A turbidity sensor development based on NL-PI observers: Experimental application to the control of a Sinaloa’s River Spirulina maxima cultivation

This article addresses the problem of controlling the growth of microalgae originating in Mexican rivers, especially in the state of Sinaloa, Culiacan River. For this purpose, a robust, high-gain nonlinear observer is proposed to estimate the unknown disturbance in the cultivation of mixotrophic microalgae with the presence of organic nutrients. Once a perturbation function related to the change of ambient light is estimated, an output feedback control for the photobioreactor is proposed, in which through Lyapunov’s convergence functions, the final boundary stability conditions are obtained. Thus, a turbidity sensor was designed for Spirulina platensis, a native microalgae of Culiacan River, which is presented using the MATLAB-Arduino programming environment. This sensor is calibrated using biomass culture and is a low-cost device. Through the numerical study, the feasibility and performance of the control and the observer are evaluated. Finally, real-time experimental evaluations are made based on the literature, studying the use of robust controllers in a photobioreactor with a mixed culture, in the presence of environmental changes in lighting.

Inverse source problem for a generalized Korteweg–de Vries equation

In this paper, we consider a seventh-order generalized Korteweg–de Vries (GKdV) equation and study the boundary stability results concerning the inverse problem of recovering a space-dependent source term. We establish a new boundary Carleman estimate for the seventh-order linear operator with the Dirichlet–Neumann type boundary conditions. Using this crucial estimate along with regularity result of the nonlinear GKdV equation, we establish a Lipschitz stability estimate of GKdV equation.

Equatorial retrograde flow in WASP-43b elicited by deep wind jets?

ABSTRACT We present WASP-43b climate simulations with deep wind jets (down to 700 bar) that are linked to retrograde (westward) flow at the equatorial day side for p < 0.1 bar. Retrograde flow inhibits efficient eastward heat transport and naturally explains the small hotspot shift and large day-night-side gradient of WASP-43b (Porb = Prot = 0.8135 d) observed with Spitzer. We find that deep wind jets are mainly associated with very fast rotations (Prot = Porb ≤ 1.5 d) which correspond to the Rhines length smaller than 2 planetary radii. We also diagnose wave activity that likely gives rise to deviations from superrotation. Further, we show that we can achieve full steady state in our climate simulations by imposing a deep forcing regime for p > 10 bar: convergence time-scale τconv = 106–108 s to a common adiabat, as well as linear drag at depth (p ≥ 200 bar), which mimics to first-order magnetic drag. Lower boundary stability and the deep forcing assumptions were also tested with climate simulations for HD 209458b (Porb = Prot = 3.5 d). HD 209458b simulations always show shallow wind jets (never deeper than 100 bar) and unperturbed superrotation. If we impose a fast rotation (Porb = Prot = 0.8135 d), also the HD 209458b-like simulation shows equatorial retrograde flow at the day side. We conclude that the placement of the lower boundary at p = 200 bar is justified for slow rotators like HD 209458b, but we suggest that it has to be placed deeper for fast-rotating, dense hot Jupiters (Porb ≤ 1.5 d) like WASP-43b. Our study highlights that the deep atmosphere may have a strong influence on the observable atmospheric flow in some hot Jupiters.

The combined effects of shear and buoyancy on phase boundary stability

We study the effects of externally imposed shear and buoyancy driven flows on the stability of a solid–liquid interface. A linear stability analysis of shear and buoyancy-driven flow of a melt over its solid phase shows that buoyancy is the only destabilizing factor and that the regime of shear flow here, by inhibiting vertical motions and hence the upward heat flux, stabilizes the system. It is also shown that all perturbations to the solid–liquid interface decay at a very modest shear flow strength. However, at much larger shear-flow strength, where flow instabilities coupled with buoyancy might enhance vertical motions, a re-entrant instability may arise.

On boundary stability of inhomogeneous 2 × 2 1-D hyperbolic systems for the C1 norm

We study the exponential stability for the C1 norm of general 2 × 2 1-D quasilinear hyperbolic systems with source terms and boundary controls. When the propagation speeds of the system have the same sign, any nonuniform steady-state can be stabilized using boundary feedbacks that only depend on measurements at the boundaries and we give explicit conditions on the gain of the feedback. In other cases, we exhibit a simple numerical criterion for the existence of basic C1 Lyapunov function, a natural candidate for a Lyapunov function to ensure exponential stability for the C1 norm. We show that, under a simple condition on the source term, the existence of a basic C1 (or Cp, for any p ≥ 1) Lyapunov function is equivalent to the existence of a basic H2 (or Hq, for any q ≥ 2) Lyapunov function, its analogue for the H2 norm. Finally, we apply these results to the nonlinear Saint-Venant equations. We show in particular that in the subcritical regime, when the slope is larger than the friction, the system can always be stabilized in the C1 norm using static boundary feedbacks depending only on measurements at the boundaries, which has a large practical interest in hydraulic and engineering applications.