The construction of discontinuous controls for non-linear dynamical system with uncertainties

The properties of the solutions of a family of non-linear parametric problems are investigated in the neighbourhood of regular and irregular values of the parameter. Rules are proposed for constructing feedback-type controlling for non-linear dynamical systems with perturbations. An example is presented.     

Parametric identification problem with a regularizer in the form of the total variation of the control

We consider a linear dynamical system, for which we need to reconstruct the control input on the basis of a noisy output. We form the corresponding family of parametric optimal control problems in which the performance criterion contains terms corresponding to the problem regularization and clearing the output signal from speckle noises. The weight coefficient multiplying the term used for noise filtration plays the role of a parameter in the family of problems. We prove a theorem that describes the properties of solutions of parametric problems in a neighborhood of a regular point, analyze the differential properties of solutions of that problem, and derive formulas for the computation of derivatives of the optimal trajectory and the optimal control with respect to a parameter. We suggest a simple method for constructing approximate solutions of perturbed optimal control problems. These results permit one to control the performance of the reconstruction of the control in the original identification problem. An illustrative example is considered.     

Contact problems for bodies with complex coatings

The paper deals with various statements and mathematical models of contact and contact-wear problems for bodies with coatings. It is shown that the mathematical models for a number of such problems can be represented as a mixed integral equation or a system of mixed integral equations with additional conditions. It is also shown that these equations contain rapidly varying or even discontinuous functions in the case of interaction between bodies of complex shape and with some surface properties. Therefore, it is necessary to use a special approach for constructing efficient analytic solutions. Its implementation is demonstrated by an example.     

Synthesis of 4-Aryl-1,5-benzodiazepine-2-carboxamides

4,N-Diaryl-1,5-benzodiazepine-2-carboxamides were synthesized by acid-catalyzed reaction of (Z)-4,N-diaryl-2-hydroxy-4-oxo-2-butenamides with o-phenylenediamine or N,N'-bis(triphenylphosphoranediyl)-o-phenylenediamine. The reaction mechanism is discussed.     

Asymptotic properties of solutions of parametric optimal control problems with varying index of the singular arcs

We consider a family of parametric linear-quadratic optimal control problems with terminal and control constraints. This family has the specific feature that the class of optimal controls is changed for an arbitrarily small change in the parameter. In the perturbed problem, the behavior of the corresponding trajectory on noncritical arcs of the optimal control is described by solutions of singularly perturbed boundary value problems. For the solutions of these boundary value problems, we obtain an asymptotic expansion in powers of the small parameter ?. The asymptotic formula starts from a term of the order of 1/? and contains boundary layers. This formula is used to justify the asymptotic expansion of the optimal control for a perturbed problem in the family. We suggest a simple method for constructing approximate solutions of the perturbed optimal control problems without integrating singularly perturbed systems. The results of a numerical experiment are presented.