Conjugacy for normal or abnormal linear Bolza problems

For certain Bolza problems with linear dynamics, two sets extending the notion of conjugate points in the calculus of variations are introduced. Independently of nonsingularity assumptions, their emptiness, in one case without normality assumptions, is shown to be equivalent to a second order necessary condition. A comparison with other notions available in the literature is given.     

Bang-bang property for time optimal control of semilinear heat equation

This paper studies the bang-bang property for time optimal controls governed by semilinear heat equation in a bounded domain with control acting locally in a subset. Also, we present the null controllability cost for semilinear heat equation and an observability estimate from a positive measurable set in time for the linear heat equation with potential.     

An existence theorem for neutral variational problems of Bolza

This note proves an existence theorem for a generalized Bolza-type problem that has time delays in both the state and velocity variables. The assumptions are stated in terms of a modification of the classical Hamiltonian, and extend ideas of Rockafellar to the delay case.     

On free boundary problems in two dimensions

Let L be a linear elliptic operator in two dimensions with analytic coefficients and of second order, andu(x, y) a solution of Lu=0 in a simply connected domain ω with rectifiable boundary Γ. Suppose ψ(x, y) analytic on ω∪Γ and L ψ≠0 there.H is shown that ifu and ψ coincide with first derivatives on an open portion Γ₀ of Γ, then Γ₀ permits the representation λ=x (θ),y=y (θ) withx(θ),y(θ)analytic functions of a real parameter θ.     

Normality of the maximum principle for nonconvex constrained Bolza problems

We consider a Bolza optimal control problem with state constraints. It is well known that under some technical assumptions every strong local minimizer of this problem satisfies first order necessary optimality conditions in the form of a constrained maximum principle. In general, the maximum principle may be abnormal or even degenerate and so does not provide a sufficient information about optimal controls. In the recent literature some sufficient conditions were proposed to guarantee that at least one maximum principle is nondegenerate, cf. [A.V. Arutyanov, S.M. Aseev, Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints, SIAM J. Control Optim. 35 (1997) 930–952; F. Rampazzo, R.B. Vinter, A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control, IMA 16 (4) (1999) 335–351; F. Rampazzo, R.B. Vinter, Degenerate optimal control problems with state constraints, SIAM J. Control Optim. 39 (4) (2000) 989–1007]. Our aim is to show that actually conditions of a similar nature guarantee normality of every nondegenerate maximum principle. In particular we allow the initial condition to be fixed and the state constraints to be nonsmooth. To prove normality we use J. Yorke type linearization of control systems and show the existence of a solution to a linearized control system satisfying new state constraints defined, in turn, by linearization of the original set of constraints along an extremal trajectory.     

A new integral representation for quasi-periodic scattering problems in two dimensions

Boundary integral equations are an important class of methods for acoustic and electromagnetic scattering from periodic arrays of obstacles. For piecewise homogeneous materials, they discretize the interface alone and can achieve high order accuracy in complicated geometries. They also satisfy the radiation condition for the scattered field, avoiding the need for artificial boundary conditions on a truncated computational domain. By using the quasi-periodic Green’s function, appropriate boundary conditions are automatically satisfied on the boundary of the unit cell. There are two drawbacks to this approach: (i) the quasi-periodic Green’s function diverges for parameter families known as Wood’s anomalies, even though the scattering problem remains well-posed, and (ii) the lattice sum representation of the quasi-periodic Green’s function converges in a disc, becoming unwieldy when obstacles have high aspect ratio.     

Exponentially fitted shape functions for advection-dominated flow problems in two dimensions

We sketch out the basic properties of three novel exponentially fitted shape functions that generalize to two-dimensional triangular elements the one-dimensional functions used in the well-known Scharfetter-Gummel method. This scheme is widely employed in the finite element approximation of the drift-diffusion equations arising in semiconductor device modelling.     

Existence theorems for general control problems of Bolza and Lagrange

The existence of solutions is established for a very general class of problems in the calculus of variations and optimal control involving ordinary differential equations or contingent equations. The theorems, while relatively simple to state, cover, besides the more classical cases, problems with considerably weaker assumptions of continuity or boundedness. For example, the cost functional may only be lower semicontinuous in the control and may approach + ∞ as one nears certain boundary points of the control region; both endpoints in the problem may be “free”. Earlier results of Cesari, Olech and the author are thereby extended.The development is based on the theory of convex integral functionals and their conjugates. The first step is to show that, for purposes of existence theory, the problem can be reduced to a simpler model where control variables are not present as such. This model, resembling a classical problem of Bolza in the calculus of variations, but where the functions are extended-real-valued, is then investigated using, above all, the conjugacy correspondence between generalized Lagrangians and Hamiltonians.     

Structure of minimizers in linear-quadratic Bolza problems of optimal control

In this paper, we study intersections of extremals in a linear-quadratic Bolza problem of optimal control. The structure of the inter-sections is described. We show that this structure implies the semipositive definiteness of the quadratic cost functional. In addition, we derive necessary and sufficient conditions for the existence of minimizers.     

Interval analysis techniques for boundary value problems of elasticity in two dimensions

In this paper we prove that the L² spectral radius of the traction double layer potential operator associated with the Lamé system on an infinite sector in R² is within 10⁻² from a certain conjectured value which depends explicitly on the aperture of the sector and the Lamé moduli of the system. This type of result is relevant to the spectral radius conjecture, cf., e.g., Problem 3.2.12 in [C.E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Reg. Conf. Ser. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1994]. The techniques employed in the paper are a blend of classical tools such as Mellin transforms, and Calderón-Zygmund theory, as well as interval analysis—resulting in a computer-aided proof.     

Methods for solving combined two level location-routing problems of realistic dimensions

Very few distribution studies have dealt with combined location-routing problems, the problems of locating depots from which customers are served by tours rather than individual trips.This paper gives a survey of methods solving combined location-routing problems. Some methods are analyzed and three new heuristic methods are developed, implemented and compared.A newspaper delivery system consisting of 4500 customers is solved. The results seem to indicate that an alternate location-allocation-savings procedure and a saving-drop procedure are promising.     

Extremal problems on triangle areas in two and three dimensions

The study of extremal problems on triangle areas was initiated in a series of papers by Erd?s and Purdy in the early 1970s. In this paper we present new results on such problems, concerning the number of triangles of the same area that are spanned by finite point sets in the plane and in 3-space, and the number of distinct areas determined by the triangles.In the plane, our main result is an O(n44/19)=O(n^2.3158) upper bound on the number of unit-area triangles spanned by n points, which is the first breakthrough improving the classical bound of O(n7/3) from 1992. We also make progress in a number of important special cases. We show that: (i) For points in convex position, there exist n-element point sets that span Ω(nlogn) triangles of unit area. (ii) The number of triangles of minimum (nonzero) area determined by n points is at most ; there exist n-element point sets (for arbitrarily large n) that span (6/π²−o(1))n² minimum-area triangles. (iii) The number of acute triangles of minimum area determined by n points is O(n); this is asymptotically tight. (iv) For n points in convex position, the number of triangles of minimum area is O(n); this is asymptotically tight. (v) If no three points are allowed to be collinear, there are n-element point sets that span Ω(nlogn) minimum-area triangles (in contrast to (ii), where collinearities are allowed and a quadratic lower bound holds).In 3-space we prove an O(n17/7β(n))=O(n^2.4286) upper bound on the number of unit-area triangles spanned by n points, where β(n) is an extremely slowly growing function related to the inverse Ackermann function. The best previous bound, O(n8/3), is an old result of Erd?s and Purdy from 1971. We further show, for point sets in 3-space: (i) The number of minimum nonzero area triangles is at most n²+O(n), and this is worst-case optimal, up to a constant factor. (ii) There are n-element point sets that span Ω(n4/3) triangles of maximum area, all incident to a common point. In any n-element point set, the maximum number of maximum-area triangles incident to a common point is O(n4/3+ε), for any ε>0. (iii) Every set of n points, not all on a line, determines at least Ω(n2/3/β(n)) triangles of distinct areas, which share a common side.     

A heuristic for solving large bin packing problems in two and three dimensions

The more-dimensional bin packing problem (BPP) considered here requires packing a set of rectangular-shaped items into a minimum number of identical rectangular-shaped bins. All items may be rotated and the guillotine cut constraint has to be respected. A straightforward heuristic is presented that is based on a method for the container loading problem following a wall-building approach and on a method for the one-dimensional BPP. 1,800 new benchmark instances are introduced for the two-dimensional and three-dimensional BPP. The instances include more than 1,500 items on average. Applied to these very large instances, the heuristic generates solutions of acceptable quality in short computation times. Moreover, the influence of different instance parameters on the solution quality is investigated by an extended computational study.     

Corner singularity for transmission problems in three dimensions

For a transmission problem for the Laplace operator the unique solvability is proved in natural Sobolev spaces in the case when edges and corners are present. The behaviour of the solution near the corner is reduced to the question when an explicitely given meromorphic family of one-dimensional integral operators on a geodesic polygon on the two sphere has a non-trivial kernel.     

Semi-linear problems in infinite dimensions

We prove existence, smoothness and ergodicity results for semilinear parabolic problems on infinite dimensional spaces assuming the Logarithmic Sobolev inequality is satisfied. As a consequence we construct a class of nonlinear Markov semigroup which are hypercontractive.     

Existence Theory for a Stochastic Bolza Problem

The direct method is used to prove the existence of optimalopen-loop controls for stochastic optimal control problems inBolza form. Instead of requiring the control set to be compact,as previous authors have done, we impose a superquadratic growthcondition on the running cost. Degeneracy of the diffusion coefficientis allowed by this approach—in fact, the main result reducesto a well known deterministic theorem when the diffusion coefficientis zero.     

Bang-bang optimal control for the dam problem

The dam problem with general geometry is considered. Fluid is drawn from the bottomS ₁ at a ratek where 0 k N, _{S 1} k M; the objective is to minimize the total pressure of the fluid in the dam. A bang-bang principle is established for any optimal controlk ₀, that is,k ₀ = 0 on a setA andk ₀ =N on the complement setS ₁ A. In the case of a rectangular dam the structure ofA is determined and the uniqueness of the minimizerk ₀ is established.This work is partially supported by National Science Foundation Grants DMS-8501397 and DMS-8420896.