We propose a generalization of the Butkovskiy's method of control with compact support [1] allowing to derive exact controllability conditions and construct explicit solutions in control problems for systems with distributed parameters. The idea is the introduction of a new state function which is supported in considered bounded time interval and coincides with the original one therein. By means of techniques of the distributions theory the problem is reduced to an interpolation problem for Fourier image of unknown function or to corresponding system of integral equalities. Treating it as infinite dimensional problem of moments, its L1, L2 and L∞-optimal solutions are constructed explicitly. The technique is explained for semilinear wave equation with distributed and boundary controls. Particular cases are discussed

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.

In this paper, we study various cosmological models involving new nonlinear forms of interaction between cold dark matter (DM) and dark energy (DE) assuming that DE is a barotropic fluid. The interactions are nonlinear either due to $log({\rho }_{de}/{\rho }_{dm})$ or $log({\rho }_{dm}/{\rho }_{de})$ parameterizations, respectively. The main purpose of this paper is to demonstrate the applicability of the forms of suggested interactions to the problem of modern cosmology known as accelerated expansion of the Universe. Using the differential age of old galaxies expressed in terms of $H\left(z\right)$ data, the peak position of baryonic acoustic oscillations (known as BAO data), the SN Ia data with strong gravitational lensing data, we obtain the best fit values of the model parameters for each case. Besides, using $Om$ analysis and $S_3$ parameter from the statefinder hierarchy analysis, we also demonstrate that the considered models are clearly different from the $\mathsf{\Lambda }$ CDM model. We obtain that the models predict Hubble parameter values consistent to the estimations from gravitational lensing, which probes the expansion out to $z\le 1.7$ . We show that, with considered models, we can also explain PLANCK 2015 and PLANCK 2018 experiment results.

The advantageous Green’s function method that originally has been developed for nonhomogeneous linear equations has been recently extended to nonlinear equations by Frasca. This article is devoted to rigorous and numerical analysis of some second-order differential equations new nonlinearities by means of Frasca’s method. More specifically, we consider one-dimensional wave equation with quadratic and hyperbolic nonlinearities. The case of exponential nonlinearity has been reported earlier. Using the method of generalized separation of variables, it is shown that a hierarchy of nonlinear wave equations can be reduced to second-order nonlinear ordinary differential equations, to which Frasca’s method is applicable. Numerical error analysis in both cases of nonlinearity is carried out for various source functions supporting the advantage of the method.