Optimal synthesis in a model problem with two-dimensional control lying in an arbitrary convex set (Q2313606)

In this paper, the optimal synthesis for the following control problem is described: \[ \frac{1}{2} \int_{0}^{\infty} x^{2}(t) dt \to \mathrm{inf}, \quad \dot x(t) = u(t), u(t) \in \Omega,\ x(0) = x_{0}, \] where \(x \in \mathbb{R}^{2}\) and \(\Omega \subset \mathbb{R}^{2}\) is a convex compact set with the origin \(O\) in the interior of \(\Omega\). The trajectory \(x(t)\) is assumed to be Lipschitz, and \(\dot x = u \in L_{\infty}\). For an arbitrary set \(\Omega\), the asymptotics of optimal trajectories and the geometric properties of the optimal synthesis are completely described, whereas for \(\Omega\) being a polygon, the problem is solved explicitly. The results of this paper are based on the two earlier papers by the first author published in [Izv. Math. 78, No. 5, 1006--1027 (2014; Zbl 1304.49004); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 78, No. 5, 167--190 (2014)] and [Proc. Steklov Inst. Math. 291, 146--169 (2015; Zbl 1338.49043); translation from Tr. Mat. Inst. Steklova 291, 157--181 (2015)].