Modes with switchings of increasing frequency in the problem of controlling a robot

Trajectories that are optimal with respect to high-speed response are constructed for a system for controlling a two-component manipulator (a robot). It is shown that when the initial conditions lie within a certain open region of the phase space, all optimal trajectories will have a segment of switchings of increasing frequency (SIF), i.e. a segment in which the control will undergo an infinite number of switchings in a finite time interval.

The synthesis of the optimal control in the R² plane containing the mode of SIF was first constructed by Fuller /1/. It was shown in /2/ that the synthesis is structurally stable in the sense that adding terms of higher order of smallness to the integrand and to the right-hand sides of the system of differential constraints does not affect the qualitative pattern of the optimal synthesis in the neighbourhood of the origin of coordinates.

The present paper explains that the synthesis in the problem of optimal control (relative to the high speed response) of the motion of the robot appears, in a certain sense, a direct product of the synthesis appearing in the Fuller problem and of the synthesis in the simplest problem of high-speed response (/3/, pp.38–47). The special aspect of the present paper consists of the proof of the proposition that switching surface is a piecewise-smooth manifold. The presence of the SIF mode is connected only with the fact that every trajectory intersects this surface an infinite number of times. In existing papers, the piecewise smoothness of the switching curve was proved for the two-dimensional problems using the SIF mode only for problems admitting of a one-parameter group of symmetries /1, 4–6/. A proof of the presence of SIF was given in /7, 8/.     

The Newsboy problem when customer demand is a compound renewal process

The Newsboy (Newsvendor) problem is probably the simplest of all stochastic inventory problems, involving a one-time purchase decision and a stochastic sales outcome. As an investment, it can be interpreted as the simplest stochastic version of the point-in, point-out investment problem of Jevons [Jevons, W.S., Theory of Political Economy, Macmillan, London 1871].     

On the equivalency of piecewise-constant control with a known number of switchings and arbitrary amplitude bounded control in a terminal problem for a linear nonstationary system

It is shown that any amplitude-bounded control of a linear terminal nonstationary system with two control coordinates can be replaced by a piecewise-constant (relaying) control with a known number of switchings, and moreover, the procedure of determining the moments of switching is reduced to a known finite number of problems of nonlinear programming in a space of two variables, which are given in the paper.Translated from Dinamicheskie Sistemy, No. 7, pp. 78–84, 1988.     

The moment problem on the Wiener space

Consider an L¹-continuous functional ? on the vector space of polynomials of Brownian motion at given times, suppose ? commutes with the quadratic variation in a natural sense, and consider a finite set of polynomials of Brownian motion at rational times, , mapping the Wiener space to R.In the spirit of Schmüdgen's solution to the finite-dimensional moment problem, we give sufficient conditions under which ? can be written in the form ∫⋅dμ for some probability measure μ on the Wiener space such that μ-almost surely, all the random variables are nonnegative.     

The maximal range problem for a bounded domain

Let be a bounded domain such that 0Ω. Denote by , the set of all complex polynomials of degree at most n. Let

where . We relate the maximal polynomial range

to the geometry of Ω.     

The Monge problem in Wiener space

We address the Monge problem in the abstract Wiener space and we give an existence result provided both marginal measures are absolutely continuous with respect to the infinite dimensional Gaussian measure γ.     

The Wiener maximum quadratic assignment problem

We investigate a special case of the maximum quadratic assignment problem where one matrix is a product matrix and the other matrix is the distance matrix of a one-dimensional point set. We show that this special case, which we call the Wiener maximum quadratic assignment problem, is NP-hard in the ordinary sense and solvable in pseudo-polynomial time.Our approach also yields a polynomial time solution for the following problem from chemical graph theory: find a tree that maximizes the Wiener index among all trees with a prescribed degree sequence. This settles an open problem from the literature.     

The Ramsey number of a graph with bounded maximum degree

The Ramsey number of a graph G is the least number t for which it is true that whenever the edges of the complete graph on t vertices are colored in an arbitrary fashion using two colors, say red and blue, then it is always the case that either the red subgraph contains G or the blue subgraph contains G. A conjecture of P. Erdös and S. Burr is settled in the affirmative by proving that for each d ≥ 1, there exists a constant c so that if G is any graph on n vertices with maximum degree d, then the Ramsey number of G is at most cn.     

Parameterized complexity of the weighted independent set problem beyond graphs of bounded clique number

The maximum independent set problem is known to be NP-hard for graphs in general, but is solvable in polynomial time for graphs in many special classes. It is also known that the problem is generally intractable from a parameterized point of view. A simple Ramsey argument implies the fixed-parameter tractability of the maximum independent set problem in classes of graphs of bounded clique number. Beyond this observation very little is known about the parameterized complexity of the problem in restricted graph families. In the present paper we develop fpt-algorithms for graphs in some classes extending graphs of bounded clique number.     

The behavior of mappings with bounded distortion when the coefficient of distortion is close to unity

Siberian Mathematical Journal -     

A problem of Földes and Puri on the Wiener process

Let be a real-valued Wiener process starting from 0, and be the right-continuous inverse process of its local time at 0. Földes and Puri [3] raise the problem of studying the almost sure asymptotic behavior of as tends to infinity, i.e. they ask: how long does stay in a tube before ``crossing very much" a given level? In this note, both limsup and liminf laws of the iterated logarithm are provided for .

     

The dominating set problem is fixed parameter tractable for graphs of bounded genus

We describe an algorithm for the dominating set problem with time complexity O((4g+40)^kn²) for graphs of bounded genus g1, where k is the size of the set. It has previously been shown that this problem is fixed parameter tractable for planar graphs. We give a simpler proof for the previous O(8^kn²) result for planar graphs. Our method is a refinement of the earlier techniques.     

The Wiener disorder problem with finite horizon

The Wiener disorder problem seeks to determine a stopping time which is as close as possible to the (unknown) time of ‘disorder’ when the drift of an observed Wiener process changes from one value to another. In this paper we present a solution of the Wiener disorder problem when the horizon is finite. The method of proof is based on reducing the initial problem to a parabolic free-boundary problem where the continuation region is determined by a continuous curved boundary. By means of the change-of-variable formula containing the local time of a diffusion process on curves we show that the optimal boundary can be characterized as a unique solution of the nonlinear integral equation.     

Hermitian matrices with a bounded number of eigenvalues

Conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex Hermitian matrices with a bounded number of distinct eigenvalues. A minimal generating system of the vanishing ideal of degenerate three by three Hermitian matrices is given, and the structure of the corresponding coordinate ring as a module over the special unitary group is determined. The method applies also for degenerate real symmetric three by three matrices. For arbitrary n   partial information on the minimal degree component of the vanishing ideal of the variety of n×n$n\times n$ Hermitian matrices with a bounded number of eigenvalues is obtained, and some known results on sum of squares presentations of subdiscriminants of real symmetric matrices are extended to the case of complex Hermitian matrices.