A NEW FINITE ELEMENT APPROXIMATION OF A STATE-CONSTRAINED OPTIMAL CONTROL PROBLEM

In this paper, we study numerical methods for an optimal control problem with pointwise state constraints. The traditional approaches often need to deal with the deltasingularity in the dual equation, which causes many difficulties in its theoretical analysis and numerical approximation. In our new approach we reformulate the state-constrained optimal control as a constrained minimization problems only involving the state, whose optimality condition is characterized by a fourth order elliptic variational inequality. Then direct numerical algorithms （nonconforming finite element approximation） are proposed for the inequality, and error estimates of the finite element approximation are derived. Numerical experiments illustrate the effectiveness of the new approach.     

A Helly-Type Theorem for Intersections of Compact Connected Sets in the Plane

For a family C of nonempty compact sets in the plane, the following conditions are equivalent:(1) Every two (not necessarily distinct) members of C have a connected union and every three (not necessarily distinct) members of C have a simply connected union.(2) C is a family of simply connected sets such that every two (not necessarily distinct) members of C have a connected intersection and every three (not necessarily distinct) members of C have a nonempty intersection.If either set of conditions is satisfied, then { C : C in C } is nonempty, simply connected, and connected. Furthermore, if the collection C is finite, then it is also true that { C : C in C } is simply connected.     

Dimensions of staircase kernels in orthogonal polygons

LetS be a simply connected orthogonal polygon in the plane. If every four points ofS see (via staircase paths inS) a common segment, then the staircase kernel ofS contains a segment as well. A parallel result holds when every four points ofS see (via staircase paths inS) a common 2-dimensional set. In each case, the number 4 is best possible. However, both results fail without the requirement that setS be simply connected.An example shows that no Helly-type analogue is possible for intersections of orthogonally convex sets in the plane.     

A family of examples showing that no Krasnosel’skii number exists for orthogonal polygons starshaped via staircase n-paths

Let S be a simply connected orthogonal polygon in the plane. A family of examples will establish the following result. For every n ≥ 2, there exists no Krasnosel’skii number h(n) which satisfies this property: If every h(n) points of S are visible via staircase n-paths from a common point, then S is starshaped via staircase n-paths.     

Enhanced para-xylene selectivity in the toluene alkylation reaction at ultralow contact time

Dramatic improvements in the para-xylene selectivity of the toluene alkylation reaction can be effected by operating the catalytic reaction at ultralow contact time. Unexpectedly, the rate of alkylation is sustained, while unwanted side reactions are suppressed. By demonstrating that contact time directly influences the fate of para-xylene, which is known to form and diffuse preferentially within the zeolite catalyst, we conclude that external mass transfer is a key parameter in controlling selectivity. Even non-optimized catalysts can be made to achieve near-perfect selectivity, without sacrificing conversion.     

The combinatorial structure of (m,n)-convex sets

LetS be a closed subset of a Hausdorff linear topological space,S having no isolated points, and letc _s (m) denote the largest integern for whichS is (m,n)-convex. Ifc _s (k)=0 andc _s (k+1)=1, then $$ c_s \left( m \right) = \sum\limits_{i = 1}^k {\left( {\begin{array}{*{20}c} {\left[ {\frac{{m + k - i}} {k}} \right]} \\ 2 \\ \end{array} } \right)} $$ . Moreover, ifT is a minimalm subset ofS, the combinatorial structure ofT is revealed.     

Intersectional properties concerning planarm-convex sets

LetC be a collection of closed sets in the plane, and letS=∩C. (1) If IncC ⊆ kerS for allC inC and if dim kerS≧1, thenS is a union of three (or fever) convex sets. In particular, the results holds when the members ofC are 3-convex sets, all having the same kernelK, provided dimK≧1. (2) IfC is a finite collection ofm-convex sets such that ∩{kerC:C inC inC} ≠ ⊘,S~ IncS is connected, and for someZ inC, lncC⊆ lncZ for allC inC, thenS ism-convex.     

AnR danalogue of Valentine’s theorem on 3-convex sets

This paper deals with anR ^danalogue of a theorem of Valentine which states that a closed 3-convex setS in the plane is decomposable into 3 or fewer closed convex sets. In Valentine’s proof, the points of local nonconvexity ofS are treated as vertices of a polygonP contained in the kernel ofS, yielding a decomposition ofS into 2 or 3 convex sets, depending on whetherP has an even or odd number of edges. Thus the decomposition actually depends onc(P′), the chromatic number of the polytopeP′ dual toP. A natural analogue of this result is the following theorem: LetS be a closed subset ofR ^d, and letQ denote the set of points of local nonconvexity ofS. We require thatQ be contained in the kernel ofS and thatQ coincide with the set of points in the union of all the (d − 2)-dimensional faces of somed-dimensional polytopeP. ThenS is decomposable intoc(P′) closed convex sets.     

A better decomposition theorem for simply connected m-convex sets

A set S in R^d is said to be m-convex, m ? 2, if and only if for every m points in S, at least one of the line segments determined by these points lies in S. For S a closed m-convex set in R², various decomposition theorems have been obtained to express S as a finite union of convex sets. However, the previous bounds may be lowered further, and we have the following result:In case S is simply connected, then S is a union of σ(m) or fewer convex sets, where $\text{σ(m) = [(m ?}\text{N}\text{)(m ?}\text{3}\text{2}\text{) +}\text{3}\text{2}\text{]}$.Moreover, this result induces an improved decomposition in the general case as well.