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.     

Solvability of the mixed initial boundary-value problem for a parabolic equation with a nonlocal condition of first kind

The spectral method is applied to solve the mixed initial boundary-value problem for a parabolic equation with nonhomogeneous boundary conditions, one of which is nonlocal. We prove existence and uniqueness of the generalized solution of this problem in the Sobolev class W ₂ ^1,0 and represent it as a biorthogonal series. We also consider optimal control by the right-hand side of the equation, which is constructed as a biorthogonal series in the root functions of the spectral problem.Translated from Nelineinaya Dinamika i Upravlenie, No. 2, pp. 209–220, 2002.     

Optimal singular and chattering modes in the problem of controlling the vibrations of a string with clamped ends

The problem of minimizing the root mean square deviation of a uniform string with clamped ends from an equilibrium position is investigated. It is assumed that the initial conditions are specified and the ends of the string are clamped. The Fourier method is used, which enables the control problem with a partial differential equation to be reduced to a control problem with a denumerable system of ordinary differential equations. For the optimal control problem in the l₂ space obtained, it is proved that the optimal synthesis contains singular trajectories and chattering trajectories. For the initial problem of the optimal control of the vibrations of a string it is also proved that there is a unique solution for which the optimal control has a denumerable number of switchings in a finite time interval.     

A problem of the bending of a plate for a doubly-connected domain bounded by polygons

The problem of the bending of an isotropic elastic plate, bounded by two convex polygons is considered. It is assumed that the internal boundary of the plate is simply supported and normal bending moments act on each section of the external contour in such a way that the angle of rotation of the middle surface of the plate is a piecewise-constant function. With respect to the complex potentials, which express the bendings of the middle surface (Goursat's formula), the problem is reduced to a Riemann-Hilbert boundary-value problem for a circular ring, the solution of which is constructed in analytic form. Estimates are given of the behaviour of these potentials in the neighbourhood of the corner points.     

Stability of a smooth flow in a Laval nozzle

A boundary-value problem for a non-linear second-order equation of mixed type in a cylindrical domain is considered. This problem simulates the development of small disturbances in a transonic flow of a chemical mixture in a Laval nozzle. The existence of a regular solution is proved with the help of a priori estimates for a corresponding linear problem and the contractive mapping theorem. The solution of the linear problem is constructed by the Galerkin method.     

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     

Optimization of a time-variable,constrained control matrix

The problem posed is to choose, in a optimal manner, a time-variable, bounded, linear transformation defining the velocity of a state point inn-dimensional space in terms of the state. The two-point boundary-value problem which arises from an application of the Pontryagin maximum principle is explicitly solvable; hence, a formula is derived showing that the optimal trajectories in state space are equiangular spirals in two-dimensional subspaces ofR ⁿ and also describing the boundary of the set of attainability. This formula is used to solve the problem of minimal-time transfer between any two given points, and the optimal control is specified both as an open-loop and a closed-loop controller. The solutions to the problem of maximizing a linear payoff function of the final state and of maximizing the angle of rotation of the state vector about the origin are also given.     

Solvability of the boundary-value problem for a mixed equation involving hyper-Bessel fractional differential operator and bi-ordinal Hilfer fractional derivative

In a rectangular domain, a boundary-value problem is considered for a mixed equation with a regularized Caputo-like counterpart of hyper-Bessel differential operator and the bi-ordinal Hilfer's fractional derivative. By using the method of separation of variables a unique solvability of the considered problem has been established. Moreover, we have found the explicit solution of initial-boundary problems for the heat equation with the regularized Caputo-like counterpart of the hyper-Bessel differential operator with the non-zero starting point.     

Numerical treatment of a mathematical model arising from a model of neuronal variability

In this paper, we describe a numerical approach based on finite difference method to solve a mathematical model arising from a model of neuronal variability. The mathematical modelling of the determination of the expected time for generation of action potentials in nerve cells by random synaptic inputs in dendrites includes a general boundary-value problem for singularly perturbed differential-difference equation with small shifts. In the numerical treatment for such type of boundary-value problems, first we use Taylor approximation to tackle the terms containing small shifts which converts it to a boundary-value problem for singularly perturbed differential equation. A rigorous analysis is carried out to obtain priori estimates on the solution of the problem and its derivatives up to third order. Then a parameter uniform difference scheme is constructed to solve the boundary-value problem so obtained. A parameter uniform error estimate for the numerical scheme so constructed is established. Though the convergence of the difference scheme is almost linear but its beauty is that it converges independently of the singular perturbation parameter, i.e., the numerical scheme converges for each value of the singular perturbation parameter (however small it may be but remains positive). Several test examples are solved to demonstrate the efficiency of the numerical scheme presented in the paper and to show the effect of the small shift on the solution behavior.     

Time-optimal steering of an object moving in a viscous medium to a desired phase state

The two-point problem of the time-optimal attainment of a desired phase state by a multidimensional dynamic object is investigated. The motion occurs in a viscous medium by means of a limited force. The open-loop and/or feedback control laws constructed by numerical-analytical methods for arbitrary initial data. An asymptotically approximate solution of the maximum principle boundary-value problem is presented for short and long time intervals. The singularities of the optimal trajectory are established for the initial and final parts of the motion. The solution obtained of the two-point problem of the optimal control of the motion of a dynamic object in a homogeneous viscous medium by means of a force of bounded modulus is compared with the known solutions in special formulations.     

Solution of the first boundary-value problem for a second-order parabolic equation in a space of generalized functions

A solution is constructed for the first boundary-value problem for a second-order parabolic equation for the case in which the right side of the equation and the boundary and initial data are generalized functions. The solution is expressed in terms of specified generalized functions and Green's function or a fundamental solution.Translated from matematicheskie Metody i Fiziko-mekhanicheskie Polya, No. 26, pp. 3–7, 1987.     

Solution of a singularly perturbed nonstationary fourth-order boundary-value problem

A difference scheme is constructed by the method of lines for a nonstationary boundary-value problem for a fourth-order equation in the space coordinate. The uniform convergence with respect to a small parameter, of the solution of the linearization scheme to a solution of the original problem is proven.Translated from Vychislitel'naya i Prikladnaya Matematika, Vol. 70, pp. 3–11, 1990.     

Integrable boundary-value problems and nonlinear Fourier harmonics

For the nonlinear Schrödinger equation, the integrable boundary-value problem on a segment is considered. The concept of nonlinear ?-harmonics similar to the ordinary Fourier harmonics in the linear case is suggested. A solution of the initial boundary-value problem on the semiaxis is constructed by means of reduction to the Cauchy problem on the whole axis. Bibliography: 11 titles.     

On the well-posedness of a weighted boundary-value problem for systems of ODEs with singularity

We consider a system of linear second-order ODEs with an irregular singular point. The matrix that relates to the equation of this system is nonnormal. That implies some complications with respect to the statement of the weighted boundary-value condition at the singular point. We propose here a well-posed weighted boundary-value problem to the considered system of ODEs and prove the existence and uniqueness of its solution.     

An Optimal Control Problem in Coefficients for a Strongly Degenerate Parabolic Equation with Interior Degeneracy

We deal with an optimal control problem in coefficients for a strongly degenerate diffusion equation with interior degeneracy, which is due to the nonnegative diffusion coefficient vanishing with some rate at an interior point of a multi-dimensional space domain. The optimal controller is searched in the class of functions having essentially bounded partial derivatives. The existence of the state system and of the optimal control are proved in a functional framework constructed on weighted spaces. By an approximating control process, explicit approximating optimality conditions are deduced, and a representation theorem allows one to express the approximating optimal control as the solution to the eikonal equation. Under certain hypotheses, further properties of the approximating optimal control are proved, including uniqueness in some situations. The uniform convergence of a sequence of approximating controllers to the solution of the exact control problem is provided. The optimal controller is numerically constructed in a square domain.     

The stress state of an elastic composite cone with centre of rotation at the apex

The torsion of a composite cone that has a centre of rotation at its apex is investigated in a spherical system of coordinates. A composite cone is a cone with one shear modulus, inserted into a conical funnel having another shear modulus and with ideal mechanical contact between its surface and the inner surface of the conical funnel. The auxiliary problem of a composite cone with its apex truncated by a spherical surface is considered first. The outer surface of such a conical body is not loaded, but a load that reduces to a torque is applied to its spherical surface. The auxiliary problem is reduced to a one-dimensional discontinuous boundary-value problem using a specially constructed integral transformation. The exact solution of this boundary-value problem is constructed. The limit is then taken in the solution obtained as the radius of the spherical surface tends to zero for the purpose of obtaining an exact solution of the problem of the torsion of a composite cone that has a centre of rotation at the apex.     

Asymptotics of Extremal Curves in the Ball Rolling Problem on the Plane

In the present paper, we study an optimal sphere rolling problem on the plane (without slew and slip) with predefined boundary-value conditions. To solve it, we use methods from the optimal control theory. The controlled system for sphere orientation is represented via the rotation quaternion. Asymptotics of extremal paths on a sphere rolling along small-amplitude sine waves is found.     

A time-decomposition algorithm for a stiff linear two-point boundary-value problem and its application to a nonlinear optimal control problem

An algorithm is proposed to solve a stiff linear two-point boundary-value problem (TPBVP). In a stiff problem, since some particular solutions of the system equation increase and others decrease rapidly as the independent variable changes, the integration of the system equation suffers from numerical errors. In the proposed algorithm, first, the overall interval of integration is divided into several subintervals; then, in each subinterval a sub-TPBVP with arbitrarily chosen boundary values is solved. Second, the exact boundary values which guarantee the continuity of the solution are determined algebraically. Owing to the division of the integration interval, the numerical error is effectively reduced in spite of the stiffness of the system equation. It is also shown that the algorithm is successfully imbedded into an interaction-coordination algorithm for solving a nonlinear optimal control problem.The authors would like to thank Mr. T. Sera and Mr. H. Miyake for their help with the calculations.     

On the solvability of a boundary-value problem for an elliptic equation

Consider a boundary-value problem for a second-order linear elliptic equation in a bounded domain. The coefficient of the required function is nonpositive everywhere in the domain, except for a small neighborhood of an interior point. The following question arises: Under what constraints on this coefficient in the given small domain do the statements on the existence and uniqueness of the solution of the first boundary-value problem remain valid?     

A mixed boundary-value problem for the wave equation in a stratified medium for high-frequency oscillations

A boundary-value problem for the wave equation in a stratified medium with mixed boundary conditions on the boundary in the case of high oscillation frequencies is considered. The Helmholtz equation for a velocity function increasing monotonically with depth is investigated. The problem is reduced to an integral equation in the high-frequency approximation, and an explicitly smooth term of its asymptotic solution is constructed.