Über Normminimale Steuerungen von L-Prozessen und Lineare Optimierung bei Normbeschränkten Steuerungen

This paper is devoted to two problems in the theory of optimal control for linear processes. The first one is characterized by a cost of the form ess sup {p(u(t)):t∈[a, b]}, whereby p denotes the distance function of a compact convex set C ° ?^m containing the origin as an interior point and u:[a, b] → ?^m represents the control. In the second problem the cost depends linear on the controls, which are limited by a bound for ess sup {p(u(t)):t∈[a, b]}. There will be proved two duality theorems leading to a method for the construction of optimal controls in the case of a strict convex C. For linear processes defined by piecewise analytic functions these controls are piecewise continuous.     

A Markovian canonical form of second-order matrix-exponential processes

Besides the fact that – by definition – matrix-exponential processes (MEPs) are more general than Markovian arrival processes (MAPs), only very little is known about the precise relationship of these processes in matrix notation. For the first time, this paper proves the persistent conjecture that – in two dimensions – the respective sets, MAP(2) and MEP(2), are indeed identical with respect to the stationary behavior. Furthermore, this equivalence extends to acyclic MAPs, i.e., AMAP(2), so that AMAP(2)≡MAP(2)≡MEP(2)$\mathrm{AMAP}(2)\equiv \mathrm{MAP}(2)\equiv \mathrm{MEP}(2)$. For higher orders, these equivalences do not hold.     

Size-determining dependencies in supramolecular organometallic host-guest chemistry

Treatment of the pentaphosphaferrocene [Cp*Fe(η⁵‐P₅)] with Cu^I halides in the presence of different templates leads to novel fullerene‐like spherical molecules that serve as hosts for the templates. If ferrocene is used as the template the 80‐vertex ball [Cp₂Fe]@[{Cp*Fe(η⁵‐P₅)}₁₂{CuCl}₂₀] ( 4 ), with an overall icosahedral C₈₀ topological symmetry, is obtained. This result shows the ability of ferrocene to compete successfully with the internal template of the reaction system [Cp*Fe(η⁵‐P₅)], although the 90‐vertex ball [{Cp*Fe(η⁵:η¹:η¹:η¹:η¹:η¹‐P₅)}₁₂(CuCl)₁₀(Cu₂Cl₃)₅{Cu(CH₃CN)₂}₅] ( 2 a ) containing pentaphosphaferrocene as a guest is also formed as a byproduct. With use of the triple‐decker sandwich complex [(CpCr)₂(μ,η⁵‐As₅)] as a template the reaction between [Cp*Fe(η⁵‐P₅)] and CuBr leads to the 90‐vertex ball [(CpCr)₂(μ,η⁵‐As₅)]@[{Cp*Fe(η⁵‐P₅)}₁₂{CuBr}₁₀{Cu₂Br₃}₅{Cu(CH₃CN)₂}₅] ( 6 ), in which the complete molecule acts as a template. However, if the corresponding reaction is instead carried out with CuCl, cleavage of the triple‐decker complex is found and the 80‐vertex ball [CpCr(η⁵‐As₅)]@[{Cp*Fe(η⁵‐P₅)}₁₂{CuCl}₂₀] ( 5 ) is obtained. This accommodates as its guest [CpCr(η⁵‐As₅)], which has only 16 valence electrons in a triplet ground state and is not known as a free molecule. The triple‐decker sandwich complex [(CpCr)₂(μ,η⁵‐As₅)] requires 53.1 kcal mol^?1 to undergo cleavage (as calculated by DFT methods) and therefore this reaction is clearly endothermic. All new products have been characterized by single‐crystal X‐ray crystallography. A favoured orientation of the guest molecules inside the host cages has been identified, which shows π???π stacking of the five‐membered rings (Cp and cyclo‐As₅) of the guests and the cyclo‐P₅ rings of the nanoballs of the hosts.