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.     

A feasible directions algorithm for time-lag optimal control problems with control and terminal inequality constraints

A computational algorithm for a class of time-lag optimal control problems involving control and terminal inequality constraints is presented. The convergence properties of the algorithm is also investigated. To test the algorithm, an example is solved.This work was partially supported by the Australian Research Grant Committee.     

An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem

In this paper, we consider a class of optimal control problems subject to equality terminal state constraints and continuous state and control inequality constraints. By using the control parametrization technique and a time scaling transformation, the constrained optimal control problem is approximated by a sequence of optimal parameter selection problems with equality terminal state constraints and continuous state inequality constraints. Each of these constrained optimal parameter selection problems can be regarded as an optimization problem subject to equality constraints and continuous inequality constraints. On this basis, an exact penalty function method is used to devise a computational method to solve these optimization problems with equality constraints and continuous inequality constraints. The main idea is to augment the exact penalty function constructed from the equality constraints and continuous inequality constraints to the objective function, forming a new one. This gives rise to a sequence of unconstrained optimization problems. It is shown that, for sufficiently large penalty parameter value, any local minimizer of the unconstrained optimization problem is a local minimizer of the optimization problem with equality constraints and continuous inequality constraints. The convergent properties of the optimal parameter selection problems with equality constraints and continuous inequality constraints to the original optimal control problem are also discussed. For illustration, three examples are solved showing the effectiveness and applicability of the approach proposed.     

Convergence of a feasible directions algorithm for a distributed optimal control problem of parabolic type with terminal inequality constraints

In this paper, we consider a class of optimal control problems with control and terminal inequality constraints, where the system dynamics is governed by a linear second-order parabolic partial differential equation with first boundary condition. A feasible direction algorithm for solving this class of optimal control problems has already been obtained in the literature. The aim of this paper is to improve the convergence result by using a topology arising in the study of relaxed controls.     

Second-order sufficient conditions for control problems with mixed control-state constraints

References 1–4 develop second-order sufficient conditions for local minima of optimal control problems with state and control constraints. These second-order conditions tighten the gap between necessary and sufficient conditions by evaluating a positive-definiteness criterion on the tangent space of the active constraints. The purpose of this paper is twofold. First, we extend the methods in Refs. 3, 4 and include general boundary conditions. Then, we relate the approach to the two-norm approach developed in Ref. 5. A direct sufficiency criterion is based on a quadratic function that satisfies a Hamilton-Jacobi inequality. A specific form of such a function is obtained by applying the second-order sufficient conditions to a parametric optimization problem. The resulting second-order positive-definiteness conditions can be verified by solving Riccati equations.The authors wish to thank K. Malanowski for helpful discussions.     

Computational algorithm for a distributed optimal control problem of parabolic type with terminal inequality constraints

A computational algorithm for optimal control problems with control and terminal inequality constraints involving first boundary-value problems of parabolic type is presented. The convergence properties are also studied.This work, which was partly supported by the Australian Research Grants Committee, was done during the period when Z. S. Wu was an Honorary Visiting Fellow in the School of Mathematics at the University of New South Wales, Australia.     

Pontryagin's principle for problems with isoperimetric constraints and for problems with inequality terminal constraints

Necessary conditions are derived for optimal control problems subject to isoperimetric constraints and for optimal control problems with inequality constraints at the terminal time. The conditions are derived by transforming the problem into the standard form of optimal control problems and then using Pontryagin's principle.     

Abort landing in the presence of windshear as a minimax optimal control problem,part 1: Necessary conditions

The landing of a passenger aircraft in the presence of windshear is a threat to aviation safety. The present paper is concerned with the abort landing of an aircraft in such a serious situation. Mathematically, the flight maneuver can be described by a minimax optimal control problem. By transforming this minimax problem into an optimal control problem of standard form, a state constraint has to be taken into account which is of order three. Moreover, two additional constraints, a first-order state constraint and a control variable constraint, are imposed upon the model. Since the only control variable appears linearly, the Hamiltonian is not regular. Thus, well-known existence theorems about the occurrence of boundary arcs and boundary points cannot be applied. Numerically, this optimal control problem is solved by means of the multiple shooting method in connection with an appropriate homotopy strategy. The solution obtained here satisfies all the sharp necessary conditions including those depending on the sign of certain multipliers. The trajectory consists of bang-bang and singular subarcs, as well as boundary subarcs induced by the two state constraints. The occurrence of boundary arcs is known to be impossible for regular Hamiltonians and odd-ordered state constraints if the order exceeds two. Additionally, a boundary point also occurs where the third-order state constraint is active. Such a situation is known to be the only possibility for odd-ordered state constraints to be active if the order exceeds two and if the Hamiltonian is regular. Because of the complexity of the optimal control, this single problem combines many of the features that make this kind of optimal control problems extremely hard to solve. Moreover, the problem contains nonsmooth data arising from the approximations of the aerodynamic forces and the distribution of the wind velocity components. Therefore, the paper can serve as some sort of user's guide to solve inequality constrained real-life optimal control problems by multiple shooting.An extended abstract of this paper was presented at the 8th IFAC Workshop on Control Applications of Nonlinear Programming and Optimization, Paris, France, 1989 (see Ref. 1).This paper is dedicated to Professor Hans J. Stetter on the occasion of his 60th birthday.     

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.     

On optimal control problems with general boundary conditions

It is shown that the necessary optimality conditions for optimal control problems with terminal constraints and with given initial state allow also to obtain in a straightforward way the necessary optimality conditions for problems involving parameters and general (mixed) boundary conditions. In a similar manner, the corresponding numerical algorithms can be adapted to handle this class of optimal control problems.This research was supported in part by the Commission on International Relations, National Academy of Sciences, under Exchange Visitor Program No. P-1-4174.The author is indebted to the anonymous reviewer bringing to his attention Ref. 9 and making him aware of the possible use of generalized inverse notation when formulating the optimality conditions.     

Sufficient optimality conditions for nonconvex control problems with state constraints

The sufficient optimality conditions of Zeidan for optimal control problems (Refs. 1 and 2) are generalized such that they are applicable to problems with pure state-variable inequality constraints. We derive conditions which neither assume the concavity of the Hamiltonian nor the quasiconcavity of the constraints. Global as well as local optimality conditions are presented.     

A unified framework for linear control problems with state-variable inequality constraints

This paper briefly reviews the literature on necessary optimality conditions for optimal control problems with state-variable inequality constraints. Then, it attempts to unify the treatment of linear optimal control problems with state-variable inequality constraints in the framework of continuous linear programming. The duality theory in this framework makes it possible to relate the adjoint variables arising in different formulations of a problem; these relationships are illustrated by the use of a simple example. This framework also allows more general problems and admits a simplex-like algorithm to solve these problems.This research was partially supported by Grant No. A4619 from the National Research Council of Canada to the first author. The first author also acknowledges the support provided by the Brookhaven National Laboratory, where he conducted his research.     

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.     

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.     

Semi-Infinite Programming Approach to Continuously-Constrained Linear-Quadratic Optimal Control Problems

Consider the class of linear-quadratic (LQ) optimal control problems with continuous linear state constraints, that is, constraints imposed on every instant of the time horizon. This class of problems is known to be difficult to solve numerically. In this paper, a computational method based on a semi-infinite programming approach is given. The LQ optimal control problem is formulated as a positive-quadratic infinite programming problem. This can be done by considering the control as the decision variable, while taking the state as a function of the control. After parametrizing the decision variable, an approximate quadratic semi-infinite programming problem is obtained. It is shown that, as we refine the parametrization, the solution sequence of the approximate problems converges to the solution of the infinite programming problem (hence, to the solution of the original optimal control problem). Numerically, the semi-infinite programming problems obtained above can be solved efficiently using an algorithm based on a dual parametrization method.     

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.     

一类连续最优控制问题的有限维近似

1　引　　言本文考虑具有状态终端约束、控制受限的非线性连续最优控制问题min　h0(x(0))+∫T0f0(x(t),u(t))dt+g0(x(T))(1.1)s.t.　x(t)=f(x(t),u(t)),　　t∈[0,T](1.2)D(x(0))=0,(1.3)E(x(T))=0,(1.4)S(u(t))≤0,　　t∈[0,T](1.5)其中,h0:Rn→R,f0:Rn×Rm→R,f:Rn×Rm→Rn,g0:Rn→R,D:Rn→Rp,E:Rn→Rq,S:Rm→Rr均为二次连续可微函数.T为终端时间(固定),p,q≤n,x(t)∈W1,∞[0,T]n,u(t)∈L∞[0,T]m分别为状态函数和控制函数.U(t)={u:S(u(t))≤0}为紧凸集.问题(1.1)—(1.5)要求寻找最佳控制u(t)使得目标函数(1.1)达到极小.…     

Differential sensitivity of solutions of convex constrained optimal control problems for discrete systems

A family of optimal control problems for discrete systems that depend on a real parameter is considered. The problems are strongly convex and subject to state and control constraints. Some regularity conditions are imposed on the constraints.The control problems are reformulated as mathematical programming problems. It is shown that both the primal and dual optimal variables for these problems are right-differentiable functions of a parameter. The right-derivatives are characterized as solutions to auxiliary quadratic control problems. Conditions of continuous differentiability are discussed, and some estimates of the rate of convergence of the difference quotients to the respective derivatives are given.     

An applied cell mapping method for optimal control problems

From the application point of view, a series of modifications are proposed for the cell mapping method discussed in Ref. 1 for the optimal control analysis of dynamical systems. The cell order around the target set is rearranged. A set of common discriminate principles is used for the selection of the optimal one among competing control strategies of the same cost. Inequality constraints of the system are taken into account. The number of elements in the set of allowable time intervals is not prescribed, but left open. These modifications seem to make the cell mapping method more efficient for analyzing feedback systems and for obtaining their global optimal control information. The algorithms presented in this paper could broaden the application of the cell mapping approach of Ref. 1 to a wider class of engineering problems.