The controllability function method

The paper is devoted to the control problem for the movement of an overhead crane with the use of a dynamic model in the form of "trolley - cargo" mechanical system and the driving force as a control parameter. To solve the system of differential equations, which describe the movement of the system taking into account constraints for the control, the controllability function method is applied. The algorithm for solving the problem is described, a program is developed as well as difficulties, which occur while implementing the method, and ways of its solution are marked. Results of constructing the control and system trajectories are also provided as an example of the program work

Horndeski Genesis: strong coupling and absence thereof

We consider Genesis in the Horndeski theory as an alternative to or completion of the inflationary scenario. One of the options free of instabilities at all cosmological epochs is the one in which the early Genesis is naively plagued with strong coupling. We address this issue to see whether classical field theory description of the background evolution at this early stage is consistent, nevertheless. We argue that, indeed, despite the fact that the effective Plank mass tends to zero at early time asymptotics, the classical analysis is legitimate in a certain range of Lagrangian parameters.Comment: 10 pages, 1 figur

Non-stationary Stochastic Optimization

We consider a non-stationary variant of a sequential stochastic optimization problem, in which the underlying cost functions may change along the horizon. We propose a measure, termed variation budget, that controls the extent of said change, and study how restrictions on this budget impact achievable performance. We identify sharp conditions under which it is possible to achieve long-run-average optimality and more refined performance measures such as rate optimality that fully characterize the complexity of such problems. In doing so, we also establish a strong connection between two rather disparate strands of literature: adversarial online convex optimization; and the more traditional stochastic approximation paradigm (couched in a non-stationary setting). This connection is the key to deriving well performing policies in the latter, by leveraging structure of optimal policies in the former. Finally, tight bounds on the minimax regret allow us to quantify the "price of non-stationarity," which mathematically captures the added complexity embedded in a temporally changing environment versus a stationary one