Generic fractal structure of the optimal synthesis in problems with affine multi-dimensional control

We consider a linear-quadratic deterministic optimal control problem where the control takes values in a two-dimensional simplex. The phase portrait of the optimal synthesis contains second-order singular extremals and exhibits modes of infinite accumulations of switchings in finite time, so-called chattering. We prove the presence of an entirely new phenomenon, namely a chaotic behaviour of the set of optimal trajectories. The set of optimal non-wandering trajectories has the structure of a Cantor set, and the dynamics of the system is described by a topological Markov chain. We compute the entropy and the Hausdorff dimension of the non-wandering set. This behaviour is generic for piece-wise smooth Hamiltonian systems in the vicinity of a junction of three discontinuity hypersurface strata.