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The present paper is concerned with Almansi's problem in the linear theory of micropolar elasticity. The problem is solved using the results from [8], [10–13].

Bilinear operators on Hilbert resolution spaces provide the setting for the analysis of linear systems with multiplicative control. Recent results on the solution of certain strictly causal linear operator equations are brought to bear in the bilinear setting. In the process a close similarity to resolvant properties of convention spectral theory is utilized. The results are interpreted in a systems context through a series of examples

An equivalence relation, rim–similarity, is defined on the set of all sequences of square matrices of a given fixed dimension. For linear discrete – time systems, theorems are presented which show that certain stability properties are invariant under n∞ similarity. General linear systems as well as those of variational type are considered. Also, sufficient conditions are given which guarantee that the uniform stability of the null solution of a non– linear system carries over to the linear variational equation with respect to this solution. All of these results are analogous to known ones for ordinary differential equations.

A brief review of the state-of-the-art of urban hydrology results from a drive for a reliable and uncomplicated method for practical application for determining urban runoff. The Linearized Subhydrographs Method is believed to be a useful tool for the design of a new system and the analysis of an existing system. This method employs the principle of mass conservation and very simple parameters and functional relationships; basically, runoff coefficient is used to compute the peak rates of runoff. Along with functional development, test applications are also examined here. In two typical storms where this method was applied, time synchronization and rate of runoff predicted are in good agreement with the recorded hydrographs.

This paper is a study of the problem of how to parametrize the set of all finite order constant linear systems. The parameters are interpreted as independent invariants for the equivalence relation which defines two systems to be equivalent when they have the same impulse response. Two kinds of canonical representations of the systems are constructed from the invariants, one of the state-space equations type and the other of the transfer function type.

Using the alternative projections theorem of J. von Neumann, formulas for determining the bilateral linear predictor of a minimal weakly stationary stochastic process are obtained. To derive these formulas it is assumed that the spectral density of the process is bounded. The known results on determination of the ordinary linear predictor of a weakly stationary stochastic process are basic to the derivation of this work. Under the extra assumption that the past and the future of the process are at positive angle, a rate of convergence for the bilinear predictor is established.

The inverse problem of determining damping from frequency response is formulated as an optimization problem. Response data are sequentially processed by an algorithm which refines the estimate of damping at each frequency point. The method is illustrated by numerical examples involving viscous damping. Other forms of damping can be considered without altering the procedure discussed here.

This paper reports a reasonably complete theory of necessary and sufficient conditions for a control to be superior with respect to a non-scalar-valued performance criterion. The latter maps into a finite-dimensional integrally closed directed partially ordered linear space. The applicability of the theory to the analysis of dynamic vector estimation problems and to a class of uncertain optimal control problems is demonstrated.

This note gives new necessary and sufficient conditions for a linear system to be passive or lossless. It will be shown that, by using the theory of operators on an indefinite J(t)-Hilbert space, both the passivity and isometric conditions for an operator to be lossless can be combined into a single condition—namely an operator is lossless if and only if it is J(t)-contractive.

The concept of a regular splitting of a real nonsingular matrix, introduced by R. Varga, is used in iterative methods for solving linear systems. The purpose of this paper is to extend this concept to rectangular linear systems \[ ( * )\qquad Ax = b, \] in two directions. Firstly, this is accomplished by replacing $A^{ - 1} $ by $A^\dag $, the Moore–Penrose inverse of A, and, secondly, by considering matrices that leave a cone invariant.Let $A \in R^{m \times n} $. The splitting $A = M - N$ is called proper if $R(A) = R(M)$ and $N(A) = N(M)$. Such is the case when A and M are nonsingular. Consider the iteration \[ ( * * )\qquad x^{i + 1} = M^ \dag Nx^i + M^ \dag b. \] It is shown that for a proper splitting, $\rho (M^ \dag N)$ (the spectral radius of $M^ \dag N$) is less than l, if and only if the iteration (**) converges to $A^ \dag b$, the best least squares approximate solution of the system (*). This approach has the advantage of avoiding the normal system $A^T Ax = A^T b$ in solving (*). Necessary an...

Plasma physicists are becoming increasingly aware of the importance of obtaining an understanding of basic plasma phenomena which cannot be explained in terms of any linear theory of small-amplitude disturbances, and which, collectively, are described as turbulent or large-amplitude phenomena. The first theoretical attempts at describing finite-amplitude effects were made over a decade ago, in the various formulations with the generic title 'quasi-linear theory'.

A general solution is derived for the linearized shallow water equations describing flow in a wide channel with constant bed slope, with a known time- and space-dependent lateral inflow, and for a known upstream hydrograph. The method used is that of the Green’s function; it has the advantage of facilitating computations and of providing a unified treatment for both the upstream inflow and the lateral inflow problem. The method can also be used to solve the linear diffusion approximation of the system.

The externally excited adaptive system (EEAS) is a two-degree-of-freedom feedback system with a nonlinearity which is saturated hard by an external periodic signal. Under certain conditions, the EEAS responds quasilinearly to command and plant disturbance signals, permitting the development of a quantitative synthesis theory for satisfying system tolerances despite large plant uncertainty. The great advantage of the EEAS is its zero sensitivity to plant gain variations, a property it shares with the self-oscillating adaptive system (SOAS). The EEAS is, however, more flexible than the SOAS in satisfying the quasilinearity constraints. The essential difference is that in the EEAS the loop transmission bandwidth is not rigorously tied to the "carrier" signal, as it is in the SOAS. There is a class of problems for which the EEAS is superior to the purely linear system, which in turn is superior to the SOAS. The superiority of the EEAS over the SOAS is especially marked in the case of significant plant disturbances, which generally vitiate a SOAS design.

Application of Laplace transforms in linear systems analysis has many well established merits. Analogously, multidimensional Laplace transforms can be employed in the analysis of non- linear analytic systems to provide similar benefits, including explicit input-output relationships. At present, one of the most serious drawbacks to the use of multidimensional transforms lies with the difficulty in reducing multidimensional kernels during inverse transformation. Herein, simplified reduction and expansion techniques for multidimensional inverse trans- formation will be presented. The labor involved in their application is comparable to that of familiar inverse transformation through use of Heaviside's expansions. Examples will help to clarify the techniques.

The theoretical development of a generalized model to predict the crimp radius of a multi-component fiber has been reported in Part I [1]. The model is based on the linear theory of elasticity and attempts to provide a first-order estimate for multi-component fibers without any restrictions on the number of components or their geometrical arrangement in the cross section. This paper illustrates the application of the formulation in [1] to a bicomponent structure. Within the limitations of the basic assumptions, the influences of dimensionless parameters (geometrical and material) on the crimp radius are investigated in some detail.

An entropy-maximizing model of a generalized pedestrian flow system is developed. This model is based on a fundamental conception of pedestrian traffic as being contained within a linear network. The model then maximizes the entropy of pedestrian flow within any given network, subject to a basic set of Kirchoff conditions. The resulting final equations are a special case of the gravity model. A general solution method is discussed and several numerical experiments with the model are described.

We consider the low Rossby number (R) flow of a stratified fluid in a long rotating channel, for which the bottom elevation varies in the downstream direction. The quasi-geostrophic response is shown to be singular at the side walls of the channel, and thus an ageostrophic analysis is necessary even for vanishingly small R. Part of the ageostrophic steady-state response is a modified quasi-geostrophic perturbation trapped near the bottom. A second component which is present even as R approaches zero is an internal Kelvin wave whose vertical wavelength adjusts so that the wave remains stationary with respect to the channel bottom and which propagates energy and momentum to infinite heights in an unbounded channel. The case of a bounded layer of fluid is also considered, and the resonance conditions are given. We also calculate the flow field when the bottom elevation varies in the cross-stream direction. We conclude that stagnation or flow reversal can be caused either by the modified quasi-geostrophic component or by the Kelvin wave and estimate the critical condition by an extrapolation of the perturbation velocity computed from linear theory.

Wave propagation in arbitrary linear systems with parameters varying in time and space according to a traveling wave law is considered by using the Lagrangean description. It is shown that the solution of this problem can be reduced to the solution of the stationary problem for a fixed inhomogeneous medium. Consequently, the reflection and transmission coefficients for a moving inhomogeneity can be expressed in terms of the analogous coefficients for the auxiliary fixed layer. Relationships connecting the energy and frequency of the interacting wave packets are obtained from which the equality of the total quantum flux through the surface enclosing the moving inhomogeneous domain to zero follows. Some particular cases are examined.

Observer theory for constant-coefficient, linear systems is extended to systems with unknown, inaccessible inputs. Two types of observers for such systems are defined, and simplified criteria for their existence are developed. A theorem regarding the structure of these observers is given which allows the application of results for systems with single unknown inputs to systems with multiple unknown inputs.

The modified Bryson-Frazier fixed interval smoothing algorithm [6], is an addendem to the Kalman filter. This algorithm when applied to the problem of fixed-lag smoothing is computationally more efficient than the algorithms recently reported in refs. [1-3]. Features of the algorithm are ease of implentation, computational efficiency, reduced storage requirements, and stability.

k-trees are a special class of perfect elimination graphs which arise in the study of sparse linear systems. We present four simple characterizations of k-trees involving cliques, paths, and separators.

Assuming lithospheric plates to be rigid, we systematically invert 68 spreading rates, 62 fracture zones trends and 10^6 earthquake slip vectors simultaneously to obtain a self-consistent model of instantaneous relative motions for eleven major plates. The inverse problem is linearized and solved iteratively by a maximum likelihood procedure. Because the uncertainties in the data are small, Gaussian statistics are shown to be adequate. The use of a linear theory permits (1) the calculation of the uncertainties in the various angular velocity vectors caused by uncertainties in the data, and (2) quantitative examination of the distribution of information within the data set. The existence of a self-consistent model satisfying all the data is strong justification of the rigid plate assumption. Slow movement between North and South America is shown to be resolvable. We then invert the trends of 20 linear island chains and aseismic ridges under the assumptions that they represent the directions of plate motions over a set of hot spots fixed with respect to each other. We conclude that these hot spots have had no significant relative motions in the last 10 My.

The discrete elements of finite dimensions which replace the structural continuum in the finite element method can always be chosen sufficiently small that the linear relations between element deformations and element stresses remain valid to the same degree of approximation as is considered acceptable in the linear theory of elasticity. This observation formed the basis for the treatment of geometrical nonlinearities by Argyris and his co-workers in their natural mode technique [1]and [2]. Here we give an alternative development of the theory. The element deformations, linearly related to nodal displacements and rotations in a local coordinate system, are expressed as analytic functions of the nodal coordinates in the global system. Then, for structures with an initially linear behaviour, the stability and postbuckling analysis is developed on the basis of the general theory founded by Koiter [3]. The theory is illustrated by the example of frame-structures. The location of the nodal points is defined in terms of the displacement vector, while the orientation of an orthogonal triad attached to each nodal point is described by means of modified angular coordinates of Euler. The accuracy of the analysis is demonstrated for a problem solved analytically by Koiter [5]and verified experimentally by Roorda [4].

The paper develops recursive techniques for off-line identification of linear and nonlinear systems. It is shown that if the system is linear and time invariant, impulse response characterization of the system coupled with an orthogonal series approximation can be utilized for the purpose stated above. The techniques of adaptive Kalman filtering are shown to be applicable, which besides permitting recursive evaluation of the coefficients, lead to a number of important advantages. In the second part of the study, the proposed method is extended for recursive identification of a class of non-linear systems which can be represented as a cascade combination of a linear dynamical system and a non-linear zero memory system. The method of Volterra series representation of such systems is utilized. Results are illustrated through numerical examples in each case.

In controller design problems, the quantitative description of the desired controlled system behavior is often given by specifying the closed loop poles. An output-variable, feedback control policy for a linear system of order n, with m outputs and p inputs, is described by u = −Gy. Matrix G contains p × m parameters which may be used to dictate the locations of the closed loop poles. The limiting condition for arbitrary pole placement, without some form of state estimation, is p × m = n since the p × m independent controller parameters may be used to specify the n poles. However, there is no systematic procedure for the design of controllers subject to limiting condition. In this paper, a technique based on modal system description is used to design static output-variable feedback controllers with limited number of inputs and outputs. It is shown that arbitrary closed loop pole placement is possible if p + m ⩾ n + 1 for sufficiently well coupled systems.

An M/G/1 service system with finite waiting room is studied. A customer is served by one server in phases, during some of which a place in the waiting room is occupied. The busy period length distribution is obtained from a system of integral equations leading to a linear system in Laplace-Stieltjes transforms. An asymptotic expression, for large intensity of arrival, for the expectation of this length is given. An efficiency measure giving the long run customer loss ratio is obtained. The model is shown to apply to an inventory and a container traffic problem.

We study the state of stress in an elastic half-plane in the presence of phase transformations caused by temperature variations at the points of the half-plane. Separately we consider the states of stress caused by the lack of homogeneity in the temperature field and the consequent volume changes taking place in the regions of phase transformations. Under the term “phase transformation” we understand the structural change in the crystal lattice which occurs when the body is heated above a certain critical temperature [1, 2], Here the purely thermal stresses are accompanied by the stresses associated with the volume change in the region undergoing phase transformations. Similar problems arise during the investigation of the stress states in the case of elastic tension and in the problems on inclusions. Such problems were studied by D.I.Sherman, Iu.A. Amen-Zade, and others. However in all the problems studied the region occupied by an inclusion was always completely contained within some external region. The present paper deals with the case in which the boundary separating the media has common points with the outer boundary of the region containing the inclusion. The stresses and strains are assumed to satisfy the conditions of the linear theory of elasticity, with the external region and the inclusion possessing the same elastic properties.

A class of singular control problems involving amplitude constraints on the controls is examined. IfL ∞ is the space of control functionsU, the control constraint setS can be identified with the unit ball inL ∞. Now, for anyn ∈ (1, ∞), an analogous problem may be set up withL n forU and the unit ball inL n forS. This modified problem is necessarily nonsingular for controllable systems. It is shown that, by takingn sufficiently large, the solution to the modified problem also solves the original problem arbitrarily closely (in a sense made precise). Behavior asn → ∞ is investigated.

Conditions are derived under which a control system described by a linear stochastic differential equation with multiplicative noise possesses a stationary probability distribution with finite pth moment. Both the Ito and Stratonovich interpretations are considered.

The dynamic analysis of complex and critical structures, such as nuclear power plants, must employ not only both theoretical and experimental analysis, but also combine these two approaches to provide a unified description of the structure. A theoretical a priori model is useful in planning a vibration test, while the data obtained from such testing can be used to modify the a priori model so that it reproduces the experimental data. This paper illustrates several mathematical methods useful in planning to placement of excitors and accelerometers; in analyzing data to identify resonant frequencies, modal damping and mode shapes; in modifying the mass and stiffness properties of the a priori model (consequently indicating improved methods of modeling in future studies); and in identifying unknown forces acting on the structure, once a good model has been obtained. Central to these methods is the use of a generalization of the inverse of a matrix — the pseudo-inverse and the availability of efficient computer software for its computation. The properties of the pseudo-inverse allow it to calculate weighted least-mean-square and weighted minimum-norm solutions to ill-conditioned linear systems. It is consequently a versatile tool in the analysis of vibrating systems where the number of measurement and forcing points are often different and the confidence in different measurements and models varies.

Since numerical calculations on a digital computer are performed on operands with a limited number of significant digits it follows that each operator in the computational arithmetic is merely an approximation of the corresponding mathematical operator. Therefore every numerical operation carried out on a computer generates a numerical error. The statistical evaluation of these errors is discussed in the first part of the paper. In the second part, the formulae obtained above are used to assess the validity of numerical results obtained in resolution of linear systems, algebraic equations and in matrix inversion.

The stability question is settled in a nonlinear formulation. The cases of simple and nonsimple elementary divisors of the characteristic matrix of a linear system have been examined. The real normal form of the Hamiltonian of the linear problem and the corresponding normalizing transformation have been found for the second case. In the first case Liapunov instability and stability, while in the second case instability and formal stability, have been proved as a function of the coefficients of the Hamilton, function.

Explicit expressions are derived for evaluation of the general time-weighted quadratic performance index associated "with a discrete-time constant linear system. The method used relies on formulating the problem in terms of Kronecker matrix products, which enable a formula for the sum of a scalar series to be applied. Similarities with the corresponding problem for continuous-time systems are demonstrated.

Let a control system be described by a continuous linear map ℒ* from the input spaceU* (some dual Banach space) into the output spaceX* (some finite-dimensional normed space). Within the class of control problems where the constraints and cost are expressed in terms of the norms on the input and output spaces, the following two have had extensive coverage: (i)minimum effort problem: find, from amongst all inputs which have corresponding outputs lying in some closed sphere inX* centered on some desired outputx d *, an output of minimum norm; and (ii)minimum deviation problem: find, from amongst all inputs lying in some closed sphere inU*, an input having corresponding output at a minimum distance fromx d *. However, thecomposite cost problem, where we seek to minimizeF(∥u*∥, ∥x d * −x*∥) over elements satisfyingx* = ℒ*u* (F a certain kind of convex functional), has not received the same attention. This paper presents results for the composite cost problem paralleling known results for the minimum effort and deviation problems. It is hoped that a gap in the literature is thereby filled. We show that (a) a solution exists, (b) the solution can be characterized in terms of some closed hyperplaneH inX, and (c)H can be computed as being an element on which some concave functional over closed hyperplanes inX achieves its maximum. The treatment allows of infinite-dimensional output spaces. We make extensive use of recently developed duality theory.

A model of a distributed-boundary control system is considered. Assume the uncontrolled system possesses an exponential asymptotically stable zero solution. We then construct suboptimal feedback controls for the distributed and boundary control problems via the direct method of Liapunov. Furthermore, existence-uniqueness of the synthesized control systems is proven by applying the theory of nonlinear semigroups and maximal dissipative sets. Applications to diffusion equations are given.

A linear theory is developed for the free vibrations of a uniform suspended cable in which the ratio of sag to span is about 1:8, or less. Both in-plane and out-of-plane motion is considered. It is shown that the analysis of the symmetric in-plane modes is heavily dependent on a parameter which allows for the effects of cable geometry and elasticity. The results of simple experiments are reported which establish the validity of the theory.

The problem of optimality of the trajectories of a linear system with a closed-loop time-optimal control is studied and solved in dimension 2.