control of an unstable reaction diffusion pde with long input delay abstract a smith predictor like design for compensation of arbitrarily long input delays is available for general controllable possibly unstable lti finite dimensional systems such a design has not been proposed previously for problems where the plant is a pde we present a design and stability analysis for a prototype problem where the plant is a reaction diffusion parabolic pde with boundary control the plant has an arbitrary number of unstable eigenvalues and arbitrarily long delay with an unbounded input operator the predictor based feedback design extends fairly routinely within the framework of infinite dimensional backstepping however the stability analysis contains interesting features that do not arise in predictor problems when the plant is an ode the unbounded character of the input operator requires that the stability be characterized in terms of the h 1 rather than the usual l 2 norm of the actuator state the analysis involves an interesting structure of interconnected pdes of parabolic and first order hyperbolic types where the feedback gain kernel for the undelayed problem becomes an initial condition in a pde arising in the compensator design for the problem with input delay space and time variables swap their roles in an interesting manner throughout the analysis

admissibility of unbounded control operators for linear systems described by dot x t ax t bu t where a generates a semigroup on the state space x and b is an unbounded operator some necessary as well as some sufficient conditions are given for b to be admissible i e for any t the state x t should be in x and should depend continuously on the input u in l p this approach begins with an axiomatic description of such a system in terms of a functional equation the results are applied to the wave equation on a bounded interval

exponential stability and spectral analysis of the pendulum system under position and delayed position feedbacks this article addresses the exponential stability and spectral analysis of the linearised pendulum system with both position and delayed position pdp for short feedbacks the semigroup approach is adopted in investigation the asymptotic spectral expression of the system is presented it is shown that the spectrum of the system is located in the left half complex plane and its real part goes to at the high frequency end when the feedback gains and the delay satisfy some additional conditions therefore exponential stability of the system follows from the spectrum determined growth property finally some numerical simulations of a planar pendulum are presented

regularity of a schrodinger equation with dirichlet control and colocated observation this paper studies the basic properties of the schrodinger equation defined on a bounded domain of r n n 2 with partial dirichlet control and colocated observation it is shown that the system is not only wellposed in the sense of d salamon but also regular in the sense of g weiss it is also shown that the corresponding feedthrough operator is zero 2005 elsevier b v all rights reserved

a proof of price s law for the collapse of a self gravitating scalar field a well known open problem in general relativity dating back to 1972 has been to prove price s law for an appropriate model of gravitational collapse this law postulates inverse power decay rates for the gravitational radiation flux through the event horizon and null infinity with respect to appropriately normalized advanced and retarded time coordinates it is intimately related both to astrophysical observations of black holes and to the fate of observers who dare cross the event horizon in this paper we prove a well defined upper bound formulation of price s law for the collapse of a self gravitating scalar field with spherically symmetric initial data we also allow the presence of an additional gravitationally coupled maxwell field our results are obtained by a new mathematical technique for understanding the long time behavior of large data solutions to the resulting coupled non linear hyperbolic system of p d e s in 2 independent variables the technique is based on the interaction of the conformal geometry the celebrated red shift effect and local energy conservation we feel it may be relevant for the problem of non linear stability of the kerr solution when combined with previous work of the first author concerning the internal structure of charged black holes which had assumed the validity of price s law our results can be applied to the strong cosmic censorship conjecture for the einstein maxwell real scalar field system with complete spacelike asymptotically flat spherically symmetric initial data under christodoulou s c0 formulation the conjecture is proven to be false

stabilization of an unstable reaction diffusion pde cascaded with a heat equation abstract we consider a control problem of an unstable reaction diffusion parabolic pde cascaded with a heat equation through a boundary where the heat influx of the heat equation is fed into the temperature of the reaction diffusion equation and the control actuator is designed at the another boundary of the heat equation a backstepping invertible transformation is used to design a suitable boundary feedback control so that the closed loop system is equivalent to a cascade of pde pde system which is shown to be exponentially stable in a suitable hilbert space with the dirichlet boundary input from the heat equation the reaction diffusion pde is shown to be exponentially stable in h 1 0 1 numerical simulations are presented to illustrate the convergence of the state of the reaction diffusion equation

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

on a class of degenerate and singular elliptic systems in bounded domains abstract this paper deals with the nonexistence and multiplicity of nonnegative nontrivial solutions to a class of degenerate and singular elliptic systems of the form div h 1 x u f u x u v in div h 2 x v f v x u v in where is a bounded domain with smooth boundary in r n n 2 and h i 0 h i l loc 1 h i i 1 2 are allowed to have essential zeroes at some points in f u f v f and is a positive parameter our proofs rely essentially on the critical point theory tools combined with a variant of the caffarelli kohn nirenberg inequality in p caldiroli r musina on a variational degenerate elliptic problem nodea nonlinear differential equations appl 7 2000 189 199

two examples of ill posedness with respect to time delays revisited we revisit two examples of elastic control systems which are stabilized by boundary feedback and then destabilized when time delays occur in the feedback in the first example a wave equation with a feedback gain greater than or equal to one arbitrary divergence rates are obtained for small time delays in the second example a euler bernoulli beam any time delay results in an arbitrary divergence rate

conditions for robustness and nonrobustness of thestability of feedback systems with respect to small delays inthe feedback loop it has been observed that for many stable feedback control systems the introduction of arbitrarily small time delays into the loop causes instability in this paper we present a systematic frequency domain treatment of this phenomenon for distributed parameter systems we consider the class of all matrix valued transfer functions which are bounded on some right half plane and which have a limit at spinfty along the real axis such transfer functions are called regular under the assumption that a regular transfer function is stabilized by unity output feedback we give sufficient conditions for the robustness and for the nonrobustness of the stability with respect to small time delays in the loop these conditions are given in terms of the high frequency behavior of the open loop system moreover we discuss robustness of stability with respect to small delays for feedback systems with dynamic compensators in particular we show that if a plant with infinitely many poles in the closed right half plane is stabilized by a controller then the stability is not robust with respect to delays we show that the instability created by small delays is itself robust to small delays three examples are given to illustrate these results

backstepping boundary control for first order hyperbolic pdes and application to systems with actuator and sensor delays abstract we consider a problem of boundary feedback stabilization of first order hyperbolic partial differential equations pdes these equations serve as a model for physical phenomena such as traffic flows chemical reactors and heat exchangers we design controllers using a backstepping method which has been recently developed for parabolic pdes with the integral transformation and boundary feedback the unstable pde is converted into a delay line system which converges to zero in finite time we then apply this procedure to finite dimensional systems with actuator and sensor delays to recover a well known infinite dimensional controller analog of the smith predictor for unstable plants we also show that the proposed method can be used for the boundary control of a korteweg de vries like third order pde the designs are illustrated with simulations

brief paper lyapunov tools for predictor feedbacks for delay systems inverse optimality and robustness to delay mismatch we consider lti finite dimensional completely controllable but possibly open loop unstable plants with arbitrarily long actuator delay and the corresponding predictor based feedback for delay compensation we study the problem of inverse optimal re design of the predictor based feedback law we obtain a simple modification of the basic predictor based controller which employs a low pass filter and has been proposed previously by mondie and michiels for achieving robustness to discretization of the integral term in the predictor feedback law the key element in our work is the employment of an infinite dimensional backstepping transformation and the resulting lyapunov function for the infinite dimensional systems consisting of the state of the ode plant and the delay state the lyapunov function allows us to quantify the lyapunov stability properties under the modified feedback the inverse optimality of the feedback and its disturbance attenuation properties for the basic predictor feedback the availability of the lyapunov function also allows us to prove robustness to small delay mismatch in both positive and negative directions

not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks we present two examples of hyperbolic partial differential equations which are stabilized by boundary feedback controls and then destabilized by small delays in these controls we show that in a general case when the controls are distributed stabilized hyperbolic systems possess nontrivial periodic solutions if small time delays are introduced into their feedbacks we also indicate by means of an example that the general case of this phenomenon is harder to demonstrate for boundary control problems

local decay of waves on asymptotically flat stationary space times author s tataru daniel abstract in this article we study the pointwise decay properties of solutions to the wave equation on a class of stationary asymptotically flat backgrounds in three space dimensions under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a t 3 local uniform decay rate for linear waves this work was motivated by open problems concerning decay rates for linear waves on schwarzschild and kerr backgrounds where such a decay rate has been conjectured by r price our results apply to both of these cases

wave equation stabilization by delays equal to even multiples of the wave propagation time for a string equation with time delays in the output feedback loop we study stability and show that the system is a riesz spectral system and prove that the spectrum determined growth condition holds for all delays when the delay is equal to the even multiples of the wave propagation time we develop the necessary and sufficient conditions for the feedback gain and time delay which guarantee the exponential stability of the closed loop system in particular we show that as the delay of even multiples is increasing to infinity the stability bound on the feedback gain decays to zero we also show that whenever the delay is an odd multiple of the wave propagation time the closed loop system is unstable the lack of robustness to a small perturbation in time delay is specifically discussed for the delay equal to two a numerical simulation for the case of the delay equal to two is presented to illustrate the convergence finally an alternative stability analysis is conducted within the framework of well posed infinite dimensional systems

finite time boundary control for delay reaction diffusion systems abstract this paper considers finite time stabilization and h performance for delay reaction diffusion systems by boundary control first a full domain controller is designed and sufficient conditions are obtained to achieve finite time stability using finite time stability lemma and wirtinger s inequality method then a boundary controller furnished with sufficient conditions to achieve finite time stability is presented when taking into consideration external noise on a delay reaction diffusion system finite horizon h boundary control with a criterion that guarantees the h performance of delay reaction diffusion systems is proposed how to handle neumann boundary conditions and mixed boundary conditions are discussed numerical simulations are carried out to verify the effectiveness of our theoretical results

on global existence for nonlinear wave equations outside of convex obstacles the authors prove global existence of small solutions to a semilinear wave equation outside of convex obstacles this extends results of christodoulou and klainerman who handled the minkowski space version the proof is a compromise of the methods of christodoulou and klainerman it relies on local estimates proved earlier by smith and sogge together with classical energy decay estimates for the wave equation of morawetz lax and phillips

uniform decay of local energy and the semi linear wave equation on schwarzschild space we provide a uniform decay estimate for the local energy of general solutions to the inhomogeneous wave equation on a schwarzschild background our estimate implies that such solutions have asymptotic behavior phi o left r 1 big t r big frac 1 2 right as long as the source term is bounded in the norm 1 frac 2m r 1 cdot 1 t r 1 l 1 big h 3 omega r 2dr d omega big in particular this gives scattering at small amplitudes for non linear scalar fields of the form square g varphi lambda varphi p varphi for all 2 p

linear systems with delayed controls a reduction linear systems with delayed control action are transformed into systems without delays under an absolute continuity condition the new system is an ordinary differential control equation in the general case the new system is a measure differential control system it is shown how the controllability stabilization and various optimization problems can be analyzed via the reduced systems

boundary controllers and observers for the linearized schrodinger equation we consider a problem of stabilization of the linearized schrodinger equation using boundary actuation and measurements we propose two different control designs first a simple proportional collocated boundary controller is shown to exponentially stabilize the system how ever the decay rate of the closed loop system cannot be prescribed the second full state feedback boundary control design achieves an arbitrary decay rate we formally view the schrodinger equa tion as a heat equation in complex variables and apply the backstepping method recently developed for boundary control of reaction advection diffusion equations the resulting controller is then sup plied with the backstepping observer to obtain an output feedback compensator the designs 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

brief paper dynamic stabilization of an euler bernoulli beam equation with time delay in boundary observation an euler bernoulli beam equation under boundary control and delayed observation is considered an observer predictor based scheme is developed to stabilize the equation on a time interval where the observation is available the state is tracked by the observer when the observation is not available due to the delay the state is estimated by the predictor the estimation is then used in proportional feedback it is shown that the state of closed loop system decays exponentially for smooth initial values

an example on the effect of time delays in boundary feedback stabilization of wave equations this note is concerned with the erect of time delays in boundary feedback stabilization schemes for wave equations the question to be addressed is whether such delays can destabilize a system which is uniformly asymptotically stable in the absence of delays