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This paper is about the strong stabilization of small amplitude gravity water waves in a vertical rectangle. The control imposes the horizontal acceleration of the water along one vertical boundary segment, as a multiple of a scalar input function u, times a function h of the position along the active boundary. The state z of the system is a vector with two components: one is the water level ζ along the top boundary and the other is its time derivative ζ. We prove that for suitable functions h, there exists a bounded feedback functional F such that the feedback u = Fz leads to a strongly stable closed-loop system. Moreover, for initial states in the domain of the semigroup generator, the norm of the solution decays like (1 + t) − 1/6. Our approach uses careful estimates on certain partial Dirichlet to Neumann and Neumann to Neumann operators associated to the rectangular domain, as well as non-uniform stabilization results due to Chill, Paunonen, Seifert, Stahn and Tomilov (2019).

We study the output feedback exponential stabilization of a one-dimensional unstable wave equation, where the boundary input, given by the Neumann trace at one end of the domain, is the sum of the ...

We consider the strong stabilization of small amplitude gravity water waves in a two dimensional rectangular domain. The control acts on one lateral boundary, by imposing the horizontal acceleration of the water along that boundary, as a multiple of a scalar input function u , times a given function h of the height along the active boundary. The state z of the system consists of two functions: the water level ζ along the top boundary, and its time derivative ζ . We prove that for suitable functions h , there exists a bounded feedback functional F such that the feedback u = F z renders the closed-loop system strongly stable. Moreover, for initial states in the domain of the semigroup generator, the norm of the solution decays like ( 1 + t ) − 1 6 . Our approach uses a detailed analysis of the partial Dirichlet to Neumann and Neumann to Neumann operators associated to certain edges of the rectangular domain, as well as recent abstract non-uniform stabilization results by Chill et al. (2019).

We consider coupled systems consisting of a well-posed and impedance passive linear system (that may be infinite dimensional), with semigroup generator $A$ and transfer function G, and an internal model controller (IMC), connected in feedback. The IMC is finite dimensional, minimal and impedance passive, and it is tuned to a finite set of known disturbance frequencies ω j , where $j$ E {1, … $n$ }, which means that its transfer function g has poles at the points $i$ ω j . We also assume that g has a feedthrough term $d$ with Re d > 0. We assume that Re G(iω j ) > 0 for all $j$ ∊ {1, … n} and the points $i$ ω j are not eigenvalues of A. We can show that the closed-loop system is well-posed and input-output stable (in particular, (I + gG)−1 E $H$ ∞ and also G(1 + gG)-l E $H$ ∞). It is also easily seen that the closed-loop system is impedance passive. We show that if $A$ has at most a countable set of imaginary eigenvalues, that are all observable, and $A$ has no other imaginary spectrum, then the closed-loop system is strongly stable. This result is illustrated with a wind turbine tower model controlled by an IMC.

We consider the mathematical model of a rigid ball moving in a viscous incompressible fluid occupying a bounded domain Ω, with an external force acting on the ball. We investigate in particular the case when the external force is what would be produced by a spring and a damper connecting the center of the ball h to a fixed point h 1 ∈ Ω . If the initial fluid velocity is sufficiently small, and the initial h is sufficiently close to h 1 , then we prove the existence and uniqueness of global (in time) solutions for the model. Moreover, in this case, we show that h converges to h 1 , and all the velocities (of the fluid and of the ball) converge to zero. Based on this result, we derive a control law that will bring the ball asymptotically to the desired position h 1 even if the initial value of h is far from h 1 , and the path leading to h 1 is winding and complicated. Now, the idea is to use the force as described above, with one end of the spring and damper at h , while other end is jumping between a finite number of points in Ω, that depend on h (a switching feedback law).

We study the global asymptotic behavior of a grid-connected constant field current synchronous generator (SG). The grid is regarded as an "infinite bus", i.e. a three-phase AC voltage source. The generator does not include any controller other than the frequency droop loop. This means that the mechanical torque applied to this generator is an affine function of its angular velocity. The negative slope of this function is the frequency droop constant. We derive sufficient conditions on the SG parameters under which there exist exactly two periodic state trajectories for the SG, one stable and another unstable, and for almost all initial states, the state trajectory of the SG converges to the stable periodic trajectory (all the angles are measured modulo $2\pi$). Along both periodic state trajectories, the angular velocity of the SG is equal to the grid frequency. Our sufficient conditions are easy to check computationally. An important tool in our analysis is an integro-differential equation called the {\em exact swing equation}, which resembles a forced pendulum equation and is equivalent to our fourth order model of the grid-connected SG. Apart from our objective of providing an analytical proof for a global asymptotic behavior observed in a classical dynamical system, a key motivation for this work is the development of synchronverters which are inverters that mimic the behavior of SGs. Understanding the global dynamics of SGs can guide the choice of synchronverter parameters and operation. As an application we find a set of stable nominal parameters for a 500 kW synchronverter., Comment: submitted in October 2015, waiting for reviews, 37 pages

We study a time-varying well-posed system resulting from the additive perturbation of the generator of a time-invariant well-posed system. The associated generator family has the form $$A+G(t)$$ , where $$G(t)$$ is a bounded operator on the state space and $$G(\cdot )$$ is strongly continuous. We show that the resulting time-varying system (the perturbed system) is well-posed and investigate its properties. In the particular case when the unperturbed system is scattering passive, we derive an energy balance inequality for the perturbed system. We illustrate this theory using it to formulate the system corresponding to an electrically conducting rigid body moving in vacuo in a bounded domain, with an electromagnetic field (both in the rigid body and in the vacuum) described by Maxwell’s equations.

We give general results about the identifiability of source terms for infinite-dimensional linear systems that are exactly observable. We allow the source term to be unbounded, i.e., not contained in the state space, but in one of a sequence of extended spaces. We show that the operator from the source term to the output function is bounded from below, in suitable norms. We apply the main result to a system described by the wave equation in a bounded $$n$$ -dimensional domain. We derive results of independent interest concerning the range of the input map of an exactly controllable system, when restricted to various spaces of smooth input functions.

This paper considers a nonlinear infinite-dimensional system Σ N obtained by the feedback interconnection of a well-posed linear system Σ P with a globally Lipschitz (memoryless) nonlinear feedback operator N. First, under mild assumptions, we establish the global existence and uniqueness of a state trajectory and an output function for Σ N , for any initial state in its state space X and any input signal of class. Then we investigate the behavior of Σ N when it is driven by a nonlinear time-invariant exosystem with well defined dynamics forward and backward in time. Under the assumption that Σ P is exponentially stable, denoting the state space of the exosystem by W , we find that there exists a continuous map Π : W → X such that regardless of initial states lim t →∞ ‖Π w (t) – x ( t )‖ = 0, where w ( t ) is the state of the exosystem and x ( t ) is the state of Σ N. In particular, when w is T -periodic, then the state of the interconnection tends to a T -periodic limit cycle. The construction of Π can be viewed as an extension of the famous center manifold theorem, which lies at the basis of nonlinear regulator theory, to a class of infinite-dimensional systems.

We study a time-varying well-posed system resulting from the additive perturbation of the generator of a time-invariant well-posed system. The associated generator family has the form A+G(t), where G(t) is a bounded operator on the state space and G(·) is strongly continuous. We show that the resulting time-varying system (the perturbed system) is well-posed and we investigate its properties. In the particular case when the unperturbed system is scattering passive, we derive an energy balance inequality for the perturbed system. If the operators G(t) are dissipative, then the perturbed system is again scattering passive. We illustrate this theory by using it to formulate the system corresponding to a conductor moving in an electromagnetic field described by Maxwell's equations.

We derive a model for a wind turbine tower in the plane of the turbine blades, which comprises an Euler–Bernoulli beam coupled with a nacelle (rigid body) and a two-mass drive-train model (with gearbox). This model has two possible control inputs: the torque created by the electrical generator and the force created by an electrically driven mass located in the nacelle. First, we consider the case of only torque control and a possibly non-uniform tower. Using the theory of coupled linear systems (one infinite dimensional and one finite dimensional) developed by us recently, we show that this wind turbine tower model is well-posed and regular on either the energy state space Xc or the domain of its generator on Xc, denoted by Xc 1. We also show that generically, this model is exactly controllable on X c 1 in arbitrarily short time. More precisely, for every T > 0, we show that if we vary a certain parameter in the model, then exact controllability in time T holds for all except three values of the parameter. In the case of using both force and torque control, we derive similar well-posedness, regularity and generic exact controllability results on a state space that is larger than Xc 1 but smaller than X c. In this second case, we assume that the tower is uniform.

The SCOLE model is a coupled system consisting of a flexible beam (modelled as an Euler–Bernoulli equation) with one end clamped and the other end linked to a rigid body. Its inputs are the force and the torque acting on the rigid body. It is well-known that the SCOLE model is not exactly controllable with L 2 input signals in the natural energy state space H c , because the control operator is bounded from the input space $${\mathbb{C}^2}$$ to H c , and hence compact. We regard the velocity and the angular velocity of the rigid body as the output signals of this system. Using the theory of coupled linear systems (one infinite-dimensional and one finite-dimensional) developed by us recently in another paper, we show that the SCOLE model is well-posed, regular and exactly controllable in arbitrarily short time when using a certain smoother state space $${\mathcal{X}\subset H^c}$$ .

We investigate two classes of time-varying well-posed linear systems. Starting from a time-invariant scattering-passive system, each of the time-varying systems is constructed by introducing a time-dependent inner product on the state space and modifying some of the generating operators. These classes of linear systems are motivated by physical examples such as the electromagnetic field around a moving object. To prove the well-posedness of these systems, we use the Lax–Phillips semigroup induced by a well-posed linear system, as in scattering theory.

We consider a hinged elastic beam described by the Rayleigh beam equation on the interval [0, pi]. We assume the presence of two sensors: one measures the angular velocity of the beam at a point xi is an element of [0, pi] and the other measures the bending (curvature) of the beam at the same point. (If xi = 0 or xi = pi, then the second output is not needed.) The corresponding operator semigroup is unitary on a suitable Hilbert state space. These two measurements are advantageous because they make the open-loop system exactly observable, regardless of the point xi. We design the actuators and the feedback law in order to exponentially stabilize this system. Using the theory of collocated static output feedback developed in our recent paper [6], we design the actuators such that they are collocated, meaning that B = C*, where B is the control operator and C is the observation operator. It turns out that if xi is an element of (0, pi), then the actuators cause a discontinuity of the bending exactly at xi (this is the price, in this example, of having collocated actuators and sensors). This obliges us to use an extension of C to define the output signal in terms of the left and right limit of the bending at xi. We prove that, for all static output feedback gains in a suitable finite range, the closed-loop system is well posed and exponentially stable. This follows from the general theory in our paper [6], whose main points are recalled here.

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

We consider infinite-dimensional linear systems without a-priori well-posedness assumptions, in a framework based on the works of M. Livsic, M. S. Brodskiǐ, Y. L. Smuljan, and others. We define the energy in the system as the norm of the state squared (other, possibly indefinite quadratic forms will also be considered). We derive a number of equivalent conditions for a linear system to be energy preserving and hence, in particular, well posed. Similarly, we derive equivalent conditions for a system to be conservative, which means that both the system and its dual are energy preserving. For systems whose control operator is one-to-one and whose observation operator has dense range, the equivalent conditions for being conservative become simpler, and reduce to three algebraic equations.

We consider feedback systems obtained from infinite-dimensional well-posed linear systems by output feedback. Thus, our framework allows for unbounded control and observation operators. Our aim is to investigate the relationship between the open-loop system, the feedback operator K and the spectrum (in particular, the eigenvalues and eigenvectors) of the closed-loop generator AK. We give a useful characterization of that part of the spectrum σ(AK) which lies in the resolvent set of A, the open-loop generator, via the “characteristic equation” involving the open-loop transfer function. We show that certain points of σ(A) cannot be eigenvalues of AK if the input and output are scalar (so that K is a number) and K≠0. We devote special attention to the case when the output feedback operator K is compact. It is relatively easy to prove that in this case, σe(A), the essential spectrum of A, is invariant, that is, it is equal to σe(AK). A related but much harder problem is to determine the largest subset of σ(A) which remains invariant under compact output feedback. This set, which we call the immovable spectrum of A, strictly contains σe(A). We give an explicit characterization of the immovable spectrum and we investigate the consequences of our results for a certain class of well-posed systems obtained by interconnecting an infinite chain of identical systems.

We solve the quadratic optimal control problem on an infinite time interval for a class of linear systems whose state space is a Hilbert space and whose operator semigroup is unitary. The difficulty is that the systems in this class, having unbounded control and observation operators, may be ill-posed. We show that there is a surprisingly simple solution to the problem (the optimal feedback turns out to be output feedback). Our approach is to use a change of variables which transforms the system into a one which, according to recent research, is known to be conservative. We show that, under a mild assumption, the transfer function of this conservative system is inner, and then it follows that the optimal control of this conservative system is trivial. We give an example with the wave equation on an n-dimensional domain, with Neumann control and Dirichlet observation of the velocity.

Let A0 be a possibly unbounded positive operator on the Hilbert space H ,w hich is boundedly invertible. Let C0 be a bounded operator from D A 1 2 0 to another Hilbert space U.W e prove that the system of equations ¨(t )+ A0z(t )+ 1 C 0C0 _ z(t )= C 0u(t); y(t )= C0 _ z(t )+ u(t); determines a well-posed linear system with input u and output y. The state of this system is x(t )= z(t) _ z(t) 2D A 1 2 0 H = X; where X is the state space. Moreover, we have the energy identity

This paper is a sequel to a paper by the second author on regular linear systems (1994), referred to here as Part I. We introduce the system operator of a well-posed linear system, which for a finite-dimensional system described by x = Ax + Bu, y = Cx + Du would be the s-dependent matrix S Σ (s) = [A-Si/C B D ]. In the general case, S Σ (s) is an unbounded operator, and we show that it can be split into four blocks, as in the finite-dimensional case, but the splitting is not unique (the upper row consists of the uniquely determined blocks A-sI and B, as in the finite-dimensional case, but the lower row is more problematic). For weakly regular systems (which are introduced and studied here), there exists a special splitting of S Σ (s) where the right lower block is the feedthrough operator of the system. Using S Σ (0), we give representation theorems which generalize those from Part I to well-posed linear systems and also to the situation when the initial time is -∞, We also introduce the Lax-Phillips semigroup T induced by a well-posed linear system, which is in fact an alternative representation of a system, used in scattering theory. Our concept of a Lax-Phillips semigroup differs in several respects from the classical one, for example, by allowing an index ω ∈ R which determines an exponential weight in the input and output spaces. This index allows us to characterize the spectrum of A and also the points where S Σ (s) is not invertible, in terms of the spectrum of the generator of? (for various values of ω). The system Σ is dissipative if and only if? (with index zero) is a contraction semigroup.

In this paper we consider a class of distributed parameter systems governed by the heat equation on bounded domains in \Bbb Rn. We consider two types of boundary inputs (actuators) and two types of boundary outputs (sensors). Allowing for any possible pairing of these, we consider a totality of four possible arrangements of our system. The first type of input (control) is through the Neumann boundary condition on a part of the boundary, together with a homogenous Neumann boundary condition on the remaining part of the boundary. For this type of input, the input space is infinite-dimensional. The second type of input (with a finite-dimensional input space) is obtained by imposing constant normal derivatives on each element of a finite partition of the boundary. The first type of output (observation) is given by evaluation (trace) of the state of the system on a part of the boundary, so that the output space is infinite-dimensional. For the second type of output (with a finite-dimensional output space), we again consider a partition of the boundary of the spatial domain (which can be different from the one considered for the inputs) and each output channel contains the average of the values of the state of the plant on one element of this partition. Our main result is that any possible combination of the aforementioned inputs and outputs provides a regular linear system.

We consider a grid connected synchronous generator (SG), having a constant field current and driven by a prime mover with constant torque. The grid is regarded as an “infinite bus”, i.e., a three-phase AC voltage source. The problem of interest is verifying if, for a given set of SG parameters, the angular velocity of the SG rotor converges to the angular velocity of the grid, irrespectively of the SG initial conditions. We pose this problem as a question of almost global asymptotic stability of a fourth order nonlinear system. In a recent work, we derived an integro-differential equation, which resembles the forced pendulum equation, and showed that the fourth order system is almost globally asymptotically stable if and only if the same holds for this equation. In this paper, it is shown that this equation is almost globally asymptotically stable if a real-valued nonlinear map defined on a finite interval is a contraction. For any given set of SG parameters, it is straight forward to check numerically if this map is a contraction. An example demonstrating our results is included.

We derive necessary conditions for exact controllability of infinite-dimensional systems described by x=Ax+Bu, where the generator A has a Riesz basis of eigenvectors and the input space is finite-dimensional. These conditions are in terms of the eigenvalues of A and the degree of unboundedness of B, but no further information about B is needed. Our results easily imply lack of exact controllability for a wide range of distributed parameter systems. It is known from partial differential equation (PDE) examples that the addition of damping into an exactly controllable second order system can destroy exact controllability. We discuss the effect of damping on such systems using our general conditions. We also give simple new proofs for results about the lack of exact controllability of hyperbolic systems in more than one space dimension.

We construct a simple example of a quadratic optimal control problem for an infinite-dimensional linear system based on a shift semigroup. This system has an unbounded control operator. The cost is quadratic in the input and the state, and the weighting operators are bounded. Despite its extreme simplicity, this example has all the unexpected features discovered recently by O. Staffans (and also by M. Weiss and G. Weiss). More precisely, in the formula linking the optimal feedback operator to the optimal cost operator, as well as in the Riccati equation, the weighting operator of the input has to be replaced by another operator, which can be derived from the spectral factorization of the Popov function.

We consider the control system x=Ax+Bu, where A generates a strongly continuous semigroup T on the Hilbert space X and the control operator B maps into the dual of D(A*), but it is not necessarily admissible for T. We prove that if the pair (A;B) is both forward and backward optimizable (our definition of this concept is slightly more general than the one in the literature), then the system is exactly controllable. This is a generalization of a well-known result called Russell's principle. Moreover the usual stabilization by state feedback u = Fx, where F is an admissible observation operator for the closed-loop semigroup, can be replaced with a more complicated periodic (but still linear) controller. The period τ of the controller has to be chosen large enough to satisfy an estimate. This controller can improve the exponential decay rate of the system to any desired value, including -∞ (dead-beat control). The corresponding control signal u, generated by alternately solving two exponentially stable homogeneous evolution equations on each interval of length τ, back and forth in time, will still be in L2. The better the decay rate that we want to achieve, the more iterations the controller needs to perform, but (unless we want to achieve -∞) the number of iterations needed on each period is finite.

We derive a model for a wind turbine tower in the plane of the turbine blades, consisting of a non-uniform SCOLE beam system coupled with a two-mass drive-train model (with gearbox). The control input of this wind turbine tower model is the torque created by the electrical generator. Using the theory of coupled linear systems developed by us recently, we show that this wind turbine tower model is well-posed and regular on either the energy state space Xc or the domain of its generator on Xc, denoted by X 1 c. We also show that generically, this model is exactly controllable on X 1 c in arbitrarily short time. More precisely, for every T > 0 we show that if we vary a certain parameter in the model, then exact controllability in time T holds for all except three values of the parameter.

International audience; Let $A$ be the generator of a strongly continuous semigroup ${\mathbb{T}}$ on the Hilbert space $X$, and let $C$ be a linear operator from $\Dscr(A)$ to another Hilbert space $Y$ (possibly unbounded with respect to $X$, not necessarily admissible). We consider the problem of estimating the initial state $z_0\in\Dscr(A)$ (with respect to the norm of $X$) from the output function $y(t)=C{\mathbb{T}}_t z_0$, given for all $t$ in a bounded interval $[0,\tau]$. We introduce the concepts of estimatability and backward estimatability for $(A,C)$ (in a more general way than currently available in the literature), we introduce forward and backward observers, and we provide an iterative algorithm for estimating $z_0$ from $y$. This algorithm generalizes various algorithms proposed recently for specific classes of systems and it is an attractive alternative to methods based on inverting the Gramian. Our results lead also to a very general formulation of Russell's principle, i.e., estimatability and backward estimatability imply exact observability. This general formulation of the principle does not require $\stline$ to be invertible. We illustrate our estimation algorithms on systems described by wave and Schrödinger equations, and we provide results from numerical simulations.

The plant to be stabilized is a system node Sigma with generating triple (A, B, C) and transfer function G, where A generates a contraction semigroup on the Hilbert space X. The control and observation operators B and C may be unbounded and they are not assumed to be admissible. The crucial assumption is that there exists a bounded operator E such that, if we replace G(s) by G(s) + E, the new system SigmaE becomes impedance passive. A trivial case is when G is already impedance passive and a special case is when Sigma has colocated sensors and actuators. Such systems include many wave, beam and heat equations with sensors and actuators on the boundary. It has been shown for many particular cases that the feedback u = -ky + v, where u is the input of the plant and k > 0, stabilizes Sigma, strongly or even exponentially. Here, y is the output of Sigma and v is the new input. Our main result is that if for some E isin L(U), SigmaE is impedance passive, and Sigma is approximately observable or approximately controllable in infinite time, then for sufficiently small k the closed-loop system is weakly stable. If, moreover, sigma(A)capiRopf is countable, then the closed-loop semigroup and its dual are both strongly stable. We illustrate our results with three classes of second order systems, only one of which has colocated actuators and sensors

We introduce two classes of well-posed linear time-varying (LTV) systems. Each of these can be constructed easily, starting from a time-invariant scattering-passive system, by introducing a time-dependent inner product on the state space and modifying some of the generating operators. These classes of LTV systems are motivated by physical examples, such as an electromagnetic field around a moving object. To prove the well-posedness of these LTV systems, we use the Lax-Phillips semigroup induced by a well-posed linear system, as in scattering theory. We modify this semigroup to obtain a Lax-Phillips type evolution family.

Let A 0 be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C 0 be a bounded operator from D(A1/2 0 ) (with the norm ||z||2 1/2 = 〈A 0 z, z〉) to another Hilbert space U. It is known that the system of equations z(t) + A 0 z(t) + 1/2 C∗ 0 C 0 Z(t) = C∗ 0 u(t), y(t) = − C 0 z(t) + u(t), determines a well-posed linear system Σ with input u and output y, input and output space U and state space X = D(A1/2 0 ) × H. Moreover, Σ is conservative, which means that a certain energy balance equation is satisfied both by the trajectories of Σ and by those of its dual system. In this paper we present various conditions which are equivalent to the exponential stability of such a systems. Among the equivalent conditions are exact controllability and exact observability. Denoting V(s) = (s2 I + s/2 C∗ 0 C 0 + A 0 )−1, we also obtain that the system is exponentially stable if and only if s → A1/2 0 V(s) is a bounded L(H)-valued function on the imaginary axis. This is also equivalent to the condition that s → sV(s) is a bounded L(H)-valued function on the imaginary axis (or equivalently, on the open right half-plane).

Let A0 be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C0 be a bounded operator from ${\cal D}(A_0^{1/2})$ (with the norm $\|z\|_{1/2}^2=\langle...

A new formula has been developed in this study, to calculate the fundamental frequency of a multistory building. This formula is derived from Dunkerley’s estimate, complies with the requirements of Uniform Building Code (UBC) and the Building Officials Code Administrators (BOCA) Code, and accounts for multistory building frames of different type and size. A coefficient Γ, function of moments of inertia and lengths of column/girder was introduced in the formula, in order to obtain a relative simple algebraic equation. To verify the validity of the new formula a thorough investigation has been conducted. Fundamental frequencies and periods have been calculated for different building frames using the new formula (Formula 1). A formula for buildings with rigid floors, Formula 2, has been derived from Formula 1. Formula 2 has been used to calculate the fundamental frequency for the same type and size of buildings used in the calculations with Formula 1, considered with rigid floors. In addition, three different empirical formulas have been used to calculate the fundamental period for the same buildings. Two of these formulas are recommended in different versions of the codes. For the same type and size of buildings the fundamental frequencies and periods have been determined by computer. Comparing with the computer values, formulas 1 and 2 give the best results and can be used in preliminary design, in vibration problems or to determine seismic lateral loads for buildings with a maximum height of 160/65 ft. The empirical formulas show large differences and have to be used with caution, only in the limited range imposed by UBC and BOCA.

First we describe the structure of an infinite-dimensional linear system with a scalar observation (measurement).

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.

We characterize the admissible unbounded input elements for diagonal semigroups on l 2 . The result verifies the necessity of a condition whose sufficience was previously established by L.F. Ho and D.L. Russell.

The interchange of traffic density between lanes moving in the same direction is investigated on the basis of a simple mathematical model. Emphasis is placed on the question of stability, i.e., attenuation of disturbances from an “equilibrium density distribution.” A solution is obtained for a system of differential difference equations with a time lag corresponding to the interaction of two lanes. This solution is directly applicable to other problems described by similar equations, such as the follow-the-leader problem. The solution is generalized to n lanes, and it is found that the inherent instability is twice as great for n approaching infinity as it is for two lanes.

In this paper we present a method for the inversion of specific heat data to find the vibrational frequency spectrum of a solid. Our results are given in terms of a power series in UJ, the frequency. In this series the coefficients depend on C v (T) and its d~rivatives at T=O. The series thus derived is valid only in an asymptotic sense, but represents an improvement over inversions by Montroll and Kroll which require a knowledge of functions formed from the data, and the evaluation of double integrals over these functions, A method for forming the successive values of Cv(n) (0) from experimental data is also presented. One of the most frequently measured physical characteristics of a solid is the specific heat at constant volume Cv (T), where T is the temperature. The behavior of this function has quite accurately been predicted by various characterizations of a lattice of atoms which have harmonic interactions. If it is assumed that the specific heat can be attributed solely to harmonic effects, then it can be written as an integral over the vibrational frequency spectrum 1

This paper considers an infinitely long line of traffic moving on a highway without traffic lights or other inhomogeneities. It is assumed that each car travels at a constant speed which is a random variable. A further assumption is that when one car overtakes another, passing is always possible and occurs without change of speed. it is shown that any initial headway distribution must relax to a negative exponential distribution in the limit of t t becoming infinite. The statistics of passing events are examined, and it is shown that the probability of passing (or being passed by) n n cars In time t t is described by a Poisson distribution.

We start from a special class of scattering passive linear infinite-dimensional systems introduced in Staffans and Weiss (SIAM J. Control and Opt., 2012). This class is called the Maxwell class of systems, because it includes the scattering formulation of Maxwell’s equations, as well as various wave and beam equations. We generalize this class by allowing a nonlinear damping term. While the system may have unbounded linear damping (for instance, boundary damping), the nonlinear damping term N is “bounded” in the sense that it defined on the whole state space (but no actual continuity assumption is made on N). We show that this new class of nonlinear infinite dimensional systems is well-posed and scattering passive. Our approach uses the theory of maximal monotone operators and the Crandall-Pazy theorem about nonlinear contraction semigroups, which we apply to a Lax-Phillips type nonlinear semigroup that represents the whole system, with input and output signals.