Solving Optimal Continuous Thrust Rendezvous Problems with Generating Functions

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.

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.

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 paper, an efficient computational algorithm for the solution of Hamiltonian boundary value problems arising from bang-bang optimal control problems is presented. For this purpose, at first, based on the Pontryagin’s minimum principle, the first order necessary conditions of optimality are derived. Then, an indirect shooting method with control parameterization, in which the control function is replaced with a piecewise constant function with values and switching points taken as unknown parameters, is presented. Thereby, the problem is converted to the solution of the shooting equation, in which the values of the control function and the switching points as well the initial values of the costate variables are unknown parameters. The important advantages of this method are that, the obtained solution satisfies the first order optimality conditions, further the switching points can be captured accurately which is led to an accurate solution of the bangbang problem. However, solving the shooting equation is nearly impossible without a very good initial guess. So, in order to cope with the difficulty of the initial guess, a homotopic approach is combined with the presented method. Consequently, no priori assumptions are made on the optimal control structure and number of the switching points, and sensitivity to the initial guess for the unknown parameters is resolved too. Illustrative examples are included at the end and efficiency of the method is reported.

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.

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].

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 objective of this paper is to provide a real-time, on-board, closed-loop guidance scheme for a minimum fuel (or, equivalently, minimum time) optimal trajectory of a launch vehicle. This is achieved through parallelization of parallel shooting method in time domain, by breaking the time of flight into number of time zones (divide and conquer). The optimal control vector over each flight time zones are obtained by solving the two point boundary value problem (TPBVP) resulting from the first order necessary condition for optimality of constrained optimization problem, in each time zones, by imposing the continuity condition for optimality between the successive time zones. The solution of the TPBVP is computationally expensive, particularly the simulation of the trajectory and the calculation of the Jacobian. A star connected network is used for implementation of parallel shooting algorithm. The Newton's method is used to generate successively approximate decision variables. The performance of thealgorithm with respect to the computation time for star connected network and the solution accuracy is done thorough the numerical simulation of the launch vehicle problem. A study on parallel computation time, speed-up and utilization is done. The algorithm gave a speed-up of 7 with 16 processors. A realistic data for a PSLV class of launch vehicle is used to simulate the closed-loop guided trajectory using the parallel shooting algorithm. A study on the effectiveness of the various nominal and off-nominal conditions on thrust and the sensor measurements are carried out to validate the closed-loop guidance. We have shown that the closed-loop guidance developed, in this paper achieves the desired height and the orbital velocity in presence of disturbances on thrust, and noise in measurements. The computation time details of the closed loop implementation on RS/6000 machine were carried out and presented.

Difference inclusions provide a discrete-time analogue of differential inclusions, which in turn play an important role in the theories of optimal control, implicit differential equations, and invariance and viability, to name a few. In this paper we: (i) introduce a framework suitable for the study of difference inclusions for which the state evolves on a manifold; (ii) use this framework to develop necessary conditions for optimality for a broad class of discrete-time problems of dynamic optimization in which the state evolves on a manifold M. The necessary conditions for optimality we derive include the case for which the state $$q_i$$ is subject to constraints $$q_i \in S_i \subseteq M$$ , for $$S_i$$ a closed set. The resulting necessary conditions for optimality appear as discrete-time versions of the Euler–Lagrange inclusion studied by Ioffe (in Trans Am Math Soc 349(7):2871–2900, 1997), Ioffe and Rockafellar (in Calc Var Partial Differ Equ 4(1):59–87, 1996), Mordukhovich (in SIAM J Control Optim 33(3):882–915, 1995), Mordukhovich (in Variational analysis and generalized differentiation II: applications. Springer, Berlin, 2006), and Vinter and Zheng (in SIAM J Control Optim 35(1):56–77, 1997) generalized in a natural way to the case in which the state is evolving on a manifold.

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.

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.

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.

In this study, optimal control of a senior coefficient in a heat conduction problem is discussed. Since such problems are severely ill-posed, it is necessary to regularize them. So, we present the regularization process which has a great importance for a optimal control problem in this study. After proving the existence and uniqueness of optimal solution, a necessary condition for optimality is stated. Then, an iterative algorithm is purposed for numerical approximations. Finally, optimal control has been approximately calculated using the obtained results. The outcomes have also been tested with numerical examples.

This paper concerns state constrained optimal control problems, in which the dynamic constraint takes the form of a differential inclusion. If the differential inclusion does not depend on time, then the Hamiltonian, evaluated along the optimal state trajectory and the co-state trajectory, is independent of time. If the differential inclusion is Lipschitz continuous, then the Hamitonian, evaluated along the optimal state trajectory and the co-state trajectory, is Lipschitz continuous. These two well-known results are examples of the following principle: the Hamiltonian, evaluated along the optimal state trajectory and the co-state trajectory, inherits the regularity properties of the differential inclusion, regarding its time dependence. We show that this principle also applies to another kind of regularity: if the differential inclusion has bounded variation with respect to time, then the Hamiltonian, evaluated along the optimal state trajectory and the co-state trajectory, has bounded variation. Two applications of these newly found properties are demonstrated. One is to derive improved conditions which guarantee the nondegeneracy of necessary conditions of optimality, in the form of a Hamiltonian inclusion. The other application is to derive new, less restrictive, conditions under which minimizers in the calculus of variations have bounded slope. The analysis is based on a new, local, concept of differential inclusions that have bounded variation with respect to the time variable, in which conditions are imposed on the multifunction involved, only in a neighborhood of a given state trajectory.

Necessary conditions for optimality in problems with nonstandard cost functionals.- Linear control problems and an existence theorem.- Optimal disturbance rejection and performance robustness in linear systems.- Necessary conditions for optimal disturbance rejection in linear systems.- Synthesis of finite-interval H ? controllers by state space methods.- Worst-case performance measures for linear control problems.- Model reduction with a finite-interval H ? criterion.