Pulse-sliding modes of controlled mechanical systems

We consider a controlled mechanical system with dry friction and positional pulse or positional discontinuous control. It can be presented in a form of Lagrange equations of the second kind
$$ A(t,q)\ddot q=g(t,q,\dot q)+Q^A(t,q,\dot q)+Q^T(t,q,\dot q)+u,\quad t\in I=[t_0,t_0+T]. \eqno{(1)} $$
The goal of the control is the motion of the system (1) in set $S=\{(t,q,\dot q)\in I\times R^n\times R^n\colon\sigma(t,q,\dot q)=0\}$ (problem of stabilization) or in the neighborhood of set $S$ (approach problem). The first problem is solved with discontinuous positional control of relay type with limited resources, for which a decomposition mode is a stable sliding mode of system (1). In case of insufficiency of resources of discontinuous control the motion of the controlled system in the neighborhood of set $S$ can be implemented under high-frequency impacts on the system in discrete time moments in the pulse-sliding mode, the uniform limit of which (an ideal pulse-sliding mode) is equal to the decomposition mode. The distinctive feature of the assigned problems is dry friction in the system (1), and said dry fiction, generally speaking, can be considered as uncontrollable discontinuous or multivalued perturbations.
Main definitions are given in the introduction of the article. In the first section the connection between ideal pulse-sliding modes of inclusion
$$ A(t,x)\dot x\in F(t,x)+u, $$
where $u$ is a positional pulse control, and sliding modes of system
$$ A(t,x)\dot x\in F(t,x)+B(t,x)\tilde u(t,x) $$
with a positional discontinuous control is considered. The second section is devoted to systems of type (1). In the third section we consider set $S$, which is important in relation to applications and is defined by the vector function $\sigma(t,q,\dot q)=\dot q-\varphi(t,q)$. For the last case more simple and informative conditions of the existence of sliding modes for a system with discontinuous controls were used. An example was considered in conclusion.