Extremum seeking feedback is a powerful method to steer a dynamical system to an extremum of a partially or completely unknown map. It often requires advanced system-theoretic tools to understand the qualitative behavior of extremum seeking systems. In this paper, a novel interpretation of extremum seeking is introduced. We show that the trajectories of an extremum seeking system can be approximated by the trajectories of a system which involves certain Lie brackets of the vector fields of the extremum seeking system. It turns out that the Lie bracket system directly reveals the optimizing behavior of the extremum seeking system. Furthermore, we establish a theoretical foundation and prove that uniform asymptotic stability of the Lie bracket system implies practical uniform asymptotic stability of the corresponding extremum seeking system. We use the established results in order to prove local and semi-global practical uniform asymptotic stability of the extrema of a certain map for multi-agent extremum seeking systems.

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Endothelial cells (ECs) regulate the barrier function of blood vessels. Here we show that basal and angiopoietin-1 (Ang-1)-regulated control of EC permeability is mediated by 2 different functional states of sphingosine kinase-1 (SK-1). Mice depleted of SK-1 have increased vascular leakiness, whereas mice transgenic for SK-1 in ECs show attenuation of leakiness. Furthermore, Ang-1 rapidly and transiently stimulates SK-1 activity and phosphorylation, and induces an increase in intracellular sphingosine-1-phosphate (S1P) concentration. Overexpression of SK-1 resulted in inhibition of permeability similar to that seen for Ang-1, whereas knockdown of SK-1 by small interfering RNA blocked Ang-1-mediated inhibition of permeability. Transfection with SK S225A , a nonphosphorylatable mutant of SK-1, inhibited basal leakiness, and both SK S225A and a dominant-negative SK-1 mutant removed the capacity of Ang-1 to inhibit permeability. These effects were independent of extracellular S1P as knockdown or inhibition of S1P 1 , S1P 2 , or S1P 3 , did not affect the Ang-1 response. Thus, SK-1 levels in ECs powerfully regulate basal permeability in vitro and in vivo. In addition, the Ang-1-induced inhibition of leakiness is mediated through activation of SK-1, defining a new signaling pathway in the Ang-1 regulation of permeability. IntroductionThe control of vascular homeostasis is mediated through the maintenance of endothelial cell (EC) monolayer integrity, which is responsible for the tonic impermeable nature of blood vessels. Altered endothelial integrity underlies changes in vascular permeability, as seen in inflammatory responses and angiogenesis, and in diseases, such as rheumatoid arthritis and atherosclerosis. Stimuli, such as thrombin, histamine, and tumor necrosis factor (TNF), are powerful inducers of EC permeability, 1 whereas the angiogenic factor angiopoietin-1 (Ang-1), acting through the tyrosine kinase receptor Tie2, and sphingosine-1-phosphate (S1P), acting through the G protein coupled receptors S1P 1 or S1P 3 , inhibit the action of these stimulators and are potent protectors of barrier function. [2][3][4] The Ang-1/Tie2 axis is essential for vascular development and, in the adult, for the formation and maintenance of EC integrity. 2 In vivo, transgenic expression of Ang-1 in mice results in leakage resistant blood vessels, 5 and systemic administration of Ang-1 inhibits vessel leak. 6 In vitro, the antipermeability and antiinflammatory effects of Ang-1 appear to function at 2 levels, one to inhibit the basal permeability of EC and the other to limit the activity of agents, such as thrombin and TNF. 3 The signaling pathways regulated in response to ligand-induced phosphorylation of Tie2 are only now being elucidated. 7 AKT is involved in the prosurvival and angiogenic activities of Ang-1. 8 Interaction of ShcA with Tie2 and phospholipase D-dependent regulation of mitogen-activated protein kinase (MAPK) are involved in Ang-1-mediated cell migration 9 and a Ras/MAPK cascade is responsible for Ang-1-induced ce...