Analysis and Control of Perturbed Density Systems

The paper investigates dynamical systems for which the derivative of some positive-definite function along the solutions of this system depends on so-called density function. In turn, such dynamical systems are called density systems. The density function sets the density of the space, where the system is evolved, and affects the behaviour of the original system. For example, this function can define (un)stable regions and forbidden regions where there are no system solutions. The density function can be used in the design of new adaptive control laws with the formulation of appropriate new control goals, e.g., stabilization in given bounded or semi-bounded sets. To design a novel adaptive control law that ensures the system outputs in a given set, systems with known and unknown parameters under disturbances are considered. All theoretical results and conclusions are illustrated by numerical simulations.