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We use Lie brackets of unbounded vector fields to consider a dissipative relation that generalizes the differential inequality which defines classic control Lyapunov functions. Under minimal regularity assumptions, we employ locally semiconcave solutions of this extended relation, called in the following degree-k control Lyapunov functions, in order to design degree-k Lyapunov feedbacks, i.e. particular discontinuous feedback laws that stabilize the underlying system to a given closed target with compact boundary, in the sample and hold sense. We also prove that this feedback construction is robust when small measurement errors and external disturbances occur.

This paper presents a new direct electron transfer based-miniaturized glucose/oxygen enzymatic fuel cell (EFC) whose operating ability has been tested in real saliva samples. The bioanode and biocathode are a graphene working electrode and a graphite counter electrode localized on the same screen printed electrode (SPE) modified with poly(vinyl alcohol) N-methyl-4(4′-formylstyryl)pyridinium methosulfate acetal (PVA-SbQ)/cellobiose dehydrogenase from Corynascus Thermophilus (CtCDH) C291Y/AuNPs and with Trametes Hirsuta laccase (ThLac)/AuNPs, respectively. In order to optimize the bioanode, several CDH immobilization procedures were adopted, such as drop-casting, use of Nafion membrane or PVA-SbQ photopolymer. The photopolymer showed the best performance in terms of stability and reliability. As biocathode a partially optimized laccase electrode was employed with the variant that the used nanomaterials allowed to reduce the overpotential of O2/H2O redox reaction catalyzed by Trametes Hirsuta Laccase (ThLac), drop-casted onto the gold nanoparticles (AuNPs) modified SPE. The performances of bioanode and biocathode were tested separately, initially immobilizing the two enzymes onto separated graphene SPEs. An efficient direct electron transfer was achieved for both elements, obtaining an apparent heterogeneous electron transfer rate constant (ks) of 0.99 ± 0.05 s−1 for CtCDH C291Y and 5.60 ± 0.05 s−1 for ThLac. Both electrodes were then assembled in a two compartment EFC obtaining a maximal power output of 5.16 ± 0.15 μW cm−2 at a cell voltage of 0.58 V and an open circuit voltage (OCV) of 0.74 V. Successively, the bioanode and biocathode were assembled in a non-compartmentalized EFC and a remarkable 50% decrease of the maximum power output at the value of 2.15 ± 0.12 μW cm−2 at cell voltage of 0.48 V and an OCV of 0.62 V at pH 6.5 was registered. In order to reduce the cell dimensions in view of its possible integration in biomedical devices, the bioanode and biocaythode were realized by immobilization of both enzymes onto the same SPE. The so miniaturized EFC delivered a maximal power output of 1.57 ± 0.07 μW cm2 and 1.10 ± 0.12 μW cm−2 with an OCV of 0.58 V and 0.41 V in a 100 μM glucose solution and in human saliva, respectively.