New necessary conditions of optimality for control problems with state-variable inequality constraints

Necessary conditions of optimality for state-variable inequality constrained problems are derived which differ from those of Bryson, Denham, and Speyer with regard to the behavior of the adjoint variables at junctions of interior and boundary arcs. In particular, it is shown that the earlier conditions under-specify the behavior of the adjoint variables at the junctions. An example is used to demonstrate that the earlier conditions may yield non-stationary trajectories.     

Numerical algorithms for simulations of a traffic model on road networks

We introduce a simulation algorithm based on a fluid-dynamic model for traffic flows on road networks, which are considered as graphs composed by arcs that meet at some junctions. The approximation of scalar conservation laws along arcs is made by three velocities Kinetic schemes with suitable boundary conditions at junctions. Here we describe the algorithm and we give an example.     

The behavior of conformal maps of domains near convex boundary arcs

The behavior of the derivatives of conformal maps of the unit disk onto simply connected domains in the complex plane whose boundaries contain convex or concave attainable arcs, as well as the behavior of the derivatives of the inverse maps, is studied. It is proved that these derivatives exist and are bounded on the corresponding arcs and near them; a criterion for their continuity at points of these arcs is stated and proved.     

Generate upper boundary vectors meeting the demand and budget for a p-commodity network with unreliable nodes

The system capacity for a single-commodity flow network is the maximum flow from the source to the sink. This paper discusses the system capacity problem for a p-commodity limited-flow network with unreliable nodes. In such a network, arcs and nodes all have several possible capacities and may fail. Different types of commodity, which are transmitted through the same network simultaneously, competes the capacities of arcs and nodes. In particular, the consumed capacity by different types of commodity varies from arcs and nodes. We first define the system capacity as a vector and then a performance index, the probability that the upper bound of the system capacity is a given pattern subject to the budget constraint, is proposed. Such a performance index can be easily computed in terms of upper boundary vectors meeting the demand and budget. A simple algorithm based on minimal cuts is thus presented to generate all upper boundary vectors. The manager can apply this performance index to measure the system capacity level for a supply-demand system.     

A problem of the bending of a plate for a doubly connected domain with a partially unknown boundary

The problem of the bending of an isotropic elastic plate, bounded by two rectangles with vertices lying on the same half-line, drawn from the common centre, is considered. The vertices of the inner rectangle are cut by convex smooth arcs (we will call the set of these arcs the unknown part of the boundary). It is assumed that normal bending moments act on each rectilinear section of the boundary contours in such a way that the angle of rotation of the midsurface of the plate is a piecewise-constant function. The unknown part of the boundary is free from external forces. The problem consists of determining the bending of the midsurface of the plate and the analytic form of the unknown part of the boundary when the tangential normal moment acting on it takes a constant value, while the shearing force and the normal bending moments and torques are equal to zero. The problem is solved by the methods of the theory of boundary-value problems of analytical functions.     

Asymptotic properties of solutions of parametric optimal control problems with varying index of the singular arcs

We consider a family of parametric linear-quadratic optimal control problems with terminal and control constraints. This family has the specific feature that the class of optimal controls is changed for an arbitrarily small change in the parameter. In the perturbed problem, the behavior of the corresponding trajectory on noncritical arcs of the optimal control is described by solutions of singularly perturbed boundary value problems. For the solutions of these boundary value problems, we obtain an asymptotic expansion in powers of the small parameter ?. The asymptotic formula starts from a term of the order of 1/? and contains boundary layers. This formula is used to justify the asymptotic expansion of the optimal control for a perturbed problem in the family. We suggest a simple method for constructing approximate solutions of the perturbed optimal control problems without integrating singularly perturbed systems. The results of a numerical experiment are presented.     

The Use of Parabolic Arcs in Matching Curved Boundaries in the Finite Element Method

If isoparametric coordinates are used to deal with curved boundariesin the finite element method, the original boundary is implicitlyreplaced by a series of parabolic or cubic arcs. The equationsof these arcs involve parameters which are the coordinates ofpoints on the curved side, and a simple procedure is outlinedfor choosing these parameters in such a way that each arc isa parabola which passes through four points of the originalcurve thus ensuring a good approximation to it.     

Boundary arcs for integral equations

Behavior of boundary arcs for control systems is investigated when the systems are governed by integral equations of the Volterra type. The main result is in the form of a maximum principle. This result is then used to obtain necessary conditions for a minimum control problem.     

An elementary sufficiency proof of an absolute minimum for a nonparametric variational problem

For a fixed endpoint, nonparametric simple integral variational problem, there is presented an expansion method proof of a sufficiency theorem for an absolute minimum. In particular, this sufficiency theorem yields readily the proof of a result of the type recently presented by Nehari (Ref. 1), but with an error in formulation and an incorrect proof. The present discussion is in a setting which permits considered arcs to be on the boundary of the set of admissible arcs; thus it contains as particular instances certain types of unilateral variational problems of a control nature.This research was supported by the National Science Foundation under Grant No. GP-36120.     

Short spirals

Previous results for planar curves having monotone curvature (spirals) and limited in length in one way or another are extended by dropping the requirements of convexity of the arc, injectivity of the projection onto the spanning chord, and continuity of the curvature. A more general class of short spiral arcs is thus introduced. The results include W. Vogt’s theorem concerning boundary angles of a convex spiral arc, necessary and sufficient conditions for its existence, and positional inequalities for such arcs. Bibliography: 8 titles.     

A domain decomposition method for approximating the conformal modules of long quadrilaterals

Summary This paper is concerned with the study of a domain decomposition method for approximating the conformal modules of long quadrilaterals. The method has been studied already by us and also by D. Gaier and W.K. Hayman, but only in connection with a special class of quadrilaterals, viz. quadrilaterals where: (a) the defining domain is bounded by two parallel straight lines and two Jordan arcs, and (b) the four specified boundary points are the four corners where the arcs meet the straight lines.Our main purpose here is to explain how the method may be extended to a wider class of quadrilaterals than that indicated above.     

Homotopy Algorithm for Optimal Control Problems with a Second-order State Constraint

This paper deals with optimal control problems with a regular second-order state constraint and a scalar control, satisfying the strengthened Legendre-Clebsch condition. We study the stability of structure of stationary points. It is shown that under a uniform strict complementarity assumption, boundary arcs are stable under sufficiently smooth perturbations of the data. On the contrary, nonreducible touch points are not stable under perturbations. We show that under some reasonable conditions, either a boundary arc or a second touch point may appear. Those results allow us to design an homotopy algorithm which automatically detects the structure of the trajectory and initializes the shooting parameters associated with boundary arcs and touch points.     

A graph theoretic approach to the investigation of system‐environment relationships†

For any empirical structure consisting of a system S and its environment E, there is an associated digraph D whose points and arcs (directed lines) correspond to the elements and relationships of the structure. The arcs of D are thus of four types: (1) internal arcs, which join two points of S; (b) external arcs, which join two points of E; (c) out‐liaisons of S, which join a point of S to one of E; and (d) in‐liaisons of S, which join a point of E to one of S. The boundary of S is defined as the subgraph of D induced by the liaisons of S and corresponds to those elements and relationships of the structure directly involved in transactions between the system and its environment. The basic structural properties of boundaries are then identified, and it is shown how the points of S and E can be stratified according to their distances to (or from) the boundary of S. Next, several results are derived concerning system‐environment relationships in structures whose digraphs are symmetric, transitive, or signed. The concept of convexity is then introduced to deal with a certain kind of segregation of a system relative to its environment. And, finally, it is shown how the adjacency matrix of D can be employed to facilitate the analysis of such structures and the calculation of various indexes of system‐environment relationships.     

Power series: Localization of singularities on the boundary of the disk of convergence

The paper contains results on localization of singularities for the power series (in terms of their coefficients) on boundary arcs of the disk of convergence.     

On the change in moduli under circular symmetrization

It is shown that under circular symmetrization of multiply-connected domains the moduli of families of curves separating two particular boundary components can be changed in any direction. This is also true for the moduli of families of arcs joining particular boundary components of a domain.Translated from Matematicheskie Zametki, Vol. 10, No. 5, pp. 527–532, November, 1971.     

Decomposition technique for minimum time trajectories

A decomposition technique is presented for minimum-time trajectories which are characterized by intermediate constraints and discontinuities. The optimization of such multiple are trajectories is usually a formidable task. One optimization method, trajectory decomposition, breaks the original trajectory at points of discontinuity into separate arcs and then optimizes each are subject to prescribed boundary conditions. This constitutes a first level of control. Each first-level solution is evaluated by a second-level controller, which iteratively specifies new are boundary conditions in order to achieve an optimum solution. Unfortunately, this two-level method cannot be applied directly to minimum-time trajectories. The two-level trajectory decomposition method is extended here to a three-level technique for treating the minimum-time trajectory. The first level again optimizes each are for specified intervention parameters. The new second level, the time interface controller, exploits certain homogeneity properties to satisfy time transversality conditions at all boundaries and to couple the first-level solution arcs in time. The third level, the state interface controller, satisfies state transversality conditions at the arc junctions iteratively while driving the trajectory to its optimum. The new three-level procedure represents a feasible decomposition because each solution trajectory in the iterative sequence is physically realizable. The technique also offers a decentralization of control effort and reduction of initial-value sensitivities. An example problem is formulated.     

Boundary behavior of derivatives of conformal mappings of simply connected domains

Conformal mappings of simply connected domains G on a disc or a half-plane are considered in the case when boundaries consist of smooth boundary arcs Γ reachable from inside of G. Sufficient conditions for existence of an angular limit of the derivative of such mappings and its bounded-ness at some given boundary point are found. A sufficient condition of existence of a bounded derivative on the region boundary is given as a corollary.     

Quasiconformal extension of quasimöbius mappings of Jordan domains

We introduce the new class of Jordan arcs (curves) of bounded rotation which includes all arcs (curves) of bounded turning. We prove that if the boundary of a Jordan domain has bounded rotation everywhere but possibly one singular point then every quasimöbius embedding of this domain extends to a quasiconformal automorphism of the entire plane.     

On the length preserving approximation of plane curves by circular arcs

A technique for the length preserving approximation of plane curves by two circular arcs is analyzed. The conditions under which this technique can be applied are extended, and certain consequences of the proved results unrelated to the approximation problem are discussed. More precisely, inequalities for the length of a convex spiral arc subject to the given boundary conditions are obtained. Conjectures on curve closeness conditions obtained using computer simulation are discussed.     

MC-based Algorithm for a telecommunication network under node and budget constraints

It is an important issue to design some performance indexes in order to measure the performance for a telecommunication network. Network analysis is an available approach to solve the performance problem for a real-life system. We construct a two-commodity stochastic-flow network with unreliable nodes (arcs and nodes all have several possible capacities and may fail) to model the telecommunication network. In which, all types of commodity are transmitted through the same network simultaneously and compete the capacities. This paper defines the system capacity as a 2-tuple vector, and then proposes a performance index, the probability that the upper bound of the system capacity equals a demand vector subject to the budget constraint. An upper boundary point is a vector representing the capacities of arcs and nodes, and is the maximal vector exactly meeting the demand vector. A simple algorithm based on minimal cuts (or named MC-based algorithm) is then presented to generate all upper boundary points in order to evaluate the performance index. The storage and computational time complexity of this algorithm are also analyzed. The performance evaluation for the multicommodity case can be extended easily.