An efficient numerical solution for time switching optimal control problems

An earlier developed theory of first and second order necessary conditions for optimality in Banach spaces which gives nondegenerate conditions also in the case of nonregular operators is applied to the optimal control problem. An extended version of the maximum principle as well as a second order optimality condition are presented which have a nontrivial form in the case of abnormal problems. >

The optimal control of a spacecraft as it transitions between specified states using continuous thrust in a fixed amount of time is studied using a recently developed technique based on Hamilton-Jacobi theory. Starting from the 1st order necessary conditions for optimality, we derive a Hamiltonian system for the state and adjoints with split boundary conditions. Then, recognizing the two point boundary value problem as a canonical transformation, we employ generating functions to find the optimal feedback control as well as the optimal trajectory. Though we formulate the optimal control problem in the context of the necessary conditions for optimality, our closed-loop solution also formally satisfies the sufficient conditions for optimality via the fundamental connection between the optimal cost function and generating functions. A solution procedure for these generating functions is posed and numerically tested on a non-linear optimal rendezvous problem in the vicinity of a circular orbit. Generating functions are developed as series expansions, and the optimal trajectories obtained from them are compared favorably with those of a numerical solution to the two point boundary value problem using a forward shooting method.

In this paper, we study a robust control problem for a class of systems gouverned by nonlinear parabolic equations with multiple time‐varying delays, in order to take account the influence of noise in data. Firstly, the control is the forcing. We formulate the robust control problem, we prove the existence and the uniqueness of the solution and we give necessary conditions of optimality. Secondly, an initial perturbation problem is considered. A robust control problem is reformulated, and the existence and the conditions of the uniqueness of the optimal solution are derived. First order necessary conditions of optimality are obtained.

We study a model for the exploitation of renewable stocks developed in [5]. In this particular control problem, the control law contains a measurable and an impulsive control component. We formulate Pontryagin's maximum principle for this kind of control problems (see [3]), proving first order necessary conditions of optimality. Manipulating the correspondent Lagrange multipliers we manage to define two special switch functions, that allow the complete description the optimal trajectories and control policies, for all possible initial conditions in the phase plane.

We consider a class of Dynkin games in the case where the underlying process evolves according to a one-dimensional but otherwise general diffusion. We establish general conditions under which both the value and the saddle point equilibrium exist and under which the exercise boundaries characterizing the saddle point strategy can be explicitly characterized in terms of a pair of standard first order necessary conditions for optimality. We also analyze those cases where an extremal pair of boundaries exists and investigate the overall impact of increased volatility on the equilibrium stopping strategies and their values.

A recent result giving the structure of the Volterra kernels in terms of Lie-brackets of vector fields allows a new formulation of some higher order necessary conditions of optimality. These conditions are equivalent to the generalized Legendre-Clebsch and Jacobson conditions.

In this study is presented an algorithm for optimal reactive dispatch problem. The algorithm is based on Newton's method and Primal-dual logarithmic barrier method. A Lagrangian function is associated with the modified problem. The first order necessary conditions for optimality are reached by Newton's method, and by updating the penalty and barrier terms. The proposed approach does not have to identify the biding constraints set and can be utilized to infeasible starting point. The effectiveness of the proposed approach has been examined by solving the Brazilian 53-bus and IEEE 162-bus systems.

First order necessary conditions for optimality are the Pontryagin Maximal Principle (PMP) or the Euler-Lagrange and Legendre-Clebsch conditions. In many problems of practical importance, particularly those when the control enters linearly, these conditions are inconclusive for determining the optimal control. In 1964 Kelley [1] discovered a generalization of the Legendre-Clebsch condition and this was extended by Robbins [2,3], Tait [4], Goh [5,6], Kopp and Moyer [7], and Kelley, Kopp and Moyer [8]. An excellent survey of these papers and related results is Gabasov and Kirillova [9].

This chapter concerns some of the most important optimal control problems related to age-dependent population dynamics. The main goal is to prove the existence of an optimal control and to obtain first order necessary conditions of optimality. These conditions allow calculation or approximation of the optimal control and the optimal value of the cost functional. We shall also present a fractional step scheme for a certain nonlinear problem. This scheme gives a very good method of approximating the optimal control.

In this paper, a finite branch-and-bound algorithm is developed for the minimization of a concave power law over a polytope. Linear terms are also included in the objective function. Using the first order necessary conditions of optimality, the optimization problem is transformed into an equivalent problem consisting of a linear objective function, a set of linear constraints, a set of convex constraints, and a set of bilinear complementary constraints. The transformed problem is then solved using a finite branch-and-bound algorithm that solves two convex problems at each of its nodes. The method is illustrated by means of an example from the literature.

The primary focus of this research is on a semi-analytical method for solving a generalized quadratic optimal control problem constrained by Ordinary Differential Equation (ODE). The solutions are obtained by applying the first order optimality conditions on the Hamiltonian function, which is a necessary condition for optimality. The Adomian Decomposition Method (ADM), which views the approximation of a non-linear equation as an infinite series that typically converges to the exact solution, is used to solve the associated general Riccati differential equation. This leads to the optimal state, control and adjoint variables which yields the optimal objective functional value. Solutions to two examples of optimal control problems constrained by ODE are given.

This chapter is devoted upon optimality, a topic in which the central result is the Pontryagin’s minimum principle. This important result is briefly approached from the classical calculus of variation. We show that the classical calculus of variation has some major limitations to modern control problems and motivate Pontryagin’s minimum principle. The Hamilton–Jacobi–Bellman method is discussed as an alternative approach to gain first-order necessary conditions for optimality. It is shown that both approaches correspond to each other under restrictive assumptions. The original Pontryagin’s minimum principle for continuous optimal control problems is not suitable for hybrid optimal control problems. However, a quite natural reformulation of the hybrid optimal control problem admits the classical theory for deduction of first-order necessary conditions in the sense of Pontryagin. The charm of this methodology is its comprehensible derivation.

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

How to verify optimal controls computed by direct shooting methods? – A tutorial

The present paper is dedicated to the investigation of optimal control problems whose behavior is described by quasilinear differential equations of first order on the plane with nonlocal Bitsadze–Samarski boundary conditions. A theorem of the existence and uniqueness of a generalized solution in the space \(C_{\mu }(\overline{G})\) is proved for quasilinear differential equations; necessary conditions of optimality are obtained in terms of the principle of maximum; the Bitsadze–Samarski boundary value problem is proved for a linear differential equation of first order; the existence of a solution in the space \(C_{\mu }^{p}(\overline{G})\) is proved and an a priori estimate is derived. A theorem on the necessary and sufficient condition of optimality is proved for a linear optimal control problem.