Mixed problems for the string vibration equation with nonlocal conditions of the general form at the right endpoint and with an inhomogeneous condition at the left endpoint

We consider four mixed problems for the string vibration equation with zero initial conditions, with a Bitsadze–Samarskii boundary condition of the general form at the right endpoint, and with an inhomogeneous Neumann or Dirichlet condition at the left endpoint. We prove the uniqueness of a generalized solution (in the sense of Il’in) of these problems and obtain an analytic representation of these solutions. The solution of each of the problems is represented in the form of a linear combination of functions constructed from the problem data, and recursion formulas for the coefficients of this linear combination are obtained.     

Transmission problems in the theory of elastic hemitropic materials

The purpose of this article is to investigate mixed transmission-boundary value problems for the differential equations of the theory of hemitropic (chiral) elastic materials. We consider the differential equations corresponding to the time harmonic dependent case, the so called pseudo-oscillation equations. Applying the potential method and the theory of pseudodifferential equations we prove uniqueness and existence theorems of solutions to the Dirichlet, Neumann and mixed transmission-boundary value problems for piecewise homogeneous hemitropic composite bodies and analyze their regularity properties. We investigate also interface crack problems and establish almost best regularity results.     

Adding machines and endpoints

In this paper we consider continuous maps on graphs. We give sufficient conditions for a point in the inverse limit space to be a local endpoint in terms of the dynamics of f. In particular we explore the relationship between the existence of adding machine dynamics and local endpoints.     

Optimal boundary displacement control of string vibrations with a nonlocal oddness condition of the first kind

We consider the problem of optimal boundary control by the displacement at left endpoint of a string in the case of a nonlocal oddness boundary condition of the first kind. We obtain a necessary and sufficient condition for the problem controllability under arbitrary initial and terminal conditions and construct a closed analytical form of the control itself under these conditions. In addition, we consider the problem of optimal boundary control by the displacement at one endpoint of the string for a given displacement mode at the other endpoint.     

Coupled intervals in the discrete calculus of variations: necessity and sufficiency

In this work we study nonnegativity and positivity of a discrete quadratic functional with separately varying endpoints. We introduce a notion of an interval coupled with 0, and hence, extend the notion of conjugate interval to 0 from the case of fixed to variable endpoint(s). We show that the nonnegativity of the discrete quadratic functional is equivalent to each of the following conditions: The nonexistence of intervals coupled with 0, the existence of a solution to Riccati matrix equation and its boundary conditions. Natural strengthening of each of these conditions yields a characterization of the positivity of the discrete quadratic functional. Since the quadratic functional under consideration could be a second variation of a discrete calculus of variations problem with varying endpoints, we apply our results to obtain necessary and sufficient optimality conditions for such problems. This paper generalizes our recent work in [R. Hilscher, V. Zeidan, Comput. Math. Appl., to appear], where the right endpoint is fixed.     

Convergence of optimal controls in some optimization problems

We consider the behavior of bundles of optimal controls when the initial state of the system goes to some given vector. We investigate three types of optimization problems: problems with fixed process length and a free right endpoint; problems with fixed process length and a fixed right endpoint; time-optimal problems. The article is a review of recent results obtained by the author. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 301–314, 2004.     

Nonnegativity and positivity of quadratic functionals in discrete calculus of variations: survey

In this paper we provide a survey of characterizations of the nonnegativity and positivity of discrete quadratic functionals which arise as the second variation for nonlinear discrete calculus of variations problems. These characterizations are in terms of (i) (strict) conjugate and (strict) coupled intervals, (ii) the conjoined bases of the associated Jacobi difference equation, and (iii) the solution of the corresponding Riccati difference equation. The results depend on the form of the boundary conditions of the quadratic functional and, basically, we distinguish three types: (a) separable endpoints with zero right endpoint (this of course includes the simplest case of both zero endpoints), (b) separable endpoints, and (c) jointly varying endpoints.     

Finite element analysis of the vibration problem of a plate coupled with a fluid

Summary. We consider the approximation of the vibration modes of an elastic plate in contact with a compressible fluid. The plate is modelled by Reissner-Mindlin equations while the fluid is described in terms of displacement variables. This formulation leads to a symmetric eigenvalue problem. Reissner-Mindlin equations are discretized by a mixed method, the equations for the fluid with Raviart-Thomas elements and a non conforming coupling is used on the interface. In order to prove that the method is locking free we consider a family of problems, one for each thickness , and introduce appropriate scalings for the physical parameters so that these problems attain a limit when . We prove that spurious eigenvalues do not arise with this discretization and we obtain optimal order error estimates for the eigenvalues and eigenvectors valid uniformly on the thickness parameter t. Finally we present numerical results confirming the good performance of the method. Received February 4, 1998 / Revised version received May 26, 1999 / Published online June 21, 2000     

Singular boundary integral equations of boundary value problems of elastic dynamics in the case of subsonic running loads

We consider boundary value problems for an elastic medium bounded by a cylindrical surface on which a load with a constant-in-time form is moving at a constant subsonic velocity. This class of problems is a model class for the dynamics of underground structures like transport tunnels and ground transport. The method of generalized functions is developed for the solution of boundary value problems. We construct dynamic analogs of Green formulas and use them to derive singular boundary integral equations, which solve the considered boundary value problems.     

Locating and capacity planning for retailers of a new supply chain to compete on the plane

This paper investigates the network design problem of a two-level supply chain (SC), which is applicable for industries such as automotive, fuel and tyre manufacturing. Models presented in this paper aim at locating retail facilities of an SC and identifying their required capacities in the presence of existing competing retailers of a rival SC. We consider feasible locating space of the retail facilities on the continuous plane with bounded constraints and static competition among the rivals of the markets with deterministic demands. The problem is used for both essential and luxury product cases; hence, we consider elastic and inelastic demands, both. The models discussed in this paper are non-linear and non-convex which are difficult to solve. We use interval branch-and-bound as optimization algorithm for small size single-retailer problems, but for large-scale, multi-retailer problems we need to have more efficient methods. Therefore, we apply a heuristic algorithm (H1), a simulated annealing (SA) algorithm, an interior point (IP) algorithm, a genetic algorithm (GA) and a pattern search algorithm for solving multi-retailer problem with elastic and inelastic demands. Computational results obtained from performing different solution approaches for both elastic and inelastic show that mostly IP, PS, and H1 methods outperform the other approaches. The computational results on a real-life case are also promising. Several extended mathematical models and an example of a typical case with details are presented in the appendices of the paper.     

A dynamic edge covering and scheduling problem: complexity results and approximation algorithms

We consider a new dynamic edge covering and scheduling problem that focuses on assigning resources to nodes in a network to minimize the amount of time required to process all edges in it. Resources need to be co-located at the endpoints of an edge for it to be processed and, therefore, this problem contains both edge covering and scheduling decisions. These new problems have motivating applications in traffic systems and military intelligence operations. We provide complexity results for the dynamic edge covering and scheduling problem over different types of networks. We then show that existing approximation algorithms for parallel machine scheduling problems can be leveraged to provide approximation algorithms for this new class of problems over certain types of networks.     

The structure of approximate solutions of variational problems without convexity

In this work we study the structure of approximate solutions of variational problems with continuous integrands f:[0,∞)×Rn×Rn→R1 which belong to a complete metric space of functions. We do not impose any convexity assumption. The main result in this paper deals with the turnpike property of variational problems. To have this property means that the approximate solutions of the problems are determined mainly by the integrand, and are essentially independent of the choice of interval and endpoint conditions, except in regions close to the endpoints.     

Generic Well-Posedness of Nonconvex Constrained Variational Problems

In our previous work, a generic well-posedness result (with respect to the variations of the integrand of the integral functional) was established for a class of nonconvex optimal control problems. In this paper, we extend this generic well-posedness result to classes of constrained variational problems in which the values at the endpoints and the constraint maps are also subject to variations. We consider constrained variational problems with constraint maps which depend on the independent variable and also on the state variable.The author is grateful to the referees for helpful comments and suggestions.     

Observability of the 1-D Wave Equation with Mixed Boundary Conditions in a Non-cylindrical Domain

We study a wave equation with mixed boundary conditions in a bounded interval with a moving endpoint. We derive precise estimates for the energy of the solution and show observability results at each endpoint. The observability constants are explicitly given and the obtained time of observability is sharp.     

Stability estimates for solutions of control problems for the Maxwell equations with mixed boundary conditions

We consider extremal problems for the time-harmonic Maxwell equations with mixed boundary conditions for the electric field. Namely, the tangential component of the electric field is given on one part of the boundary, and an impedance boundary condition is posed on the other part. We prove the solvability of the original mixed boundary value problem and the extremal problem. We obtain sufficient conditions on the input data ensuring the stability of solutions of specific extremal problems under certain perturbations of both the performance functional and some functions occurring in the boundary value problem.     

Boundary control by displacements at one endpoint of an inhomogeneous rod for fixed other endpoint in the case of coinciding times of passage of the wave through either of the inhomogeneity parts

In the present paper, we consider a boundary control by displacements at one endpoint of an inhomogeneous rod that has two parts of different densities and elasticities with fixed other endpoint for the case in which the times of passage of the wave through each of these inhomogeneity parts coincide. We find a closed analytic form of the boundary control by displacements bringing the stick from the original rest state into a given terminal state specified by given terminal displacement and velocity.     

Optimality conditions for discrete optimal control problems with equality and inequality type of constraints

We consider nonlinear discrete optimal control problems with variable endpoints and with equality and inequality type of constraints on control. We derive new nontrivial first and second-order necessary optimality conditions.      

Maximum principle for optimal control problems with intermediate constraints

We consider optimal control problems with constraints at intermediate points of the trajectory. A natural technique (propagation of phase and control variables) is applied to reduce these problems to a standard optimal control problem of Pontryagin type with equality and inequality constraints at the trajectory endpoints. In this way we derive necessary optimality conditions that generalize the Pontryagin classical maximum principle. The same technique is applied to so-called variable structure problems and to some hybrid problems. The new optimality conditions are compared with the results of other authors and five examples illustrating their application are presented.