Optimal boundary control by a force at one end of a string with a given force mode at the other end

We consider the problem of boundary control by a force applied to one end of a string in the case of a given force mode at the other end. The problem is studied in the sense of the generalized solution of the corresponding mixed initial-boundary value problem in the Sobolev space. We also solve the problem of choosing an optimal boundary control in the set of all admissible controls. The generalized solution of the mixed initial-boundary value problem is constructed in closed form, and its uniqueness is proved.     

Optimal boundary control by displacement at one end of a string under a given elastic force at the other end

The problem of optimal boundary control by displacement at one end of a string under a specified force mode at the other end is studied in the sense of a generalized solution of the corresponding mixed initial-boundary value problem from a Sobolev space. The problem of choosing an optimal boundary control from an infinite number of admissible controls is solved. A generalized solution of the mixed initial-boundary value problem is constructed explicitly and the uniqueness of the solution is proved.     

Optimal boundary force control at one end of a string for a given displacement mode at the other end

We solve the problem of optimal boundary force control at one end of a string for the case of a given displacement mode at the other end. The problem is studied in the sense of a generalized solution of the corresponding mixed initial-boundary value problem in the Sobolev space. We also solve the problem of choosing an optimal boundary control from infinitely many feasible controls. The generalized solution of the mixed initial-boundary value problem is constructed in closed form, and the uniqueness of the solution is proved.     

A Lower Bound for the Penalty Parameter in the Exact Minimax Penalty Function Method for Solving Nondifferentiable Extremum Problems

In the paper, we consider the exact minimax penalty function method used for solving a general nondifferentiable extremum problem with both inequality and equality constraints. We analyze the relationship between an optimal solution in the given constrained extremum problem and a minimizer in its associated penalized optimization problem with the exact minimax penalty function under the assumption of convexity of the functions constituting the considered optimization problem (with the exception of those equality constraint functions for which the associated Lagrange multipliers are negative—these functions should be assumed to be concave). The lower bound of the penalty parameter is given such that, for every value of the penalty parameter above the threshold, the equivalence holds between the set of optimal solutions in the given extremum problem and the set of minimizers in its associated penalized optimization problem with the exact minimax penalty function.     

Reducibility of minimax to minisum 0–1 programming problems

We consider 0–1 programming problems with a minimax objective function and any set of constraints. Upon appropriate transformations of its cost coefficients, such a minimax problem can be reduced to a linear minisum problem with the same set of feasible solutions such that an optimal solution to the latter will also solve the original minimax problem.Although this reducibility applies for any 0–1 programming problem, it is of particular interest for certain locational decision models. Among the obvious implications are that an algorithm for solving a p-median (minisum) problem in a network will also solve a corresponding p-center (minimax) problem.It should be emphasized that the results presented will in general only hold for 0–1 problems due to intrinsic properties of the minimax criterion.     

Diffraction of a plane electromagnetic wave obliquely incident upon the edge of a wedge with a dielectric coating

The diffraction of a plane electromagnetic wave obliquely incident upon the edge of a coated wedge is considered. The generalized impedance boundary conditions (GIBC's) on the wedge's faces are used to simulate the effect of the coatings. To insure the well-posedness of the problem, special contact conditions (CC's) on the edge are additionally imposed. By using Sommerfeld integrals, the problem is reduced to a system of coupled functional equations, which is solved by the perturbation method. It is shown that, for a certain range of the angles of oblique incidence, the solution can be represented in the form of convergent series that are Neumann series for linear equations with contracting operators. Nonuniform asymptotics of the wave field for regions outside a neighborhood of the edge of the wedge are constructed. Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 218, 1994, pp. 72–95. Translated by L. G. Vardapetyan and M. A. Lyalinov.     

Control optimization in a hyperbolic linear system

An optimal control problem is considered for a distributed-parameter Goursat-Darboux system with controlled boundary conditions. For the numerical solution of the problem, an algorithm based on separability and minimax theorems is constructed, which reduces the problem to finding the maximum of a concave functional defined in the class of one-variable functions.     

An explicit solution of the pseudo-hyperbolic initial boundary value problem in a multiply connected region

An explicit solution of the pseudo-hyperbolic initial boundary value problem with a mixed boundary condition has been constructed. The problem describes the propagation of non-stationary internal waves in a stratified and rotational fluid. The generation of waves is caused by small oscillations of double-sided plates beginning at time t = 0. Dynamic pressure is specified on one set of plates and this yields the first boundary condition. Normal velocities are specified on another set of plates and this leads to an analogue of the second boundary condition with time derivatives. The solution has been obtained by the method of non-classical time-dependent dynamic potentials. The uniqueness of the solution has been studied.     

Generalized thermoelastic waves in a layer with a boundary condition that is nonlinear with respect to temperature

A solution for the unsteady, generalized problem of thermoelasticity is constructed and investigated for a plane layer on one of whose surfaces there is given a generalized boundary condition of the third kind with a temperature-dependent heat-transfer coefficient.Translated from Matematicheskie Melody i Fiziko-mekhanicheskie Polya, No. 26, pp. 31–35, 1987.     

The interaction of punches on the face of an elastic wedge

The interaction of two punches, which are elliptic in plan, on the face of an elastic wedge is investigated in a three-dimensional formulation for different types of boundary conditions on the other face. The wedge material is assumed to be incompressible. An asymptotic solution is obtained for punches which are relatively distant from one another and from the edge of the wedge. For the case when the punches are arranged relatively close to the edge of the wedge (or reach the edge, the contact area is unknown) the numerical method of boundary integral equations is used. The mutual effect of the punches is estimated by means of calculations. The asymptotic solution of the generalized Galin problem, concerning the effect of a concentrated force applied on the edge of the three-dimensional wedge on the contact pressure distribution under a circular punch relatively far from the edge, is obtained.     

Boundary Integral Equations for Mixed Boundary Value Problems in R3

Both exterior and interior mixed Dirichlet-Neumann problems in R³ for the scalar Helmholtz equation are solved via boundary integral equations. The integral equations are equivalent to the original problem in the sense that the traces of the weak seolution satisfy the integral equations, and, conversely, the solution of the integral equations inserted into Green's formula yields the solution of the mixed boundary value problem. The calculus of pseudodifferential operators is used to prove existence and regularity of the solution of the integral equations. The regularity results — obtained via Wiener-Hopf technique — show the explicit “edge” behavior of the solution near the submanifold which separates the Dirichlet boundary from the Neumann boundary.     

Control of a family of nonlinear dynamic systems under measurements with bounded disturbances

We consider a control synthesis problem for nonlinear dynamic systems under parametric uncertainty and bounded measurement noises. Because of bounded disturbances in measurements of the state vector and the nonlinearity in the control object, the initially formulated control synthesis problem for a family of nonlinear systems as a generalized Zubov problem is transformed into a symbiosis of generalized Zubov–Bulgakov problems. The main result of the paper is the analytic solution of a minimax synthesis problem, which yields a constructive method for finding an invariant set.     

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.     

Construction of a generalized solution to an equation that preserves the Bellman type in a given domain of the state space

A Cauchy problem is considered for a Hamilton-Jacobi equation that preserves the Bellman type in a spatially bounded strip. Sufficient conditions are obtained under which there exists a continuous generalized (minimax/viscosity) solution to this problem with a given structure in the strip. A construction of this solution is presented.     

Extension of the variational formulation of the problem for a rigid-plastic medium to velocity fields with slip-type discontinuities

Sets of velocity fields containing slip-type discontinuities at the boundary of the rigid-plastic medium, as well as within it, and the functionals defined on these sets, are described. It is shown that the exact lower bounds of the variational problems for these functionals are equal to the coefficient of the critical load. The minimax problem with saddle point constructed here is regarded as an extension of the classical minimax problem of the theory of critical loads.     

Optimal boundary displacement control at one end of a string with a medium exerting resistance at the other end

We study a problem of optimal boundary control of vibrations of a one-dimensional elastic string, the objective being to bring the string from an arbitrary initial state into an arbitrary terminal state. The control is by the displacement at one end of the string, and a homogeneous boundary condition containing the time derivative is posed at the other end. We study the corresponding initial-boundary value problem in the sense of a generalized solution in the Sobolev space and prove existence and uniqueness theorems for the solution. An optimal boundary control in the sense of minimization of the boundary energy is constructed in closed analytic form.     

Boundary layer resolution in hierarchical plate modelling

The dimensional reduction of an elliptic model boundary value problem on a thin, plate-like domain of thickness 2d is analysed. A class of lower-dimensional models with variable model orders in the interior and near the boundary of the plate is investigated and it is shown that for a sufficiently large model order in a O(d∣ln d∣)-neighbourhood of the edge the hierarchical models compensate for the boundary layers of the exact solution—in the sense that their asymptotic rate of convergence as d → 0 is the same as the optimal one for compatible, layer-free solutions.     

Multicriterion structure/control design for optimal maneuverability and fault tolerance of flexible spacecraft

A multicriterion design problem for optimal maneuverability and fault tolerance of flexible spacecraft is considered. The maneuverability index reflects the time required to perform rest-to-rest attitude maneuvers for a given set of angles, with the postmaneuver spillover within a specified bound. The performance degradation is defined to reflect the maximum possible attitude error after maneuver due to the effect of faults. The fault-tolerant design is to minimize the worst performance degradation from all admissible faults by adjusting the design of the spacecraft. It is assumed that admissible faults can be specified by a vector of real parameters. The multicriterion design for optimal maneuverability and fault tolerance is shown to be well defined, leading to a minimax problem. Analysis for this nonsmooth problem leads to closed-form expressions of the generalized gradient of the performance degradation function with respect to the fault parameters and structural design variables. Necessary and sufficient conditions for the optimum are derived, and the closed-form expressions of the generalized gradients are applied for their interpretation. The bundle method is applicable to this minimax problem. Approximate methods which efficiently solve this minimax problem with relatively little computational difficulties are presented. Numerical examples suggest that it is possible to improve the fault tolerance substantially with relatively little loss in maneuverability.     

Optimal m-stage Runge-Kutta schemes for steady-state solutions of hyperbolic equations

A numerical scheme is developed to find optimal parameters and time step of m-stage Runge-Kutta (RK) schemes for accelerating the convergence to -steady-state solutions of hyperbolic equations. These optimal RK schemes can be applied to a spatial discretization over nonuniform grids such as Chebyshev spectral discretization. For each m given either a set of all eigenvalues or a geometric closure of all eigenvalues of the discretization matrix, a specially structured nonlinear minimax problem is formulated to find the optimal parameters and time step. It will be shown that each local solution of the minimax problem is also a global solution and therefore the obtained m-stage RK scheme is optimal. A numerical scheme based on a modified version of the projected Lagrangian method is designed to solve the nonlinear minimax problem. The scheme is generally applicable to any stage number m. Applications in solving nonsymmetric systems of linear equations are also discussed. © 1993 John Wiley & Sons, Inc.     

Singular solutions with exponential growth to Protter’s problems

Four-dimensional boundary value problems which were formulated by Proter for the nonhomogeneous wave equation are studied. They can be considered as multidimensional versions of the Darboux problems in ?². Protter’s problem is not well posed in the frame of classical solvability. On the other hand, it is known that the unique generalized solution may have a strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic cone and does not propagate along the cone. Some known results suggest that the solution may have at most exponential growth. We construct an infinitely smooth right-hand side function such that the corresponding generalized solution to Protter’s problem has an exponential singularity.