The perturbed time-optimal problem of controlling the final position of a material point by means of a limited force (Q1892377)

An algorithm is proposed for the solution of the perturbed optimal control problem \(t_f\to \min\) subject to the constraints \(x''= u+ \varepsilon f(x, x')\); \(x(0)= x_0\), \(x' (0)= x_0'\) and \(x(t_f)= 0\), \(x' (t_f)\in D\subset \mathbb{R}^n\), \(|u(t) |\leq u_0\), where \(x(t)\in \mathbb{R}^n\) and \(\varepsilon \geq 0\) is a small parameter. The dependence of the optimal time and optimal solution on the small parameter is also investigated.