We present sufficient conditions for the existence of a periodic solution for a class of systems describing the periodically forced motion of a massive point on a compact surface with a boundary.Keywords: periodic solution, Euler-Poincaré characteristic, nonlinear system Brief introductionIn 1922, G. Hamel proved [1] that equations describing motion of a periodically forced pendulum have at least one periodic solution. Since then, many results concerning periodic solutions in pendulum-like systems have been obtained by various authors including results for a one-dimensional forced pendulum [2], a result by M. Furi and M. P. Pera [3] who showed that a frictionless spherical pendulum also have forced oscillations and a work by V. Benci and M. Degiovanni [4] in which motion of a massive point on a compact boundaryless surface with friction is studied and sufficient conditions for existence of forced oscillations are presented. As far as we know, the case of compact surface with boundary is far less developed.However, surfaces with boundaries naturally appear in various mechanical systems. For instance, in a book [5] by R. Courant and H. Robbins the authors consider the problem which states that for an inverted planar pendulum placed on a floor of a train carriage, for any law of motion for the train, there always exists at least one initial position such that the pendulum, starting its motion from this position with zero generalized velocity, moves without falling for an arbitrary long time. Here, the compact surface is a half-circle and its boundary is the two-pointed set.Topological ideas, which lie in the basis of the above result, can be rigorously justified [6] -in [5] some details are omitted -and generalized for different types of systems. Moreover, it was proved [6] that for an inverted pendulum with a periodic law of motion for its pivot point, there exists a periodic solution along which the pendulum never becomes horizontal, i.e. it never falls. This was obtained as an application of a topological theorem by R. Srzednicki, K. Wójcik, and P. Zgliczyński [7].In the current paper we further develop this result [6] and present sufficient conditions for the existence of a periodic solution for a class of systems describing periodically forced

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Consider a periodically forced nonlinear system which can be presented as a collection of smaller subsystems with pairwise interactions between them. Each subsystem is assumed to be a massive point moving with friction on a compact surface, possibly with a boundary, in an external periodic field. We present sufficient conditions for the existence of a periodic solution for the whole system. The result is illustrated by a series of examples including a chain of strongly coupled pendulums in a periodic field.Keywords: periodic solution, Euler-Poincaré characteristic, nonlinear system, coupled pendulums, nonlinear lattice Brief introductionThe phenomenon of forced oscillations has been studied since at least 1922, when G. Hamel proved [1] that the equation describing the motion of a periodically forced pendulum have at least one periodic solution. Various results, which generalize and develop [1], have appeared in the literature since this paper; see, for example, [2,3,4,5,6]. Forced oscillations in a system of coupled planar pendulums and its generalizations has been studied in [7,8] under the assumption of certain symmetry properties for the forcing terms. The existence and multiplicity of periodic solutions for coupled systems also discussed in [9].At the same time, a nonlinear lattice is a cornerstone model in nonlinear physics and it is widely used for analytical and computational purposes. See, for example, [10,11,12,13]. In our work we present a result which lies in the intersection of the both mentioned research areas and can be useful for studying forced oscillations in a nonlinear lattice and its generalizations. The result given in the paper continues a previously reported result [14].In the paper the following systems are considered. Let us have several compact smooth manifolds, possibly with boundaries, with non-zero Euler-Poincaré characteristics. Suppose that for each manifold there is a massive point moving on it with viscous friction. All points are in an external periodic field and may interact with each other. It is allowed that the interactions may be of different types and also may be arbitrarily strong. We present sufficient conditions for the existence of a periodic solution for such a system.

We consider a classical problem of control of an inverted pendulum by means of a horizontal motion of its pivot point. We suppose that the control law can be non-autonomous and non-periodic w.r.t. the position of the pendulum. It is shown that global stabilization of the vertical upward position of the pendulum cannot be obtained for any Lipschitz control law, provided some natural assumptions. Moreover, we show that there always exists a solution separated from the vertical position and along which the pendulum never becomes horizontal. Hence, we also prove that global stabilization cannot be obtained in the system where the pendulum can impact the horizontal plane (for any mechanical model of impact). Similar results are presented for several analogous systems: a pendulum on a cart, a spherical pendulum, and a pendulum with an additional torque control.