Showing papers on "Asymptotic analysis published in 1994"
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TL;DR: In this paper, a singular, abnormal minimizer for the Lagrange problem with linear velocity constraints and quadratic definite Lagrangian was constructed, which is a counterexample to a theorem that has appeared several times in the differential geometry literature.
Abstract: This paper constructs the first example of a singular, abnormal minimizer for the Lagrange problem with linear velocity constraints and quadratic definite Lagrangian, or, equivalently, for an optimal control system of linear controls, with $k$ controls, $n$ states, and a running cost function that is quadratic positive-definite in the controls. In the example, $k=2, n=3$, and the system is completely controllable. The example is stable: if both the control law and cost are perturbed, the singular minimizer persists. Its importance is due, in part, to the fact that it is a counterexample to a theorem that has appeared several times in the differential geometry literature. There, the problem is called the problem of finding minimizing sub-Riemannian geodesics, and it has been claimed that all minimizers are normal Pontryagin extremals [The Mathematical Theory of Optimal Processes, Wiley-Interscience, New York, 1962]. (If the number of states equals the number of controls, then the problem is that of finding Riemannian geodesics.) The main difficulty is proving minimality. To do this, the length (cost) of the abnormal is compared with all competing normal extremals. A detailed asymptotic analysis of the differential equations governing the normals shows that they are all longer.
194 citations
Journal Article•
171 citations
TL;DR: The primal trajectory has an asymptotic ray and the dual trajectory converges to an interior dual feasible solution, which converges exponentially fast to a particular dual optimal solution.
Abstract: We consider the linear program min{cźx: Axźb} and the associated exponential penalty functionfr(x) = cźx + rΣexp[(Aix ź bi)/r]. Forr close to 0, the unconstrained minimizerx(r) offr admits an asymptotic expansion of the formx(r) = x* + rd* + ź(r) wherex* is a particular optimal solution of the linear program and the error termź(r) has an exponentially fast decay. Using duality theory we exhibit an associated dual trajectoryź(r) which converges exponentially fast to a particular dual optimal solution. These results are completed by an asymptotic analysis whenr tends to ź: the primal trajectory has an asymptotic ray and the dual trajectory converges to an interior dual feasible solution.
140 citations
TL;DR: An analysis for the class of so-called subspace fitting algorithms shows that an overall optimal weighting exists for a particular array and noise covariance error model and concludes that no other method can yield more accurate estimates for large samples and small model errors.
Abstract: The principal sources of estimation error in sensor array signal processing applications are the finite sample effects of additive noise and imprecise models for the antenna array and spatial noise statistics While the effects of these errors have been studied individually, their combined effect has not yet been rigorously analyzed The authors undertake such an analysis for the class of so-called subspace fitting algorithms In addition to deriving first-order asymptotic expressions for the estimation error, they show that an overall optimal weighting exists for a particular array and noise covariance error model In a companion paper, the optimally weighted subspace fitting method is shown to be asymptotically equivalent with the more complicated maximum a posteriori estimator Thus, for the model in question, no other method can yield more accurate estimates for large samples and small model errors Numerical examples and computer simulations are included to illustrate the obtained results and to verify the asymptotic analysis for realistic scenarios >
89 citations
TL;DR: The discrete coagulation-fragmentation equations are a model for the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones as mentioned in this paper.
Abstract: The discrete coagulation-fragmentation equations are a model for the kinetics of cluster growth in which clusters can coagulate via binary interactions to form larger clusters or fragment to form smaller ones. The assumptions made on the fragmentation coefficients have the physical interpretation that surface effects are important. Our results on the asymptotic behavior of solutions generalize the corresponding results of Ball, Carr, and Penrose for the Becker-Doring equation.
82 citations
TL;DR: Asymptotic behavior for the vorticity equations in dimensions two and three has been studied in this paper, where the authors consider the case where the equations are expressed as partial differential equations.
Abstract: (1994). Asymptotic behavior for the vorticity equations in dimensions two and three. Communications in Partial Differential Equations: Vol. 19, No. 5-6, pp. 827-872.
80 citations
TL;DR: In this paper, the authors developed semiclassical treatment of the quantum baker's map paying special attention to the discrete and finite nature of its Hilbert space, and the asymptotic analysis of the true quantum objects themselves.
79 citations
TL;DR: In this paper, the authors show how to obtain the full asymptotic expansion for solutions of integrable wave equations to all orders, as t→∞, without relying on an a priori ansatz for the form of the solution.
Abstract: The authors show how to obtain the full asymptotic expansion for solutions of integrable wave equations to all orders, as t→∞. The method is rigorous and systematic and does not rely on an a priori ansatz for the form of the solution.
76 citations
TL;DR: In this article, the self-focusing of gravity-capillary surface waves modeled by the Davey-Stewartson equations is analyzed and a sharp upper bound for the initial amplitude of the wave that prevents singularity formation is derived based on dynamic rescaling and asymptotic analysis.
71 citations
TL;DR: In this article, a study on the notion of asymptotic convergence for the Bubnov-Galerkin method in context of both finite and boundary element approximations is presented.
Abstract: The following notes are devoted to a study on the notion of the asymptotic convergence for the Bubnov-Galerkin Method in context of both finite and boundary element approximations.
64 citations
TL;DR: In this article, the numerical behavior of elliptic problems with a small parameter is studied and an example concerning the computation of elastic arches is analyzed using this mathematical framework, and conditions ensuring that a numerical scheme will converge uniformly with respect to the small parameter are given.
Abstract: In this paper we study the numerical behaviour of elliptic problems in which a small parameter is involved and an example concerning the computation of elastic arches is analyzed using this mathematical framework. At first, the statements of the problem and its Galerkin approximations are defined and an asymptotic analysis is performed. Then we give general conditions ensuring that a numerical scheme will converge uniformly with respect to the small parameter. Finally we study an example in computation of arches working in linear elasticity conditions. We build one finite element scheme giving a locking behaviour, and another one which does not.
TL;DR: In this article, an asymptotic analysis of on-axis intensity fluctuation variance of a finite-sized light beam is performed starting from the functional integral representation for the field in a random medium.
Abstract: Asymptotic analysis of on-axis intensity fluctuation variance of a finite-sized light beam is performed starting from the functional integral representation for the field in a random medium. Collimated, diverging and converging beams with different diffraction properties are considered. A complete set of asymptotes is obtained using the main and additional coherence channels approach to the fourth moment of field analyses. We study the conditions of weak and strong fluctuation regimes for all these types of beams and obtain relatively simple formulae for on-axis intensity variance, which are valid for the case of stratified turbulence. It is shown that scintillations in the vicinity of the beam focus are governed by double-scattering processes for weak scintillations and double scattering from coherence channels for strong fluctuations.
TL;DR: It is proved that the variance of the internal path length in a symmetric digital search tree under the Bernoulli model is asymptotically equal to N.26600 + N\cdot\delta(\log_2 N), where $N$ is the number of stored records and $\delta(x)$ is a periodic function of mean zero and a very small amplitude.
Abstract: This paper studies the asymptotics of the variance for the internal path length in a symmetric digital search tree under the Bernoulli model. This problem has been open until now. It is proved that the variance is asymptotically equal to $N\cdot0.26600 +N\cdot\delta(\log_2 N),$ where $N$ is the number of stored records and $\delta(x)$ is a periodic function of mean zero and a very small amplitude. This result completes a series of studies devoted to the asymptotic analysis of the variances of digital tree parameters in the symmetric case. In order to prove the previous result a number of nontrivial problems concerning analytic continuations and some others of a numerical nature had to be solved. In fact, some of these techniques are motivated by the methodology introduced in an influential paper by Flajolet and Sedgewick.
TL;DR: In this article, the problem of the rigorous justification of the global asymptotic results obtained via the Isomonodromy Method is considered, taking the pure imaginary solutions of the second Painleve equation as an example.
Abstract: During the last ten years considerable progress in the theory of classical Painleve equations has been achieved. This progress is based on the so-called Isomonodromy Method which makes the Painleve transcendents as effective in modern nonlinear analysis as the usual special functions are in linear analysis. In this paper, the problem of the rigorous justification of the global asymptotic results which are obtained via Isomonodromy Method, is considered. Taking the pure imaginary solutions of the second Painleve equation as an example, we discuss in detail three different rigorous methodologies of their asymptotic analysis including derivation of the corresponding connection formulae.
TL;DR: In this paper, a new representation for the gamma function is proposed, which can be used to derive a number of properties of the asymptotic expansion of the Gamma function, including explicit and realistic error bounds.
Abstract: Recent work of Berry & Howls, which reformulated the method of steepest descents, is exploited to derive a new representation for the gamma function. It is shown how this representation can be used to derive a number of properties of the asymptotic expansion of the gamma function, including explicit and realistic error bounds, the Berry transition between different asymptotic representations across a Stokes line, and asymptotic estimates for the late coefficients.
TL;DR: In this article, an asymptotic expansion of the canard point is obtained, with very good agreement with the numerical results of Peng et al. They showed that the discontinuous canard transition suggested by Peng et.
Abstract: The Edblom–Orban–Epstein (EOE) reaction, involving iodate, sulphite and ferrocyanide ions may have oscillations in a continuous stirred tank reactor. A simplification of a 10-variable model, including two state variables, was analysed numerically by Peng et al. ( Phil. Trans. R. Soc. Lond . A337, 275 (1991)). They found that within a very small range of the inflow concentration, the amplitude of of the oscillations varied drastically. Such behaviour is well understood in singular perturbations systems, where it is known as a canard explosion . Apparently, the EOE equations do not belong to this class. We show that the two-dimensional EOE equations can be recast as a singular perturbation problem. An asymptotic expansion of the the canard point is obtained, with very good agreement with the numerical results of Peng et al . To deal with the canard explosion in systems which are not of singular perturbation type, Peng et al . introduce a new definition of the canard point, based on change of curvature of the limit cycle. We discuss the new definition, and show that it may essentially agree with the definition based on a singular perturbation approach, but is much less fit for analytical computations. We show that the discontinuous canard transition suggested by Peng et al . violates a continuity theorem in the theory of ordinary differential equations.
TL;DR: In this article, the authors unify and extend previous work, by obtaining necessary and sucient conditions for the asymptotic stability as t! 1 of solutions of quasi-variational systems.
TL;DR: In this paper, a parallel root locus theory for distributed parameter systems is presented for quite general, nonconstant coefficient, even order differential operators on a finite interval with control and output boundary conditions representative of a choice of colocated point actuators and sensors.
Abstract: In this paper, a fairly complete parallel of the finite-dimensional root locus theory is presented for quite general, nonconstant coefficient, even order ordinary differential operators on a finite interval with control and output boundary conditions representative of a choice of colocated point actuators and sensors. Root-locus design methods for linear distributed parameter systems have also been studied for some time and the primary difficulties in rigorously interpreting root locus conclusions for distributed parameter systems are well known. First, the transfer function of a distributed parameter system may not be meromorphic at infinity so that many of the standard Rouche arguments, required even in the lumped case to determine the asymptotic behavior of the root loci, are not generally valid. Another difficulty is that the infinitesimal generator in the state-space model for a closed-loop system may not be selfadjoint, accretive or even satisfy the spectrum determined growth condition. Thus, regardless of whether the root loci---interpreted as closed-loop eigenvalues---lie in the open left-half plane, additional analysis would be required to conclude that the closed-loop system would be asymptotically stable. Formulating the systems in the classical format of a boundary control problem, the asymptotic analysis of the root loci can be based on the pioneering work by Birkhoff on eigenfunction expansions for boundary value problems, work that predated and indeed motivated the development of spectral theory in Hilbert space. Birkhoff's work also contains an asymptotic expansion of eigenfunctions in the spatial variable, generalizing the earlier Sturm--Liouville theory for second-order operators. By further extending this general asymptotic analysis to also include expansions in the gain parameter, a rigorous treatment of the open and closed loop transfer functions and of the corresponding return difference equation can be presented. The asymptotic analysis of the return difference equation forms the basis for both our rigorous formulation of the basic problem and its solution.
TL;DR: The concept of a dynamic job shop is introduced by interpreting the system as a directed graph, and the structure of the system dynamics is characterized for its use in the asymptotic analysis.
Abstract: This paper presents an asymptotic analysis of hierarchical production planning in a general manufacturing system consisting of a network of unreliable machines producing a variety of products. The concept of a dynamic job shop is introduced by interpreting the system as a directed graph, and the structure of the system dynamics is characterized for its use in the asymptotic analysis. The optimal control problem for the system is a state-constrained problem, since the number of parts in any buffer between any two machines must remain nonnegative. A limiting problem is introduced in which the stochastic machine capacities are replaced by corresponding equilibrium mean capacities, as the rate of change in machine states approaches infinity. The value function of the original problem is shown to converge to that of the limiting problem, and the convergence rate is obtained. Furthermore, near-optimal controls for the original problem are constructed from near-optimal controls of the limiting problem, and an error estimate is obtained on the near optimality of the constructed controls. >
TL;DR: In this paper, the authors developed an asymptotic theory for dynamic analysis of anisotropic inhomogeneous plates within the framework of three-dimensional elasticity, where the inhomogeneities are considered to be in the thickness direction.
TL;DR: Assemien et al. as mentioned in this paper studied the dependence of the constants appearing in some Sobolev embeddings and showed that inertial effects are negligible at the first order for the asymptotic behavior of a thin film flow.
Abstract: Assemien, A., Bayada, G. and M. Chambat, Inertial effects in the asymptotic behavior of a thin film flow, Asymptotic Analysis 9 (1994) 177-20S. We study the dependence with respect to the domain of the constants appearing in some Sobolev embeddings. It shows that the inertial effects are negligible at the first order for the asymptotic behavior of a thin film flow. The influence of the inertial effects is obtained at the second order in the asymptotic expansions of the pressure and velocity, with full convergence argument for chosen boundary conditions.
TL;DR: In this paper, the results of extensive numerical studies for a large class of functions f(x) associated with strong-coupling lattice approximations are reported, and it is conjectured that for large n, Pnn(∞)∼f( ∞)+B/ln n.
Abstract: A difficult and long‐standing problem in mathematical physics concerns the determination of the value of f(∞) from the asymptotic series for f(x) about x=0. In the past the approach has been to convert the asymptotic series to a sequence of Pade approximants {Pnn(x)} and then to evaluate these approximants at x=∞. Unfortunately, for most physical applications the sequence {Pnn(∞)} is slowly converging and does not usually give very accurate results. In this paper the results of extensive numerical studies for a large class of functions f(x) associated with strong‐coupling lattice approximations are reported. It is conjectured that for large n, Pnn(∞)∼f(∞)+B/ln n. A numerical fit to this asymptotic behavior gives an accurate extrapolation to the value of f(∞).
TL;DR: In this paper, the optimal remainder terms in the well-known asymptotic series solutions of homogeneous linear differential equations of the second order in the neighbourhood of an irregular singularity of rank one are obtained.
Abstract: Re-expansions are found for the optimal remainder terms in the well-known asymptotic series solutions of homogeneous linear differential equations of the second order in the neighbourhood of an irregular singularity of rank one. The re-expansions are in terms of generalized exponential integrals and have greater regions of validity than the original expansions, as well being considerably more accurate and providing a smooth interpretation of the Stokes phenomenon. They are also of strikingly simple form. In addition, explicit asymptotic expansions for the higher coefficients of the original asymptotic solutions are obtained.
TL;DR: The boundary value problem e/G is a version of the Berman's problem and its solutions describe a laminar flow in a channel with porous walls as mentioned in this paper, and when 0 < e
Abstract: The boundary-value problem e/G") = [/ /'" /' /"], /(0) = /"(0) = /'(I) = 0, /(I) = 1, is a version of Berman's problem and its solutions describe a laminar flow in a channel with porous walls. When 0 < e
TL;DR: In this paper, an asymptotic analysis of hierarchical manufacturing systems with stochastic demand and machines subject to breakdown and repair as the rate of change in machine states approaches infinity is presented.
TL;DR: Bardos et al. as discussed by the authors proposed an algorithm for the computation of the outgoing solution of the Helmholtz equation in an exterior domain, where an artificial boundary with absorbing boundary condition is inserted, and the periodic response is characterized as the unique minimum of a convex functional.
Abstract: Bardos, C . and J. Rauch, Variational algorithms for the Helmholtz equation using time evolution and artificial boundaries, Asymptotic Analysis 9 (1994) 101-117. This paper is devoted to the mathematical analysis of some algorithms for the computation of the outgoing solution of the Helmholtz equation in an exterior domain. In a first approximation an artificial boundary with absorbing boundary condition is inserted. One then computes the periodic response in this bounded dissipative setting to a periodic forcing term. The response is characterized as the unique minimum of a convex functional. The functional is computed from solution of the time dependent problem in the artificially bounded domain. One such algorithm is due to Glowinski who proposed the functional J2 described below, it has been implemented by Bristeau et al. [5]. We propose a different functional J1 which is unconditionally coercive while the coerciveness of J 2 depends in a subtle way on the geometry of the domain. It is coercive for non-trapping obstacles'. The coerciveness property is essential for the convergence of the numerical method.