Boundary Control of PDEs via Curvature Flows: the View from the Boundary, II

We describe some results on the exact boundary controllability of the wave equation on an orientable two-dimensional Riemannian manifold with nonempty boundary. If the boundary has positive geodesic curvature, we show that the problem is controllable in finite time if (and only if) there are no closed geodesics in the interior of the manifold. This is done by solving a parabolic problem to construct a convex function. We exhibit an example for which control from a subset of the boundary is possible, but cannot be proved by means of convex functions. We also describe a numerical implementation of this method.     

Solutions of the helmholtz equation, concentrated near a plane periodic boundary

The existence of solutions of the Helmholtz equation, exponentially decreasing with distance from a periodic boundary in the upper half-plane, is proved. These solutions exist for a special form of the boundary under the Dirichlet or Neumann boundary conditions. In either case, the boundary has the form of a chain of resonators joined with the upper half-plane by narrow splits. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 83–96. Translated by V. Yu. Gotlib     

Euler's problem,Euler's method,and the standard map; or,the discrete charm of buckling

Summary We explore the relation between the classical continuum model of Euler buckling and an iterated mapping which is not only a mathematical discretization of the former but also has an exact, discrete mechanical analogue. We show that the latter possesses great numbers of “parasitic” solutions in addition to the natural discretizations of classical buckling modes. We investigate this rich bifurcational structure using both mechanical analysis of the boundary value problem and dynamical studies of the initial value problem, which is the familiar standard map. We use this example to explore the links between discrete initial and boundary value problems and, more generally, to illustrate the complex relations among physical systems, continuum and discrete models and the analytical and numerical methods for their study.     

Existence of periodic solutions of the generalized burgers system : PMM vol. 43, no. 6, 1979. pp. 992–997

The boundary value problem for Burgers equations of compressible fluid is considered. Proof is given of the existence of periodic solution as the limit of solutions of initial boundary value problems in which the instant of initial data definition tends to minus infinity.     

Approximate solution of nonlinear problems of optimal control of oscillatory processes using the method of canonical separation of motions : PMM vol.40, n 6, 1976, pp. 1024–1033

An asymptotic method of solving certain problems of optimal control of motion of the standard type systems with rotating phase is developed. It is assumed that the controls enter only the small perturbing terms, and that the fixed time interval over which the process is being considered is long enough to ensure that the slow variables change essentially. Assuming also that the system and the controls satisfy the necessary requirements of smoothness, the method of canonical averaging [1] is used to construct a scheme for deriving a simplified boundary value problem of the maximum principle. The structure of the set of solutions of the boundary value problem is investigated and a scheme for choosing the optimal solution with the given degree of accuracy in the small parameter is worked out. The validity of the approximate method of solving the boundary value problem is proved. The method suggested in [2] for constructing a solution in the first approximation for similar problems of optimal control is developed.     

On a variational problem with unknown boundaries and the determination of optimal shapes of elastic bodies : PMM vol. 39, n 6, 1975, pp. 1082–1092

The optimization problem is considered for a partial differential equation of elliptic type. The boundary of the domain in which the equation is given emerges as the control function and is to be determined from the condition of the extremum of the integral of the solution of the boundary value problem. Seeking the extremals is reduced to solving a va national problem without differential constraints. Necessary conditions for optimality are obtained, and shapes of elastic bars possessing the maximum stiffness under torsion are found with their aid.     

The boundary of the Krein space tracial numerical range, an algebraic approach and a numerical algorithm

In this article, tracial numerical ranges associated with matrices in an indefinite inner product space are investigated. The boundary equations of these sets are obtained, and the case of the boundary being a polygon is studied. As an application, a numerical algorithm for plotting the tracial numerical range of an arbitrary complex matrix is presented. Our approach uses the elementary idea that the boundary may be traced by computing the supporting lines.     

APS boundary conditions, eta invariants and adiabatic limits

We prove an adiabatic limit formula for the eta invariant of a manifold with boundary. The eta invariant is defined using the Atiyah-Patodi-Singer boundary condition and the underlying manifold is fibered over a manifold with boundary. Our result extends the work of Bismut-Cheeger to manifolds with boundary.

     

Analytic discs, plurisubharmonic hulls, and non-compactness of the -Neumann operator

We show that a complex manifold M in the boundary of a smooth bounded pseudoconvex domain Ω in is an obstruction to compactness of the -Neumann operator on Ω, provided that at some point of M, the Levi form of bΩ has the maximal possible rank n−1−dim(M) (i.e. the boundary is strictly pseudoconvex in the directions transverse to M). In particular, an analytic disc is an obstruction, provided that at some point of the disc, the Levi form has only one zero eigenvalue (i.e. the eigenvalue zero has multiplicity one). We also show that a boundary point where the Levi form has only one zero eigenvalue can be picked up by the plurisubharmonic hull of a set only via an analytic disc in the boundary. Research supported in part by NSF grant number DMS-0100517.     

Method of successive approximations for three-dimensional laminar boundary layer problems (locally self-similar case) : PMM vol. 37, n6, 1973, pp. 974–983

We present an analytical method for the computation of problems of incompressible boundary layer theory based on an application of the method of successive approximations. The system of equations is reduced to a form suitable for integration. Parameters characterizing the external flow and the body geometry are contained only in the coefficients of the system and do not enter into the boundary conditions. The transformed momentum equations are integrated across the boundary layer from a current value to infinity with the boundary conditions taken into account. If the integration is made from zero to infinity, then the equations pass over into the Kármán relations. Integrating the system of equations a second time, using the boundary conditions at the wall, we obtain a system of nonlinear integro-differential equations. To solve this system of equations we apply the method of successive approximations. To satisfy the boundary Conditions at infinity we introduce, at each step of the iterations, unknown “governing” functions. From the conditions at the outer side of the boundary layer we obtain additional equations for their determination. With the iterational algorithm formulated in this way, the boundary conditions, both on the body and at the outer side of the boundary layer; are satisfied automatically.We consider a locally self-similar approximation. In this case, relative to the “governing” functions, we obtain an algebraic system of equations. We write out the solution in the first approximation. The results obtained in the first approximation are compared with the results of finite-difference computations for a wide range of problems. The results obtained in this paper are compared with those obtained in [1] for the flow in the neighborhood of a stagnation point. An indication is given of the nonuniqueness of the solutions of the three-dimensional boundary layer equations.     

A shortwave source near a smooth,convex hypersurface and the spectral function of the Laplace operator on a Riemannian manifold

The Tauberian theorem of B. M. Levitan reduces the question of the asymptotics of the spectral function of the Laplace operator on a smooth Riemannian manifold with boundary to the problem of constructing the asymptotics of a Green function possessing certain additional properties. The paper is devoted to the construction of the appropriate Green function for the case of a geodesically concave boundary.     

Existence of solutions to the nonlinear,singular second order Bohr boundary value problems

This paper investigates the existence of solutions for nonlinear systems of second order, singular boundary value problems (BVPs) with Bohr boundary conditions. A key application that arises from this theory is the famous Thomas–Fermi equations for the model of the atom when it is in a neutral state. The methodology in this paper uses an alternative and equivalent BVP, which is in the class of resonant singular BVPs, and thus this paper obtains novel results by implementing an innovative differential inequality, Lyapunov functions and topological techniques. This approach furnishes new results in the area of singular BVPs for a priori bounds and existence of solutions, where the BVP has unrestricted growth conditions and subject to the Bohr boundary conditions. In addition, the results can be relaxed and hold for the non-singular case too.     

Boundaries on spacetimes: causality,topology, and group actions

The future causal boundary on a spacetime serves to explicate the causal behavior of the spacetime at future infinity. The purely causal nature of this boundary has a categorically universal nature, the category being that of chronological sets. There is an associated topology with any chronological set, replicating the appropriate topology for a spacetime. Adding the future causal boundary (and using this topology) provides a quasi-compactification. The boundary for a product spacetime can be detailed in terms of the Riemannian factor M.      

The Axisymmetric Deformation of a Thin,or Moderately Thick,Elastic Spherical Cap

A refined shell theory is developed for the elastostatics of a moderately thick spherical cap in axisymmetric deformation. This is a two-term asymptotic theory, valid as the dimensionless shell thickness tends to zero.The theory is more accurate than “thin shell” theory, but is still much more tractable than the full three-dimensional theory. A fundamental difficulty encountered in the formulation of shell (and plate) theories is the determination of correct two-dimensional boundary conditions, applicable to the shell solution, from edge data prescribed for the three-dimensional problem. A major contribution of this article is the derivation of such boundary conditions for our refined theory of the spherical cap. These conditions are more difficult to obtain than those already known for the semi-infinite cylindrical shell, since they depend on the cap angle as well as the dimensionless thickness. For the stress boundary value problem, we find that a Saint-Venant-type principle does not apply in the refined theory, although it does hold in thin shell theory. We also obtain correct boundary conditions for pure displacement and mixed boundary data. In these cases, conventional formulations do not generally provide even the first approximation solution correctly. As an illustration of the refined theory, we obtain two-term asymptotic solutions to two problems, (i) a complete spherical shell subjected to a normally directed equatorial line loading and (ii) an unloaded spherical cap rotating about its axis of symmetry.     

The sommerfeld half-plane problem revisited,II the factoring of a matrix of analytic functions

A half-plane under plane wave excitation obeys a Dirichlet boundary condition on one side and a Neumann boundary condition on the other. These boundary conditions contrast the ones used by A. Sommerfeld in his classical paper. The present problem leads to a system of integral equations of the Wiener-Hopf type which may be solved by a matrix factoring method suggested by A. E. Heins in 1950.     

Classical Solvability of a Model Problem in a Half-Space, Related to the Motion of an Isolated Mass of a Compressible Fluid

For an arbitrary finite time interval, the unique solvability of a linear half-space problem is obtained in Hölder classes of functions. The problem arises as the result of the linearization of a free boundary problem for the Navier--Stokes system governing the unsteady motion of a finite mass of a compressible fluid. The boundary conditions in the linear problem are noncoercive because of the surface tension acting on the free boundary. This fact presents the main difficulty in the problem, while the differential system in itself is parabolic in the sense of Petrovskii. The principal idea of the investigation is to reduce the noncoercive problem to a coercive one with zero coefficient of the surface tension. Bibliography: 6 titles.     

Stabilization of the trial method for the Bernoulli problem in case of prescribed Dirichlet data

We apply the trial method for the solution of Bernoulli's free boundary problem when the Dirichlet boundary condition is imposed for the solution of the underlying Laplace equation, and the free boundary is updated according to the Neumann boundary condition. The Dirichlet boundary value problem for the Laplacian is solved by an exponentially convergent boundary element method. The update rule for the free boundary is derived from the linearization of the Neumann data around the actual free boundary. With the help of shape sensitivity analysis and Banach's fixed‐point theorem, we shed light on the convergence of the respective trial method. Especially, we derive a stabilized version of this trial method. Numerical examples validate the theoretical findings.Copyright © 2014 John Wiley & Sons, Ltd.     

A new family of first-order boundary conditions for the Maxwell system: derivation,well-posedness and long-time behavior

This work deals with the time-dependent Maxwell system in the case of TE-polarized electromagnetic waves, when associated with a family of first-order local boundary conditions. The boundary conditions are derived by using a micro-diagonalization method, actuated by the standard one of M.E. Taylor and involving pseudodifferential technics. The conditions differ from an arbitrary function and any of them leads to a well-posed mixed problem that is described by a continuous semi-group. The arbitrary function can be seen as a parameter and an asymptotic analysis in time shows that it can be chosen so that the resulting boundary condition is absorbing: the system is related to an energy functional that converges towards zero as time tends to infinity. By involving an invariant space for the Maxwell system, the limit state can be explicitly written as a solution to a boundary-value problem depending on the initial data. The long time behavior of the solution is then completely analyzed.     

The Rotation of a Gravitating Sphere in a Monatomic Gas,II

The purpose of this work is to study the fluid motion caused by the high speed rotation of a gravitating sphere in a monatomic gas. It has been possible to find a stable steady solution only for very small Prandtl number, which can be interpreted to mean an optically thick gas. The flow is characterized by a flat radial jet in the equatorial plane and a viscous boundary layer on the spherical surface which, in some cases, lies beneath a thermal boundary layer. That the outer region must be hydrostatic puts very stringent constraints on the associated velocity field which necessitate still another boundary layer on the sphere. This last layer is shown to be unstable to small disturbances in certain temperature ranges. Finally, a similar solution that exists for order one Prandtl number must be disregarded because this last boundary layer is always unstable.     

Problem with Nonlocal Conditions,Specified on Parts of the Boundary Characteristics and on the Degeneracy Segment,for the Gellerstedt Equation with Singular Coefficient

Russian Mathematics - For the Gellerstedt equation with a singular coefficient, we investigate a boundary value problem with nonlocal conditions, given on parts of the boundary characteristics, and...