Terminal control of a singularly perturbed linear system, with constraints on the right endpoint of the trajectory is considered. The asymptotic behaviour of the solution is analysed and on that basis an algorithm is proposed for the asymptotic allocation of optimal control switching points. A computational procedure is outlined which utilizes the asymptotic approximations to obtain an exact solution for any given value of the small parameter.
The problem of approximating a function satisfying a Lipschitz condition in the interval [a, b] is considered. A single-step optimal stochastic algorithm for selecting the data points is constructed.
The second term in the asymptotic expansion of the contribution from a nondegenerate stationary point is determined for the multidimensional case.
The problem of determining the field of permeability of an oil-bearing layer from the variations of the pressure and the yield of liquid at the boring wells can be stated as a problem for the minimum value of a functional and can be approximately solved by means of the methods of optimal control theory. The construction of numerical schemes is discussed. The results of a numerical solution of the problem enable one to draw conclusions regarding the uniqueness of the solution.
A one-dimensional model problem concerning the diffusion of a domain of turbulent perturbations in a homogeneous fluid is considered. It is assumed that the process takes place in a half-strip and is characterized by the equation of balance for the turbulent energy. A theorem on the existence of a solution is proved.
An approach to the programming of a new algorithm involving a certain numbering of the unknowns is proposed, the use of which does not require permutation of rows and columns of the system to ensure stability of the solution algorithm. A method is proposed for solving the corresponding system of linear algebraic equations, whose matrix is not positive-definite but is of a special form.
In the context of the numerical treatment of the two-dimensional Navier-Stokes equations, the boundary conditionsatasolid surface may be realized in different ways, some of which are examined. The approach considered here assumes that the equations of the system are solved separately. It is shown that then, irrespective of the specific formulation of the Navier-Stokes equations for an incompressible viscous liquid — in terms of velocity-pressure, velocity-vorticity or vorticity-stream function — the boundary conditions can be realized in an algorithmically universal way, based on a two-parameter formula previously proposed to approximate vorticity on a wall.
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