Vol 213, No 4 (2022)

The topological bifurcations of Liouville foliations on invariant $3$-manifolds that are induced by attaching toric $\Theta$-handles are investigated. It is shown that each marked molecule (Fomenko-Zieschang invariant) can be realized on an invariant submanifold of a closed symplectic manifold with contact singularities which is obtained by attaching toric $\Theta$-handles sequentially to a set of symplectic manifolds, while these latter have the structures of locally trivial fibrations over $S^1$ associated with atoms. Bibliography: 10 titles.

A second-order quasilinear elliptic equation with a measure of special form on the right-hand side is considered. Restrictions on the structure of the equation are imposed in terms of a generalized $N$-function such that the conjugate function obeys the $\Delta_2$-condition and the corresponding Musielak-Orlicz space is not necessarily reflexive. In an arbitrary domain satisfying the segment property, the existence of an entropy solution of the Dirichlet problem is proved. It is established that this solution is renormalized. Bibliography: 29 titles.

The time minimization problem with admissible control in a half-disc is considered on the group of motions of a plane. The control system under study provides a model of a car on the plane that can move forwards or rotate in place. Optimal trajectories of such a system are used to detect salient curves in image analysis. In particular, in medical image analysis such trajectories are used for tracking vessels in retinal images. The problem is of independent interest in geometric control theory: it provides a model example when the set of values of the control parameters contains zero at the boundary. The problem of controllability and existence of optimal trajectories is studied. By analysing the Hamiltonian system of the Pontryagin maximum principle the explicit form of extremal controls and trajectories is found. Optimality of the extremals is partially investigated. The structure of the optimal synthesis is described. Bibliography: 33 titles.