On elastic bodies with thin rigid inclusions and cracks

This paper is concerned with the analysis of equilibrium problems for two‐dimensional elastic bodies with thin rigid inclusions and cracks. Inequality‐type boundary conditions are imposed at the crack faces providing a mutual non‐penetration between the crack faces. A rigid inclusion may have a delamination, thus forming a crack with non‐penetration between the opposite faces. We analyze variational and differential problem formulations. Different geometrical situations are considered, in particular, a crack may be parallel to the inclusion as well as the crack may cross the inclusion, and also a deviation of the crack from the rigid inclusion is considered. We obtain a formula for the derivative of the energy functional with respect to the crack length for considering this derivative as a cost functional. An optimal control problem is analyzed to control the crack growth. Copyright © 2010 John Wiley & Sons, Ltd.     

Junction problem for elastic and rigid inclusions in elastic bodies

An equilibrium problem for an elastic body is considered. It is assumed that the body has a thin elastic inclusion and a thin rigid inclusion. We analyze a junction problem assuming that the inclusions have a joint point. Different equivalent problem formulations are discussed, and existence of solutions is proved. A set of junction conditions is found. We investigate a convergence to infinity and to zero of a rigidity parameter of the elastic inclusion. A delamination of the elastic inclusion is also investigated. In this case, inequality‐type boundary conditions are imposed at the crack faces to prevent a mutual penetration between crack faces. Copyright © 2015 John Wiley & Sons, Ltd.     

The conjunction problem for thin elastic and rigid inclusions in an elastic body

Under consideration is the conjunction problem for a thin elastic and a thin rigid inclusions that are in contact at one point and placed in an elastic body. Depending on what kind of conjunction conditions are set at the contact point of inclusions, we consider the two cases: the case of no fracture, where, as the conjunction conditions, we take the matching of displacements at the contact point and preservation of the angle between the inclusions, and the case with a fracture in which only the matching of displacements is assumed. At the point of conjunction, we obtain the boundary conditions for the differential formulation of the problem. On the positive face of the rigid inclusion, there is delamination. On the crack faces, some nonlinear boundary conditions of the type of inequalities are set, that prevent mutual penetration of the faces. The existence and uniqueness theorems for the solution of the equilibrium problem are proved in both cases.     

Existence of an optimal shape of the thin rigid inclusions in the Kirchhoff-Love plate

An optimal control problem is studied for the elliptic system of equations describing the equilibrium of the Kirchhoff-Love plate with an delaminated thin rigid inclusion. It is required to minimize the mean-square integral deviation of the bending moment from a function defined on the exterior boundary. The shape of the inclusion is chosen as the control function. The solvability of this problem is established.     

On an optimal control problem

In this paper, we applied the finite differences method to the solution of variational problem of an inverse problem for the Scrödinger equation with a final functional. These types of problems arise in various fields in quantum-mechanical, nuclear physics and modern physics [2, 11]. Also, we prove two estimates for the differences scheme and convergence speed of difference approximations according to the functional. The inverse problems for the Schrödinger equation having different variational formulation were investigated in [7, 12, 13].     

The stressed state of a thin coating on an elastic inclusion of optimal shape in an elastic medium

We examine the properties of the stressed state of the thin intermediate zone separating a nonhomogeneous inclusion from the matrix. Explicit expressions are given for the components of the stress tensor in the thin coating depending on the properties of the phases and the load parameters.Simferopol' University. Translated from Dinamicheskie Sistemy, No. 10, pp. 26–30, 1992.     

On sensitivity in an optimal control problem

We give a new criterion for the existence of value in differential games. The method of proof involves Lipschitz differential games and hence extends to games with more general dynamics. The connection between using measurable control functions or simply constants is clarified.     

On an optimal control problem for weakly nonlinear hyperbolic equations

The correctness of the problem of optimal control by the coefficients for a weakly nonlinear hyperbolic equation is studied, and a necessary condition of optimality is established.     

On the existence of an optimal element for a delay control problem

The existence theorems of the optimal element are proved for a nonlinear control problem with constant delay in phase coordinates and with general functional. Here element implies the collection of delay parameter and initial function, initial moment and vector, control and finally moment.     

On an optimal shape design problem to control a thermoelastic deformation under a prescribed thermal treatment

A shape optimization problem concerned with thermal deformation of elastic bodies is considered. In this article, measure theory approach in function space is derived, resulting in an effective algorithm for the discretized optimization problem. First the problem is expressed as an optimal control problem governed by variational forms on a fixed domain. Then by using an embedding method, the class of admissible shapes is replaced by a class of positive Borel measures. The optimization problem in measure space is then approximated by a linear programming problem. The optimal measure representing optimal shape is approximated by the solution of this finite-dimensional linear programming problem. Numerical examples are also given.     

On an optimal control problem for a hyperbolic partial differential equation

Existence of solutions is proved for a minimum problem for a distributed-parameter control system described by a linear, hyperbolic partial differential equation. The cost function is an integral depending on boundary controls.This research was supported by the Consiglio Nazionale delle Ricerche, Rome, Italy.     

On the limit matrix obtained in the homogenization of an optimal control problem

A new formulation for the limit matrix occurring in the cost functional of an optimal control problem on homogenization is obtained. It is used to obtain an upper bound for this matrix (in the sense of positive definite matrices).     

On an optimal control problem for a distributed parameter control process with unbounded coefficients

Let $$\begin{gathered} z_{xy} + A\left( {x,y} \right)z_x + B\left( {x,y} \right)z_y + C\left( {x,y} \right)z = U\left( {x,y} \right), \hfill \\ a.e. in \Delta = ]0,a[ \times ]0,b[, \hfill \\ \end{gathered}$$ be a distributed parameter control process, where the target consists of the boundary data on the two sides of \(\bar \Delta\) ( \(\bar \Delta\) is the closure of Δ), which do not contain the origin. For such a process, Pulvirenti and Santagati (Refs. 1–5) have investigated various control and optimal control problems, subject to the following assumption for the coefficients:A,B,C,A _x ,B _y ∈C°( \(\bar \Delta\) ). In this paper, the above process is considered under more general assumptions on the coefficients, which now need not be bounded functions. A minimization problem, where the cost is an integral depending on the controls (which are the functionU and the boundary data on the two sides of \(\bar \Delta\) which contain the origin), is studied. Some existence theorems are established. To this aim, we first prove existence, uniqueness, and continuous dependence on the controls of the response of the process.     

On the optimal structural design for a nonconservative,elastic stability problem

The design of a cantilever column under a follower load is considered with the aim of maximizing the critical value of the load. The optimality condition is derived, and a modified Ritz method is used to determine an approximate solution for the bending stiffness. Results are obtained numerically for the case of a sandwich column with constant bending stiffness in each of two segments. It is found that, for the same structural weight, the optimal design yields a critical load significantly higher than that for a uniform column.This research was supported in part by the US Army Research Office–Durham and in part by the United States Navy under Grant No. NONR N00014-67-A-0191-0009.     

On an optimal control problem with an integral functional of a rational control function

We present the solution of an optimization problem with integral performance functional that is nonlinear with respect to the control and contains a discounting parameter in the class of programmed controls under two-sided control constraints. The optimal control is found in the form of a function of time (a program). On the basis of the theoretical results, we perform numerical experiments with model and real data.     

Solving an optimal control problem for parabolic equations

We consider optimal boundary control of a distributed-parameter system. The system state is described by two parabolic equations of second order, where the coefficients of one equation depend on the gradient of the solution of the second equation. An existence and uniqueness theorem is proved for the optimal control in this problem and the necessary conditions of optimality are derived.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 90–98, 1986.     

A ritz method for an optimal control problem

We generalize and simplify the proofs of the basic papers of Bosarge and Johnson (Refs. 1-3) on a variational procedure for approximating the solution of thestate regular problem. We derive generala priori error bounds for this procedure and apply these results to obtain asymptotic error bounds for the special case of spline-type approximations.     

An optimal design problem for submerged bodies

The problem of finding the shape of a smooth body submerged in a fluid of finite depth which minimizes added mass or damping is considered. The optimal configuration is sought in a suitably constrained class so as to be physically meaningful and for which the mathematical problem of a submerged body with linearized free surface condition is uniquely solvable. The problem is formulated as a constrained optimization problem whose cost functional (e.g. added mass) is a domain functional. Continuity of the solution of the boundary value problem with respect to variations of the boundary is established in an appropriate function space setting and this is used to establish existence of an optimal solution. A variational inequality is derived for the optimal shape and it is shown how finite dimensional approximate solutions may be found.     

On a student-related optimal control problem

In a recent paper (Ref. 1), Cheng and Teo discussed some further extensions of a student-related optimal control problem which was originally proposed by Raggettet al. (Ref. 2) and later on modified by Parlar (Ref. 3). In this paper, we treat further extensions of the problem.This paper is a modified and improved version of Ref. 4. It is based, in part, on research sponsored by NSF.