Projected dynamical systems in a complementarity formalism
The authors consider the so-called gradient complementarity systems of differential equations \[ \dot x(t)=- F(x(t))+ [H(x(t))]^T u(t), \] where \(H\) is the Jacobian of \(h\), \(h(x,t): \mathbb{R}^n\to \mathbb{R}^p\), \(0\leq y(t)\perp u(t)\geq 0\). The authors state that such equations belong to the class of projected dynamical equations of the form \[ \dot x(t)= \Pi_K(x(t)- F(x(t))), \] where \(F\) is a vector field, \(K\) is a closed convex set, and \(\Pi_K\) is a projection operator, that prevents the solution from moving outside the constrained set \(K\). The establishment of a connection between the areas of projected dynamical systems and complementarity systems facilitates the transfer of techniques from one domain to the other.
scientific article; zbMATH DE number 1547199
scientific article