Some necessary conditions for optimality for a class of optimal control problems which are linear in the control variable

For Meyer type optimal control problems the principle of optimality states that every subarc of an optimal solution that extends to the final time furnishes again a solution to the same optimal control problem, only with different initial conditions. Through a simple counter example it is shown in this paper that this principle is not satisfied, in general, for solutions obtained from a standard collocation approach. Introduction The principle of optimality [1] is one of the most intuitive and yet most fundamental theorems in the theory of optimal control. Its basic statement is that for Meyer type optimal control problems every subarc of an optimal solution that extends to the final time furnishes again a solution to the same optimal control problem, only with different initial conditions. This theorem has significant implications. For example, it is the starting point for the derivation of the Hamilton-Jacobi-Bellman equations [2] and it is the underlying principle in Bellman's dynamic programming approach [1], [3]. The present paper shows that the principle of optimality is not satisfied, in general, for solutions obtained from a standard collocation approach[4]. For a simple example whose collocation solution can be generated analytically it is shown that further improvements are possible along an optimal collocation solution if the design parameters along a number of leading nodes are frozen and the remainder of the trajectory is re-optimized. I. Continuous Optimal Control Problem In this section of the paper we introduce a simple class of optimal control problems without state and control constraints. In Section 1.2 we present the associated necessary conditions for optimality [2], [5]. For ease of notation we assume that the right-hand side of the 'Supervising Engineer, Member AIAA, t Senior Engineer, Member AIAA, *17 Research Drive, Hampton, VA 23666. ^Graduate Student, Member AIAA, Department, 304 Town Engineering, Ames, IA 50011 equations of motion does not depend explicitly on time t. This does not represent a loss of generality, as it is always possible to transform explicit time dependence away through the introduction of an additional, timelike state variable. 1. Problem Formulation Let us consider the following class of optimal control problems: min «e(Pwc[*o,*/]), (1) subject to the conditions ±(t) = /(*(*),«(*)), tf.(z(*oMo) = 0, (2)

In this paper, we consider a class of optimal switching control problems with multiple time-delays and a cost on changing control and subject to terminal state constraints. A computational method involving three stages is developed to solve this class of optimal control problems. First, by parameterizing the control function with piecewise-constant functions, the optimal switching control problem is approximated by a sequence of finite-dimensional optimization problems, where the original switching times, the control heights and the control switching times are decision variables. Second, by introducing new variables, the total variation of the control variables is transformed into an equivalently smooth function. Third, we convert the constrained optimization problem into one only with box constraints by an exact penalty function method. The gradients of the cost functional are then derived, which can be combined with any gradient-based optimization method to determine the optimal solution. Finally, a numerical example is given to illustrate the effectiveness of the proposed algorithm.

This paper is mainly concerned with a class of optimal control problems of systems governed by the nonlinear dynamic systems on time scales. Introducing the reasonable weak solution of nonlinear dynamic systems, the existence of the weak solution for the nonlinear dynamic systems on time scales and its properties are presented. Discussing L1-strong-weak lower semicontinuity of integral functional, we give sufficient conditions for the existence of optimal controls. Using integration by parts formula and Hamiltonian function on time scales, the necessary conditions of optimality are derived respectively. Some examples on continuous optimal control problems, discrete optimal control problems, mathematical programming and variational problems are also presented for demonstration.

A method for solving a class of optimal control problems which are linear in the control variable

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

In this paper, we consider a class of optimal control problems in which the system is to be determined in an optimal way. This class of problems involves the choice of a fixed number of switching time points which divide the system’s time horizon into a number of time periods. For each of these time periods, a subsystem is selected, from a finite number of given candidate subsystems, to run during that time periods. The choice of the switching points and the selection of the subsystems are carried out in such a way that a given cost functional is minimized. We consider only problems involving ordinary differential equations over a finite time horizon. A computational method is developed for solving these problems. In our method, the candidate subsystems are combined into a single system by considering their ‘linear combination’. By introducing suitable constraints on the coefficients of the linear combination and using a time rescaling technique, the original problem is transformed into an equivalent optimal control problem with system parameters. An algorithm is proposed for solving this transformed problem and the required gradient formulae are derived. To show the effectiveness of the method, a numerical example is solved.

In this article, we consider a class of optimal control problems governed by state equations of Kobayashi-Warren-Carter-type. The control is given by physical temperature. The focus is on problems in dimensions less than or equal to 4. The results are divided into four Main Theorems, concerned with: solvability and parameter dependence of state equations and optimal control problems; the first-order necessary optimality conditions for these regularized optimal control problems. Subsequently, we derive the limiting systems and optimality conditions and study their well-posedness.

In this paper, we focus on a new class of optimal control problems governed by a simple integral cost functional and isoperimetric-type constraints (constant level sets of some simple integral functionals). By using the notions of a variational differential system and adjoint equation, necessary optimality conditions are established for a feasible solution in the considered optimization problem. More precisely, under simplified hypotheses and using a modified Legendrian duality, we establish a maximum principle for the considered optimization problem.

This paper introduces a numerical technique for solving a class of optimal control problems containing nonlinear dynamical system and functional of state variables. This numerical method consists of two major parts. In the first part, using linear combination property of intervals, we convert the nonlinear dynamical system into an equivalent linear system. And in the second part, which we are dealing with a linear dynamical system, using Legendre expansions for approximating both the state and associated control together with discretizing the constraints over the Chebyshev-Gauss-Lobatto points, the optimal control problem is transformed into a corresponding NLP problem which is diretly solved. The proposed idea is illustrated by several numerical examples.

In the paper, the method of straight lines approximately solves one class of optimal control problems for systems, the behavior of which is described by a nonlinear equation of parabolic type and a set of ordinary differential equations. Control is carried out using distributed and lumped parameters. Distributed control is included in the partial differential equation, and lumped controls are contained both in the boundary conditions and in the right-hand side of the ordinary differential equation. The convergence of the solutions of the approximating boundary value problem to the solution of the original one is proved when the step of the grid of straight lines tends to zero, and on the basis of this fact, the convergence of the approximate solution of the approximating optimal problem with respect to the functional is established. A constructive scheme for constructing an optimal control by a minimizing sequence of controls is proposed. The control of the process in the approximate solution of a class of optimization problems is carried out on the basis of the Pontryagin maximum principle using the method of straight lines. For the numerical solution of the problem, a gradient projection scheme with a special choice of step is used, this gives a converging sequence in the control space. The numerical solution of one variational problem of the mentioned type related to a one-dimensional heat conduction equation with boundary conditions of the second kind is presented. An inequality-type constraint is imposed on the control function entering the right-hand side of the ordinary differential equation. The numerical results obtained on the basis of the compiled computer program are presented in the form of tables and figures. The described numerical method gives a sufficiently accurate solution in a short time and does not show a tendency to «dispersion». With an increase in the number of iterations, the value of the functional monotonically tends to zero

In this work, we discuss normal forms of necessary conditions of optimality (NCO) for optimal control problems subject to pathwise state constraints and for problems in the calculus of variations with inequality constraints. It is known that standard forms of the NCO may fail to provide information that is useful to identify optimal solutions, namely when the multiplier associated with the objective function takes the value zero. The normal forms of the NCO guarantee that the conditions remain always informative, which is of importance in critical applications where decisions based on optimization are taken, such as autonomous systems. Based on a previous nondegenerate maximum principle for optimal control problems, we extend the strengthness of these conditions to normality while applying them to the particular case of calculus of variations problems. We compare our results with existent normal forms of NCO for dynamic optimization problems and conclude that, when applied to calculus of variations problems, we may say that, under similar conditions, we can apply such result to a wider class of problems, having less regularity on the data.

A Numerical Method for a Class of Mixed Switching and Impulsive Optimal Control Problems

In optimal control theory, it is well known that the costate arc and the associated maximized Hamiltonian function can be interpreted in terms of gradients of the value function, evaluated along the optimal state trajectory. Such relations have been referred to as `sensitivity relations? in the literature. In this paper, we announce new sensitivity relations for state constrained optimal control problems. For the class of optimal control problems considered there is no guarantee that the co-state arc is unique; a key feature of the results is that they assert `some? choice of co-state arc can be made, for which the sensitivity relations are valid. The proof technique is to introduce a new optimal control problem that possesses a richer set of control variables than the original problem. The introduction of the additional control variables in effect enlarges the class of variations with respect to which the state trajectory under consideration is a minimizer; the extra information obtained is precisely the desired set of sensitivity relations.

A new approach is proposed for constructing a relaxation sequence of admissible controls in the class of optimal control problems with constraints. The approach is based on the construction of a system of non-local conditions for improving the admissible control in the form of a fixed point problem of the control operator. To build the conditions for improving the admissible control, we apply the transition to an auxiliary optimization problem based on the well-known principle of extension. Sufficient conditions for the optimality of admissible control and the existence of a minimizing sequence of admissible controls in the considered class of problems with constraints are substantiated. A comparative analysis of the computational efficiency of the proposed iterative method of fixed points with the exact implementation of constraints in model and test optimal control problems is carried out.

In this paper, we consider a class of optimal control problems involving continuous inequality state constraints. This class of optimal control problems can be solved using the technique developed in a paper by Goh and Teo, where a simple constraint transcription is used to convert continuous inequality state constraints into equivalent equality terminal state constraints. However, that constraint transcription has the disadvantage that the equality terminal state constraints so obtained do not satisfy the usual constraint qualification. Thus, convergence is not guaranteed and some oscillation may exist in numerical computation. The aim of this paper is to use a new constraint transcription together with the concept of control parametrization to devise a new computational algorithm for solving this general class of constrained optimal control problems. This new algorithm is much more stable numerically, as we have successfully overcome the above-mentioned disadvantages.