The subject of this paper is the analysis of sparse state feedback design procedures for linear discrete-time systems. By sparsity we mean the presence of zero rows in the gain matrix; this requirement is natural in the engineering practice when designing ``economy'' control systems which make use of a small amount of control inputs. Apart from the design of stabilizing sparse controllers, the linear-quadratic regulation problem is considered in the sparse formulation. Also, we consider a regularization scheme typical to the $\ell_1$-optimization theory. The efficiency of the approach is illustrated via numerical examples.

We consider the problem of stabilization of discrete-time bilinear control systems. Using the linear matrix inequality technique and quadratic Lyapunov functions, we formulate a method for the construction of the so-called stabilizability ellipsoid having the property that the trajectories of the closed-loop system emanating from the points in the ellipsoid asymptotically tend to the origin. The proposed approach allows for an efficient construction of nonconvex domains of stabilizability of discrete-time bilinear control systems. The results are extended to the robust statement of the problem where the system matrix is subjected to structured uncertainties.

The article is concerned with the study of asymptotic behavior of solutions of the Burgers equation and its generalizations with initial value --- boundary problem on a finite interval, with constant boundary conditions. Since these equations take a dissipation into account, it is naturally to presuppose that any initial profile will evolve to an invariant time-independent solution with the same boundary values. Yet the answer happens to be slightly more complex. There are three possibilities: the initial profile may regularly decay to an invariant solution; or a Heaviside-type gap develops through a dispersive shock and multi-oscillations; or, exotically, an asymptotic limit is a 'frozen multi-oscillation' piecewise-differentiable solution, composed of different smooth invariant solutions.

The investigations of the influence of aerodynamics on the motion of an artificial two solid satellite-link a reconditioned by the expansion of their field of application to orbital large extension tether systems. Aerodynamic effects are quite significant for such large space systems. The paper considers the classification of cylindrical phase space limit cycles and also researches the form of the limit cycle for the equation of the relative motion of two bodies orbital link with the influence of the gravitational effect, aerodynamic pressure, aero-gradient and dissipative factors, depending on the eccentricity growth and aerodynamic parameter. We use the associated motion equations taking into account gravity gradient and aerodynamic factors. An assessment of the influence of eccentricity and aerodynamic parameter changes on the behavior of limit cycles is also given in the paper. The research applies well-known methods of nonlinear mechanics: Lagrange equations of the first kind, the phase plane method, a numerical implementation of the point mapping method and stability theory methods. *