Theory of differential inclusions and its application in mechanics

The following chapter deals with systems of differential equations with discontinuous right-hand sides. The key question is how to define the solutions of such systems. The most adequate approach is to treat discontinuous systems as systems with multivalued right-hand sides (differential inclusions). In this work three well-known definitions of solution of discontinuous system are considered. We will demonstrate the difference between these definitions and their application to different mechanical problems. Mathematical models of drilling systems with discontinuous friction torque characteristics are considered. Here, opposite to classical Coulomb symmetric friction law, the friction torque characteristic is asymmetrical. Problem of sudden load change is studied. Analytical methods of investigation of systems with such asymmetrical friction based on the use of Lyapunov functions are demonstrated. The Watt governor and Chua system are considered to show different aspects of computer modeling of discontinuous systems

Urinary pH: its regulation and relevance in urolithiasis metaphylaxis

Urolithiasis is a common multifactorial disease characterized by a high recurrence rate. This review is devoted to the urine pH as one of the main factors determining its lithogenic properties. It affects the excretion of lithogenic substances and stone formation inhibitors, the solubility, and the crystallization of substances involved in stone formation. The urine pH significantly affects the solubility of uric acid in urine, which decreases at a pH < 5.5. This explains the high incidence of uric acid concretions in patients with metabolic syndrome. Their insulin resistance leads to a decrease in the excretion of ammonium ions in the proximal tubules, leading to persistent urine acidification. The activity of many transport processes involved in the processing of calcium, citrates and phosphates is sensitive to changes in systemic or local pH. The data on the effect of urine pH on the solubility of calcium oxalate remain contradictory. At the same time, there is no doubt about the determining role of urine pH in the excretion of citrate, the most important stone formation inhibitor. The alkaline urine pH promotes the formation of concretions containing calcium phosphates. In conditions of constantly elevated urine pH in patients with persistent urease-producing urinary tract infection, a rapid growth of "infectious" concretions occurs. The review summarizes information on the causes of the decrease and increase in the urine pH, as well as the possibilities of medicinal and non-medicinal methods of modifying the urine pH during the prevention of stone formation recurrence

Absolute stability of time-varying delay Lurie indirect control systems with unbounded coefficients

This paper investigates the absolute stability problem of time-varying delay Lurie indirect control systems with variable coefficients. A positive-definite Lyapunov-Krasovskii functional is constructed. Some novel sufficient conditions for absolute stability of Lurie systems with single nonlinearity are obtained by estimating the negative upper bound on its total time derivative. Furthermore, the results are generalised to multiple nonlinearities. The derived criteria are especially suitable for time-varying delay Lurie indirect control systems with unbounded coefficients. The effectiveness of the proposed results is illustrated using simulation examples