Optimal control of an oscillator of variable rigidity (Q1908445)

In this paper the following optimal control problem \hskip17mm \(t_1\to \text{inf}\), \hskip17mm \(\ddot x+ (1- \varepsilon u) x= 0\), \hskip17mm \(x(0)= x_0\), \(\dot x_0= \dot x(0)= y_0\), \(x^2(t_1)+ \dot x^2(t_1)= r^2\), \hskip17mm \(0\leq u\leq 1\), \(0< \varepsilon< 1\) is studied. In the above written equations the factor \((1- \varepsilon u)\) corresponds to the rigidity of the oscillator, \(u\) is the control and \(\varepsilon\) is a fixed parameter. The phase portrait on the \((x, \dot x= y)\) plane is constructed and the entire switching curve for both \(x^2_0+ y^2_0> r^2\) and \(x^2_0+ y^2_0< r^2\). Investigating the switching curve asymptotics for \(r\to 0\) it is proven that no limiting curve exists for the switching curve family \(r\to 0\).