A systematic method for establishment of exact and approximate controllability of dynamic systems with nonlinear state constraints within infinite time is developed. Using the recently developed Green's function approach, we derive necessary and sufficient conditions for exact controllability, as well as sufficient conditions for approximate controllability. Conditions for null‐controllability and lack of controllability are established as well. Several possibilities of deriving explicit form for resolving controls are described. Possibilities of heuristic determination of resolving controls is studied in details. The advantages and drawbacks of the method are revealed on specific examples of nonlinear wave and diffusion equations in unbounded domains.
The accelerated expansion of the large-scale universe can be explained in various ways. There are various modifications, and each of them makes an attempt to give its own explanation of the physics behind it. It is well known that modern cosmology is full of various phenomenological assumptions to obtain comprehensive results comparable with observational data. General Relativity is the main theory of gravity and proposed modifications compared to it, giving a hope to find explanations of phenomenological assumptions. f(T) theory of gravity is one of the options. In this paper, we will consider a particular example of f(T) theory and study the effects of various interactions on a cosmological model. Phase space analysis is used to have a qualitative understanding of the late-time behavior of the suggested cosmological models. During our study, we found that among phenomenological models suggested in this paper, we have cosmological models being in good agreement with the observational data. Moreover, study of the behavior of the deceleration parameter q showed a phase transition from a decelerated expanding universe to the accelerated (recent) expanding universe. On the other hand, for the parameters of the models giving the mentioned phase transition, we have estimated the present day values of statefinder parameters (r, s).
We suggest to apply the Bubnov-Galerkin method to solving control problems for bilinear systems. We reduce the solution of a control problem to a finite-dimensional system of linear problem of moments. We show a specific example of applying this procedure and give its numerical solution.
The problem of finite, partially glued to a fixed rigid base rod longitudinal vibrations damping by optimizing adhesive structural topology is investigated. Vibrations of the rod are caused by external load, concentrated on free end of the rod, the other end of which is elastically clamped. The problem is mathematically formulated as a boundary-value problem for onedimensional wave equation with attenuation and variable controlled coefficient. The intensity of adhesion distribution function is taken as optimality criterion to be minimized. Structure of adhesion layer, optimal in that sense, is obtained as a piecewise-constant function. Using Fourier real generalized integral transform, the problem of unknown function determination is reduced to determination of certain switching points from a system of nonlinear, in general, complex equations. Some particular cases are considered.
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