First-order Necessary Conditions For Optimality Research Articles

Analysis and Approximation of the Velocity Tracking Problem for Navier--Stokes Flows with Distributed Control

We consider the mathematical formulation, analysis, and the numerical solution of a time-dependent optimal control problem associated with the tracking of the velocity of a Navier--Stokes flow in a bounded two-dimensional domain through the adjustment of a distributed control. The existence of optimal solutions is proved and the first-order necessary conditions for optimality are used to derive an optimality system of partial differential equations whose solutions provide optimal states and controls. Semidiscrete-in-time and fully discrete space-time approximations are defined and their convergence to the exact optimal solutions is shown. A gradient method for the solution of the fully discrete equations is examined, as are its convergence properties. Finally, the results of some illustrative computational experiments are presented.

A Feasible Directions Algorithm for Optimal Control Problems with State and Control Constraints: Convergence Analysis

In this paper we describe an optimization algorithm for the computation of solutions to optimal control problems with control, state, and terminal constraints. Inequality and equality constraints are dealt with by means of feasible directions and exact penalty approaches, respectively. We establish a general convergence property of the algorithm which makes no reference to the existence of accumulation points; in this analysis the compactness of the space of relaxed controls is used only to guarantee boundedness of the sequence of penalty parameters. We also demonstrate that relaxed accumulation points of sequences generated by the algorithm satisfy standard first-order necessary conditions of optimality. The algorithm contains a number of computation saving features, including an $\varepsilon$-active strategy for dealing with the infinite dimensional inequality constraints. Our convergence analysis provides techniques for studying the convergence properties of related optimization algorithms in which direction-finding subproblems involve the approximation of directional derivatives of the Chebyshev functional associated with state constraints. A companion paper provides details of implementation and numerical examples.

Optimum beam design using finite elements

The optimum structural design of beams for several benchmark problems are considered. The penalty function method is employed to impose the inequality constraints and the Newton-Raphson algorithm has been used to solve the first-order necessary condition for optimality. The finite element procedure shows to be very effective in converging to the exact optimum solution.

Numerical solution of constrained optimal control problems with parameters

This article presents a numerical solution technique for constrained optimal control problems that contain parameters. Here, the state, control, and parameter inequality constraints are accommodated via an extended penalty function. This penalty function takes on large values when the constraints are violated and small values when the constraints are satisfied. Using the calculus of variation it is shown that the first-order necessary conditions for optimality are in the form of a two-point boundary-value problem involving differential and algebraic equations (BVP-DAE). A multiple shooting/continuation method is developed for solving this BVP-DAE. Two examples are presented to demonstrate the effectiveness of the solution approach developed in the paper.

Optimal-power-flow solution by Newton's method applied to an augmented Lagrangian function

The paper describes a new approach to the optimal-power-flow problem based on Newton's method which it operates with an augmented Lagrangian function associated with the original problem. The function aggregates all the equality and inequality constraints. The first-order necessary conditions for optimality are reached by Newton's method, and by updating the dual variables and the penalty terms associated with the inequality constraints. The proposed approach does not have to identify the set of binding constraints and can be utilised for an infeasible starting point. The sparsity of the Hessian matrix of the augmented Lagrangian is completely exploited in the computational implementation. Tests results are presented to show the good performance of this approach.

Optimal lateral-escape maneuvers for microburst encounters during final approach

This paper is concerned with the optimization of lateral-escape trajectories in a microburst wind field for an aircraft on final approach. The objective is to minimize the peak value of altitude drop. An extensive numerical effort has been undertaken to investigate the characteristics of open-loop extremal solutions for various locations of the microburst. When a sufficiently large aerodynamic roll-angle limit is specified and the center of the microburst is not too far offset from the centerline extension of the approach runway, typically three trajectories can be found that satisfy the first-order necessary conditions of optimality for a given set of boundary conditions, namely, a trajectory that passes the microburst center to the left, a trajectory passing the center to the right, and a trajectory passing right through the center. The results bear out that a lateral-escape maneuver, in which an aircraft is turned away from the microburst center, may significantly improve an aircraft's surv ivability, in comparison to an escape maneuver that is restricted to a vertical plane. One of the most striking observations in this study is that, in contrast to nonturning escape maneuvers, lateral-escape maneuvers often exhibit a climb, rather than a descent, in the initial phase. The insight obtained from the present study may help the development of near-optimal lateral-escape guidance strategies for onboard application.

Optimality Conditions for Control Problems Governed by Variational Inequalities

This paper deals with the theory of first-order necessary conditions of optimality for nonsmooth control problems governed by variational inequalities. We present a general approach in which necessary conditions of first-order are obtained by means of differentiation tools based on epi-convergence of second-order differential quotients.

FIRST-ORDER NECESSARY CONDITIONS IN THE PROBLEM OF OPTIMAL CONTROL OF A DIFFERENTIAL INCLUSION WITH PHASE CONSTRAINTS

Nondegenerate first-order necessary conditions for optimality are obtained for the problem (1.1)-(1.4) under different assumptions about controllability at the endpoints. These necessary conditions are obtained in the Hamiltonian form of Clarke [1]. With the help of a smoothing technique [2] the perturbation method in [3] is used to carry the main results in [4] (there the case when the support function depends smoothly on the variable is considered) over to the more natural class of problems with locally Lipschitz support function . Bibliography: 19 titles.

Time-optimal reorientation maneuvers for a combat aircraft

Results are presented from a study of time-optimal attitude reorientation maneuvering for an aircraft with thrust-vectoring capability. A detailed mathematical model for attitude motions of the aircraft is developed. First-order necessary conditions for optimality are applied and a family of extremal solutions obtained for two classes of reorientation maneuvers of practical interest. The thrust-vectoring power is varied parametrically, and thus an estimate of the reduction in maneuvering time due to the thrust-vectoring enhancement of the aircraft is obtained. ARIOUS analyses have shown that emerging trends in missile and radar technologies will have a profound effect on design requirements for air-superiority vehicles. It appears that, in close-in combat, vehicles with the ability to operate at extreme angles of attack will have a decided advantage.1'4 Designing such supermaneuverable aircraft is a complex and challenging task, and future design efforts may well lead to radically new concepts. At present, however, the preferred approach is to provide a form of thrust vectoring in which propulsive power is used to generate control moments.5'6 At high angles of attack a, especially in the post-stall region, the effectiveness of the aerodynamic control surfaces decreases rapidly with increase in a.. Thus, thrust-vectoring generated (propulsive) moments must be used to maintain control of the aircraft at high a. At low a the propulsive moments can supplement the aerodynamic control surfaces and thus increase the agility of the aircraft. In addition, thrust vectoring can be used for control of the aircraft in case of mechanical failure or malfunction of the aerodynamic control surfaces. Several research programs that focus on utilizing thrust-vectoring control are currently under way. Among these is the F/A-18 based High Angle-of-attack Research Vehicle (HARV) program.7 The flight mechanics issues can be divided into two areas: 1) analysis of changes in the velocity vector and 2) analysis of changes in the body attitude. This paper presents some results obtained in a study that focused on understanding the nature of aircraft minimum-time fuselage-reorientation maneuvering. In the study, data for the HARV were used. Accordingly, the numerical results correspond to this aircraft. However, the discussion and methodology in analyzing the results presented in the next sections are quite general. A mathematical model of the aircraft and the interacting environment (the aerodynamic forces and moments) is derived in Sec. II. The basic ideas and assumptions in the development of the mathematical model are discussed in detail. In Sec. Ill a class of optimal control problems of interest is formulated and a set of necessary conditions for optimality derived by using the Minimum Principle.8'10 This set of necessary conditions is cast into a numerical multipoint boundary-value problem (MPBVP). Subsequently, a homotopy method and the related procedure for solving the MPBVPs (and thus generat

Optimal control of two-time-scale systems with state-variable inequality constraints

The established necessary conditions for optimality in nonlinear control problems that involve state-variable inequality constraints are applied to a class of singularly perturbed systems. The distinguishing feature of this class of two-time-scal e systems is a transformation of the state-variable inequality constraint, present in the full-order problem, to a constraint involving states and controls in the reduced problem. It is shown that, when a state constraint is active in the reduced problem, the boundary-layer problem can be of finite time in the stretched time scale. Thus, the usual requirement for asymptotic stability of the boundary-layer system is not applicable and cannot be used to construct approximate boundary-layer solutions. Several alternative solution methods are explored and illustrated with simple examples. I. Introduction S TATE-VARIABLE inequality constraints are commonly encountered in the study of dynamic systems. The study of rigid body aircraft dynamics and control is certainly no exception. For instance, a maximum allowable value of dynamic pressure is usually prescribed for aircraft with supersonic capability. This limit is required to insure that the vehicle's structural integrity is maintained. Given a typical statespace description of the vehicle dynamics, this limit constitutes an inequality constraint on vehicle state. State inequality constraints have been studied extensively by researchers in the field of optimal control. First-order necessary conditions for optimality when general functions of state are constrained have been obtained.13 However, the construction of solutions via this set of conditions proves difficult. Most practitioners seeking an open-loop control solution rely on direct approaches to optimization that employ penalty functions for satisfaction of state inequality constraints.4 As a rule, algorithms employing such methods are computationally intense and slow to converge. Consequently, they are not well suited for real-time implementation. As discussed in the literature, the use of singular perturbation techniques in the study of aircraft trajectory optimization can, through order reduction, lead to both open- and closedloop solutions that are computationally efficient. These methods can sometimes be used to circumvent difficulties associated with enforcing a state inequality constraint as well.5 As an example, consider the minimum time intercept problem of Ref. 6 that employs a model of the F-8 aircraft. A nearoptimal feedback solution is obtained via singular perturbation theory that includes consideration of an inequality constraint on dynamic pressure. In the zero-order reduced solution, algebraic constraints are obtained when the perturbation parameter e, which premultiplies the so-called fast dynamic equations, is set to zero. These constraints can be used to eliminate the fast states (in this case altitude and flight-path angle) from the reduced problem. One can choose, however, to retain one or more of the fast states and to eliminate instead

On the Identification of Local Minimizers in Inertia-Controlling Methods for Quadratic Programming

The verification of a local minimizes of a general (i.e., nonconvex) quadratic program is in general an NP-hard problem. The difficulty concerns the optimality of certain points (which we call dead points) at which the first-order necessary conditions for optimality are satisfied, but strict complementarity does not hold. Inertia-controlling quadratic programming (ICQP) methods form an important class of methods for solving general quadratic programs. We derive a computational scheme for proceeding at a dead point that is appropriate for a general ICQP method.

Optimal rigid body motions, part 1: Approximate formulation

Optimal rigid body angular motions are investigated in the absence of direct control over one of the angular velocity components, via an approximate dynamic model. An analysis of first-order necessary conditions for optimality with the proposed model reveals that, over a large range of boundary conditions, there are, in general, several distinct extremal solutions. A classification in terms of subfamilies of extremal solutions is presented. Second-order necessary conditions are investigated to establish local optimality for the candidate minimizers.

Algorithms for relaxed control problems with delay

Three convergent procedures are presented. These determine a minimizing point of a relaxed optimal control problem defined by delay-differential equations with the control restricted to a compact convex subset of ℝ m In the first of these procedures, a strong variational algorithm due to Virk (1985 a) is extended to relaxed controls. The second uses a steepest descent approach. It is shown that both these procedures generate accumulation controls satisfying first-order necessary conditions of optimality. In the third procedure, the relaxed controls generated at each iteration are approximated using ordinary controls. When this is done, we show that limit points satisfy optimality conditions to within delta. Numerical performances of all three algorithms are also presented.

Convergence and optimality of modified two-step algorithm for integrated system optimization and parameter estimation

This paper investigates convergence and optimality properties of the modified two-step algorithm for on-line determination of the optimum steady-state operating point of an industrial process. Mild sufficient conditions are derived for the convergence and feasibility of the algorithm. It is shown that every point within the solution set of the algorithm satisfies first-order necessary conditions for optimality, and that every optimal solution belongs to this set. It is also shown that there are advantages to be gained by using a linear mathematical model of the process within the implementation of the algorithm.

On conditions for optimality of a class of nondifferentiable functions

The purpose of this paper is to derive first-order necessary conditions for optimality of a class of nondifferentiable functions. The first-order necessary conditions for optimality for the minimax function and thel 1-function can be considered as special cases of the present method. Furthermore, the optimality conditions obtained are used to obtain threshold values for the controlling parameters of a class of exact penalty functions.

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.

A SECOND ORDER OPTIMALITY PRINCIPLE FOR A TIME-OPTIMAL PROBLEM

In this paper, a general necessary optimality condition of second order is proved for a time-optimal problem. Necessary optimality conditions of second order are applied, in general, in studying singular optimal regimes which can not be found with the aid of the first-order necessary condition for optimality, i.e. with the aid of the Pontrjagin maximum principle.Bibliography: 4 titles.

Necessary conditions of optimality for a differential game with bounded state variables

First-order necessary conditions of optimality are examined for a two-person zero-sum deterministic differential game (an extension of Isaacs' War of Attrition and Attack [4]) with bounded state variables. Both players use closed-loop strategies. Multiplier conditions associated with the state constraints are shown to play a key role in the determination of optimal strategies.