Time-optimality problem for a nonlinear controlled model
The paper considers the following nonlinear control system: \[ \begin{aligned} \dot x_1&=\omega_1y\phi(x_1,x_2),\\ \dot x_2&= \omega_2(1-y)\phi(x_1,x_2),\\ \dot y &=\frac{\dot \phi(x_1,x_2)}{\phi(x_1,x_2)} (u-y), \end{aligned} \] where \(\omega_1\) and \(\omega_2\) are positive constants satisfying \(\omega_1+\omega_2=1\), \(\phi(x_1,x_2)=\min(x_1,x_2)\), and \(u\) is the control function, \(0\leq u\leq 1\). This system describes a certain process originating from biology. First, the authors distinguish the range of initial data such that for any admissible control, the corresponding trajectory enters a certain domain and remains there all the time. In this domain, the system considered simplifies as follows: \[ \begin{aligned} \dot x_1&=\omega_1yx_1,\\ \dot x_2&=\omega_2(1-y)x_1, \\ \dot y&=\omega_1y(u-y). \end{aligned} \] Second, the authors prove the controllability result for the above reduced system in the class of bang-bang controls with one switching point. Finally, they prove that the resolving control in the controllability problem is in fact time-optimal.
scientific article; zbMATH DE number 2099443