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This research proposes a procedure for identifying dynamic routing policies in stochastic transportation networks. It addresses the problem of maximizing the probability of arriving on time. Given a current location (node), the goal is to identify the next node to visit so that the probability of arriving at the destination by time t or sooner is maximized, given the probability density functions for the link travel times. The Bellman principle of optimality is applied to formulate the mathematical model of this problem. The unknown functions describing the maximum probability of arriving on time are estimated accurately for a few sample networks by using the Picard method of successive approximations. The maximum probabilities can be evaluated without enumerating the network paths. The Laplace transform and its numerical inversion are introduced to reduce the computational cost of evaluating the convolution integrals that result from the successive approximation procedure.

The Gauss principle of least constraint is derived from a new point of view. Then, an extended principle of least constraint is derived to cover the case of nonideal constraints. Finally, a version of the principle for general underdetermined systems is adumbrated. Throughout, the notion of generalized inverses of matices plays a prominent role.

A linear-quadratic optimization problem is formulated in a dynamic programming manner. An updating formula for obtaining the solutions to such a problem is provided and illustrated using a few simple examples. This updating formula is also compared to a well-known updating formula for obtaining the inverses of symmetric positive-definite matrices. Numerical results are given.

This paper shows how the output derived from the α Qβ R algorithm can be used to calculate various statistical quantities needed to evaluate linear models. In particular, we show how to calculate standard statistical quantities like the coefficient of determination R2, the F-statistics, and the t-statistics. These quantities serve as a measure of how well the model fits the data.

When a mechanical system is subjected to equality constraints, use of the chain rule of differentiation and of generalized inverses of matrices enables us to write the most general possible equation of motion, no use being made of any physical principles, Eq. (8). Then employment of standard physical principles enables us to further interpret the terms in this general equation of motion.

In this paper we obtain the explicit equations of motion for mechanical systems under nonideal constraints without the use of generalized inverses. The new set of equations is shown to be equivalent to that obtained using generalized inverses. Examples demonstrating the use of the general equations are provided.

In this paper, we provide a set of results for the sequential determination of the {1, 4}-generalized inverse of a matrix. This inverse is of importance in areas where the minimal norm solution of a system of algebraic equations is desired.

In this paper, we present the general structure for the explicit equations of motion for general mechanical systems subjected to holonomic and non-holonomic equality constraints. The constraints considered here need not satisfy D'Alembert's principle, and our derivation is not based on the principle of virtual work. Therefore, the equations obtained here have general applicability. They show that in the presence of such constraints, the constraint force acting on the system can always be viewed as made up of the sum of two components. The explicit form for each of the two components is provided. The first of these components is the constraint force that would have existed, were all the constraints ideal; the second is caused by the non-ideal nature of the constraints, and though it needs specification by the mechanician and depends on the particular situation at hand, this component nonetheless has a specific form. The paper also provides a generalized form of D'Alembert's principle which is then used to obtain the explicit equations of motion for constrained mechanical systems where the constraints may be non-ideal. We show an example where the new general, explicit equations of motion obtained in this paper are used to directly write the equations of motion for describing a non-holonomically constrained system with non-ideal constraints. Lastly, we provide a geometrical description of constrained motion and thereby exhibit the simplicity with which Nature seems to operate.

This paper presents the general form of the explicit equations of motion for mechanical systems. The systems may have holonomic and/or nonholonomic constraints, and the constraint forces may or may not satisfy D’Alembert’s principle at each instant of time. The explicit equations lead to new fundamental principles of analytical mechanics.

This paper examines multiple scattering processes from the points of view of the source function of radiative transfer and the emergence probability of photon diffusion. It extends previous results to the case of layered slabs, i.e., inhomogeneous plane-parallel media. It derives equations for basic functions of multiple scattering and photon diffusion and obtains relationships among them. The invariant imbedding method is used to physically and analytically derive initial-value problems for functions which also satisfy integral equations. The initial-value problems are useful for numerically solving direct problems, and for tackling inverse problems of remote sensing.

Since its inception about 200 years ago, Lagrangian mechanics has been based upon the Principle of D’Alembert. There are, however, many physical situations where this confining principle is not suitable, and the constraint forces do work. To date, such situations are excluded from general Lagrangian formulations. This paper releases Lagrangian mechanics from this confinement, by generalizing D’Alembert’s principle, and presents the explicit equations of motion for constrained mechanical systems in which the constraints are nonideal. These equations lead to a simple and new fundamental view of Lagrangian mechanics. They provide a geometrical understanding of constrained motion, and they highlight the simplicity with which Nature seems to operate.

Consider the family of Fredholm integral equations u(t,@c)=g(t)+@c@!^1"0k(t,y)u(y,@c)dy, where @c is sufficiently small to guarantee a solution, and the Cauchy system u"@c(t,@c)=@!^1"0K(t,y,@c)u(y,@c)dy,K"@c(t,y,@c)=@!^1"0K(t,y^',@c)K(y^',y,@c)dy^',0=

This paper deals with mechanical systems subjected to a general class of non-ideal equality con- straints. It provides the explicit equations of motion for such systems when subjected to such nonideal, holonomic and/or nonholonomic, constraints. It bases Lagrangian dynamics on a new and more general principle, of which D'Alembert's principle then becomes a special case applicable only when the constraints become ideal. By expanding its perview, it allows Lagrangian dynamics to be directly applicable to many situations of practical importance where non-ideal constraints arise, such as when there is sliding Coulomb friction. One of the central problems in the field of mechanics is the determination of the equations of motion pertinent to con- strained systems. The problem dates at least as far back as Lagrange (1787), who devised the method of Lagrange mul- tipliers specifically to handle constrained motion. Realizing that this approach is suitable to problem-specific situations, the basic problem of constrained motion has since been worked on intensively by numerous scientists, including Volterra, Boltzmann, Hamel, Novozhilov, Whittaker, and Synge, to name a few. About 100 years after Lagrange, Gibbs (1879) and Appell (1899) independently devised what is today known as the Gibbs-Appell method for obtaining the equations of motion for constrained mechanical systems with nonintegrable equal- ity constraints. The method relies on a felicitous choice of quasicoordinates and, like the Lagrange multiplier method, is amenable to problem-specific situations. The Gibbs-Appell ap- proach relies on choosing certain quasicoordinates and elimi- nating others, thereby falling under the general category of elimination methods (Udwadia and Kalaba 1996). The central idea behind these elimination methods was again first devel- oped by Lagrange when he introduced the concept of gener- alized coordinates. Yet, despite their discovery more than a century ago, the Gibbs-Appell equations were considered by many, up until very recently, to be at the pinnacle of our under- standing of constrained motion; they have been referred to by Pars (1979) in his opus on analytical dynamics as ''probably the simplest and most comprehensive equations of motion so far discovered.'' Dirac considered Hamiltonian systems with constraints that were not explicitly dependent on time; he once more attacked the problem of determining the Lagrange multipliers of the Hamiltonian corresponding to the constrained dynamical sys- tem. By ingeniously extending the concept of Poisson brack- ets, he developed a method for determining these multipliers in a systematic manner through the repeated use of the con- sistency conditions (Dirac 1964; Sudarshan and Mukunda 1974). More recently, an explicit equation describing con- strained motion of both conservative and nonconservative dy- namical systems within the confines of classical mechanics was developed by Udwadia and Kalaba (1992). They used as their starting point Gauss's principle (1829) and considered general bilateral constraints that could be both nonlinear in the generalized velocities and displacements and explicitly depen- 1

Results for the recursive determination of different types of generalized inverses of a matrix are presented for the case of the addition of a block-column matrix of arbitrary size. Using a unifying underlying theme, results for the generalized inverse, least-square generalized inverse, minimum norm generalized inverse, and Moore-Penrose inverse are included.

International audience; Gauss' principle of least constraint is solved in a sequential fashion via dynamic programming in this paper. The solution itself constitutes a new principle for constrained motion, which we may name the Bellman-Gauss principle for constrained motion.

In this paper we apply the theory of linear associative memories in producing initial parameter estimates for nonlinear iterative approaches. We also propose the use of FEED (Fast and Efficient Evaluation of Derivatives) to evaluate partial derivatives of functions encountered in nonlinear estimation. Suggested methods are presented in the context of calibrating spatial interaction models and are illustrated through numerical examples.

Time-dependent radiation and energy transport problems are important in atmospheric science, medicine, biochemistry, and other areas. To determine external energy fields, direct problems (in which parameters are known) can be solved computationally by numerical integration followed by the numerical inversion of Laplace transforms. On the other hand, this paper treats inverse problems of estimating transport parameters on the basis of external observations of radiant intensity. These problems are approached using associative memory neural networks whose associated least squares problem is solved using a new dynamic programming algorithm. The quality of the estimates in the presence of noise in measurements is studied.

A simple proof of the Greville formula for the recursive computation of the Moore–Penrose (MP) inverse of a matrix is presented. The proof utilizes no more than the elementary properties of the MP inverse.

Starting from the principle of virtual work, this paper states and establishes an extended version of D’Alembert’s Principle. Using this extended principle and elementary linear algebra, it develops, from first principles, the explicit equation of motion for constrained mechanical systems. The results are compared with the authors’ previous results. The approach points to new ways of extending these results.

This work presents a study of jumping through a biomechanical model of a leg, which is subjected to an inequality constraint. The activation and deactivation of an equality constraint reproduce the inequality constraint. Within the framework of the generalized inverse equations of motion, it is shown that the activation of this additional constraint can be implemented by Greville's formulae.

This paper deals with the description of constrained motion within the context of classical dynamics. An alternative, and simpler, proof for the recently developed new equation of motion for constrained systems is presented. The interpretation of this equation leads to new principles of analytical dynamics. We show how these results relate to Lagrange's formulation of constrained motion. New results related to the existence, uniqueness, and explicit determination of the Lagrange multipliers are provided. The approach developed herein is compared with those of Gibbs and Appell, and that of Dirac. Three examples of the application of the new equation are provided to illustrate their use.

Parameter estimation problems for nonlinear dynamical systems are typically formulated as nonlinear optimization problems. For such problems, one has the usual difficulty that standard successive approximation schemes generally require good initial parameter estimates in order to converge to the truth. The linear associative memory method has demonstrated its effectiveness in obtaining useful initial parameter estimates for simple nonlinear dynamical systems. No work, however, has yet been done to apply this method to a chaotic system. This paper initiates such a study using the logistic map, which is capable of generating mathematical chaos. Supervised training was conducted between system parameters and system outputs to construct optimal memory matrices. Untrained system outputs were then used together with the memory matrices to estimate system parameters. Very accurate parameter estimates were obtained for noise-free system outputs. Good parameter estimates were obtained for system outputs corrupted by noise. A “rule of thumb” is suggested that can be used to aid in a successful search for true parameter values if the initial training range is not located “near” them.

In decision theory, incommensurabilities among conflicting decision criteria are typically handled by multicriteria optimization methods such as Pareto efficiency and mean-variance analysis. In econometrics and statistics, where conflicting model criteria replace conflicting decision criteria. probability assessments are routinely used to transform disparate model discrepancy terms into apparently commensurable quantities. This tactic has both strengths and weaknesses. On the plus side, it permits the construction of a single real-valued measure of theory and data incompatibility in the form of a likelihood function or a posterior probability distribution. On the minus side, the amalgamation of conceptually distinct model discrepancy terms into a single real-valued incompatibility measure can make it difficult to untangle the true source of any diagnosed model specification problem. This paper discusses recent theoretical and empirical work on a multicriteria ‘flexible least squares” (FLS) approach to model specification and estimation. The basic FLS objective is to determine the “cost-efficient frontier’, that is, the set of estimates that are minimally incompatible with a specified set of model criteria. The relation of this work to previous work in econometrics, statistics, and system science is also clarified.

(1995). On The Generalized Inverse Form of the Equations of Constrained Motion. The American Mathematical Monthly: Vol. 102, No. 9, pp. 821-825.

A general procedure for obtaining the exact control forces to be exerted on a mechanical system to maintain constraints is introduced in this paper. It is illustrated through a simple example. The discussion extends to Nature's choice of control forces and its relation to the general control forces.

The newly developed equation of motion for constrained mechanical systems are derived using D'Alembert's principle.

If the matrixA is not of full rank, there may be many solutions to the problem of minimizing ‖Ax−b‖ overx. Among such vectorsx, the unique one for which ‖x‖ is minimum is of importance in applications. This vector may be represented asx=A+b. In this paper, the functional equation technique of dynamic programming is used to find the shortest solution to the least-squares problem in a sequential fashion. The algorithm is illustrated with an example.

A current assessment of the generalized inverse theory of constrained motion of discrete mechanical systems is presented, and applications are described.

This paper discusses new analytical and computational aspects of detecting and tracking dim objects (targets). It describes a filter that processes an image from a detector array and updates a function that measures the degree of belief that there is a target present in the individual detectors of the array. The nonlinear filter is derived using an evidential approach based on dynamic programming and fuzzy sets. In numerical experiments with computer simulation, targets whose signals are only one-half the noise level (−3 db) are detected and tracked. The filter works with multiple targets in two dimensions and can be extended in complexity of target motion, noise, clutter, and geometry. This fuzzy filter can also be viewed as a neural network. The computing load of this filter is very low, depending only on the number of detectors, not the number of tracks. This is important when there are many targets. The estimated degree of belief functions can be utilized by fuzzy controllers.

Die Arbeit bringt eine geometrische Interpretation der expliziten Bewegungsgleichungen fur mechanische Systeme mil Zwangsbedingungen. Dies fuhrt zu einem geometrischen Prinzip der analytischen Mechanik. This paper provides a geometrical interpretation of the explicit equations of motion for constrained mechanical systems. This leads to a geometric principle of analytical mechanics.

Associative memory techniques are drawn from the artificial intelligence literature, and have demonstrated considerable utility for parameter identification in dynamical systems. Previous turning point forecasts constructed by LeSage are compared to forecasts generated by associative memories and simple autoregressive models. Both the associative memories and the autoregressions perform as well or better than the more complicated econometric procedures described by LeSage, with the exception of West and Harrison's (1989) dynamic linear model specification. Extensions are suggested.

In this paper we formulate Lagrangian mechanics as a constrained quadratic minimization problem. This quadratic minimization problem is then solved using the theory of generalized inverses of matrices thereby obtaining the explicit equations of motion of constrained, discrete mechanical systems. The approach extends the boundaries of Lagrangian mechanics in that we provide a general formulation for describing the constrained motion of such systems without either the use of Lagrange multipliers or the use of quasi-coordinates. An important feature of the approach is that we do not require prior knowledge of the specific set of constraints to accomplish this formulation. This makes the equations presented here more generally useful, and perhaps more aesthetic, than the Gibbs-Appell equations which require a felicitous choice of problem-specific quasi-coordinates. The new equations of motion presented here are applicable to both the holonomic and nonholonomic constraints that Lagrangian mechanics deals with. They are obtained in terms of the usual generalized coordinates used to describe the constrained system. Furthermore, they can be integrated by any of the currently available numerical integration methods, thus yielding analytical and/or computational descriptions of the motions of constrained mechanical systems.

In this paper we develop an analytical set of equations to describe the motion of discrete dynamical systems subjected to holonomic and/or nonholonomic Pfaffian equality constraints. These equations are obtained by using Gauss’s Principle to recast the problem of the constrained motion of dynamical systems in the form of a quadratic programming problem. The closed-form solution to this programming problem then explicitly yields the equations that describe the time evolution of constrained linear and nonlinear mechanical systems. The direct approach used here does not require the use of any Lagrange multipliers, and the resulting equations are expressed in terms of two different classes of generalized inverses—the first class pertinent to the constraints, the second to the dynamics of the motion. These equations can be numerically solved using any of the standard numerical techniques for solving differential equations. A closed-form analytical expression for the constraint forces required for a given mechanical system to satisfy a specific set of nonholonomic constraints is also provided. An example dealing with the position tracking control of a nonlinear system shows the power of the analytical results and provides new insights into application areas such as robotics, and the control of structural and mechanical systems.

The problem of passive ranging is complex, yet important. This paper formulates it as a nonlinear least squares problem which is solved via the Newton-Raphson technique. We use the FEED method for rapid prototyping and the automatic evaluation of partial derivatives. The paper presents two significant results. 1. The approach leads to rapidly convergent and accurate estimates of position for a variety of different noise models. 2. The use of FEED has led to a new and exact solution to the question of evaluating the effect of noise on parameter estimates without the need to perform Monte Carlo computational experiments. Nonlinear methods such as this require preliminary parameter estimates, for which we suggest associative memory neural networks.

This paper presents a new method for identification of parameters in nonlinear structural and mechanical systems in which the initial guesses of the unknown parameter vectors may be far from their true values. The method uses notions from the field of artificial neural nets and, using an initial set of training parameter vectors, generates in an adaptive fashion other relevant training vectors to yield identification of the parameter vector in a recursive fashion. The simplicity and power of the technique are illustrated by considering three highly nonlinear systems. It is shown that the technique presented here yields excellent estimates with only a limited amount of response data, even when each element of the set comprising the initial training parameter vectors is far from its true value—in fact, sufficiently far that the usual recursive identification schemes fail to converge.

The explicit general equations of motion for constrained discrete dynamical systems are obtained. These new equations lead to a simple and new fundamental view of lagrangian mechanics.

This paper proposes the use of associative memories for obtaining preliminary parameter estimates for nonlinear systems. For each parameter vector r, in a selected training set, the system equations are used to determine a vector s, of system outputs, An associative memory matrix M is then constructed which optimally, in the least squares sense, associates each system output vector s, with its corresponding parameter vector I-,. Given any observed system output vector s*, an estimate i for the system parameters is obtained by setting i = Ms*. Numerical experiments are reported which indicate the effectiveness of this approach, especially for the nonlinear associative memory case in which the training vectors s, include not only the system output levels but also products of these levels. Training with noisy output vectors is shown to improve the accuracy of the parameter estimates when the

Many optimization procedures presume the availability of an initial approximation in the neighborhood of a local or global optimum. Unfortunately, finding a set of good starting conditions is itself a nontrivial proposition. We describe a procedure for identifying approximate solutions to constrained optimization problems. Recurrent neural network structures are interpreted in the context of linear associative memory matrices. A recurrent associative memory (RAM) is trained to map the inputs of closely related transportation linear programs to optimal solution vectors. The procedure performs well when training cases are selected according to a simple rule, identifying good heuristic solutions for representative test cases. Modest infeasibilities exist in some of these estimated solutions, but the basic variables associated with true optimums are usually apparent. In the great majority of cases, rounding identifies the true optimum.