Hierarchical exact controllability of semilinear parabolic equations with distributed and boundary controls

We present some exact controllability results for parabolic equations in the context of hierarchic control using Stackelberg–Nash strategies. We analyze two cases: in the first one, the main control (the leader) acts in the interior of the domain and the secondary controls (the followers) act on small parts of the boundary; in the second one, we consider a leader acting on the boundary while the followers are of the distributed kind. In both cases, for each leader, an associated Nash equilibrium pair is found; then, we obtain a leader that leads the system exactly to a prescribed (but arbitrary) trajectory. We consider linear and semilinear problems.

Non-equilibrium pair interactions in colloidal dispersions

We study non-equilibrium pair interactions between microscopic particles moving through a model shear-thinning fluid. Prior efforts to model pair interactions in non-Newtonian fluids have largely focused on constitutive models derived from polymer-chain kinetic theories focusing on conformational degrees of freedom, but neglecting the details of microstructural evolution beyond a single polymer length scale. To elucidate the role of strong structural distortion in mediating pair interactions in Brownian suspensions, we formulate and solve a Smoluchowski equation describing the detailed evolution of the particle configuration between and around a pair of microscopic probes driven at fixed velocity by an external force through a colloidal dispersion. To facilitate analysis, we choose a model system of Brownian hard spheres that do not interact hydrodynamically; while simple, this ‘freely draining’ model permits insight into connections between microstructure and rheology. The flow induces a non-equilibrium particle density gradient that gives rise to both viscous drag and an interactive force between the probes. The drag force acts to slow the centre-of-mass velocity of the pair, while the interactive force arising from osmotic pressure gradients can lead to attraction or repulsion, as well as deterministic reorientation of the probes relative to the external force. The degree to which the microstructure is distorted, and the shape of that distortion, depend on the arrangement of the probes relative to one another and their orientation to the driving force. It also depends on the magnitude of probe velocity relative to the Brownian velocity of the suspension. When only thermal fluctuations set probe velocity, the equilibrium depletion attraction is recovered. For weak forcing, long-ranged interactions mediated via the bath-particle flux give rise to entropic forces on the probes. The linear response is a viscous drag that slows forward motion; only the weakly nonlinear response can produce relative motion–attraction, repulsion or reorientation of the probes. We derive entropic coupling tensors, similar in ethos to pair hydrodynamic tensors, to describe this behaviour. The structural symmetry that permits this analogy is lost when forcing becomes strong, revealing instabilities in system behaviour. Far from equilibrium, the interactive force depends explicitly on the initial probe separation, orientation and strength of forcing; widely spaced probes interact through the distorted microstructure, whereas the behaviour of closely spaced probes is largely set by excluded-volume effects. In this regime, a pair of closely spaced probes sedimenting side-by-side tend to attract and reorient to permit alignment of their line-of-centres with the flow, while widely spaced probes fall without reorienting. Our results show qualitative agreement with experimental observations and provide a potential connection to the observed column instability in shear-thinning fluids.

The quantum pinch effect in semiconducting quantum wires: A bird’s-eye view

Those who measure success with culmination do not seem to be aware that life is a journey not a destination. This spirit is best reflected in the unceasing failures in efforts for solving the problem of controlled thermonuclear fusion for even the simplest pinches for over decades; and the nature keeps us challenging with examples. However, these efforts have permitted researchers the obtention of a dense plasma with a lifetime that, albeit short, is sufficient to study the physics of the pinch effect, to create methods of plasma diagnostics, and to develop a modern theory of plasma processes. Most importantly, they have impregnated the solid state plasmas, particularly the electron–hole plasmas in semiconductors, which do not suffer from the issues related with the confinement and which have demonstrated their potential not only for the fundamental physics but also for the device physics. Here, we report on a two-component, cylindrical, quasi-one-dimensional quantum plasma subjected to a radial confining harmonic potential and an applied magnetic field in the symmetric gauge. It is demonstrated that such a system, as can be realized in semiconducting quantum wires, offers an excellent medium for observing the quantum pinch effect at low temperatures. An exact analytical solution of the problem allows us to make significant observations: Surprisingly, in contrast to the classical pinch effect, the particle density as well as the current density display a determinable maximum before attaining a minimum at the surface of the quantum wire. The effect will persist as long as the equilibrium pair density is sustained. Therefore, the technological promise that emerges is the route to the precise electronic devices that will control the particle beams at the nanoscale.

EXISTENCE OF EQUILIBRIUM FOR GAMES WITH UPPER SEMICONTINUOUS CORRESPONDENCES, UNDER UNCERTAINTY

In this paper we prove the existence of equilibrium pairs for the new model of a Bayesian free abstract economy which extends Kim and Lee's deterministic model of a free abstract economy (2006). Our existence theorems are proved for the case of upper semicontinuous correspondences. We also define a model of a general Bayesian abstract economy and introduce the new notion of equilibrium pair. We prove the existence of equilibrium pair and equilibrium for this type of abstract economy.