Time-scaling symmetry and Zeno solutions
The article deals with the analysis of dynamical systems, described with the differential equation \[ \dot x(t) = f(x(t)); \] the function \(f(\cdot)\) is not assumed to be continuous. However, it is assumed that this system exhibits time-scaling symmetry with respect to a given spatial transformation group \({\mathcal F} = \{F_\alpha: {\mathbb R}^n \to {\mathbb R}^n, \;\alpha \in (0,\infty)\}\) if, for any solution \(x(t)\) and any \(\alpha > 0\) the function \(T_\alpha x(t) = F_\alpha(x(\alpha t))\) is also a solution (on the corresponding time-interval). As examples of such systems the well known Fuller system \[ \dot x_1(t) = x_2(t), \quad \dot x_2(t) = - \text{sgn} \, (x_1(t) + \frac12 m x_2^2(t) \;\text{sgn} \, x_2(t)) \] (with the group \(F_\alpha = \text{diag} \, (\alpha^{-2},\alpha^{-1})\)), Filippov system \[ \dot x_1(t) = \text{sgn} \, x_1(t) - 2 \;\text{sgn} \, x_2(t), \quad \dot x_2(t) = 2 \;\text{sgn} \, x_1(t) + \text{sgn} \, x_2(t) \] (with the group \(F_\alpha = \text{diag} \, (\alpha^{-1},\alpha^{-1})\)) and Pogromsky -- Heemels -- Nijmeijer system \[ \dot x_1(t) = x_2(t), \quad \dot x_2(t) = x_3(t), \quad \dot x_3(t) = - \text{sgn} \, x_1(t) \] are considered. The first result in the article is the statement that, for an invariant set \(S\) of the system, the spatial transformation group restricted to \(S\) commutes with the Poincaré return map; the latter is defined by the formula \(Rx_0 = x(\tau(x_0),x_0)\), \(\tau(x_0)\) is the first return time of \(x_0\). The second result is concerned with the system that is described with the equation \[ \dot x(t) = f(x) = Ax(t) - b \;\text{sgn} \, (c^Tx(t)); \] under this it is assumed that there exists a matrix \(K\) such that \(AK - KA = A\), \(Kb = - b\), \(c^TK = \lambda c^T\) with \(\lambda < 0\) and \(c^TA^{-\lambda-1}b = 1\). If there exists \(\alpha > 1\) such that the matrix \(I + e^{-A} \alpha^K\) is invertible, \[ h(\alpha) \doteq c^T(I + e^{-a}\alpha^K)^{-1} \int_0^1 e^{-As} \, ds \, b = 0, \] and the polynomial from the following equation \[ c^T x(t) = \sum_{j=1}^{m-1} \frac1{j!} c^T A^j x_0 t^j - \frac1{(-\lambda)!} t^{-\lambda}, \quad x_0 = \int_0^1 e^{-As} \, ds \, b, \] is positive on the interval \((0,1)\), then the system admits a right Zeno solution; if the same conditions hold for \(\alpha \in (0,1)\), then the system admits a left Zeno solution. The definitions of the right and left Zeno solutions are absent.
scientific article; zbMATH DE number 5563967
scientific article