Continuous and discrete embedded optimal control problems and their application to the analysis of Clebsch optimal control problems and mechanical systems

Nonlinear discrete time and continuous time optimal control problems with terminal constraints are solved using a new evolutionary approach which seeks the control history directly by evolutionary computation. Unlike methods that use the first order necessary conditions to determine the optimum, the main advantage of the present method is that it does not require the development of a Hamiltonian formulation and consequently, it eliminates the requirement to solve the adjoint problem which usually leads to a difficult two-point boundary value problem. The method is verified on two benchmark problems. The first problem is the discrete time velocity direction programming problem with the effects of gravity, thrust and drag and a terminal constraint on the final vertical position. The second problem is a continuous time optimal control problem in rocket dynamics, the Goddard’s problem. The solutions of both problems compared favorably with published results based on gradient methods.

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.

This paper presents the results of investigations on the application to optimal control problems of a new optimization technique based on Liapunov concepts. In this method, the actual search is carried out on a positive-definite Liapunov Type of Function (LTF) of the squared values of the gradients and other components of an augmented objective function. The distinguishing feature of this method is that the solution of all types of optimal control problems can be obtained by a single unconstrained optimization of the LTF. It is shown that the simple Gauss-Newton Least Square method is very well suited for this particular structure as the CPU time is lower than for the variable metric methods.

Existence, uniqueness, and stability of solutions to singular linear quadratic optimal control problems

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.

We investigate a discrete optimal control problem described by an ordinary difference equation with nonlocal boundary conditions and a Moiseyev-type functional (with an untyped quality criterion). Under various assumptions, the necessary first-order optimality conditions are proved.

In this research, we investigate an optimal control problem governed by elliptic PDEs with Dirichlet boundary conditions on complex connected domains, which can be utilized to model the cooling process of concrete dam pouring. A new convergence result for two-dimensional Dirichlet boundary control is proven with the Fourier finite volume element method. The Lagrange multiplier approach is employed to find the optimality systems of the Dirichlet boundary optimal control problem. The discrete optimal control problem is then obtained by applying the Fourier finite volume element method based on Galerkin variational formulation for optimality systems, that is, using Fourier expansion in the azimuthal direction and the finite volume element method in the radial direction, respectively. In this way, the original two-dimensional problem is reduced to a sequence of one-dimensional problems, with the Dirichlet boundary acting as an interval endpoint at which a quadratic interpolation scheme can be implemented. The convergence order of state, adjoint state, and Dirichlet boundary control are therefore proved. The effectiveness of the method is demonstrated numerically, and numerical data is provided to support the theoretical analysis.

The author summarizes the theoretical results in "A discrete optimal control problem for descriptor systems," [IEEE Trans. Automat. Control, vol.34, pp.177-181, 1989] and "Authors' reply to 'Comments on 'A discrete optimal control problem for descriptor systems"," [ibid., to appear], and uses them in the computer-aided optimization problem for descriptor systems. An efficient algorithm for computing the optimal solution sequence of the optimization problem is developed and an example of the macroeconomic systems is shown.

In this paper we consider autonomous discrete time optimal control problems. We discuss the reduction to finite cost and the representation formula, the existence of optimal solutions on infinite horizon and their structure, and the structure of optimal solutions on finite intervals.

Mathematical formulation and numerical solutions of an optimal Dirichlet boundary control problem for the Boussinesq equations are considered. The solution of the optimal control problem is obtained by adjusting of the temperature on the boundary. We analyze finite element approximations. A gradient method for the solution of the discrete optimal control problem is presented and analyzed. Finally, the results of some computational experiments are presented.

A discrete optimal control problem can always be treated as a conventional mathematical program but the dimension of the latter depens on N, the number of stages, and can, therefore, be forbiddingly high. In this paper we exploit the structure of the optimal control problem and present a version of a conventional sequential quadratic programming algorithm which merely requires, at each iteration, the solution of N quadratic sub-problems of dimension m (equal to that of the control) rather than the solution of one quadratic sub-problem of dimension Nm as required by the mathematical programming formulation.

Optimal control of systems with noisy memory and BSDEs with Malliavin derivatives

Discrete control problems with phase constraints

The modeling of the double curvature shell shaping technology using CAE-systems is considered. To ensure high accuracy and quality of parts, it is necessary to perform a complete analysis of full-size theoretical models. When setting up an automated shaping of parts with a stretch forming, it is necessary to develop a control program and an electronic model of a punch. On condition the specified punch, the problem of optimal control is posed to find the best trajectory of clamps movement in the equipment. The value of material damage in plasticity is considered as an optimization criterion. A discrete optimal control problem is formulated, which is solved by the dynamic programming method. The space of solutions to optimal control problems must include non-monotonic deformation trajectories that take into account partial unloading. The article presents the implementation of a numerical method for optimizing the deformation trajectory of a panel on a stretch-forming punch in the finite element system MSC.Marc.

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)