Convergence of a feasible directions algorithm for relaxed controls in time-lag systems

In this paper, we consider a class of time-lag optimal control problems involving control and terminal inequality constraints. A feasible direction algorithm has been obtained by Teo, Wong, and Clements for solving this class of optimal control problems. It was shown that anyL accumulation points of the sequence of controls generated by the algorithm satisfy a necessary condition for optimality. However, suchL accumulation points need not exist. The aim of this paper is to prove a convergence result, which ensures that the sequence of controls generated by the algorithm always has accumulation points in the sense of control measure, and these accumulation points satisfy a necessary condition for optimality for the corresponding relaxed problem.This work was done when the first author was on sabbatical leave at the School of Mathematics, University of New South Wales, Australia.     

Solving elliptic control problems with interior point and SQP methods: control and state constraints

We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. Both boundary control and distributed control problems are considered with boundary conditions of Dirichlet or Neumann type. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. Necessary conditions of optimality are discussed both for the continuous and the discretized control problem. It is shown that the recently developed interior point method LOQO of [35] is capable of solving these problems even for high discretizations. Four numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang–bang controls.     

A transformation technique for optimal control problems with partially linear state inequality constraints

This paper considers optimal control problems involving the minimization of a functional subject to differential constraints, terminal constraints, and a state inequality constraint. The state inequality constraint is of a special type, namely, it is linear in some or all of the components of the state vector.A transformation technique is introduced, by means of which the inequality-constrained problem is converted into an equality-constrained problem involving differential constraints, terminal constraints, and a control equality constraint. The transformation technique takes advantage of the partial linearity of the state inequality constraint so as to yield a transformed problem characterized by a new state vector of minimal size. This concept is important computationally, in that the computer time per iteration increases with the square of the dimension of the state vector.In order to illustrate the advantages of the new transformation technique, several numerical examples are solved by means of the sequential gradient-restoration algorithm for optimal control problems involving nondifferential constraints. The examples show the substantial savings in computer time for convergence, which are associated with the new transformation technique.This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-76-3075, and by the National Science Foundation, Grant No. MCS-76-21657.     

An exact penalty function algorithm for optimal control problems with control and terminal equality constraints,part 1

The presence of control constraints, because they are nondifferentiable in the space of control functions, makes it difficult to cope with terminal equality constraints in optimal control problems. Gradient-projection algorithms, for example, cannot be employed easily. These difficulties are overcome in this paper by employing an exact penalty function to handle the cost and terminal equality constraints and using the control constraints to define the space of permissible search directions in the search-direction subalgorithm. The search-direction subalgorithm is, therefore, more complex than the usual linear program employed in feasible-directions algorithms. The subalgorithm approximately solves a convex optimal control problem to determine the search direction; in the implementable version of the algorithm, the accuracy of the approximation is automatically increased to ensure convergence.This work was supported by the United Kingdom Science Research Council, by the US Army Research Office, Contract No. DAAG-29-73-C-0025, and by the National Science Foundation, Grant No. ENG-73-08214-A01.     

An exact penalty function algorithm for optimal control problems with control and terminal equality constraints,part 2

In Part 1 of this paper, implementable and conceptual versions of an algorithm for optimal control problems with control constraints and terminal equality constraints were presented. It was shown that anyL accumulation points of control sequences generated by the algorithms satisfy necessary conditions of optimality. Since such accumulation points need not exist, it is shown in this paper that control sequences generated by the algorithms always have accumulation points in the sense of control measure, and these accumulation points satisfy optimality conditions for the corresponding relaxed control problem.This work was supported by the United Kingdom Science Research Council, by the US Army Research Office, Contract No. DAA-29-73-C-0025, and by the National Science Foundation, Grant No. ENG-73-08214-A01.     

Nonlinearly constrained time-delayed optimal control problems

A computational algorithm for solving a class of optimal control problems involving terminal and continuous state constraints of inequality type was developed in Ref. 1. In this paper, we extend the results of Ref. 1 to a more general class of constrained time-delayed optimal control problems, which involves terminal state equality constraints as well as terminal state inequality constraints and continuous state constraints. Two examples have been solved to illustrate the efficiency of the method.     

Numerical solution of positive control problem via linear programming

The reachability problem for linear time-invariant discrete-time control systems with sign-restricted input is considered. The time-optimal control is constructed by an iterative procedure. Each step of the iteration is defined as a linear programming problem. This problem is solved by the simplex algorithm. The initial feasible solution for the simplex algorithm is provided by the preceding step of the iteration. The inversion of the basis matrix is reduced to a bordering procedure. The structural stability of the solution is investigated.     

Function-space quasi-Newton algorithms for optimal control problems with bounded controls and singular arcs

Two existing function-space quasi-Newton algorithms, the Davidon algorithm and the projected gradient algorithm, are modified so that they may handle directly control-variable inequality constraints. A third quasi-Newton-type algorithm, developed by Broyden, is extended to optimal control problems. The Broyden algorithm is further modified so that it may handle directly control-variable inequality constraints. From a computational viewpoint, dyadic operator implementation of quasi-Newton methods is shown to be superior to the integral kernel representation. The quasi-Newton methods, along with the steepest descent method and two conjugate gradient algorithms, are simulated on three relatively simple (yet representative) bounded control problems, two of which possess singular subarcs. Overall, the Broyden algorithm was found to be superior. The most notable result of the simulations was the clear superiority of the Broyden and Davidon algorithms in producing a sharp singular control subarc.This research was supported by the National Science Foundation under Grant Nos. GK-30115 and ENG 74-21618 and by the National Aeronautics and Space Administration under Contract No. NAS 9-12872.     

Modified quasilinearization algorithm for optimal control problems with nondifferential constraints and general boundary conditions

This paper considers the numerical solution of the problem of minimizing a functionalI, subject to differential constraints, nondifferential constraints, and general boundary conditions. It consists of finding the statex(t), the controlu(t), and the parameter so that the functionalI is minimized while the constraints are satisfied to a predetermined accuracy.The modified quasilinearization algorithm (MQA) is extended, so that it can be applied to the solution of optimal control problems with general boundary conditions, where the state is not explicitly given at the initial point.The algorithm presented here preserves the MQA descent property on the cumulative error. This error consists of the error in the optimality conditions and the error in the constraints.Three numerical examples are presented in order to illustrate the performance of the algorithm. The numerical results are discussed to show the feasibility as well as the convergence characteristics of the algorithm.This work was supported by the Electrical Research Institute of Mexico and by CONACYT, Consejo Nacional de Ciencia y Tecnologia, Mexico City, Mexico.     

Remarks on the algorithm of sakawa for optimal control problems

A general optimal control problem for ordinary differential equations is considered. For this problem, some improvements of the algorithm of Sakawa are discussed. We avoid any convexity assumption and show with an example that the algorithm is even applicable in cases in which no optimal control exists.     

A new augmented penalty function technique for optimal control problems

This note presents an extension of the Miele—Cragg-Iyer-Levy augmented function method for finite-dimensional optimization problems to optimal control problems. A numerical study is provided.     

Optimization Techniques for Solving Elliptic Control Problems with Control and State Constraints: Part 1. Boundary Control

We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. In a first part we consider boundary control problems with either Dirichlet or Neumann conditions. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. It is shown that a recently developed interior point method is able to solve these problems even for high discretizations. Several numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang-bang and singular controls. The necessary conditions of optimality are checked numerically in the presence of active control and state constraints.     

SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control

Parametric nonlinear optimal control problems subject to control and state constraints are studied. Two discretization methods are discussed that transcribe optimal control problems into nonlinear programming problems for which SQP-methods provide efficient solution methods. It is shown that SQP-methods can be used also for a check of second-order sufficient conditions and for a postoptimal calculation of adjoint variables. In addition, SQP-methods lead to a robust computation of sensitivity differentials of optimal solutions with respect to perturbation parameters. Numerical sensitivity analysis is the basis for real-time control approximations of perturbed solutions which are obtained by evaluating a first-order Taylor expansion with respect to the parameter. The proposed numerical methods are illustrated by the optimal control of a low-thrust satellite transfer to geosynchronous orbit and a complex control problem from aquanautics. The examples illustrate the robustness, accuracy and efficiency of the proposed numerical algorithms.     

A two-stage feasible directions algorithm for nonlinear constrained optimization

We present a feasible directions algorithm, based on Lagrangian concepts, for the solution of the nonlinear programming problem with equality and inequality constraints. At each iteration a descent direction is defined; by modifying it, we obtain a feasible descent direction. The line search procedure assures the global convergence of the method and the feasibility of all the iterates. We prove the global convergence of the algorithm and apply it to the solution of some test problems. Although the present version of the algorithm does not include any second-order information, like quasi-Newton methods, these numerical results exhibit a behavior comparable to that of the best methods known at present for nonlinear programming. Research performed while the author was on a two years appointment at INRIA, Rocquencourt, France, and partially supported by the Brazilian Research Council (CNPq).     

Local convergence of an algorithm for solving optimal control problems

In this paper, a method is proposed for the numerical solution of optimal control problems with terminal equality constraints. The multiplier method is employed to deal with the terminal equality constraints. It is shown that a sequence of control functions, which converges to the optimal control, is obtained by the alternate update of control functions and multipliers.The authors wish to thank Dr. N. Fujii for his most valuable comments and suggestions.     

A virtual control concept for state constrained optimal control problems

A linear elliptic control problem with pointwise state constraints is considered. These constraints are given in the domain. In contrast to this, the control acts only at the boundary. We propose a general concept using virtual control in this paper. The virtual control is introduced in objective, state equation, and constraints. Moreover, additional control constraints for the virtual control are investigated. An error estimate for the regularization error is derived as main result of the paper. The theory is illustrated by numerical tests.     

A control reduced primal interior point method for a class of control constrained optimal control problems

A primal interior point method for control constrained optimal control problems with PDE constraints is considered. Pointwise elimination of the control leads to a homotopy in the remaining state and dual variables, which is addressed by a short step pathfollowing method. The algorithm is applied to the continuous, infinite dimensional problem, where discretization is performed only in the innermost loop when solving linear equations. The a priori elimination of the least regular control permits to obtain the required accuracy with comparatively coarse meshes. Convergence of the method and discretization errors are studied, and the method is illustrated at two numerical examples. Supported by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin. This paper appeared as ZIB Report 04-38.     

带1-范数约束的分裂可行问题的投影算法

本文主要研究带1-范数约束的分裂可行问题的求解算法.用一种交替投影算法,求得了问题的解,提出松弛交替投影算法,改进了直接往闭凸集上投影这一不足,并证明了该算法的收敛性.     

Solving nonlinear inequalities in a finite number of iterations

This paper describes a modified Newton algorithm for solving a finite system of inequalities in a finite number of iterations.This research was supported by NSF Grant No. ENG-73-08214-AO1, by NSF-RANN Grant No. ENV-76-04264, and by the United Kingdom Science Research Council.