A Weak Invariance Principle and Asymptotic Stability for Evolution Equations with Bounded Generators

If V is a Lyapunov function of an equation du/dt u’ Zu in a Banach space thenasymptotic stability of an equilibrium point may be easily proved if it is known that sup(V’) < 0 onsufficiently small spheres centered at the equilibrium point. In this paper weak asymptotic stability isproved for a bounded infinitesimal generator Z under a weaker assumption V’ < 0 (which aloneimplies ordinary stability only) if some observability condition, involving Z and the Frechet derivativeof V’, is satisfied. The proof is based on an extension of LaSalle’s invariance principle, which yieldsconvergence in a weak topology and uses a strongly continuous Lyapunov funcdon. The theory isillustrated with an example of an integro-differential equation of interest in the theory of chemicalprocesses. In this case strong asymptotic stability is deduced from the weak one and explicit sufficientconditions for stability are given. In the case of a normal infinitesimal generator Z in a Hilbertspace, strong asymptotic stability is proved under the following assumptions Z* + Z is weaklynegative definite and Ker Z 0 }. The proof is based on spectral theory

Treatment of cheese whey wastewater by combined electrochemical processes

This study shows the good performance of a sequential electrochemical methodology, consisting in electrocoagulation (EC) followed by an electrochemical advanced oxidation process (EAOP), to treat raw cheese whey wastewater at laboratory and pre-pilot scales. In EC, different electrode materials like Fe, Al and stainless steel (AISI 304 and ASI 316L) were tested. Among EAOPs, photoelectro-Fenton (PEF) and electrochemical oxidation (EO) with active anodes like Pt or DSA((R)) and non-active ones like boron-doped diamond (BDD) were studied. At both scales, the optimum anode/cathode combination in EC was Fe/AISI 304, which yielded the highest total organic carbon (TOC) removal of 22.0-27.0%. This is due to various effects on organic compounds: (i) coagulation promoted by Fe(OH)(3) flocs, (ii) cathodic reduction, and (iii) oxidation with generated active chlorine. At small scale, the resulting wastewater was further treated by PEF at pH 3.0. The highest TOC removal was achieved using the BDD, owing to the great oxidation power of hydroxyl radicals. In contrast, total nitrogen was abated much more rapidly with active anodes because of the attack of active chlorine on N-compounds. At pre-pilot scale, the post-treatment of conditioned wastewater made by EO with a BDD/Pt flow cell combined with UVA irradiation yielded the highest TOC removal, i.e., 49.1%. The high energy consumed by the UVA lamp would be a drawback at industrial scale, which could be overcome by using sunlight