Optimization of Mayer Problem with Sturm–Liouville-Type Differential Inclusions

The present paper studies a new class of problems of optimal control theory with Sturm–Liouville-type differential inclusions involving second-order linear self-adjoint differential operators. Our main goal is to derive the optimality conditions of Mayer problem for differential inclusions with initial point constraints. By using the discretization method guaranteeing transition to continuous problem, the discrete and discrete-approximation inclusions are investigated. Necessary and sufficient conditions, containing both the Euler–Lagrange and Hamiltonian-type inclusions and “transversality” conditions are derived. The idea for obtaining optimality conditions of Mayer problem is based on applying locally adjoint mappings. This approach provides several important equivalence results concerning locally adjoint mappings to Sturm–Liouville-type set-valued mappings. The result strengthens and generalizes to the problem with a second-order non-self-adjoint differential operator; a suitable choice of coefficients then transforms this operator to the desired Sturm–Liouville-type problem. In particular, if a positive-valued, scalar function specific to Sturm–Liouville differential inclusions is identically equal to one, we have immediately the optimality conditions for the second-order discrete and differential inclusions. Furthermore, practical applications of these results are demonstrated by optimization of some “linear” optimal control problems for which the Weierstrass–Pontryagin maximum condition is obtained.     

The Strong Convergence of Subgradients of Convex Functions Along Directions: Perspectives and Open Problems

In this note, some questions concerning the strong convergence of subgradients of convex functions along a given direction are recalled and posed. It is shown that some open problems in literature are linked to that of the existence of limits of subgradients from subdifferentials along a given segment.     

Vortex Shedding: A Review on Flat Plate

Flat plates, both single and in tandem or side by side arrangement, are widely used in many engineering applications. Despite vast investigations of the flow structures and wakes downstream of these bluff bodies, this unsteady phenomenon yet remains a fundamental issue in many industrial applications. This paper reviews the state of the art concerning the flow over flat plates in different arrangements focusing on plates normal to the flow. Turbulent wake regions are discussed for the flat plates in side by side or tandem arrangement. Numerical studies are reviewed with emphasis on the realized turbulent models. The effect of the chosen turbulence model on the prediction of the wake region is discussed.     

Consistency requirements and pattern methods in cost sharing problems with technological cooperation

Using the discrete cost sharing model with technological cooperation, we investigate the implications of the requirement that demand manipulations must not affect the agents’ shares. In a context where the enforcing authority cannot prevent agents (who seek to reduce their cost shares) from splitting or merging their demands, the cost sharing methods used must make such artifices unprofitable. The paper introduces a family of rules that are immune to these demand manipulations, the pattern methods. Our main result is the characterization of these methods using the above requirement. For each one of these methods, the associated pattern indicates how to combine the technologies in order to meet the agents’ demands. Within this family, two rules stand out: the public Aumann–Shapley rule, which never rewards technological cooperation; and the private Aumann–Shapley rule, which always rewards technology providers. Fairness requirements imposing natural bounds (for the technological rent) allow to further differentiate these two rules.     

Static Charge Susceptibility in the <Emphasis Type="Italic">t-J-V</Emphasis> Model

We describe the static charge susceptibility and correlation function of the charge density in the twodimensional t-J-V model based on the method of equations of motion for the relaxation functions of the Hubbard operators. We obtain the dependence of the susceptibility and correlation function on the hole concentration and temperature. Charge density waves can develop if the intersite Coulomb interaction is sufficiently strong.     

Repdigits as sums of three Pell numbers

In this paper, all base 10 repdigits expressible as sums of three Pell numbers are found.