Structural stability in the design of optimal trajectories in Fuller's problem (Q1823449)

The author considers the following problem (P): \[ \int^{\infty}_{0}x^ 2(1+\epsilon f(x))dt\to \inf, \] \(\dot x=y\), \(\dot y=u\in [-1,1]\), \(z=(x,y)\), \(z(0)=z_ 0\in \{z:\) \(| x| <\rho\), \(| y| <\rho \}\), \(x(\cdot)\in W^{0,2}_{0,\infty}[0,\infty)\), \(f(\cdot)\in C^ 3\), \(f(-x)=f(x)\). If \(\epsilon =0\) and \(\rho =\infty\) (the Fuller problem), the optimal control has the form \(u(z)=-sgn(x+ay^ 2sgn y)\), where \(a\approx 0.4446\). The optimal trajectories cross the switching curve an infinite number of times and reach the point (0,0) in finite time [\textit{A. T. Fuller}, Autom. Remote Control, Vol. I, 510-519 (Moscow 1960), (Butterworths, London 1961)]. The main result is as follows: If \(\epsilon >0\) and \(\rho >0\) are sufficiently small then the synthesis in (P) is analogous to that in the Fuller problem, and the switching curve has the form \(x=-h(y)y^ 2sgn y\), \(| h(y)-a| <\rho\).