Feedback stabilization of real analytic systems in the plane

We investigate the local feedback stabilization of single input control affine analytic systems in the plane. New necessary and sufficient conditions for local stabilization with feedback laws of the form u = v(x₁,x₂), x2), (v/x₁(0, 0))² + (v/x₁ (0, 0))² 0≠ 0, v(0, 0) = 0, are obtained by using Lyapunov's stability theorems on two-dimensional analytic systems. If the sufficient conditions are satisfied, we also provide explicit feedback laws.     

Diffusions, their derivatives and expansions in Wiener chaos

Using the Isobe–Sato formula we identify derivatives of the random variable c(x_T), where x_t is a diffusion given by some SDE. It appears that the derivative is propagated by a system of linear ODEs driven by some functionals of the sample x_(·). This leads to a new integral representation of the kernels of chaos expansion which seems to be more convenient for performing numerical simulations.     

Stabilizability of two-dimensional linear systems via switched output feedback

The problem of stabilizing a second-order SISO LTI system of the form , y=Cx with feedback of the form u(x)=v(x)Cx is considered, where v(x) is real-valued and has domain which is all of . It is shown that, when stabilization is possible, v(x) can be chosen to take on no more than two values throughout the entire state-space (i.e., v(x){k₁,k₂} for all x and for some k₁,k₂), and an algorithm for finding a specific choice of v(x) is presented. It is also shown that the classical root locus of the corresponding transfer function C(sI-A)^-1B has a strong connection to this stabilization problem, and its utility is demonstrated through examples.     

Rational Solutions of Riccati-like Partial Differential Equations

When factoring linear partial differential systems with a finite-dimensional solution space or analysing symmetries of nonlinear ODEs, we need to look for rational solutions of certain nonlinear PDEs. The nonlinear PDEs are called Riccati-like because they arise in a similar way as Riccati ODEs. In this paper we describe the structure of rational solutions of a Riccati-like system, and an algorithm for computing them. The algorithm is also applicable to finding all rational solutions of Lie’s system { ∂_xu + u² + a₁u + a₂v + a₃, ∂_yu + uv + b₁u + b₂v + b₃, ∂_xv + uv + c₁u + c₂v + c₃, ∂_yv + v² + d₁u + d₂v + d₃},where a₁, . . . , d₃are rational functions of x and y.     

A Primal-Dual Approximation Algorithm for Partial Vertex Cover: Making Educated Guesses

We study the partial vertex cover problem. Given a graph G=(V,E), a weight function w:V→R ⁺, and an integer s, our goal is to cover all but s edges, by picking a set of vertices with minimum weight. The problem is clearly NP-hard as it generalizes the well-known vertex cover problem. We provide a primal-dual 2-approximation algorithm which runs in O(nlog n+m) time. This represents an improvement in running time from the previously known fastest algorithm. Our technique can also be used to get a 2-approximation for a more general version of the problem. In the partial capacitated vertex cover problem each vertex u comes with a capacity k _u . A solution consists of a function x:V→ℕ₀ and an orientation of all but s edges, such that the number of edges oriented toward vertex u is at most x _u k _u . Our objective is to find a cover that minimizes ∑ _v∈V x _v w _v . This is the first 2-approximation for the problem and also runs in O(nlog n+m) time. Research supported by NSF Awards CCR 0113192 and CCF 0430650, and the University of Maryland Dean’s Dissertation Fellowship.     

A comparison of the average case numerical condition of the power and bernstein polynomial bases

The relative numerical condition of a root x ₀, of arbitrary multiplicity, of a polynomial p(x) in the power and Bernstein bases is considered. The polynomial equation p(x)=0 and the linear algebraic equation that defines the transformation between the bases are used to show that the relative numerical condition of x ₀ in the bases is strongly dependent on the numerical condition of this equation. Furthermore, as the multiplicity of x _o increases for a given polynomial order, the relative numerical condition of x ₀ approaches unity. Computational examples that illustrate the theoretical results are presented.     

Langford strings are square-free

Given a set V _n ={v1, …v _n } of n symbols, a sequence x = x1x2…x _m is called a weak Langford string if x, ? Vn for 1≦i≦m, and any two consecutive occurrences of V _j in x are separated by precisely j characters of x, 1≦ j≦n. Proving subsequent conjectures of P[acaron] un [5] and Marcus and P[acaron] un [4], we show that every weak Langford string is square-free. As a consequence, all Langford languages are finite. Some related questions are raised.     

An efficient method for computing leading eigenvalues and eigenvectors of large asymmetric matrices

An efficient method for computing a given number of leading eigenvalues (i.e., having largest real parts) and the corresponding eigenvectors of a large asymmetric matrixM is presented. The method consists of three main steps. The first is a filtering process in which the equationx = Mx is solved for an arbitrary initial conditionx(0) yielding:x(t)=e ^Mt x(0). The second step is the construction of (n+1) linearly independent vectorsv _m =M ^m x, 0mn orv _m =e ^mMt x ( being a short time interval). By construction, the vectorsv _m are combinations of only a small number of leading eigenvectors ofM. The third step consists of an analysis of the vectors {v _m } that yields the eigenvalues and eigenvectors. The proposed method has been successfully tested on several systems. Here we present results pertaining to the Orr-Sommerfeld equation. The method should be useful for many computations in which present methods are too slow or necessitate excessive memory. In particular, we believe it is well suited for hydrodynamic and mechanical stability investigations.     

Non-linear filtering of counting processes driven by Ornstein-Uhlenbeck processes

The intensity of the doubly stochastic Poisson process (DSPP) considered in this paper is a linear function of a first-order Gauss-Markov process x ₁, (Ornstein-Uhlenbeck process).

By observing a DSPP realization and by analysing the conditional characteristic function of x ₁, we intend to find a non-linear recursive filter that gives an estimation of the intensity.

The expression of the centred conditional moments up to any order is established recursively, and a practical numerical algorithm is developed on the basis of a suboptimal non-linear filter. Consideration of the centred odd moments is also justified. The results of the numerical simulations are presented and enable a comparison to be made between the behaviour of the suboptimal non-linear filter and that of an adapted linear filter. The number of centred conditional moments to be retained in the formulation of the suboptimal non-linear filter is discussed.

Finally, numerical simulation results are given and commented on.     

On the joint nonlinear filtering-smoothing of diffusion processes

The smoothing of diffusions dx_t = f(x_t) dt + σ(x_t) dw_t, measured by a noisy sensor dy_t = h(x_t) dt + dv_t, where w_t and v_t are independent Wiener processes, is considered in this paper. By focussing our attention on the joint p.d.f. of (x_τ x_t), 0 ≤ τ < t, conditioned on the observation path {y_s, 0 ≤ s ≤ t}, the smoothing problem is represented as a solution of an appropriate joint filtering problem of the process, together with its random initial conditions. The filtering problem thus obtained possesses a solution represented by a Zakai-type forward equation. This solution of the smoothing problem differs from the common approach where, by concentrating on the conditional p.d.f. of x_τ alone, a set of ‘forward and reverse’ equations needs to be solved.     

Development of a multiplier method for dynamic response optimization problems

The multiplier method is studied for optimum design of mechanical and structural systems subjected to dynamic loads. Certain key parameters in the algorithm are identified and extensive numerical experiments are conducted to see their effect on the performance of the method. Several mathematical programming problems, and static and dynamic response structural design problems are used to evaluate the method. Some new numerical procedures are proposed and evaluated to improve performance of the method. As a result of this study, a better understanding of the multiplier method has been achieved, and the effect of various parameters and procedures of the algorithm is better understood.Notation Number of equality constraints - COST Cost function value at the solution point - CPU Total CPU time on DN10000 - f(x) Cost function - g(x) Constraint vector of dimension m×1 - IFAIL Number of failed problems - A parameter used in the algorithm - L-BFGS Unconstrained minimization program that uses limited memory BFGS method - m Total number of constraints - n Number of design variables - v Number of degrees of freedom - NF Average number of function evaluations - NG Average number of gradient evaluations - NIT Average number of unconstrained minimizations - Parameter vector of dimension m×1 used in the definition of augmented Lagrange functional - r Penalty parameter vector of dimension m×1 - TRDDB Unconstrained minimization program that computes trust region step using the double dogleg method - u Lagrange multiplier vector of dimension m×1 - x Design variable vector of dimension n×1 - x _i Lower bound onx _i - x _ui Upper bound onx _i - (x) Augmented Lagrangian - P(x) Penalty function     

Bessel function of the first kind with complex argument

A new method of computing integral order Bessel functions of the first kind J_n(z) when either the absolute value of the real part or the imaginary part of the argument z = x + iy is small, is described. This method is based on computing the Bessel functions from asymptotic expressions when x∼ 0 (or y ∼ 0). These expansions are derived from the integral definition of Bessel functions. This method is necessary because some existing algorithms and methods fail to give correct results for small x small y. In addition, our overall method of computing Bessel functions of any order and argument is discussed and the logarithmic derivative is used in computing these functions. The starting point of the backward recurrence relations needed to evaluate the Bessel function and their logarithmic derivatives are investigated in order to obtain accurate numerical results. Our numerical method, together with established techniques of computing the Bessel functions, is easy to implement, efficient, and produces reliable results for all z.     

Cost-Based Arc Consistency for Global Cardinality Constraints

A global cardinality constraint (gcc) is specified in terms of a set of variables X={x ₁,...,x _p} which take their values in a subset of V={v ₁,...,v _d}. It constrains the number of times each value v _iV is assigned to a variable in X to be in an interval [l _i,u _i]. A gcc with costs (costgcc) is a generalization of a gcc in which a cost is associated with each value of each variable. Then, each solution of the underlying gcc is associated with a global cost equal to the sum of the costs associated with the assigned values of the solution. A costgcc constrains the global cost to be less than a given value. Cardinality constraints with costs have proved very useful in many real-life problems, such as traveling salesman problems, scheduling, rostering, or resource allocation. For instance, they are useful for expressing preferences or for defining constraints such as a constraint on the sum of all different variables. In this paper, we present an efficient way of implementing arc consistency for a costgcc. We also study the incremental behavior of the proposed algorithm.     

A cubically convergent iteration method for multiple roots of f(x)=0

This paper describes a cubically convergent iteration method for finding the multiple roots of nonlinear equations, f(x)=0, where f:?→? is a continuous function. This work is the extension of our earlier work [P.K. Parida, and D.K. Gupta, An improved regula-falsi method for enclosing simple zeros of nonlinear equations, Appl. Math. Comput. 177 (2006), pp. 769–776] where we have developed a cubically convergent improved regula-falsi method for finding simple roots of f(x)=0. First, by using some suitable transformation, the given function f(x) with multiple roots is transformed to F(x) with simple roots. Then, starting with an initial point x ₀ near the simple root x* of F(x)=0, the sequence of iterates {x _n }, n=0, 1, … and the sequence of intervals {[a _n , b _n ]}, with x*∈{[a _n , b _n ]} for all n are generated such that the sequences {(x _n ?x*)} and {(b _n ?a _n )} converges cubically to 0 simultaneously. The convergence theorems are established for the described method. The method is tested on a number of numerical examples and the results obtained are compared with those obtained by King [R.F. King, A secant method for multiple roots, BIT 17 (1977), pp. 321–328.].     

Variable Precision Algorithm for the Numerical Computation of the Fermi–Dirac Function ℱj(x) of Order j=−3/2

The purpose of this technical note is to present a piecewise Chebyshev expansion for the numerical computation of the Fermi–Dirac function _–3/2(x), –<x<. The variable precision algorithm we given automatically adjusts the degrees of the Chebyshev expansions so that _–3/2(x) can be efficiently computed to d significant decimal digits of accuracy, for a user specified value of d in the range 1d15.     

Why clustering in function approximation? Theoretical explanation

Function approximation is a very important practical problem: in many practical applications, we know the exact form of the functional dependence y=f(x₁,…,x_n) between physical quantities, but this exact dependence is complicated, so we need a lot of computer space to store it, and a lot of time to process it, i.e., to predict y from the given x_i. It is therefore necessary to find a simpler approximate expression g(x₁,…,x_n)≈f(x₁,…,x_n) for this same dependence. This problem has been analyzed in numerical mathematics for several centuries, and it is, therefore, one of the most thoroughly analyzed problems of applied mathematics. There are many results related to approximation by polynomials, trigonometric polynomials, splines of different type, etc. Since this problem has been analyzed for so long, no wonder that for many reasonable formulations of the optimality criteria, the corresponding problems of finding the optimal approximations have already been solved. Lately, however, new clustering‐related techniques have been applied to solve this problem (by Yager, Filev, Chu, and others). At first glance, since for most traditional optimality criteria, optimal approximations are already known, the clustering approach can only lead to non‐optimal approximations, i.e., approximations of inferior quality. We show, however, that there exist new reasonable criteria with respect to which clustering‐based function approximation is indeed the optimal method of function approximation. © 2000 John Wiley & Sons, Inc.     

The zero divisor problem of multivariable stochastic adaptive control

Stochastic adaptive minimum variance control algorithms require a division by a function of a recursively computed parameter estimate at each instant of time. In order that the analysis of these algorithms is valid, zero divisions must be events of probability zero. This property is established for the stochastic gradient adaptive control algorithm under the condition that the initial state of the system and all finite segments of its random disturbance process have a joint distribution which is absolutely continuous with respect to Lebesgue measure. This result is deduced from the following general result established in this paper: a non-constant rational function of a finite set of random variables {x₁},x_n} is absolutely continuous with respect to Lebesgue measure if the joint distribution function of {x₁,…,x_n} has this property.     

Radio number of ladder graphs

Let G be a connected graph with diameter diam(G). The radio number for G, denoted by rn(G), is the smallest integer k such that there exists a function f: V(G)→{0, 1, 2, …, k} with the following satisfied for all vertices u and v:|f(u)?f(v)|≥diam (G)?d _G (u, v)+1, where d _G (u, v) is the distance between u and v in G. In this paper, we determine the radio number of ladder graphs.