Network models for nonlocal traffic flow

We present a network formulation for a traffic flow model with nonlocal velocity in the flux function. The modeling framework includes suitable coupling conditions at intersections to either ensure maximum flux or distribution parameters. In particular, we focus on 1-to-1, 2-to-1 and 1-to-2 junctions. Based on an upwind type numerical scheme, we prove the maximum principle and the existence of weak solutions on networks. We also investigate the limiting behavior of the proposed models when the nonlocal influence tends to infinity. Numerical examples show the difference between the proposed coupling conditions and a comparison to the Lighthill-Whitham-Richards network model.

Circuit Optimization Study According to the Maximum Principle

The minimization of the processor time of designing can be formulated as a problem of time minimization for transitional process of dynamic system. A special control vector that changes the internal structure of the equations of optimization procedure serves as a principal tool for searching the best strategies with the minimal CPU time. In this case a well-known maximum principle of Pontryagin is the best theoretical approach for finding of the optimum structure of control vector. Practical approach for realization of the maximum principle is based on the analysis of behavior of a Hamiltonian for various strategies of optimization. The possibility of applying the maximum principle to the problem of optimization of electronic circuits is analyzed. It is shown that in spite of the fact that the problem of optimization is formulated as a nonlinear task, and the maximum principle in this case isn't a sufficient condition for obtaining a minimum of the functional, it is possible to obtain the decision in the form of local minima. The relative acceleration of the CPU time for the best strategy found by means of maximum principle compared with the traditional approach is equal two to three orders of magnitude.

An Optimal Inventory Pricing and Ordering Strategy Subject to Stock and Price Dependent Demand

This article considers the deterministic optimal control problem of profit maximization for inventory replenished at a variable rate and depleted by demand which is assumed to vary with price and stock availability. Optimal policies for the inventor, product order rate and price are derived using the maximum principle. Bounds on the maximum price possible are also derived.

Bounds for rotating Rayleigh–Bénard convection at large Prandtl number

Bounds are derived for rotating Rayleigh–Bénard convection with free slip boundaries as a function of the Rayleigh, Taylor and Prandtl numbers ${\textit {Ra}}$ , ${\textit {Ta}}$ and ${\textit {Pr}}$ . At infinite ${\textit {Pr}}$ and ${\textit {Ta}} > 130$ , the Nusselt number ${\textit {Nu}}$ obeys ${\textit {Nu}} \leqslant \frac {7}{36} \left ({4}/{{\rm \pi} ^2} \right )^{1/3} {\textit {Ra}} {\textit {Ta}}^{-1/3}$ , whereas the kinetic energy density $E_{kin}$ obeys $E_{kin} \leqslant ({7}/{72 {\rm \pi}}) \left ({4}/{{\rm \pi} } \right )^{1/3} {\textit {Ra}}^2 {\textit {Ta}}^{-2/3}$ in the frame of reference in which the total momentum is zero, and $E_{kin} \leqslant ({1}/{2{\rm \pi} ^2})({{\textit {Ra}}^2}/{{\textit {Ta}}})({\textit {Nu}}-1)$ . These three bounds are derived from the momentum equation and the maximum principle for temperature and are extended to general ${\textit {Pr}}$ . The extension to finite ${\textit {Pr}}$ is based on the fact that the maximal velocity in rotating convection at infinite ${\textit {Pr}}$ is bound by $1.23 {\textit {Ra}} {\textit {Ta}}^{-1/3}$ .

ALGORITHM FOR CONTROLLING THE SPECTRUM OF EIGENFREQUENCIES AND VIBRATION MODES BY CHARDING THE GEOMETRIC PARAMETERS OF MAST SESTEMS

The article considers a model of a mast with six levels of fastening of cables. The main attention in the work is considered to the methods of control of the natural frequency spectrum, due to the use of methods of sensitivity analysis and optimization. The above task is achieved by varying the cross-sectional area of the pipes - racks. Automation of computational processes is provided by programming the built-in module in the Revit program. For more convenient and faster control of the natural frequency spectrum, the algorithm described above was written in a free add-on for Revit - Dynamo. With the help of so-called nodes, an application was created that took data from the depicted 3D model Revit and performed calculations. This allows you to easily use optimality conditions similar to the maximum principle. The sensitivity analysis for the first and second own is carried out in the work. The mechanism of their management within the limits of the investigated model is shown. The relations in the case of the problem of finding the natural frequency extremum with a given number are given, provided that the total amount of varied bands is fixed. The numerical control algorithm is based on the necessary optimality conditions in the form of the maximum principle for rod models. A variant of varying the area of the belts along the height of the mast is proposed. The sensitivity analysis for the first and second natural frequencies is carried out and its use for construction of effective computational process is shown. Based on the results of the work, a working software algorithm was created for fast analysis of mast oscillations on extensions. Graphs of zones of possible change of the first and second frequencies are resulted. The distribution of the cross-sectional area for frequencies is shown. To compare the results of natural frequency calculations on other calculation models, the first and second natural frequencies of bending oscillations were calculated by the finite element method in the SCAD complex. The errors for the points of the curves (constant in the height of the mast area of the belts) do not exceed 10%. It should be noted that the consideration of optimization problems of the above type on the basis of finite element models is quite difficult; for them it is not possible to formulate the necessary conditions of optimality similar to the principle of maximum.

Value Functions and Optimality Conditions for Nonconvex Variational Problems with an Infinite Horizon in Banach Spaces

We investigate the value function of an infinite horizon variational problem in the infinite-dimensional setting. First, we provide an upper estimate of its Dini–Hadamard subdifferential in terms of the Clarke subdifferential of the Lipschitz continuous integrand and the Clarke normal cone to the graph of the set-valued mapping describing dynamics. Second, we derive a necessary condition for optimality in the form of an adjoint inclusion that grasps a connection between the Euler–Lagrange condition and the maximum principle. The main results are applied to the derivation of the necessary optimality condition of the spatial Ramsey growth model.

Qualitative analysis for the nonlinear fractional Hartree type system with nonlocal interaction

In the present paperwe study the existence of nontrivial solutions of a class of static coupled nonlinear fractional Hartree type system. First, we use the direct moving plane methods to establish the maximum principle(Decay at infinity and Narrow region principle) and prove the symmetry and nonexistence of positive solution of this nonlocal system. Second, we make complete classification of positive solutions of the system in the critical case when some parameters are equal. Finally, we prove the existence of multiple nontrivial solutions in the critical case according to the different parameters ranges by using variational methods. To accomplish our results we establish the maximum principle for the fractional nonlocal system.

Deployment the Spoke of a Large-Sized Transformable Refl ector Using a Sequential Optimization Algorithm

The use of large-sized opening reflectors close-packed in spacecraft is associated with spreading the spokes at a given angle, extending the fragments of the spokes and adjusting the shape of the radio-reflecting mesh. The problem of optimizing these processes with automatic output of the reflector to the deployed working state is urgent. The optimal control problem of spreading spokes of a large-sized space-based reflector with respect to bending vibrations is investigated in the article. The optimization process is complicated by ensuring convergence of iterative procedure for control finding. The bending vibrations of the spokes complicate the task of spreading. That makes it difficult to fix spokes when reaching the stops. In this paper the mathematical model of spoke dynamics is improved with respect to spoke’s bend change in length and in time, the model takes into account the presence of stop and retainer devices and an actuator. It is proposed to consider a hierarchy of two target composed functions and develop an algorithm for sequential optimization for a smooth exit to the stops. It is suggested to include the terminal condition for the angular spreading rate in the first criterion. A study was carried out using mathematical simulation for the process of turning the spoke by a given angle at small values of the angular velocity at the final moment of time taking into account bending vibrations. The exact values of the weight coefficients included in the target composed functions are found. Weight coefficients influence on transient processes is investigated. The performance of the algorithm was checked when the value of the optimization interval was changed. The comparison of the results of simulation modeling with control options using the PID controller, application of an algorithm with a predictive model and an algorithm with optimal correction of the control structure, revealed by means of the maximum principle, was carried out. The results of simulation modeling foe spokes spreading process using the sequential optimization algorithm demonstrate the achievement of the required accuracy with permissible tolerance residual vibrations. The developed algorithm of sequential optimization forms control in a real time and it is recommended to use it in more complex solutions under random disturbances using measurement process and optimization of observation intervals.

Near Optimality of Linear Delayed Doubly Stochastic Control Problem

In this paper, we study a kind of near optimal control problem which is described by linear quadratic doubly stochastic differential equations with time delay. We consider the near optimality for the linear delayed doubly stochastic system with convex control domain. We discuss the case that all the time delay variables are different. We give the maximum principle of near optimal control for this kind of time delay system. The necessary condition for the control to be near optimal control is deduced by Ekeland’s variational principle and some estimates on the state and the adjoint processes corresponding to the system.