Optimal control of nonlinear evolution inclusions

In this paper, we study the optimal control of nonlinear evolution inclusions. First, we prove the existence of admissible trajectories and then we show that the set that they form is relatively sequentially compact and in certain cases sequentially compact in an appropriate function space. Then, with the help of a convexity hypothesis and using Cesari's approach, we solve a general Lagrange optimal control problem. After that, we drop the convexity hypothesis and pass to the relaxed system, for which we prove the existence of optimal controls, we show that it has a value equal to that of the original one, and also we prove that the original trajectories are dense in an appropriate topology to the relaxed ones. Finally, we present an example of a nonlinear parabolic optimal control that illustrates the applicability of our results.This research was supported by NSF Grant No. DMS-88-02688.     

Discontinuous local semiflows for Kurzweil equations leading to LaSalle's invariance principle for differential systems with impulses at variable times

In this paper, we consider an initial value problem for a class of generalized ODEs, also known as Kurzweil equations, and we prove the existence of a local semidynamical system there. Under certain perturbation conditions, we also show that this class of generalized ODEs admits a discontinuous semiflow which we shall refer to as an impulsive semidynamical system. As a consequence, we obtain LaSalle's invariance principle for such a class of generalized ODEs. Due to the importance of LaSalle's invariance principle in studying stability of differential systems, we include an application to autonomous ordinary differential systems with impulse action at variable times.     

An approach for solving of a moving boundary problem

In this paper we shall study moving boundary problems, and we introduce an approach for solving a wide range of them by using calculus of variations and optimization. First, we transform the problem equivalently into an optimal control problem by defining an objective function and artificial control functions. By using measure theory, the new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; then we obtain an optimal measure which is then approximated by a finite combination of atomic measures and the problem converted to an infinite-dimensional linear programming. We approximate the infinite linear programming to a finite-dimensional linear programming. Then by using the solution of the latter problem we obtain an approximate solution for moving boundary function on specific time. Furthermore, we show the path of moving boundary from initial state to final state.     

On a constructive approximation of the efficient outcomes in bicriterion vector optimization

For bicriterion quasiconvex optimization problems, we present a constructive procedure for an approximation of the efficient outcomes. Performing this procedure we can estimate the accuracy of the approximation. Conversely, if we prescribe an accuracy for the approximation, we can calculate the number of points which have to be computed by a certain scalarization method to remain under the given accuracy. Finally, we give a numerical example.     

Solving some optimal path planning problems using an approach based on measure theory

In this paper we solve a collection of optimal path planning problems using a method based on measure theory. First we consider the problem as an optimization problem and then we convert it to an optimal control problem by defining some artificial control functions. Then we perform a metamorphosis in the space of problem. In fact we define an injection between the set of admissible pairs, containing the control vector function and a collision-free path defined on free space and the space of positive Radon measures. By properties of this kind of measures we obtain a linear programming problem that its solution gives rise to constructing approximate optimal trajectory of the original problem. Some numerical examples are proposed.     

An inner approximation method incorporating a branch and bound procedure for optimization over the weakly efficient set

In this paper, we consider an optimization problem which aims to minimize a convex function over the weakly efficient set of a multiobjective programming problem. From a computational viewpoint, we may compromise our aim by getting an approximate solution of such a problem. To find an approximate solution, we propose an inner approximation method for such a problem. Furthermore, in order to enhance the efficiency of the solution method, we propose an inner approximation algorithm incorporating a branch and bound procedure.     

A geometrical construction of inverse points with respect to an ellipse

In this note, we introduce the inverse point respect to an ellipse, which generalizes the classical inversion in a circle. Specifically, we show a geometrical construction of inverse points with respect to an ellipse. For this, we extend the classical method to construct the inverse of a point respect to a circle.     

An axiomatic description of the Strambach planes

In this paper we generalize the construction of Strambach planes to an arbitrary ordered Euclidean field, we describe collineation groups of the obtained structures, and we give an axiomatic characterization of them.     

紧度量空间上的连续在上映射是粘合映射

本文讨论了有关粘合映射的一个问题 ,证明了 ,如果X是紧致度量空间 ,Y是度量空间 ,则由X到Y的连续在上映射是粘合映射 .并给出了一个反例 ,说明 :如果去掉紧致性条件 ,则定理的结论不再成立 .     

On essential sets and essential components of efficient solutions for vector optimization problems

In this paper, we first introduce the notions of an essential set and an essential component of the set of efficient solutions for continuous vector optimizations on a nonempty compact subset of a metric space. Then we show that for each of these vector optimizations, each set of all efficient solutions corresponding to the same optimal values is essential. Basing on this result, we give full characterizations of an essential point, an essential set and an essential component, respectively. As an application, we prove that for continuous quasiconvex vector optimization problems on a nonempty compact subset of a metric vector space, each component of the set of efficient solutions is essential even though the efficient solution set is not connected.     

Optimal control of a two-dimensional contact problem

We consider a frictionless contact problem with unilateral constraints for a 2D bar. We describe the problem, then we derive its weak formulation, which is in the form of an elliptic variational inequality of the first kind. Next, we establish the existence of a unique weak solution to the problem and prove its continuous dependence with respect to the applied tractions and constraints. We proceed with the study of an associated control problem for which we prove the existence of an optimal pair. Finally, we consider a perturbed optimal control problem for which we prove a convergence result.     

An exact penalty method for weak linear bilevel programming problem

In this paper, we present an exact penalty method, which is different from the existing penalty method, for solving weak linear bilevel programming problem. Then, we establish an existence result of solutions for such a problem. Finally, we propose an algorithm and give two examples to illustrate its feasibility.     

Bigeneric initial ideals, diagonal subalgebras and bigraded Hilbert functions

In this paper we study some problems concerning bigraded ideals. By introducing the concept of bigeneric initial ideal, we answer an open question about diagonal subalgebras and we give a necessary condition for a function to be the bigraded Hilbert function of a bigraded algebra. Moreover, we give an upper bound for the regularity of a bistable ideal in terms of the degrees of its generators.     

Perturbation des Resolvantes Triangulaires et Balayage des Couples Excessifs

In this work, we start by developing an elementary potential theory associated to a triangular kernel. Then we study the perturbation of triangular resolvent, which enable us to characterize, with an elementary way, the excessive couples relatively to a triangular resolvent U. Finally, we study the balayage of U-surmedians couples and the triangular resolvents in duality.     

随机环境中的马氏链的不变测度与遍历性

本文考虑了一类特殊的随机环境的马氏链。假设随机“Doeblin”条件成立，我们证明了随机环境的马氏链的不变测度存在，且任何初始分布以指数收敛速度到些不变测度。进一步的，存在关于绕积算子遍历的不变测度。最后，我们得到了随机马氏链的强大数定律。     

基于微分算子的边缘检测法在基因芯片图像识别中的应用

一种基于梯度算子和拉普拉斯算子检测医学基因芯片荧光数字图像边缘的方法．数字图像经过图像增强等预处理以后，突出了我们感兴趣的数字图像特征，根据数字图像灰度变化的趋势，用灰度变化的一阶导数(梯度算子)检测图象中的边缘区域；用灰度变化的二阶导数确定图像边缘的中心位置．最后根据某些预定的准则把满足这些准则的象素组成一条边缘．     

Asymptotical feedback controllability of continuous-time probabilistic logic control networks

In this article, we study the controllability of continuous-time probabilistic logic control networks (CT-PLCNs) under sampled-data feedback controls (SDFCs). First, we demonstrate that the concept of finite-time controllability with probability one for discrete-time probabilistic logic control networks cannot be generalized to CT-PLCNs. Then, we propose the concepts of asymptotical feedback reachability and asymptotical feedback controllability for CT-PLCNs. Based on the invariant subsets, we prove that a target state is asymptotically feedback reachable if and only if the target state is a control equilibrium point and any initial state has an admissible path to the target state. Moreover, we introduce the concept of reachability matrix and propose an easily verifiable criterion for asymptotical feedback reachability expressed in terms of the reachability matrix. Based on these, we prove that a CT-PLCN is asymptotically feedback controllable if and only if every state is a control equilibrium point and there is an admissible path between any pair of initial and target states. The relation between controllability and stabilizability is also discussed. We prove that a CT-PLCN is asymptotically feedback controllable if and only if every state is asymptotically feedback stabilizable. For a controllable CT-PLCN, we propose an algorithm of designing a stabilizing SDFC for any given target state. Additionally, we discuss the asymptotical feedback controllability of CT-PLCNs under time-varying nonuniform SDFCs. Finally, an illustrative example is presented to explain the proposed methods and verify the controllability criteria.     

Representations of torsion-free arithmetic matroids

We study the representability problem for torsion-free arithmetic matroids. After introducing a “strong gcd property” and a new operation called “reduction”, we describe and implement an algorithm to compute all essential representations, up to equivalence. As a consequence, we obtain an upper bound to the number of equivalence classes of representations. In order to rule out equivalent representations, we describe an efficient way to compute a normal form of integer matrices, up to left-multiplication by invertible matrices and change of sign of the columns (we call it the “signed Hermite normal form”). Finally, as an application of our algorithms, we disprove two conjectures about the poset of layers and the independence poset of a toric arrangement.     

The explicit solution of the profit maximization problem with box-constrained inputs

In this paper we study the profit-maximization problem, considering maximum constraints for the general case of m-inputs and using the Cobb-Douglas model for the production function. To do so, we previously study the firm’s cost minimization problem, proposing an equivalent infimal convolution problem for exponential-type functions. This study provides an analytical expression of the production cost function, which is found to be a piece-wise potential. Moreover, we prove that this solution belongs to class C¹. Using this cost function, we obtain the explicit expression of maximum profit. Finally, we illustrate the results obtained in this paper with an example.