Observer-based output feedback control design for a fractional ODE and a fractional PDE cascaded system.

This paper considers a new bi-directional cascaded system of a fractional ordinary differential equation (FODE) and a fractional partial differential equation (FPDE) which interacts at an intermediate point. The space-dependent coefficients, interaction between the FODE and FPDE at an intermediate point and the presence of fractional calculus makes the FODE-FPDE cascaded system, representative. In this note, we first apply an invertible integral transformation to convert the system into a FODE-FPDE coupled system, as the target system, which is Mittag-Leffler stable. Using the backstepping method and under some assumptions of the space-dependent coefficients, we work out the kernel functions in the transformation and we design a boundary controller. Also, by the invertibility of the transformation, we show the Mittag-Leffler stability of the closed-loop system via the Lyapunov approach. Second, we propose an observer for which we prove that it can well estimate the original cascaded system. Then, we design an output feedback boundary control law and show that the closed-loop system is Mittag-Leffler stable under the designed output feedback control law. Finally, we present some numerical illustrations to show the correctness of the theoretical results.

Electronic effects on atom tunneling: conformational isomerization of monomeric para-substituted benzoic acid derivatives.

We present the first generation and spectroscopic identification of the higher-lying E conformer of the simplest aromatic carboxylic acid, benzoic acid (1a), as its O-deuterated isotopologue (E)-d(1)-1a using matrix-isolation techniques; the parent (E)-1a could not be observed because of fast H-tunneling to the more stable conformer (Z)-1a. Even deuterated (E)-d(1)-1a converts quickly back to (Z)-d(1)-1a through D-tunneling with a half-life (τ) of ∼12 min in Ar at 11 K. Tunneling computations using an Eckart barrier in conjunction with a CCSD(T)/cc-pVTZ//MP2/cc-pVDZ + ZPVE intrinsic reaction path revealed that τ of (E)-1a is only ∼10(-5) min, in marked contrast to those of simple aliphatic acids, which are in the range of minutes. The electronic substituent effects on D-tunneling in para-substituted benzoic acid derivatives (p-X-PhCOOD, d(1)-1) were systematically studied in Ar matrices at 11 K to derive the first Hammett relationships for atom tunneling. σ-Electron donors (X = alkyl) increase the half-life of d(1)-1, while σ-acceptor/π-donor groups (X = OD, NH(2), halogen) and to an even greater extent a σ-/π-acceptor group (X = NO(2)) decrease τ. The latter finding is in line with the smaller E-to-Z reaction barriers and narrower reaction widths for the isomerization. Tunneling substituent constants (σ(t)) for this conformational isomerization were derived experimentally and computationally.

Understanding the torquoselectivity in 8pi-electrocyclic cascade reactions: synthesis of fenestradienes versus cyclooctatrienes.

Unusual and novel polycyclic cyclooctatrienes, fenestradienes, and fenestrenes form readily from trienynes depending on the structure of the starting trienynes and the reaction conditions. The experimentally observed high torquoselectivities and complete diastereoselectivities of the 8pi-electrocyclization products have been thoroughly studied using density functional computations at B3PW91/6-31G(d,p). The different P- and M-helical topologies for the Mobius aromatic transition structures are the origin of the observed torquoselectivities in the cyclooctatrienes. The P-helical topologies direct the newly formed single bonds into a favorable equatorial position of the neighboring cycloalkane moieties (X = ring size) that retain their most stable conformation. The M-helical transition structures lead to an axial connection for the smaller rings (X = 4-6) and an equatorial connection for the seven- and eight-membered cycloalkanes. This leads to unfavorable conformations for the larger cycloalkane moieties. Experiments and computations show that for trienynes involving small neighboring cycloalkane groups (X = 4-6) M-helical topology is preferred toward cyclooctatrienes and in the following the corresponding fenestradienes can be formed as kinetic or even thermodynamic products; they convert to their more stable cyclooctatriene valence isomers derived from P-helical transition structures at higher temperatures. For larger cycloalkane moieties with more conformational flexibility only cyclooctatrienes with torquoselectivities derived from P-helical transition structures form.

Non-Kekulé N-substituted m-phenylenes: N-centered diradicals versus zwitterions.

The relationship between the structures and electronic ground state properties of non-Kekulé-type N-substituted m-phenylenes was studied utilizing density functional theory (DFT), with the aim of determining the factors that lead to ground state triplet diradicals. At B3LYP/6-311G(d,p)//B3LYP/6-311G(d,p) we identified octahydropyridoquinoline with an exceptionally large singlet-triplet energy separation of +27.9 kcal mol(-1), in favor of the triplet. The high-spin structures can readily be obtained by interruption of the full cyclic pi-delocalization that avoids cross-conjugation of the nitrogen radical centers. Introduction of additional heteroatoms, on the other hand, preferentially stabilizes the singlet zwitterionic resonance contributors in these systems. The identified diradicals show strong ferromagnetic exchange interactions between two radical centers.

3,5,7,9-Substituted hexaazaacridines: toward structures with nearly degenerate singlet-triplet energy separations.

Toward the goal of preparing stable, neutral open-shell systems, we synthesized a novel series of p-phenyl-substituted 3,5,7,9-hexaazaacridine and 3,5,7,9-hexaazaanthracene derivatives. The effects of substitution on the molecular electronic properties were probed both experimentally and computationally [B3LYP/6-311G(d,p)//B3LYP/6-31G(d,p)]. While the experimentally prepared structures already have small (20-25 kcal/mol) singlet-triplet energy gaps, systems with even smaller (<9 kcal/mol) singlet-triplet energy separations can be realized through systematic variation of the substituent numbers, types, and patterns. Hexaazaanthracenes show generally smaller singlet-triplet energy gaps than hexaazaacridines. Nitrogen-bonded sigma- and pi-acceptor substituents that cause positive inductive and mesomeric effects as well as carbon-bonded sigma-donor substituents make substituted hexaazaanthracenes promising candidates for purely organic high-spin systems.