One presents a survey of the works on the theory of optimal control by deterministic objects, described by systems of ordinary differential or difference equations, where the investigation is carried out with the aid of the methods of the theory of singular perturbations. One analyzes the possibility of the application of the theory of singular perturbations to the investigation of control problems with large amplification coefficient in the feedback circuit, to the description of sliding regimes in systems with variable structure, and to the construction of effective numerical algorithms for the solution of optimization problems.

Singular perturbations in optimal control problems

Asymptotic approaches in mechanics: New parameters and procedures

ContentsIntroductionChapter 1. Variational elliptic inequalities ??1.1. First example ??1.2. Variational inequalities ??1.3. Variational inequalities and many-valued operators ??1.4. Examples ??1.5. The method of penalization ??1.6. The method of regularization ??1.7. Interpretation of variational inequalities and multipliers ??1.8. Regularity ??1.9. Comparison and convexity properties of solutions Chapter 2. Parabolic evolution inequalities (I) ??2.1. First examples ??2.2. The variational formulation. The strong solution ??2.3. The variational formulation. The weak solution ??2.4. Examples ??2.5. The method of elliptic regularization ??2.6. The method of penalization ??2.7. Some generalizations ??2.8. Quasi-stationary problems ??2.9. Regularity ??2.10. Remarks Chapter 3. Parabolic evolution inequalities (II). The flow of a non-Newtonian fluid ??3.1. Statement of the problem ??3.2. The main results ??3.3. The principle of the proof Chapter 4. Inequalities of the second order with respect to?t ??4.1. First example ??4.2. The general variational problem ??4.3. Examples ??4.4. Methods for existence proofs Chapter 5. Variational inequalities for Maxwell operators ??5.1. Statement of the problem ??5.2. Results Chapter 6. Other problems ??6.1. Coupled phenomena ??6.2. A problem with a decay ??6.3. Behaviour at infinity. Almost periodic solution Chapter 7. General remarks on the numerical approximation of solutions of inequalities ??7.1. Discrete schemes for evolution inequalities ??7.2. Analysis of stability ??7.3. The solution of stationary problems Chapter 8. Singular layer phenomena ??8.1. Generalities ??8.2. An example Chapter 9. Open problems References

Partial differential inequalities

The paper is devoted to the studies of viscous flows caused by a vibrating boundary. The fluid domain is a half-space, its boundary is a nondeformable plane that exhibits purely tangential vibrations. Such a simple geometrical setting allows us to study general boundary velocity fields and to obtain general results. From a practical viewpoint, such boundary conditions may be seen as the tangential vibrations of the material points of a stretchable plane membrane. In contrast to the classical boundary layer theory, we aim to build a global solution. To achieve this goal we employ the Vishik‐Lyusternik approach, combined with two-timing and averaging methods. Our main result is: we obtain a uniformly valid in the whole fluid domain approximation to the global solutions. This solution corresponds to general boundary conditions and to three different settings of the main small parameter. Our solution always include the inner part and outer part that both contain oscillating and non-oscillating components. It is shown that the nonoscillating outer part of the solution is governed either by the full Navier‐Stokes equations or the Stokes equations (both with the unit viscosity) and can be interpreted as a steady or unsteady streaming. In contrast to the existing theories of a steady streaming, our solutions do not contain any secular (infinitely growing with the inner normal coordinate) terms. The examples of the spatially periodic vibrations of the boundary and the angular torsional vibrations of an infinite rigid disc are

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Viscous Flows in a Half Space Caused by Tangential Vibrations on Its Boundary

The Vishik-Lyusternik method for elliptic boundary-value problems in regions with conical points

The asymptotic method of Lyusternik and Vishik has now been greatly expanded and has found important applications in many fields of mechanics The principal virtues of the method lie in its conceptual simplicity and in the fact that it can be applied to broad classes of partial differential equations with a small parameter occurring in the highest derivatives The present paper has two chapters In the first the method is illustrated by the example of the degeneration into elliptical equations of elliptical equations of higher orders, and the basic papers in which the method is developed and extended are surveyed The cases when the boundary layer functions are defined by partial differential equations (parabolic, elliptic and hyperbolic boundary layers), and also the phenomenon of the interior boundary layer, are illustrated in greater detail Recommendations are made for the application of the Newton diagram method in boundary layer problems when there is a small parameter entering arbitrarily Among the most interesting applications of the method we note the problem of the passage past a muffled body of the ultrasonic flow of a viscous gas (Markov and Chudov); number of problems in the non-linear theory of thin flexible plates and shells (Srubshchik and Yudovich); and problems in the motion of a rigid body with cavities containing a viscous fluid (Moiseev, Chernous'ko, and others) The second chapter consists mainly of the author's results on differential equations in a Banach space containing a small parameter in the leading derivatives Here the Lyusternik-Vishik method has been very little applied so far despite its undoubted advantages in this type of problem Of special interest are the cases when the boundary problem is situated on the spectrum The results of the chapter can be applied, in particular, to systems of ordinary differential equations, to integral-differential equations, and also to parabolic equations

https://iopscience.iop.org/article/10.1070/RM1970v025n04ABEH001262/pdf

V A Trenogin

Papers

The development and applications of the asymptotic method of lyusternik and vishik