This paper addresses distributed adaptive optimal resource allocation problems over weight-balanced digraphs. By leveraging state-of-the-art adaptive coupling designs for multiagent systems, two adaptive algorithms are proposed, namely a directed-spanning-tree-based algorithm and a node-based algorithm. The benefits of these algorithms are that they require neither sufficiently small or unitary step sizes, nor global knowledge of Laplacian eigenvalues, which are widely required in the literature. It is shown that both algorithms belong to a class of uncertain saddle-point dynamics, which can be tackled by repeatedly adopting the Peter-Paul inequality in the framework of Lyapunov theory. Thanks to this new viewpoint, global asymptotic convergence of both algorithms can be proven in a unified way. The effectiveness of the proposed algorithms is validated through numerical simulations and case studies in IEEE 30-bus and 118-bus power systems.

The Ti₃C₂T_x film with metallic conductivity and high pseudo-capacitance holds profound promise in flexible high-rate supercapacitors. However, the restacking of Ti₃C₂T_x sheets hinders ion access to thick film electrodes. Herein, a mild yet green route has been developed to partially oxidize Ti₃C₂T_x to TiO₂/Ti₃C₂T_x by introducing O₂ molecules during refluxing the Ti₃C₂T_x suspension. The subsequent etching away of these TiO₂ nanoparticles by HF leaves behind numerous in-plane nanopores on the Ti₃C₂T_x sheets. Electrochemical impedance spectroscopy shows that longer oxidation time of 40 min yields holey Ti₃C₂T_x (H-Ti₃C₂T_x) with a much shorter relax time constant of 0.85 s at electrode thickness of 25 µm, which is 89 times smaller than that of the pristineTi₃C₂T_x film (75.58 s). Meanwhile, H-Ti₃C₂T_x film with 25 min oxidation exhibits less-dependent capacitive performance in film thickness range of 10–84 µm (1.63–6.41 mg cm⁻²) and maintains around 60% capacitance as the current density increases from 1 to 50 A g⁻¹. The findings clearly demonstrate that in-plane nanopores not only provide more electrochemically active sites, but also offer numerous pathways for rapid ion impregnation across the thick Ti₃C₂T_x film. The method reported herein would pave way for fabricating porous MXene materials toward high-rate flexible supercapacitor applications.

In this article, the time-varying formation and time-varying formation tracking problems are solved for linear multiagent systems over digraphs without the knowledge of the eigenvalues of the Laplacian matrix associated with the digraph. The solution to these problems relies on an approach that generalizes the directed spanning tree (DST) adaptive method, which was originally limited to consensus problems. Necessary and sufficient conditions for the existence of solutions to the formation problems are derived. Asymptotic convergence of the formation errors is proved via graph theory and Lyapunov analysis.