Asymptotic Solution of a Boundary-Value Problem for Linear Singularly-Perturbed Functional Differential Equations Arising in Optimal Control Theory

The Hamiltonian boundary-value problem, associated with a singularly-perturbed linear-quadratic optimal control problem with delay in the state variables, is considered. A formal asymptotic solution of this boundary-value problem is constructed by application of the boundary function method. The justification of this asymptotic solution is done. The asymptotic solution of the Hamiltonian boundary-value problem is constructed and justified assuming boundary-layer stabilizability and detectability.     

Asymptotic solution of a singularly perturbed set of functional-differential equations of Riccati type encountered in the optimal control theory

The boundary-value problem for the set of functional-differential equations with partial derivatives of Riccati type, associated with a singularly perturbed linear-quadratic optimal control problem with delay in state, is considered. The expression for a solution of the problem, which transforms it to the explicit singular perturbation form, is proposed. An asymptotic solution of this problem is constructed. Received August 7, 1997     

Optimal control of delay systems via block pulse functions

The concept of coefficient shift matrix is introduced to represent delay variables in block pulse series. The optimal control of a linear delay system with quadratic performance index is then studied via block pulse functions, which convert the problems into the minimization of a quadratic form with linear algebraic equation constraints. The solution of the two-point boundary-value problem with both delay and advanced arguments is circumvented. The control variable obtained is piecewise constant.     

The tricomi problem for a nonlinear mixed-type equation with functional retarded,advanced, and concentrated deviation

We study the boundary-value problem for a nonlinear mixed-type equation with the Lavrent’ev–Bitsadze operator in the main part and a functional delay or advance in the lowest terms. We construct a general solution to the equation under consideration and prove the unique solvability of the problem.     

On solvability of non-linear semi-periodic boundary-value problem for system of hyperbolic equations

We study a non-linear semi-periodic boundary-value problem for a system of hyperbolic equations with mixed derivative. At that, the semi-periodic boundary-value problem for a system of hyperbolic equations is reduced to an equivalent problem, consisting of a family of periodic boundary-value problems for ordinary differential equations and functional relation. When solving a family of periodic boundary-value problems of ordinary differential equations we use the method of parameterization. This approach allowed to establish sufficient conditions for the existence of an isolated solution of non-linear semi-periodic boundary-value problem for a system of hyperbolic equations.     

Uniformly continuous dependence of a solution to a controlled functional operator equation on a shift of control

We establish sufficient conditions for the uniform (with respect to the set of admissible controls) continuous dependence of a solution to a controlled functional operator equation on a shift of control along a vector of independent variables. The shift of control may mean, in particular, some time delay (or outstripping) of the control. We illustrate the use of general results by an example of a mixed boundary-value problem associated with a wave equation.     

Criteria of unique solvability of nonlocal boundary-value problem for systems of hyperbolic equations with mixed derivatives

We consider nonlocal boundary-value problem for a system of hyperbolic equations with two independent variables. We investigate questions of existence of unique classical solution to problem under consideration. In terms of initial data we propose criteria of unique solvability and suggest algorithms of finding of solutions to nonlocal boundary-value problem. As an application we give conditions of solvability of periodic boundary-value problem for a system of hyperbolic equations.     

Solvability of nonlinear boundary-value problems arising in modeling plasma diffusion across a magnetic field and its equilibrium configurations

We study the simplest one-dimensional model of plasma density balance in a tokamak type system, which can be reduced to an initial boundary-value problem for a second-order parabolic equation with implicit degeneration containing nonlocal (integral) operators. The problem of stabilizing nonstationary solutions to stationary ones is reduced to studying the solvability of a nonlinear integro-differential boundary-value problem. We obtain sufficient conditions for the parameters of this boundary-value problem to provide the existence and the uniqueness of a classical stationary solution, and for this solution we obtain the attraction domain by a constructive method.     

Singularly perturbed Dirichlet boundary-value problem for a stationary system in the linear elasticity theory

We consider a singularly perturbed Dirichlet boundary-value problem for an elliptic operator of the linear elasticity theory in a bounded domain with a small cavity. The main result is the proof of the theorem about the convergence of eigenelements of the perturbed boundary-value problem to eigenelements of the corresponding limiting boundary-value problem, when the parameter ? which defines the diameter of the small cavity tends to zero.     

关于耦合型对流-扩散方程组的奇异摄动边值问题

本文考虑下述耦合型对流-扩散方程组的奇异摄动边值问题:本文提出两种方法:一种是初值化解法,用这种方法,原始问题转化成一系列没有扰动的一阶微分方程或方程组的初值问题,从而得到一个渐近展开式;第二种是边值化解法,用这种方法,原始问题转化成一组没有边界层现象的边值问题,从而可以得到精确解和使用经典的数值方法去得到具有关于摄动参数ε一致的高精度数值解.     

On Quantum Stochastic Differential Equations as Dirac Boundary-Value Problems

We prove that a single-jump unitary quantum stochastic evolution is unitarily equivalent to the Dirac boundary-value problem on the half-line in an extended space. It is shown that this solvable model can be derived from the Schrödinger boundary-value problem for a positive relativistic Hamiltonian on the half-line as the inductive ultrarelativistic limit corresponding to the input flow of Dirac particles with asymptotically infinite momenta. Thus the problem of stochastic approximation can be reduced to a quantum mechanical boundary-value problem in the extended space. The problem of microscopic time reversibility is also discussed in the paper.     

Solvability of Linear Matrix Boundary-Value Problem

We find solvability conditions and give a construction of generalized Green operator for a linear matrix boundary-value problem. We suggest an operator which reduces a linear matrix equation to a standard linearNoetherian boundary-value problem. To solve a linearmatrix systemwe use an operatorwhich reduces a linear matrix equation to a linear algebraic equation with rectangular matrix.     

Boundary conditions of the second-order differential equation and the Riccati equation

The conversion of a second-order linear ordinary differential equation with variable coefficients into a Riccati equation depends on whether the second-order problem is an initial-value or two-point boundary-value problem. The distinction is critical in determining the initial condition for the Riccati equation. If the second-order problem is an initial-value problem, the choice of the Riccati transformation depends on whether a zero initial condition for the function or its derivative is specified. If the problem is a two-point boundary-value problem, special methods must be introduced as described in the paper.     

Convergence of the method of lines for one quasilinear parabolic equation with delay

We investigate the applicability of the method of lines to delayed equations. We consider a boundary-value problem for a quasilinear parametric equation with delay in the right-hand side. The scheme of the method of lines is shown to be second-order convergent when the delay is constant and nonnegative and the function f is Lipschitzian in the third argument in the class W₂ ²(Q_T).Translated from Vychislitel'naya i Prikladnaya Matematika, No. 62, pp. 20–26, 1987.     

Continuous approximations of discontinuous nonlinearities of semilinear elliptic-type equations

We obtain new variational principles of the existence of strong and semiregular solutions of principal boundary-value problems for elliptic-type second-order equations with discontinuous nonlinearity. We study a problem of proximity between the sets of solutions of an approximating problem with nonlinearity continuous in phase variable and solutions of the initial boundary-value problem with discontinuous nonlinearity. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 2, pp. 224–233, February, 1999.     

New conditions for averaging of nonlinear dirichlet problems in perforated domains

We study the problem of averaging Dirichlet problems for nonlinear elliptic second-order equations in domains with fine-grained boundary. We consider a class of equations admitting degeneration with respect to the gradients of solutions. We prove a pointwise estimate for solutions of the model nonlinear boundary-value problem and construct an averaged boundary-value problem under new structural assumptions concerning perforated domains. In particular, it is not assumed that the diameters of cavities are small as compared to the distances between them.     

Solving some problems in the theory of thin shells of revolution with complex boundary conditions

An approach is proposed to solving linear boundary-value problems for shells of revolution that are closed in the circumferential direction, with complex boundary conditions in which the coefficients of the solving functions depend on the circumferential coordinate. The approach relies on reduction of the boundary-value problem to a number of boundary-value problems for systems of ordinary differential equations and systems of algebraic equations. We solve a specific problem for the stressed state of a conical shell with one of its ends supported by an elastic foundation with a variable modulus.Institute of Mechanics, Ukrainian Academy of Sciences. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 85–93, 1989.     

Weakly nonlinear boundary-value problem in a special critical case

We investigate the problem of the determination of conditions for the existence of solutions of weakly nonlinear Noetherian boundary-value problems for systems of ordinary differential equations and the construction of these solutions. We consider the special critical case where the equation for finding the generating solution of a weakly nonlinear Noetherian boundary-value problem turns into an identity. We improve the classification of critical cases and construct an iterative algorithm for finding solutions of weakly nonlinear Noetherian boundary-value problems in the special critical case.     

Trapped modes of oscillation of a liquid for surface-piercing bodies in oblique waves

The linear problem of the harmonic oscillations of an ideal incompressible heavy liquid with a free surface in the presence of two and more infinitely long partially submerged cylindrical bodies of arbitrary cross-section is considered. It is proved that there are configurations of the bodies which provide examples of the non-uniqueness of the boundary-value problem in the case of an arbitrary frequency of the oscillations and an arbitrary non-zero angle between the generatrix of the cylinders and the direction of propagation of the surface waves. In the case of these configurations, the homogeneous boundary-value problem has non-trivial solutions with a finite energy integral, which describe trapped modes of oscillation of the liquid.     

Local convexity conditions for reachability tubes of distributed control systems

For a nonlinear functional operator equation describing a wide class of controlled initial boundary-value problems we introduce the notion of an abstract reachability set analogous to the notion of a reachability tube. We obtain local sufficient conditions for the convexity of such a set. We consider a mixed boundary-value problem associated with a semilinear hyperbolic equation of the second order in a rather general form as an example illustrating the reduction of a controlled initial boundary-value problem to the studied equation, as well as the verification of the stated assumptions.