval thanks to a time varying dirichlet feedback law on the left boundary recently backstepping approach has been successfully used to prove the null controllability of the corresponding linearized system instead of carleman inequalities we use the adding an integrator technique to gain regularity on boundary control term which clears the difficulty from getting stabilization in small time

exact controllability and stabilizability of the korteweg de vries equation in this paper we consider distributed control of the system described by the korteweg de vries equation i atu uaxu e3 u f on the interval 0 0 with periodic boundary conditions ii au 27r t ku o t k 0 1 2 where the distributed control f f x t is restricted so that the volume f u x t dx of the solution is conserved both exact controllability and stabilizibility questions are studied for the system in the case of open loop control if the control f is allowed to act on the whole spatial domain 0 27r it is shown that the system is globally exactly controllable i e for given t 0 and functions q x x with the same volume one can alway find a control f so that the system i ii has a solution u x t satisfying u x 0 0 x u x t x if the control f is allowed to act on only a small subset of the domain 0 27r then the same result still holds if the initial and terminal states b and 0 have small amplitude in a certain sense in the case of closed loop control the distributed control f is assumed to be generated by a linear feedback law conserving the volume while monotonically reducing f u x t 2dx the solutions of the resulting closed loop system are shown to have uniform exponential decay to a constant state as in the open loop control case a small amplitude assumption is needed if the control is allowed to act on only a small subdomain the smoothing property of the periodic linear kdv equation discovered recently by bourgain has played an important role in establishing the exact controllability and stabilizability results presented in this paper

controllability of the korteweg de vries equation from the right dirichlet boundary condition in this paper we consider the controllability of the korteweg de vries equation in a bounded interval when the control operates via the right dirichlet boundary condition while the left dirichlet and the right neumann boundary conditions are kept to zero we prove that the linearized equation is controllable if and only if the length of the spatial domain does not belong to some countable critical set when the length is not critical we prove the local exact controllability of the nonlinear equation

boundary feedback stabilization of the korteweg de vries burgers equation posed on a finite interval abstract this paper studies the boundary feedback stabilization problem of the korteweg de vries burgers equation posed on a finite interval a linear boundary feedback control law is proposed then the resulting closed loop nonlinear system is shown not only to be globally well posed but also to be exponentially stable in the space h s 0 l with s 0 3 the sharp kato smoothing property of the associated linear problem is revealed through the boundary integral operator which plays a key role in our analysis

reduction of constrained mechanical systems and stability of relative equilibria a mechanical system with perfect constraints can be described under some mild assumptions as a constrained hamiltonian system m h d w m thephase space is a symplectic manifold h thehamiltonian a smooth function onm d theconstraint submanifold a submanifold ofm andw theprojection bundle a vector sub bundle oft d m the reduced tangent bundle alongd we prove that when these data satisfy some suitable conditions the time evolution of the system is governed by a well defined differential equation ond we define constrained hamiltonian systems with symmetry and prove a reduction theorem application of that theorem is illustrated on the example of a convex heavy body rolling without slipping on a horizontal plane two other simple examples show that constrained mechanical systems with symmetry may have an attractive or repulsive set of relative equilibria

boundary layer phenomena for differential delay equations with state dependent time lags iii in this paper we begin a study of the differential delay equation varepsilon x t x t f x t r r r x t

stability of infinite dimensional sampled data systems suppose that a static state feedback stabilizes a continuous time linear infinite dimensional control system we consider the following question if we construct a sampled data controller by applying an idealized sample and hold process to a continuous time stabilizing feedback will this sampled data controller stabilize the system for all sufficiently small sampling times here the state space x and the control space u are hilbert spaces the system is of the form x t ax t bu t where a is the generator of a strongly continuous semigroup on x and the continuous time feedback is u t fx t the answer to the above question is known to be yes if x and u are finite dimensional spaces in the infinite dimensional case if f is not compact then it is easy to find counterexamples therefore we restrict attention to compact feedback we show that the answer to the above question is yes if b is a bounded operator from u into x moreover if b is unbounded we show that the answer yes remains correct provided that the semigroup generated by a is analytic we use the theory developed for static state feedback to obtain analogous results for dynamic output feedback control

generalized sampled data stabilization of well posed linear infinite dimensional systems we consider well posed linear infinite dimensional systems the outputs of which are sampled in a generalized sense using a suitable weighting function under certain natural assump tions on the system the weighting function and the sampling period we show that there exists a generalized hold function such that unity sampled data feedback renders the closed loop system exponentially stable in the state space sense as well as l 2 stable in the input output sense to illustrate our main result we describe an application to a structurally damped euler bernoulli beam

boundary control of an anti stable wave equation with anti damping on the uncontrolled boundary abstract much of the boundary control of wave equations in one dimension is based on a single principle passivity under the assumption that control is applied through neumann actuation on one boundary and the other boundary satisfies a homogeneous dirichlet boundary condition we have recently expanded the scope of tractable problems by allowing destabilizing anti stiffness a robin type condition on the uncontrolled boundary where the uncontrolled system has a finite number of positive real eigenvalues in this paper we go further and develop a methodology for the case where the uncontrolled boundary condition has anti damping which makes the real parts of all the eigenvalues of the uncontrolled system positive and arbitrarily high i e the system is anti stable exponentially stable in negative time using a conceptually novel integral transformation we obtain extremely simple explicit formulae for the gain functions for the case with only boundary sensing available at the same end with actuation we design backstepping observers which are dual to the backstepping controllers and have explicit output injection gains we then combine the control and observer designs into an output feedback compensator and prove the exponential stability of the closed loop system

output feedback boundary control of a ginzburg landau model of vortex shedding an exponentially convergent observer is designed for a linearized ginzburg landau model of vortex shedding in viscous flow past a bluff body measurements are restricted to be taken collocated with the actuation which is applied on the cylinder surface the observer is used in conjuction with a state feedback boundary controller designed in previous work to attenuate vortex shedding while the theoretical results apply to the linearized system under sufficiently smooth initial data that satisfy the boundary conditions simulations demonstrate the performance of the linear output feedback scheme on the nonlinear plant model

geometric formulation of mechanical systems subjected to time dependent one sided constraints a geometrical setting for time dependent impulsive constraints is given this description permits us to treat in a systematic way a wide family of mechanical systems subjected to time dependent constraints like moving walls etc if the impulsive constraints remain then an affine projector can be defined giving the new initial data in terms of the old ones several examples are discussed

boundary feedback stabilization of an unstable heat equation in this paper we study the problem of boundary feedback stabilization for the unstable heat equation ut x t uxx x t a x u x t this equation can be viewed as a model of a heat conducting rod in which not only is the heat being diffused mathematically due to the diffusive term uxx but also the destabilizing heat is generating mathematically due to the term a u with a 0 we show that for any given continuously differentiable function a and any given positive constant l we can explicitly construct a boundary feedback control law such that the solution of the equation with the control law converges to zero exponentially at the rate of l this is a continuation of the recent work of boskovic krstic and liu ieee trans automat control 46 2001 pp 2022 2028 and balogh and krstic european j control 8 2002 pp 165 176

convergence of solutions for an equation with state dependent delay abstract a result of smith and thieme shows that if a semiflow is strongly order preserving then a typical orbit converges to the set of equilibria for the equation with state dependent delay t x t f x t r x t where 0 and f and r are smooth real functions with f 0 0 and f 0 we construct a semiflow which is monotone but not strongly order preserving we prove a convergence result under a monotonicity condition different from the strong order preserving property and apply it to the above equation to obtain generic convergence

reduction of constrained systems with symmetries a general model is proposed for constrained dynamical systems on a symplectic manifold which covers among others the description of lagrangian and hamiltonian systems with nonholonomic constraints and the canonical description of mechanical systems with a singular lagrangian the reduction properties of these systems in the presence of symmetry are investigated within this general framework

non holonomic lagrangian systems in jet manifolds a geometrical setting in terms of jet manifolds is developed for time dependent non holonomic lagrangian systems an almost product structure on the evolution space is constructed in such a way that the constrained dynamics is obtained by projection of the free dynamics a constrained poincare cartan 2 form is defined if the non holonomic system is singular a constraint algorithm is constructed special attention is devoted to caplygin systems and a reduction theorem is proved