Some control problems solved via a sliding manifold approach (Q1355036)

The nonlinear control system \[ {dx\over dt} =f(t,x,u), \quad x(0)= x_0 \tag{1} \] is considered, \(0\leq t\leq T\). Here \(x=x(t) \in\mathbb{R}^n\), \(u=u(t) \in\mathbb{R}^m\), \(m\leq n\). Let \(s: \mathbb{R}^n \to\mathbb{R}^m\) be a continuously differential function and \(S=\{x \in\mathbb{R}^n: s(x)=0\}\) be ``a sliding manifold''. A classical problem of the control of a variable structure system (Utkin and others) is to steer and then hold the state vector \(x(t)\) on the sliding manifold \(S\) by using feedback control \(u=u(t,x)\) which is discontinuous in \(S\). The aim of control is to solve this problem ``approximatively'' by using a different class of smooth feedback controls. Let \[ g(t,x,u) ={\partial \over \partial t} s(t,x)+ \left({\partial \over \partial x} s(t,x)\right) f(t,x,u), \] \[ \varepsilon {du \over dt} =g(t,x,u), \quad u(0)=u_0. \tag{2} \] Under some assumptions the following result is obtained. The solution \([x(t,\varepsilon), u(t,\varepsilon)]\) of (1) and (2) has the properties \[ \begin{aligned} x(t,\varepsilon) \to x_0(t) \quad & \text{uniformly in } [0,T]\\ u(t, \varepsilon) \to u_0(t) \quad & \text{uniformly in } [t_1,T],\;0<t_1<T, \end{aligned} \] where \([x_0(t),\;u_0(t)]\) is the solution of the reduced system (1) and \(g(t,x,u) \equiv 0\). Some other approximation results are obtained. A linear tracking problem and the control problem of satellite are considered.