Necessary Optimality Conditions in Isoperimetric Constrained Optimal Control Problems

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.

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 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 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 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 general, optimal control problems rely on numerically rather than analytically solving methods, due to their nonlinearities. The direct method, one of the numerically solving methods, is mainly to transform the optimal control problem into a nonlinear optimization problem with finite dimensions, via discretizing the objective functional and the forced dynamical equations directly. However, in the procedure of the direct method, the classical discretizations of the forced equations will reduce or affect the accuracy of the resulting optimization problem as well as the discrete optimal control. In view of this fact, more accurate and efficient numerical algorithms should be employed to approximate the forced dynamical equations. As verified, the discrete variational difference schemes for forced Birkhoffian systems exhibit excellent numerical behaviors in terms of high accuracy, long-time stability and precise energy prediction. Thus, the forced dynamical equations in optimal control problems, after being represented as forced Birkhoffian equations, can be discretized according to the discrete variational difference schemes for forced Birkhoffian systems. Compared with the method of employing traditional difference schemes to discretize the forced dynamical equations, this way yields faithful nonlinear optimization problems and consequently gives accurate and efficient discrete optimal control. Subsequently, in the paper we are to apply the proposed method of numerically solving optimal control problems to the rendezvous and docking problem of spacecrafts. First, we make a reasonable simplification, i.e., the rendezvous and docking process of two spacecrafts is reduced to the problem of optimally transferring the chaser spacecraft with a continuously acting force from one circular orbit around the Earth to another one. During this transfer, the goal is to minimize the control effort. Second, the dynamical equations of the chaser spacecraft are represented as the form of the forced Birkhoffian equation. Then in this case, the discrete variational difference scheme for forced Birkhoffian system can be employed to discretize the chaser spacecraft's equations of motion. With further discretizing the control effort and the boundary conditions, the resulting nonlinear optimization problem is obtained. Finally, the optimization problem is solved directly by the nonlinear programming method and then the discrete optimal control is achieved. The obtained optimal control is efficient enough to realize the rendezvous and docking process, even though it is only an approximation of the continuous one. Simulation results fully verify the efficiency of the proposed method for numerically solving optimal control problems, if the fact that the time step is chosen to be very large to limit the dimension of the optimization problem is noted.

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.

The optimization problem consists in finding an element that minimizes or maximizes the goal (or cost) functional on the set of admissible controls. The paper is devoted to the optimization problem for a pseudoparabolic equation with control in coefficients of the variational equality as well as on the right-hand side. We consider optimal control problem (parabolic initial boundary value problem) of a viscoelastic plate. The bending of a plate is described by means of the Kirchhoff model. We assume that a nonhomogeneous and anisotropic viscoelastic plate occupying a domain Ω × (—H, H) of the space ℝ3 is loaded be a transversal distributed force perpendicular to the middle plane of the plate. The viscoelastic reinforced plate with welded bars rests on a unilateral elastic foundation. The role of the control variables is played by the thickness of the viscoelastic plate and the stiffness characteristics of the elastic Winkler medium, respectively. We consider the desired moments field (the bending moments and torque) in time T as the cost functional. The control variables have to belong to a set of Lipschitz-continuous functions and to a set of continuously differentiable functions, respectively. The state problem is modelled by a pseudoparabolic variational equality, where the control variables influence both the coefficients of the linear strictly monotone operator and the Winkler functional. We derive there some apriori estimates of the state function which are useful for the problem of the existence of the optimal thickness function and the Winkler modulus, respectively, and for unicity of the optimal controls. On the basis of a general existence theorem for a class of pseudoparabolic optimization problems with the state variational equation, we prove the existence of at least one solution of the optimal control problem.

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.

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 this paper, we consider the numerical solution of a class of optimal control problems involving variable time points in their cost functions. The control enhancing transform is first used to convert the optimal control problem with variable time points into an equivalent optimal control problem with fixed multiple characteristic time (MCT). Using the control parametrization technique, the time horizon is partitioned into several subintervals. Let the partition points also be taken as decision variables. The control functions are approximated by piecewise constant or piecewise linear functions in accordance with these variable partition points. We thus obtain a finite dimensional optimization problem. The control parametrization enhancing control transform (CPET) is again used to convert approximate optimal control problems with variable partition points into equivalent standard optimal control problems with MCT, where the control functions are piecewise constant or piecewise linear functions with pre-fixed partition points. The transformed problems are essentially optimal parameter selection problems with MCT. The gradient formulae for the objective function as well as the constraint functions with respect to relevant decision variables are obtained. Numerical examples are solved using the proposed method.

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.

In this article, it is considered a class of optimal control problems constrained by differential and integral constraints are called canonical form. A modified measure theoretical approach is introduced to solve this class of optimal control problems.

This paper is devoted to applications of modern methods of variational analysis to constrained optimization and control problems generally formulated in infinite-dimensional spaces. The main focus is on the study of problems with nonsmooth structures, which require the usage of advanced tools of generalized differentiation. In this way we derive new necessary optimality conditions in optimization problems with functional and operator constraints and then apply them to optimal control problems governed by discrete-time inclusions in infinite dimensions. The principal difference between finite-dimensional and infinite-dimensional frameworks of optimization and control consists of the “lack of compactness” in infinite dimensions, which leads to imposing certain “normal compactness” properties and developing their comprehensive calculus, together with appropriate calculus rules of generalized differentiation. On the other hand, one of the most important achievements of the paper consists of relaxing the latter assumptions for certain classes of optimization and control problems. In particular, we fully avoid the requirements of this type imposed on target endpoint sets in infinite-dimensional optimal control for discrete-time inclusions.

An iteration method to solve a class of optimal control problems with integral functional and phase constraints is developed in the paper.