Section: Research Program

Theory of homogeneous systems

Homogeneity is a property of mathematical objects, such as functions or vector fields, to be scaled in a consistent manner with respect to a scaling operation (called a dilation) applied to their argument (a kind of symmetry). The first rise of homogeneity deals with homogeneous polynomials investigated by L. Euler in 18^th century. In 50s and 60s more generic notions of homogeneity (weighted and coordinate-free or geometric) have been introduced by V.I. Zubov and his group. For example, a function $f:{ℝ}^n\to ℝ$ is called homogeneous (in Euler's sense) if

for some $\nu \ge -1$ called the degree of homogeneity of $f$ (a parameter of symmetry). Such a type of symmetry leads to the scaling of trajectories of resultant dynamical systems, e.g. for

denote a solution corresponding to the initial condition $x_0$ by $X(t,x_0)$, then

So homogeneous systems possess several important and useful properties: their local behavior is the same as global one, the rate of convergence to the origin can be identified by degree of homogeneity, the stability is robust to various perturbations. There are also plenty of researches performed in the last 30 years and the members of Non-A POST team extended these notions of homogeneity to discontinuous systems (in a geometric framework), time-delay systems, partial differential equations, time-varying systems, and recently to discrete-time models (together with the concept of local homogeneity). They also proposed plenty of control and estimation algorithms based on homogeneity.

Advantages of homogeneous algorithms taking into account the above mentioned criteria:

A1) The rate of convergence in homogeneous systems can be qualified by its degree (finite-time and fixed-time for negative and positive degrees, respectively).

A2) Due to symmetry of these systems they admit special discretization tools (also developed by the members of Non-A POST team), which make simpler they realization for on-line scenarios.

A3) The internal symmetry of these dynamics makes them inherently robust with respect to external perturbations, measurement noises and delays, which is especially important in networked systems.