Linearized impulsive rendezvous problem

The solution of the problem of impulsive minimization of a weighted sum of characteristic velocities of a spacecraft subject to linear equations of motion is presented without the use of calculus of variations or optimal control theory. The geometric structure of the set of boundary points associated with an optimal primer vector is found to be a simplex composed of convex conical sets. Eachk-dimensional open face of the simplex consists of boundary points having nondegeneratek-impulse solutions. This geometric structure leads to a simple proof that at mostn-impulses are required to solve a problem inn-dimensional space. This work is applied to the problem of planar rendezvous of a spacecraft with a satellite in Keplerian orbit using the Tschauner-Hempel equations of motion, with special emphasis on four-impulse solutions. Primer vectors representing four-impulse solutions are sought out and found for elliptical orbits, but none were found for orbits of higher eccentricity. For highly eccentric elliptical orbits, degenerate fiveimpulse solutions were found. In this situation, computer simulations reveal vastly different optimal trajectories having identical boundary conditions and cost.     

Optimal power-limited rendezvous of a spacecraft with bounded thrust and general linear equations of motion

The solution of the fixed-time optimal power-limited rendezvous with a general linear system of ordinary differential equations and a bound on the magnitude of the applied thrust is presented. Necessary and sufficient conditions for thrust saturation in an optimal solution are included.Because of the generality of the linear system of equations of motion, controllability considerations are required for a complete solution of this problem. It is shown that the condition of controllability can be defined completely in terms of a class of primer vectors associated with this problem. Moreover, it is shown that two distinct versions of the primer vector appear in this problem. Therefore, there is not a unique primer vector associated with every rendezvous problem.The work is applied to the problem of the rendezvous of a spacecraft near a satellite in circular orbit. The optimal rendezvous trajectory is determined by the interaction of a primer vector and the bound on the thrust magnitude. The results of computer simulations are presented graphically.     

Optimal Starting Conditions for the Rendezvous Maneuver, Part 1: Optimal Control Approach

We consider the three-dimensional rendezvous between two spacecraft: a target spacecraft on a circular orbit around the Earth and a chaser spacecraft initially on some elliptical orbit yet to be determined. The chaser spacecraft has variable mass, limited thrust, and its trajectory is governed by three controls, one determining the thrust magnitude and two determining the thrust direction. We seek the time history of the controls in such a way that the propellant mass required to execute the rendezvous maneuver is minimized. Two cases are considered: (i) time-to-rendezvous free and (ii) time-to-rendezvous given, respectively equivalent to (i) free angular travel and (ii) fixed angular travel for the target spacecraft. The above problem has been studied by several authors under the assumption that the initial separation coordinates and the initial separation velocities are given, hence known initial conditions for the chaser spacecraft. In this paper, it is assumed that both the initial separation coordinates and initial separation velocities are free except for the requirement that the initial chaser-to-target distance is given so as to prevent the occurrence of trivial solutions. Analyses performed with the multiple-subarc sequential gradient-restoration algorithm for optimal control problems show that the fuel-optimal trajectory is zero-bang, namely it is characterized by two subarcs: a long coasting zero-thrust subarc followed by a short powered max-thrust braking subarc. While the thrust direction of the powered subarc is continuously variable for the optimal trajectory, its replacement with a constant (yet optimized) thrust direction produces a very efficient guidance trajectory: Indeed, for all values of the initial distance, the fuel required by the guidance trajectory is within less than one percent of the fuel required by the optimal trajectory. For the guidance trajectory, because of the replacement of the variable thrust direction of the powered subarc with a constant thrust direction, the optimal control problem degenerates into a mathematical programming problem with a relatively small number of degrees of freedom, more precisely: three for case (i) time-to-rendezvous free and two for case (ii) time-to-rendezvous given. In particular, we consider the rendezvous between the Space Shuttle (chaser) and the International Space Station (target). Once a given initial distance SS-to-ISS is preselected, the present work supplies not only the best initial conditions for the rendezvous trajectory, but simultaneously the corresponding final conditions for the ascent trajectory.     

Minimum-fuel rocket trajectories involving intermediate-thrust arcs

The optimal trajectories in the neighborhood of an optimal intermediate-thrust arc are investigated for the minimumfuel orbit rendezvous problem with fixed specific impulse. Since such an arc is singular, the thrust acceleration magnitude being the singular control component, a second-variation analysis leads to the identification of a field of neighboring, singular arcs in a state space of dimension four rather than six, provided that a suitable Jacobi condition is met. A given neighboring initial six-dimensional state vector does not generally lie on a neighboring singular arc, and junction onto the appropriate singular arc must be accomplished by a short period of strong variations in the acceleration. This contributes an addition to the fuel expenditure which is of order 5/2 rather than 2 in the initial state displacement. The minimization of this higher-order cost, in the case of bounded acceleration, leads to an unsymmetrical version of Fuller's problem, whose solution requires an infinite number of switches between maximum and zero thrust during the short period. For unbounded thrust, the junction simplifies to either coast-impulse-singular trajectories or impulse-coast-impulse-singular trajectories. The neighboring singular arc meets the final condition in 4 dimensions, rather than 6 dimensions, and rendezvous must be completed by another, terminal short period of strong variations in the acceleration. Implications for midcourse guidance near a singular arc are discussed.     

Optimal Trajectories for Spacecraft Rendezvous

The efficient execution of a rendezvous maneuver is an essential component of various types of space missions. This work describes the formulation and numerical investigation of the thrust function required to minimize the time or fuel required for the terminal phase of the rendezvous of two spacecraft. The particular rendezvous studied concerns a target spacecraft in a circular orbit and a chaser spacecraft with an initial separation distance and separation velocity in all three dimensions. First, the time-optimal rendezvous is investigated followed by the fuel-optimal rendezvous for three values of the max-thrust acceleration via the sequential gradient-restoration algorithm. Then, the time-optimal rendezvous for given fuel and the fuel-optimal rendezvous for given time are investigated. There are three controls, one determining the thrust magnitude and two determining the thrust direction in space. The time-optimal case results in a two-subarc solution: a max-thrust accelerating subarc followed by a max-thrust braking subarc. The fuel-optimal case results in a four-subarc solution: an initial coasting subarc, followed by a max-thrust braking subarc, followed by another coasting subarc, followed by another max-thrust braking subarc. The time-optimal case with fuel given and the fuel-optimal case with time given result in two, three, or four-subarc solutions depending on the performance index and the constraints. Regardless of the number of subarcs, the optimal thrust distribution requires the thrust magnitude to be at either the maximum value or zero. The coasting periods are finite in duration and their length increases as the time to rendezvous increases and/or as the max allowable thrust increases. Another finding is that, for the fuel-optimal rendezvous with the time unconstrained, the minimum fuel required is nearly constant and independent of the max available thrust. Yet another finding is that, depending on the performance index, constraints, and initial conditions, sometime the initial application of thrust must be delayed, resulting in an optimal rendezvous trajectory which starts with a coasting subarc. This research has been supported by NSF under Grant CMS-0218878.     

Optimal bounded-thrust space trajectories based on linear equations

Necessary and sufficient conditions for solution of the general minimum-fuel linear bounded-thrust spacecraft trajectory problem are presented in terms of fundamental matrix solutions and their inverses. This work is rigorous, generalizes and unifies many known results for specific problems, and also presents a new necessary condition. Finally, an application is presented for a spacecraft rendezvous near a general Keplerian orbit in which the linearized equations of motion are nonautonomous. A fundamental matrix solution is found and inverted, solving this class of problems.     

Optimal Starting Conditions for the Rendezvous Maneuver,Part 2: Mathematical Programming Approach

In a companion paper (Part 1, J. Optim. Theory Appl. 137(3), [2008]), we determined the optimal starting conditions for the rendezvous maneuver using an optimal control approach. In this paper, we study the same problem with a mathematical programming approach. Specifically, we consider the relative motion between a target spacecraft in a circular orbit and a chaser spacecraft moving in its proximity as described by the Clohessy-Wiltshire equations. We consider the class of multiple-subarc trajectories characterized by constant thrust controls in each subarc. Under these conditions, the Clohessy-Wiltshire equations can be integrated in closed form and in turn this leads to optimization processes of the mathematical programming type. Within the above framework, we study the rendezvous problem under the assumption that the initial separation coordinates and initial separation velocities are free except for the requirement that the initial chaser-to-target distance is given. In particular, we consider the rendezvous between the Space Shuttle (chaser) and the International Space Station (target). Once a given initial distance SS-to-ISS is preselected, the present work supplies not only the best initial conditions for the rendezvous trajectory, but simultaneously the corresponding final conditions for the ascent trajectory.     

SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control

Parametric nonlinear optimal control problems subject to control and state constraints are studied. Two discretization methods are discussed that transcribe optimal control problems into nonlinear programming problems for which SQP-methods provide efficient solution methods. It is shown that SQP-methods can be used also for a check of second-order sufficient conditions and for a postoptimal calculation of adjoint variables. In addition, SQP-methods lead to a robust computation of sensitivity differentials of optimal solutions with respect to perturbation parameters. Numerical sensitivity analysis is the basis for real-time control approximations of perturbed solutions which are obtained by evaluating a first-order Taylor expansion with respect to the parameter. The proposed numerical methods are illustrated by the optimal control of a low-thrust satellite transfer to geosynchronous orbit and a complex control problem from aquanautics. The examples illustrate the robustness, accuracy and efficiency of the proposed numerical algorithms.     

Existence and Multiple Solutions of the Minimum-Fuel Orbit Transfer Problem

In this paper, the well-known problem of piloting a rocket with a low thrust propulsion system in an inverse square law field (say from Earth orbit to Mars orbit or from Earth orbit to Mars) is considered. By direct methods, it is shown that the existence of a fuel-optimal solution of this problem can be guaranteed, if one restricts the admissible transfer times by an arbitrarily prescribed upper bound. Numerical solutions of the problem with different numbers of thrust subarcs are presented which are obtained by multiple shooting techniques. Further, a general principle for the construction of such solutions with increasing numbers of thrust subarcs is given. The numerical results indicate that there might not exist an overall optimal solution of the Earth-orbit problem with unbounded free transfer time.     

Numerical solution of the optimal boundary control of transverse vibrations of a beam

Boundary control is an effective means for suppressing excessive structural vibrations. By introducing a quadratic index of performance in terms of displacement and velocity, as well as the control force, and an adjoint problem, it is possible to determine the optimal control. This optimal control is expressed in terms of the adjoint variable by utilizing a maximum principle. With the optimal control applied, the determination of the corresponding displacement and velocity is reduced to solving a set of partial differential equations involving the state variable, as well as the adjoint variable, subject to boundary, initial, and terminal conditions. The set of equations may not be separable and analytical solutions may only be found in special cases. Furthermore, the computational effort to determine an analytic solution may also be excessive. Herein a numerical algorithm is presented, which easily solves the optimal boundary control problem in the space‐time domain. An example of a continuous system is analyzed. This is the case of the vibrating cantilever beam. Using a finite element recurrence scheme, numerical solutions are obtained, which compare the behavior of the controlled and uncontrolled systems. Also, the analytic solution to the problem is compared with the results obtained using the numerical scheme presented. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 558–568, 1999     

Balance function for the optimal control problem

The balance-function concept for transforming constrained optimization problems into unconstrained optimization problems, for the purpose of finding numerical iterative solutions, is extended to the optimal control problem. This function is a combination orbalance between the penalty and Lagrange functions. It retains the advantages of the penalty function, while eliminating its numerical disadvantages. An algorithm is developed and applied to an orbit transfer problem, showing the feasibility and usefulness of this concept.These results are part of the author's doctoral thesis written under Professors H. Lo and D. Alspaugh of Purdue University.     

A numerical method for nonconvex multi-objective optimal control problems

A numerical method is proposed for constructing an approximation of the Pareto front of nonconvex multi-objective optimal control problems. First, a suitable scalarization technique is employed for the multi-objective optimal control problem. Then by using a grid of scalarization parameter values, i.e., a grid of weights, a sequence of single-objective optimal control problems are solved to obtain points which are spread over the Pareto front. The technique is illustrated on problems involving tumor anti-angiogenesis and a fed-batch bioreactor, which exhibit bang–bang, singular and boundary types of optimal control. We illustrate that the Bolza form, the traditional scalarization in optimal control, fails to represent all the compromise, i.e., Pareto optimal, solutions.     

Homogenization and Asymptotics for Small Transaction Costs: The Multidimensional Case

In the context of the multi-dimensional infinite horizon optimal consumption investment problem with small proportional transaction costs, we prove an asymptotic expansion. Similar to the one-dimensional derivation in our accompanying paper, the first order term is expressed in terms of a singular ergodic control problem. Our arguments are based on the theory of viscosity solutions and the techniques of homogenization which leads to a system of corrector equations. In contrast with the one-dimensional case, no explicit solution of the first corrector equation is available and we also prove the existence of a corrector and its properties. Finally, we provide some numerical results which illustrate the structure of the first order optimal controls.     

Asymptotic properties of solutions of parametric optimal control problems with varying index of the singular arcs

We consider a family of parametric linear-quadratic optimal control problems with terminal and control constraints. This family has the specific feature that the class of optimal controls is changed for an arbitrarily small change in the parameter. In the perturbed problem, the behavior of the corresponding trajectory on noncritical arcs of the optimal control is described by solutions of singularly perturbed boundary value problems. For the solutions of these boundary value problems, we obtain an asymptotic expansion in powers of the small parameter ?. The asymptotic formula starts from a term of the order of 1/? and contains boundary layers. This formula is used to justify the asymptotic expansion of the optimal control for a perturbed problem in the family. We suggest a simple method for constructing approximate solutions of the perturbed optimal control problems without integrating singularly perturbed systems. The results of a numerical experiment are presented.     

Asymptotic estimates for a solution of a singular perturbation optimal control problem on a closed interval under geometric constraints

An optimal control problem is considered for solutions of a boundary value problem for a second-order ordinary differential equation on a closed interval with a small parameter at the second derivative. The control is scalar and satisfies geometric constraints. General theorems on approximation are obtained. Two leading terms of an asymptotic expansion of the solution are constructed and an error estimate is obtained for these approximations.     

A computational approach for understanding immune response to multiple epitopes based on optimal control formulation

The immune system does not response in equal probability to every epitope of an invader. We investigate the immune system’s decision making process using optimal control principles. Mathematically, this formulation requires the solution of a two-point boundary-value problem, which is a challenging task especially when the control variables are bounded. In this work, we develop a computational approach based on the shooting technique for bounded optimal control problems. We then utilize the computational approach to carry out extensive numerical studies on a simple immune response model of two competing controls. Numerical solutions demonstrate that the results of optimal control depend on the objective function, the limitations on control inputs, as well as the amounts of peptides. Moreover, the state space of peptides can be divided into different regions according the properties of the solutions. The developed algorithm not only provides a useful tool for understanding decision making strategies of the immune system but can also be utilized to solve other complex optimal control problems.     

On a Fourier Method of Embedding Domains Using an Optimal Distributed Control

We propose a domain embedding method to solve second order elliptic problems in arbitrary two-dimensional domains. This method can be easily extended to three-dimensional problems. The method is based on formulating the problem as an optimal distributed control problem inside a rectangle in which the arbitrary domain is embedded. A periodic solution of the equation under consideration is constructed easily by making use of Fourier series. Numerical results obtained for Dirichlet problems are presented. The numerical tests show a high accuracy of the proposed algorithm and the computed solutions are in very good agreement with the exact solutions.     

On variational perturbations of control problems: Minimum-time problem and minimum-effort problem

In order to obtain numerical solutions for an abstract optimal control problem, one approximates the abstract operations in a computationally feasible manner. After having found an approximate optimal solution, the question is whether a sequence of these approximate optimal solutions converges to an optimal solution of the original problem. In this work, we are concerned with this type of convergence on the time-optimal control problem for a class of linear systems with distributed parameters and on the minimum-effort problem.     

A New Mixed Iterative Algorithm to Solve the Fuel-Optimal Linear Impulsive Rendezvous Problem

The optimal fuel impulsive time-fixed rendezvous problem is reviewed. In a linear setting, it may be reformulated as a non-convex polynomial optimization problem for a pre-specified fixed number of velocity increments. Relying on variational results previously published in the literature, an improved mixed iterative algorithm is defined to address the issue of optimization over the number of impulses. Revisiting the primer vector theory, it combines variational tests with sophisticated numerical tools from algebraic geometry to solve polynomial necessary and sufficient conditions of optimality. Numerical examples under circular and elliptic assumptions show that this algorithm is efficient and can be integrated into a rendezvous planning tool.     

On the choice of auxiliary linear operator in the optimal homotopy analysis of the Cahn-Hilliard initial value problem

Analytical solutions for the Cahn-Hilliard initial value problem are obtained through an application of the homotopy analysis method. While there exist numerical results in the literature for the Cahn-Hilliard equation, a nonlinear partial differential equation, the present results are completely analytical. In order to obtain accurate approximate analytical solutions, we consider multiple auxiliary linear operators, in order to find the best operator which permits accuracy after relatively few terms are calculated. We also select the convergence control parameter optimally, through the construction of an optimal control problem for the minimization of the accumulated L ²-norm of the residual errors. In this way, we obtain optimal homotopy analysis solutions for this complicated nonlinear initial value problem. A variety of initial conditions are selected, in order to fully demonstrate the range of solutions possible.