Asymptotic stabilization of coupled oscillators with dry friction by feedback control
Consider the system \[ \dot{x}=y,\quad \dot{y}\in -\nabla f(x)-\partial\Phi(y) -u, \quad t\geq 0, \tag{1} \] where \(x,y \in X=\mathbb R^{m}\), \(f\in C^{1}(X,\mathbb R)\), \(\Phi(y)= a|y| +\Psi(y)\), \(\Psi\in C^{1}(X,\mathbb R)\) is a convex function with \(\nabla\Psi(0) =0\), \(a\geq 0\), \(u\) is a control. Let \(M=\{z: \nabla f(z)=0\}\), \(v(x) = \nabla f(x)/| \nabla f(x)|\) if \(x\in X\setminus M\) and \(v(x)=\{v\in X:|v|\leq 1\}\) if \(x\in M\). Theorem. Suppose that \(\lim_{|x|\to+\infty}f(x)=+\infty\). Define \[ u=u(x,y) =a\max\{1-|y|,0\}v(x),\quad (x,y)\in X\times X. \] Then, for any solution \(x(\cdot), y(\cdot)\) of (1), one has \[ \lim_{t\to\infty}(d(x(t),M) +|y(t)|) =0; \] if \(M_{c}:= f^{-1}(c)\cap M\) is finite for any \(c\in \mathbb R\), then any solution \(x(\cdot), y(\cdot)\) approaches an equilibrium \((z_{c},0)\in\{(z,0)\in X\times X: z\in M_{c}\}\).
scientific article; zbMATH DE number 6118936