Wiener estimates for a class of systems of parabolic variational inequalities

Summary Wiener estimates at a point for parabolic diagonal systems of parabolic variational inequalities with obstacle are proved by a Green function method.This paper was written while the first Author was visiting the Department of Mathematics of Linköping University.     

Duality applied to contact problems with friction

The duality theory of Mosco, Capuzzo-Dolcetta, and Matzeu for variational and quasi-variational inequalities is extended. Then it is applied to two problems of contact with friction of an elastic body with a rigid foundation. The more realistic normal compliance condition is used in place of Signorini's condition on the contact surface.We (A.M. and M.S.) are grateful for the financial support and hospitality of the Department of Mechanical Engineering in the Linköping Institute for Technology during our two weeks stay in August 1988. In addition M.S. was partially supported by the Oakland University Faculty Research Award.     

Sufficient conditions for variational problems with delayed argument

Various first-order and second-order sufficient conditions of optimality for calculus of variations problems with delayed argument are formulated. The cost functionals are not required to be convex. A second-order sufficient condition is shown to be related to the existence of solutions of a Riccati-type matrix differential inequality.     

A sublogarithmic convex hull algorithm

We present a parallel algorithm for finding the convex hull of a sorted set of points in the plane. Our algorithm runs inO(logn/log logn) time usingO(n log logn/logn) processors in theCommon crcw pram computational model, which is shown to be time and cost optimal. The algorithm is based onn ^1/3 divide-and-conquer and uses a simple pointer-based data structure.Part of this work was done when the last three authors were at the Department of Computer and Information Science, Linköping University. The research of the second author was supported by the Academy of Finland.     

Optimality conditions for the control of a double-membrane complex system by a modal decomposition technique

The optimal of damping out the oscillations of an elastically rectangular double-membrane system by means of point-wise actuators is solved analytically. The membrane is clamped along the boundaries. The motion of the system is initiated by given initial displacement and velocity conditions. The basic control problem is to minimize the deflection and the velocity of displacements at a specified time with the minimum expenditure of actuation energy. A quadratic performance functional is chosen as the cost functional which comprises the functionals of the deflection, velocity and the point-wise actuators. Necessary and sufficient conditions of optimality are investigated. The necessary conditions of optimality are obtained from a variational approach and formulated in the form of degenerate integrals which lead to explicit optimal control laws for the actuators. Numerical results are given for various problem parameters and the efficiency of the control mechanism is investigated.     

Stability and sensitivity of solutions to optimal control problems for systems with control appearing linearly

We consider a family of optimal control problems for systems described by nonlinear ordinary differential equations with control appearing linearly. The cost functionals and the control constraints are convex. All data depend on a vector parameter.Using the concept of the second-order sufficient optimality conditions it is shown that the solutions of the problems, as well as the associated Lagrange multipliers, are locally Lipschitz continuous and directionally differentiable functions of the parameter.     

Stability estimates in identification problems for the convection-diffusion-reaction equation

Identification problems for the stationary convection-diffusion-reaction equation in a bounded domain with a Dirichlet condition imposed on the boundary of the domain are studied. By applying an optimization method, these problems are reduced to inverse extremum problems in which the variable diffusivity and the volume density of substance sources are used as control functions. Their solvability is proved for an arbitrary weakly lower semicontinuous cost functional and particular cost functionals. An analysis of the optimality system is used to establish sufficient conditions on the input data under which the solutions of particular extremum problems are unique and stable with respect to small perturbations in the cost functional and in one of the functions involved in the boundary value problem.     

Polyhedral annexaton,dualization and dimension reduction technique in global optimization

We demonstrate how the size of certain global optimization problems can substantially be reduced by using dualization and polyhedral annexation techniques. The results are applied to develop efficient algorithms for solving concave minimization problems with a low degree of nonlinearity. This class includes in particular nonconvex optimization problems involving products or quotients of affine functions in the objective function.This work was completed while the author was visiting the Department of Mathematics of Linköping University.     

Coefficient inverse extremum problems for stationary heat and mass transfer equations

A technique is developed for analyzing coefficient inverse extremum problems for a stationary model of heat and mass transfer. The model consists of the Navier-Stokes equations and the convection-diffusion equations for temperature and the pollutant concentration that are nonlinearly related via buoyancy in the Boussinesq approximation and via convective heat and mass transfer. The inverse problems are stated as the minimization of certain cost functionals at weak solutions to the original boundary value problem. Their solvability is proved, and optimality systems describing the necessary optimality conditions are derived. An analysis of the latter is used to establish sufficient conditions ensuring the local uniqueness and stability of solutions to the inverse extremum problems for particular cost functionals.     

Characterization of the Darboux point for particular classes of problems

A minimal sufficient condition for global optimality involving the Darboux point, analogous to the minimal sufficient condition of local optimality involving the conjugate point, is presented. The Darboux point is then characterized for optimal control problems with linear dynamics, cost functionals with a general terminal state term and an integrand quadratic in the state and control, and general terminal conditions. The Darboux point is shown to be the supremum of a sequence of conjugate points. If the terminal state term is quadratic, along with a scalar quadratic boundary condition, then the Darboux point is also the time at which the Riccati matrix becomes unbounded, giving a characterization of the unboundedness of the Riccati matrix at points which are not in general conjugate points.This research was supported by the National Science Foundation under Grant No. GK-30115.This is Definition 2.1 of Ref. 1.     

Vortex control in channel flows using translational invariant cost functionals

The use of translation invariant cost functionals for the reduction of vortices in the context of shape optimization of fluid flow domain is investigated. Analytical expressions for the shape design sensitivity involving different cost functionals are derived. Channel flow problems with a bump and an obstacle as possible control boundaries are taken as test examples. Numerical results are provided in various graphical forms for relatively low Reynolds numbers. Striking differences are found for the optimal shapes corresponding to the different cost functionals, which constitute different quantification of a vortex.     

拟线性椭圆变分不等式的障碍最优控制问题

对拟线性椭圆变分不等式的障碍最优控制问题(即以障碍为控制变量)进行了研究．指标泛函为Lagrange型,其中含有控制变量二阶导数的p次幂,这使得最优性条件的推导颇为不易．对所考虑的问题给出了最优控制的存在性定理以及必要条件．     

A global optimization approach for the linear two-level program

Linear two-level programming deals with optimization problems in which the constraint region is implicity determined by another optimization problem. Mathematical programs of this type arise in connection with policy problems to which the Stackelberg leader-follower game is applicable. In this paper, the linear two-level programming problem is restated as a global optimization problem and a new solution method based on this approach is developed. The most important feature of this new method is that it attempts to take full advantage of the structure in the constraints using some recent global optimization techniques. A small example is solved in order to illustrate the approach.The paper was completed while this author was visiting the Department of Mathematics of Linköping University.     

Vortex control of instationary channel flows using translation invariant cost functionals

The use of translation invariant cost functionals for the reduction of vortices in the context of shape optimization of fluid flow domains is investigated. Analytical expressions for the shape design sensitivity involving different cost functionals are derived. Instationary channel flow problems with a bump and an obstacle as possible control boundaries are taken as test examples. Numerical results are provided in various graphical forms for relatively low Reynolds numbers. Striking differences are found for the optimal shapes corresponding to the different cost functionals, which constitute different quantification of a vortex.     

The Legendre condition of the fractional calculus of variations

Fractional operators play an important role in modelling nonlocal phenomena and problems involving coarse-grained and fractal spaces. The fractional calculus of variations with functionals depending on derivatives and/or integrals of noninteger order is a rather recent subject that is currently in fast development due to its applications in physics and other sciences. In the last decade, several approaches to fractional variational calculus were proposed by using different notions of fractional derivatives and integrals. Although the literature of the fractional calculus of variations is already vast, much remains to be done in obtaining necessary and sufficient conditions for the optimization of fractional variational functionals, existence and regularity of solutions. Regarding necessary optimality conditions, all works available in the literature concern the derivation of first-order fractional conditions of Euler–Lagrange type. In this work, we obtain a Legendre second-order necessary optimality condition for weak extremizers of a variational functional that depends on fractional derivatives.     

First and second-order optimality conditions using approximations for vector equilibrium problems with constraints

We consider various kinds of solutions to nonsmooth vector equilibrium problems with functional constraints. By using first and second-order approximations as generalized derivatives, we establish both necessary and sufficient optimality conditions. Our first-order conditions are shown to be applicable in many cases, where existing ones cannot be used. The second-order conditions are new.     

On the [28, 7, 12] binary self-complementary codes and their residuals

Recently Jungnickel and Tonchev have shown that there exist at least four inequivalent binary selfcomplementary [28, 7, 12] codes and have asked if there are other [28, 7] codes with weight distributionA ₀=A ₂₈=1,A ₁₂=A ₁₆=63. In the present paper we give a negative answer: these four codes are, up to equivalence, the only codes with the given parameters. Their residuals are also classified.This work has been done during the visit of the author as a guest researcher to the Department of Electrical Engineering at Linköping University, Sweden. The research was partially supported by the Bulgarian National Science Foundation under contracts No. 37/1987 and No. 876/1988.     

Modelling of topological derivatives for contact problems

The problem of topology optimization is considered for free boundary problems of thin obstacle types. The formulae for the first term of asymptotics for energy functionals are derived. The precision of obtained terms is verified numerically. The topological differentiability of solutions to variational inequalities is established. In particular, the so-called outer asymptotic expansion for solutions of contact problems in elasticity with respect to singular perturbation of geometrical domain depending on small parameter are derived by an application of nonsmooth analysis. Such results lead to the topological derivatives of shape functionals for contact problems. The topological derivatives are used in numerical methods of simultaneous shape and topology optimization. Partially supported by the grant 4 T11A 01524 of the State Committee for the Scientific Research of the Republic of Poland     

A variational approach to optimal meshes

Summary. A variational approach for the optimization of triangular or tetrahedral meshes is presented. Starting from some very basic assumptions we will rigorously demonstrate that the functional controlling optimality is of a certain type related to energy functionals in non linear elasticity. It will be proved that these functionals attain their minima over admissible sets of mesh deformations which respect boundary conditions. In addition the injectivity of the deformed mesh is discussed. Thereby it is possible to construct suitable meshes for various numerical applications. Received March 14, 1994 / Revised version received August 8, 1994     

Variational inequalities with varying obstacles: The general form of the limit problem

When minimizing a functional of the calculus of variations with varying constraints of obstacle type, it may happen that, in the limit problem (when it exists), the constraint is no longer of obstacle type; it may take a “relaxed” form. This situation is analysed in this paper. Following De Giorgi's Γ-convergence approach, a class of functionals is constructed which is compact and which contains the obstacle constraint functionals. In this sense, this class of problems has been closed.