Automatic solution of optimal-control problems. IV. gradient method

An automatic method for obtaining the numerical solution of first-order nonlinear optimal-control problems is described. The nonlinear two-point boundary-value problem is solved using the gradient method for obtaining successive approximations of the solution. The derivatives required for the solution of the problem are computed automatically using the table method. The user of the program need only input the integrand of the cost functional and the Hamiltonian and specify the initial conditions and the terminal time. None of the derivatives usually associated with Pontryagin's maximum principle and the gradient method need be calculated by hand. Examples are given with numerical results.     

The Euler-Jacobi equation in variational calculus

The use of the method of the Euler-Jacobi equation is considered in the study of a quadratic functional defined on a cone. Such functionals occur in the variation of optimal-control problems. Several concepts are introduced with the aid of which the Euler-Jacobi equation is extended and the application of this method is justified also in the case that the equation is not a linear differential equation.Translated from Matematicheskie Zametki, Vol. 20, No. 6, pp. 847–858, December, 1976.     

Optimal control of variational inequalities. Approximation theory and numerical realization

An optimal-control problem of a variational inequality of the elliptic type is investigated. The problem is approximated by a family of finite-dimensional problems and the convergence of the approximated optimal controls is shown. The finite-dimensional problems, being nonsmooth, are to be optimized by a bundle algorithm, which requires an element of Clarke's generalized gradient of the minimized function. A simple algorithm which yields this element is proposed. Some numerical experiments with a simple model problem have also been carried out.     

Automatic solution of optimal-control problems. I. simplest problem in the calculus of variations

An automatic method for obtaining the numerical solution for the simplest problem in the calculus of variations is described. The nonlinear two-point boundary-value Euler-Lagrange equation is solved using the Newton-Raphson method for obtaining successive approximations of the solution. The derivatives required for the solution of the problem are computed automatically using the table method. The user of the program need only input the integrand of the objective function in the calculus-of-variations problem and specify the boundary conditions. None of the derivatives usually associated with the Euler-Lagrange equation and the Newton-Raphson method need be calculated by hand. An example is given with numerical results. The automatic solution of the simplest problem in the calculus of variations in this paper is considered to be the first step in the automatic solution of more general optimal-control problems.     

Optimal control laws for the model of information diffusion in a social group

The article investigates two models of information diffusion in a social group. The dynamics of the process is described by a one-dimensional controlled Riccati differential equation. Our two models differ from the original model of K. V. Izmodenova and A. P. Mikhailov in the choice of the functional being optimized. Two different choices of the optimand functional are considered. The optimal control problems are solved by the Pontryagin maximum principle. It is shown that the optimal control program is a relay function of time with at most one switching point. Conditions on the problem parameters are proposed that are easy to check and guarantee the existence of an optimal-control switching point. The theoretical analysis leads to a one-dimensional convex minimization problem to find the optimal-control switching point. The article also describes an alternative approach to the construction of the optimal solution, which does not resort to the maximum principle and instead utilizes a special representation of the optimand functional and works with reachability sets that are independent of the functional. For the two models considered in this article optimal feedback controls are derived from the programmed optimal controls.     

On sample size control in sample average approximations for solving smooth stochastic programs

We consider smooth stochastic programs and develop a discrete-time optimal-control problem for adaptively selecting sample sizes in a class of algorithms based on variable sample average approximations (VSAA). The control problem aims to minimize the expected computational cost to obtain a near-optimal solution of a stochastic program and is solved approximately using dynamic programming. The optimal-control problem depends on unknown parameters such as rate of convergence, computational cost per iteration, and sampling error. Hence, we implement the approach within a receding-horizon framework where parameters are estimated and the optimal-control problem is solved repeatedly during the calculations of a VSAA algorithm. The resulting sample-size selection policy consistently produces near-optimal solutions in short computing times as compared to other plausible policies in several numerical examples.     

A priori and a posteriori error analysis of edge stabilization Galerkin method for the optimal control problem governed by convection-dominated diffusion equation

In this paper, we study an edge-stabilization Galerkin approximation scheme for the constrained optimal-control problem governed by convection-dominated diffusion equation. The method uses least-square stabilization of the gradient jumps across element edges. A priori and a posteriori error estimates are derived for both the state, co-state and the control. The theoretical results are illustrated by two numerical experiments.     

Constant Sustainable Consumption Rate in Optimal Growth with Exhaustible Resources

The problem of optimal growth with an exhaustible resource deposit under R. M. Solow's criterion of maximum sustainable consumption rate, previously formulated as a minimum-resource-extraction problem, is shown to be a Mayer-type optimal-control problem. The exact solution of the relevant firstorder necessary conditions for optimality is derived for a Cobb-Douglas production function, whether or not the constant unit resource extraction cost vanishes. The related problem of maximizing the terminal capital stock over an unspecified finite planning period is investigated for the development of more efficient numerical schemes for the solution of multigrade-resource deposit problems. The results for this finite-horizon planning problem are also important from a theoretical viewpoint, since they elucidate the economic content of the optimal growth paths for infinite-horizon problems.     

Design of linear regulators with energy constraints

In the usual design of linear-quadratic optimal-control systems, the regulator performance is obtained for several different values of the constant Lagrange multiplier q. The Lagrange multiplier determines the amount of control energy expended. If the energy is to be constrained, then the value of q must be found such that the energy constraint is satisfied. In this paper a method is described for determining simultaneously the optimal trajectory and the value of q which satisfies the energy constraint.     

Sufficient conditions for time optimal heating of rigid bodies by internal heat sources

We derive sufficient conditions for the time optimality of the control of heating of rigid bodies by internal heat sources under constraints on phase coordinates. We numerically solve the optimal-control problem in the case of heating of an unbounded plats with constraints on temperature and thermoelastic stresses.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 56–60, 1987.     

Analytical solutions in the problem of the optimal control of the rotation of an axisymmetric body

The problem of the optimal control of the rotation of an axisymmetric rigid body is investigated. An integral functional, characterizing the power consumption to carry out a manoeuvre is chosen as the criterion, and the boundary conditions for the angular velocity vector are arbitrary. The principal moment of the applied external forces serves as the control. The necessary conditions of the maximum principle are used to solve the problem in the case of a fixed completion time. New non-trivial first integrals are established for the canonical system of direct and conjugate differential equations obtained, which enable the set of all extremals to be parametrized. Hence, the optimal-control problem is reduced to a problem of non-linear mathematical programming. It is shown that there cannot be more than two different solutions in the latter, and a family of boundary conditions is established when the optimum rotation is determined in a uniquely explicit form.     

Imprecise parameters for near-optimal control of stochastic SIV epidemic model

The change of parameters may influence the dynamic behaviors of epidemic diseases. Biological system parameters can also be changed due to diverse uncertainties such as lack of data and errors in the statistical approach. The problem of how to define and decide the optimal-control strategies of epidemic diseases with imprecise parameters deserves further researches. The paper presents a stochastic susceptible, infected, and vaccinated (SIV) system that includes imprecise parameters. Firstly, we give the method of parameter estimates of the SIV model. Then, by using Ekeland's principle and Hamiltonian function, we obtain the sufficient conditions and necessary conditions of near-optimal control of the SIV epidemic model with imprecise parameters. At last, numerical examples prove our theoretical results.     

Optimal Forest Harvesting with Ordered Site Access

A long-standing problem in forestry management is the optimal harvesting of a growing population of trees to maximize the resulting discounted aggregate net revenue. For an ongoing forest, the trees are harvested and replanted repeatedly; for a once-and-for-all forest, there is no replanting after a single harvest. In this paper, we outline a new formulation for the optimal-harvest problem which avoids difficulties associated with functional-differential equations or partial differential equations of state in the relevant optimal-control problem encountered in recent studies of the ongoing-forest problem. Our new formulation is based on the observation that tree logging is necessarily ordered by practical and/or regulatory considerations (e.g., it is illegal to cut the younger trees first in some jurisdictions); random access to tree sites does not occur in practice. The new formulation is described here for the simpler problem of a once-and-for-all forest. New results for nonuniform initial age distributions and variable unit harvest costs for this simpler problem are reported herein; results for an ongoing forest will be reported in [10]. The new model is also of interest from a control-theoretic viewpoint, as it exhibits the unique feature of having time as a state variable, in contrast to its usual role as an independent variable in conventional control problems.     

A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control

Simple directly verifiable conditions are derived under whichthere exists a state trajectory satisfying a specified stateconstraint. The conclusions differ from the kind of informationprovided by viability and invariance-type theorems, insofaras an estimate is provided of the distance (in the supremumnorm) of the state trajectory from a specified state trajectory,in terms of the degree to which the specified state trajectoryviolates the state constraint. The constructions involved inthe existence proof are related to ones previously employedby Soner to establish continuity properties of a value functionarising in infinite-horizon state-constrained optimal control,but the accompanying analysis contains refinements to ensurea sub-Lipschitz property of the value function considered here.It is expected that this existence result will have a numberof implications for systems theory and optimal control. Herewe show how it leads to a non-degenerate maximum principle forstate-constrained optimal-control problems, in situations wherethe standard necessary conditions give no useful informationabout minimizers. Email: rampazzo{at}pdmat1.unipd.it Email: r.vinter{at}ic.ac.uk     

Optimal Release Times in Single-Stage Manufacturing Systems with Blocking: Optimal Control Perspective

This paper concerns a due-date matching problem in a single-stage manufacturing system. Given a finite sequence of jobs and their service order, and given the delivery due date of each job, the problem is to choose the jobs release (arrival) times so as to match as closely as possible their completion times to their respective due dates. The system is modelled as a deterministic single-server FIFO queue with an output buffer for storing jobs whose service is completed prior to their due dates. The output buffer has a finite capacity; when it is full, the server is being blocked. Associated with each job there is a convex cost function penalizing its earliness as well as tardiness. The due-date matching problem is cast as an optimal control problem, whose objective is to minimize the sum of the above cost functions by the choice of the jobs arrival (release) times. Time-box upper-bound and lower-bound constraints are imposed on the jobs output (delivery) times. The optimal-control setting brings to bear on the development of fast and efficient algorithms having intuitive geometric appeal and potential for online implementation.Communicated by W. B. GongResearch supported in part by the National Science Foundation under Grant ECS-9979693 and by the Georgia Tech Manufacturing Research Center under Grant B01-D06.     

An asymptotic initial value method for second order singular perturbation problems of convection-diffusion type with a discontinuous source term

In this paper a numerical method is presented to solve singularly perturbed two points boundary value problems for second order ordinary differential equations consisting a discontinuous source term. First, in this method, an asymptotic expansion approximation of the solution of the boundary value problem is constructed using the basic ideas of a well known perturbation method WKB. Then some initial value problems and terminal value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial value problems are happened to be singularly perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples provided to illustrate the method.     

Convergence of algorithms for perturbed optimization problems

Infinite-dimensional optimization problems occur in various applications such as optimal control problems and parameter identification problems. If these problems are solved numerically the methods require a discretization which can be viewed as a perturbation of the data of the optimization problem. In this case the expected convergence behavior of the numerical method used to solve the problem does not only depend on the discretized problem but also on the original one. Algorithms which are analyzed include the gradient projection method, conditional gradient method, Newton's method and quasi-Newton methods for unconstrained and constrained problems with simple constraints.     

Helly’s Principle and Its Application to an Infinite-Horizon Optimal Control Problem

In this paper, a numerical method is presented to solve singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a discontinuous source term. First, an asymptotic expansion approximation of the solution of the boundary-value problem is constructed using the basic ideas of the well-known WKB perturbation method. Then, some initial-value problems and terminal-value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial-value problems and terminal-value problems are singularly-perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples are provided to illustrate the method.     

A critical review of discrete filled function methods in solving nonlinear discrete optimization problems

Many real life problems can be modeled as nonlinear discrete optimization problems. Such problems often have multiple local minima and thus require global optimization methods. Due to high complexity of these problems, heuristic based global optimization techniques are usually required when solving large scale discrete optimization or mixed discrete optimization problems. One of the more recent global optimization tools is known as the discrete filled function method. Nine variations of the discrete filled function method in literature are identified and a review on theoretical properties of each method is given. Some of the most promising filled functions are tested on various benchmark problems. Numerical results are given for comparison.     

A meshless method for nonhomogeneous polyharmonic problems using method of fundamental solution coupled with quasi-interpolation technique

This paper proposes a meshless method based on coupling the method of fundamental solutions (MFS) with quasi-interpolation for the solution of nonhomogeneous polyharmonic problems. The original problems are transformed to homogeneous problems by subtracting a particular solution of the governing differential equation. The particular solution is approximated by quasi-interpolation and the corresponding homogeneous problem is solved using the MFS. By applying quasi-interpolation, problems connected with interpolation can be avoided. The error analysis and convergence study of this meshless method are given for solving the boundary value problems of nonhomogeneous harmonic and biharmonic equations. Numerical examples are also presented to show the efficiency of the method.