A novel matrix method for coupled vibration and damping effect analyses of liquid-filled circular cylindrical shells with partially constrained layer damping under harmonic excitation

A novel matrix method is further developed for a liquid-filled circular cylindrical shell with partially constrained layer damping (CLD), which consists of treating liquid domain with Bessel function approach, and shell domain with transfer matrix equation based on a new set of first order matrix differential equation. In order to indicate its advantage to the finite element method (FEM), free vibration analysis on such an empty shell under clamped–clamped boundary are carried out by using the present method together with FEM. Meanwhile, coincident result is yielded for the liquid-filled shell by adopting the present method with by other transfer matrix method. Finally, a series of valuable numerical results are obtained by this method.     

Optimal constrained bidding

Consider a bidder who, subject only to a constraint on his choice of bidding strategy, will choose a strategy that maximizes his expected gross receipts. The bidder's choice of strategy may be constrained by an upper limit on exposure, an upper limit on expected expenditures, or a lower limit on expected profits. We prove that constraining the bidder only by an appropriate upper limit on expected expenditures in effect limits him to expected profit maximizing strategies; in many examples, expected profit maximizing strategies can not be induced by solely limiting the bidder's exposure.     

Optimal scheduling of parallel machines with constrained resources

This paper analyzes a manufacturing system consisting of parallel machines, which produce one product-type with controllable production rates subject to continuously-divisible, time-dependent resources. The objective is to produce the required amount of product-type units by a due date while minimizing inventory, backlog and production related costs over a production horizon. With the aid of the maximum principle, a number of analytical rules of the optimal scheduling is derived whereby the continuous-time scheduling is reduced to discrete sequencing and timing. As a result, a polynomial-time algorithm is developed for solving the problem.     

On models for continuous facility location with partial coverage

This paper presents a new concept of partial coverage distance, where demand points within a given threshold distance of a new facility are covered in the traditional sense, while non-covered demand points are penalized an amount proportional to their distance to the covered region. Two single facility location models, based on the minisum and minimax criteria, are formulated with the new distance function, and the structure of the models is analysed.     

Optimal control of an elastic string with viscous damping

We study bilinear optimal control of a wave equation with one spatial dimension. The problem describes oscillations of an elastic string with viscous damping, and the damping coefficient is taken as the control. The objective functional involves driving the state solution close to a desired profile and incurring a cost on the control. The optimal control is characrerized in terms of an optimality system.     

Optimal coverage of convex regions

The problem of the coverage of convex regions with polygons and quadratic configurations of minimal volume is considered. The regions are presented as inequality constraints of a linear or nonlinear programming problem. It is shown that the problem of the optimal coverage with an arbitrary polygon can be reduced to a convex one of coverage with a multidimensional rectangle. If, however, rotation of the coordinate system is allowed, an additional nonconvex problem must be solved. It is also shown that, to find the minimal covering hypersphere or hyperellipsoid, one has to solve two convex programming problems. Algorithms and examples illustrating the feasibility of the proposed methods are presented.This work was supported by funds for the promotion of research at the Technion under Contract No. 190-515.The authors would like to express their indebtedness to Prof. Aharon Ben-Tal from Technion and to Prof. L. C. W. Dixon of the Hatfield Polytechnic for several valuable suggestions on an earlier draft of this paper.     

Forced vibrations of a viscoelastic damping layer

Forced vibrations of a viscoelastic damping layer are investigated. The equations obtained are reduced to normal modes, and the solutions are determined for nonresonance and resonance cases of vibrations.Moscow Institute of Electronic Machine Construction. Translated from Mekhanika Polimerov, No. 1, pp. 124–132, January–February, 1972.     

Optimal damping control and nonlinear parabolic systemes

An optimal control problem for a parabolic equation when the control parameter is the zero order coefficient of the differential operator is considered. An optimality system is derived. Under a certain sign condition, the problem is solved completely, by proving uniqueness and providing a constructive existence proof for the nonlinear parabolic optimality system.     

Optimal damping ratio for linear second-order systems

In this paper, it is shown that the optimal damping ratio for linear second-order systems that results in minimum-time no-overshoot response to step inputs is of bang-bang type. The optimal damping ratio is zero at the outset and is switched to some maximum value at an appropriate instant of time. The switching time is shown to be a function of the maximum damping ratio and the system natural frequency. Furthermore, it is shown that the larger the maximum damping ratio is, the shorter it takes for the system to reach the desired set point. Finally, it is shown that, if the optimal damping ratio is switched as a function of the system state, then the minimum-time no-overshoot criterion is satisfied, irrespective of the magnitude of the uncertainty in the value of the system natural frequency.     

Optimal control problems for Kirchhoff type equation with a damping term

Optimal control problems are studied for the equation of Kirchhoff type with a damping term. The Gâteaux differentiability of solution mapping on control variables is proved and the various types of necessary optimality conditions corresponding to the distributive and terminal value observations are established.     

Optimal damping of perturbations in a system with unknown initial conditions

We consider the perturbation damping problem for a system in which, along with an external perturbation bounded in the L ₂-norm, there is an initial perturbation caused by unknown nonzero initial conditions. We state necessary and sufficient conditions for the existence of an optimal control law minimizing the maximum L ₂-norm of the system output for all L ₂-bounded external perturbations and bounded initial states and synthesize this control law.     

Optimal liquidation under partial information with price impact

We study the optimal liquidation problem in a market model where the bid price follows a geometric pure jump process whose local characteristics are driven by an unobservable finite-state Markov chain and by the liquidation rate. This model is consistent with stylized facts of high frequency data such as the discrete nature of tick data and the clustering in the order flow. We include both temporary and permanent effects into our analysis. We use stochastic filtering to reduce the optimal liquidation problem to an equivalent optimization problem under complete information. This leads to a stochastic control problem for piecewise deterministic Markov processes (PDMPs). We carry out a detailed mathematical analysis of this problem. In particular, we derive the optimality equation for the value function, we characterize the value function as continuous viscosity solution of the associated dynamic programming equation, and we prove a novel comparison result. The paper concludes with numerical results illustrating the impact of partial information and price impact on the value function and on the optimal liquidation rate.     

Optimal feedback control for constrained mechanical systems

An approach to minimize the control costs and ensuring a stable deviation control is the Riccati controller and we want to use it to control constrained dynamical systems (differential algebraic equations of Index 3). To describe their discrete dynamics, a constrained variational integrators [1] is used. Using a discrete version of the Lagrange-d’Alembert principle yields a forced constrained discrete Euler-Lagrange equation in a position-momentum form that depends on the current and future time steps [2]. The desired optimal trajectory (q_opt, p_opt) and according control input u_opt is determined solving the discrete mechanics and optimal control (DMOC) algorithm [3] based on the variational integrator. Then, during time stepping of the perturbed system, the discrete Riccati equation yields the optimal deviation control input u_R. Adding u_opt and u_R to the discrete Euler-Lagrange equation causes a structure preserving trajectory as both DMOC and Riccati equations are based on the same variational integrator. Furthermore, coordinate transformations are implemented (minimal, redundant and nullspace) enabling the choice of different coordinates in the feedback loop and in the optimal control problem. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal constrained selection of a measurable subset

Necessary and sufficient conditions are given for a class of optimization problems involving optimal selection of a measurable subset from a given measure space subject to set function inequality constraints. Results are developed firstly for the case where the set functions involved possess a differentiability property and secondly where a type of convexity is present. These results are then used to develop numerical methods. It is shown that in a special case the optimal set can be obtained via solution of a fixed point problem in Euclidean space.     

Optimal damping of selected eigenfrequencies using dimension reduction

We consider linear vibrational systems described by a system of second‐order differential equations of the form , where M and K are positive definite matrices, representing mass and stiffness, respectively. The damping matrix D is assumed to be positive semidefinite. We are interested in finding an optimal damping matrix that will damp a certain (critical) part of the eigenfrequencies. For this, we use an optimization criterion based on the minimization of the average total energy of the system. This is equivalent to the minimization of the trace of the solution of the corresponding Lyapunov equation AX + XA^T = ?GG^T, where A is the matrix obtained from linearizing the second‐order differential equation, and G depends on the critical part of the eigenfrequencies to be damped. The main result is the efficient approximation and the corresponding error bound for the trace of the solution of the Lyapunov equation obtained through dimension reduction, which includes the influence of the right‐hand side GG^T and allows us to control the accuracy of the trace approximation. This trace approximation yields a very accelerated optimization algorithm for determining the optimal damping. Copyright © 2011 John Wiley & Sons, Ltd.     

Optimal inspection intervals for safety systems with partial inspections

The introduction of International Standard IEC 61508 and its industry-specific derivatives sets demanding requirements for the definition and implementation of life-cycle strategies for safety systems. Compliance with the Standard is important for human safety and environmental perspectives as well as for potential adverse economic effects (eg, damage to critical downstream equipment or a clause for an insurance or warranty contract). This situation encourages the use of reliability models to attain the recommended safety integrity levels using credible assumptions. During the operation phase of the safety system life cycle, a key decision is the definition of an inspection programme, namely its frequency and the maintenance activities to be performed. These may vary from minimal checks to complete renewals. This work presents a model (which we called ρβ model) to find optimal inspection intervals for a safety system, considering that it degrades in time, even when it is inspected at regular intervals. Such situation occurs because most inspections are partial, that is, not all potential failure modes are observable through inspections. Possible reasons for this are the nature and the extent of the inspection, or potential risks generated by the inspection itself. The optimization criterion considered here is the mean overall availability A _o , but also taking into account the requirements for the safety availability A _s . We consider several conditions that ensure coherent modelling for these systems: sub-systems decomposition, k-out-of-n architectures, diagnostics coverage (observable/total amount of failure modes), dependent and independent failures, and non-negligible inspection times. The model requires an estimation for the coverage and dependent-failure ratios for each component, global failure rates, and inspection times. We illustrate its use through case studies and compare results with those obtained by applying previously published methodologies.     

Optimal investment under partial information

We consider the problem of maximizing terminal utility in a model where asset prices are driven by Wiener processes, but where the various rates of returns are allowed to be arbitrary semimartingales. The only information available to the investor is the one generated by the asset prices and, in particular, the return processes cannot be observed directly. This leads to an optimal control problem under partial information and for the cases of power, log, and exponential utility we manage to provide a surprisingly explicit representation of the optimal terminal wealth as well as of the optimal portfolio strategy. This is done without any assumptions about the dynamical structure of the return processes. We also show how various explicit results in the existing literature are derived as special cases of the general theory.     

Optimal policies for constrained average-cost Markov decision processes

We give mild conditions for the existence of optimal solutions for a Markov decision problem with average cost, under m constraints of the same kind, in Borel actions and states spaces. Moreover, there is an optimal policy that is a convex combination of at most m+1 deterministic policies.     

Optimal generation scheduling for constrained multi-area hydrothermal power systems with cascaded reservoirs

The optimization problem in this paper is targeted at large-scale hydrothermal power systems. The thermal part of the system is a multi-area power pool with tie-line constraints, and the hydro part is a set of cascaded hydrostations. The objective is to minimize the operation cost of the thermal subsystem. This is an integer nonlinear optimization process with a large number of variables and constraints. In order to obtain the optimal solution in a reasonable time, we decompose the problem into thermal and hydro subproblems. The coordinator between these subproblems is the system Lagrange multiplier. For the thermal subproblem, in a multi-area power pool, it is necessary to coordinate the area generations for reducing the operation cost without violating tie limits. For the hydro subsystem, network flow concepts are adopted to coordinate water usage over the entire study time span, and the reduced gradient method is used to overcome the linear characteristic of the network flow method in order to obtain the optimal solution. In this study, load forecasting errors and forced outages of generating units are incorporated in system reliability requirements. Three case studies for the proposed method are presented.