Parallel processing of optimal-control problems by dynamic programming

A method is presented for the parallel evaluation of optimal-control problems using an iterative method of dynamic programming. The procedure is based on the application of the interaction prediction principle by which the global optimal-control problem is reduced to the optimization of subproblems. Each of these problems is locally optimized using successive approximations of single-variable state-increment dynamic-programming problems. Examples of the parallel solution of multivariable, constrained, nonlinear problems for which the computation is carried out on a microcomputer are given.     

Near-optimal solutions to one-dimensional cutting stock problems

This paper describes a set of heuristic procedures for efficiently generating good solutions to one-dimensional cutting stock problems in which (i) there are multiple stock lengths with constraints on their availability, (ii) it is desirable to cut the trim into as few pieces as possible and (iii) it is difficult because of the problem's structure, to round fractional, LP solutions to good feasible cutting plans. The point of departure for the procedures is the column generation technique of Gilmore and Gomory. The computational experience reported here suggests that the heuristics are both effective and efficient.     

Numerical solutions for the one-dimensional heat-conduction equation using a spreadsheet

We show how to use a spreadsheet to calculate numerical solutions of the one-dimensional time-dependent heat-conduction equation. We find the spreadsheet to be a practical tool for numerical calculations, because the algorithms can be implemented simply and quickly without complicated programming, and the spreadsheet utilities can be used not only for graphics, printing, and file management, but also for advanced mathematical operations.We implement the explicit and the Crank-Nicholson forms of the finite-difference approximations and discuss the geological applications of both methods. We also show how to adjust these two algorithms to a nonhomogeneous lithosphere in which the thermal properties (thermal conductivity, density, and radioactive heat generation) change from the upper crust to the lower crust and to the mantle.The solution is presented in a way that can fit any spreadsheet (Lotus-123, Quattro-Pro, Excel). In addition, a Quattro-Pro program with macros that calculate and display the thermal evolution of the lithosphere after a thermal perturbation is enclosed in an appendix.     

Numerical solutions of non-linear kinetic equations for a one-dimensional evaporation-condensation problem

Numerical solutions have been obtained for both the nonlinear Boltzmann equation (for two collision laws) and the Krook equation for a one-dimensional evaporation-condensation problem for a range of parameters. Our calculations of rate indicate that the linear prediction underestimates the non-equilibrium hindrance effect due to intermolecular collisions. The Krook evaporation rates are lower than the corresponding Boltzmann results and, thus, differ even more from the linear values. The Boltzmann evaporation rates for a gas of hard spheres are lower than those for Maxwelian molecules at lower values of Knudsen number. The microscopic and macroscopic properties obtained from the Krook solutions differ appreciably from the corresponding Boltzmann results.     

Numerical verification of solutions for elasto-plastic torsion problems

In this paper, we consider a numerical technique which enables us to verify the existence of solutions for the elasto-plastic torsion problems governed by the variational inequality. Based upon the finite element approximations and the explicit a priori error estimates for a simple problem, we present an effective verification procedure that through numerical computation generates a set which includes the exact solution. This paper is an extension of the previous paper [1] in which we mainly dealt with the obstacle problems, but some special techniques are utilized to verify the solutions for nondifferentiable nonlinear equations concerned with the present problem. A numerical example is illustrated.     

Numerical solutions for constrained time-delayed optimal control problems

In this paper, time-delayed optimal control problems governed by delayed differential equation are solved. Two different techniques based on integration and differentiation matrices are considered. The time-delayed term of the problem has been approximated by Chebyshev interpolating polynomials. On this basis, the optimal control problem can be solved as a mathematical programming problem. The example illustrates the robustness, accuracy and efficiency of the proposed numerical techniques.     

Numerical solutions for system of third-order boundary value problems

We develop a numerical method for computing approximations for the solutions of a system of third order boundary value problems associated with odd order obstacle problems. Such a problem arise in physical oceanography and can be studied in the framework of variational inequality theory. We study the convergence analysis of the present method and we show that it gives numerical results which are better than the other available results. Numerical example is presented to illustrate the applicability of the new method.     

Numerical methods for transient solutions of machine repair problems

This paper reviews and compares two types of numerical methods of computing transient probabilities of finite Markovian queues (particularly the machine repair problem). A brief review of each method is followed by numerical examples of small and moderate size machine repair problems. The results demonstrate the feasibility of applying numerical techniques for obtaining transient solutions to Markovian queueing problems.     

Application of general discrete orthogonal polynomials to optimal-control systems

The shift-transformation matrix of general discrete orthogonal polynomials is introduced. General discrete orthogonal polynomials are adopted to obtain the modified discrete Euler-Lagrange equations. Then general discrete orthogonal polynomials are applied to simplify the discrete Euler-Lagrange equations into a set of linear algebraic ones for the approximation of state and control variables of digital systems. An example is included to demonstrate the simplicity and applicability of the method. Also, a comparison of the results obtained via several classical discrete orthogonal polynomials for the same problem is given.     

Turnpike solutions in optimal control problems for quantum-mechanical systems

We consider the problem of optimal impulse control for a quantum-mechanical spin sequence as a system of oscillators. We study this problem with a double transformation to a derivative problem in which phase variables are oscillator amplitudes. As a result, we get the turnpike solutions. For special (turnpike) sets of boundary conditions they correspond exactly to generalized solutions of the original problem, while for other conditions they can be used as initial approximations for iterative procedures. This problem is a generalization of the special case of two oscillators which we study exhaustively and use as an illustrative example.     

Numerical verification of solutions for obstacle problems using a Newton-like method

This paper is a continuation of the preceding study [1] in which we described a method which automatically proves the existence of solutions for variational inequalities by computer. We newly formulate a verification method using a Newton-like method. This approach enables us to remove the restriction in the previous paper to the retraction property of the operator in a neighborhood of the solution. We show some numerical examples which confirm that the method is really applicable to problems which have no retraction property.     

Bicompact monotonic schemes for a multidimensional linear transport equation

Bicompact difference schemes, previously proposed by the authors for linear one-dimensional transport equations are generalized to the multidimensional case by using a coordinate-wise splitting of the multidimensional problem. The scheme stencil for each of the spatial directions is minimal and consists of two points. The schemes are efficient and can be solved by the running calculation method. The proposed difference schemes have the fourth-order approximation in space variables and first- or third-order time approximation for smooth solutions. The schemes for solving multidimensional problems have inherited the monotonicity property of one-dimensional bicompact schemes. Numerical examples are given illustrating the actual accuracy order of bicompact schemes for smooth solutions and the scheme monotonicity for discontinuous solutions.     

A general one-dimensional search algorithm for computing approximate solutions of some non-linear optimal control problems

Barr and Gilbert (1966, 1969 b) have presented computing algorithms for converting a brood class of optimal control problems (including minimum time, and fixed-time minimum fuel, energy and effort problems) to a sequence of optimal regulator problems, using a one dimensional search of the cost variable. These Barr and Gilbert algorithms, which use quadratic programming algorithms by the same authors (1969 a) to solve the resulting optimal regulator problems, are restricted to dynamic equations linear in state by virtue of using the convexity and compactness (Neustadt 1963) and contact function (Gilbert 1966) of the reachable set

This paper extends the above approach to a class of terminal cost optimal control problems similar to those considered by Barr and Gilbert (including quite general control constraints, but only allowing initial and final state constraints), having differential equations non-linear instate and control (where the convexity-compactness results do not hold), by converting each such problem to a sequence of optimal regulator problems, with non-linear differential equations. These, in turn, are solved by one of the author's earlier algorithms (Katz 1974) that makes use of the above convexity, compactness, and contact function results by repeatedly linearizing the regulator problems. The approach of this paper differs from that of Halkin (1964 b), in that Halkin directly linearizes the original problem (e.g. converting a non-linear minimum fuel problem to a linear minimum fuel problem) and then solves the linearized version by a doubly iterative procedure

The computing algorithm presented here is based on the definition of an appropriate approximate solution of the terminal cost problem. A local-minimum convergence proof is given, which is weak in the sense that it assumes convergence of the substep algorithm (Katz 1974) for non-linear optimal regulator problems, whose convergence has not been proved. A subsequent paper (Katz and Wachtor, to appear) shows good convergence of the (overall) terminal cost problem algorithm in examples having singular arcs, with no prior knowledge of the solution or its singular nature, other than an initial upper bound on the cost.     

Unidimensional curve fitting search schemes revisited for multidimensional optimization algorithms

This paper describes two efficient alternatives to the well-known curve-fitting techniques, namely, the quadratic fitting method and the cubic interpolation method, which are the most popular unidimensional search schemes in nonlinear optimization algorithms. These new alternatives are tested using typical unidimensional test functions as also typical multidimensional test functions in conjunction with Powell and Broyden algorithms. In terms of equivalent function evaluations and CPU time, the usual criteria for comparison, the performance of these new techniques is found to be extemely good.     

Numerical schemes for nonlinear predictor feedback

This paper focuses on a specific aspect of the implementation problem for predictor-based feedback laws: the problem of the approximation of the predictor mapping. It is shown that the numerical approximation of the predictor mapping by means of a numerical scheme in conjunction with a hybrid feedback law that uses sampled measurements can be used for the global stabilization of all forward complete nonlinear systems that are globally asymptotically stabilizable and locally exponentially stabilizable in the delay-free case. Explicit formulae are provided for the estimation of the parameters of the resulting hybrid control scheme.     

Positional solutions of Hamilton-Jacobi equations in control problems for discrete-continuous systems

We develop a canonical global optimality theory based on operating with the set of solutions for the Hamilton-Jacobi inequalities that parametrically depend on the initial (or final) position. These solutions, called positional L-functions (of Lyapunov type), naturally arise in the studies of control problems for discrete-continuous (hybrid, impulse) systems; an important prototype of such problems are classical optimal control problems with general end constraints on the trajectory. We analyze sufficient optimality conditions with this new class of L-functions and invert the maximum principle into a sufficient condition for nonlinear problems of optimal impulse control.