Numerical Computation of Optimal Feed Rates for a Fed-Batch Fermentation Model

In this paper, we consider a model for a fed-batch fermentation process which describes the biosynthesis of penicillin. First, we solve the problem numerically by using a direct shooting method. By discretization of the control variable, we transform the basic optimal control problem to a finite-dimensional nonlinear programming problem, which is solved numerically by a standard SQP method. Contrary to earlier investigations (Luus, 1993), we consider the problem as a free final time problem, thus obtaining an improved value of the penicillin output. The results indicate that the assumption of a continuous control which underlies the discretization scheme seems not to be valid. In a second step, we apply classical optimal control theory to the fed-batch fermentation problem. We derive a boundary-value problem (BVP) with switching conditions, which can be solved numerically by multiple shooting techniques. It turns out that this BVP is sensitive, which is due to the rigid behavior of the specific growth rate functions. By relaxation of the characteristic parameters, we obtain a simpler BVP, which can be solved by using the predicted control structure (Lim et al., 1986). Now, by path continuation methods, the parameters are changed up to the original values. Thus, we obtain a solution which satisfies all first-order and second-order necessary conditions of optimal control theory. The solution is similar to the one obtained by direct methods, but in addition it contains certain very small bang-bang subarcs of the control. Earlier results on the maximal output of penicillin are improved.     

Optimal Interplanetary Orbital Transfers via Electrical Engines

The Hohmann transfer theory, developed in the 19th century, is the kernel of orbital transfer with minimum propellant mass by means of chemical engines. The success of the Deep Space 1 spacecraft has paved the way toward using advanced electrical engines in space. While chemical engines are characterized by high thrust and low specific impulse, electrical engines are characterized by low thrust and hight specific impulse. In this paper, we focus on four issues of optimal interplanetary transfer for a spacecraft powered by an electrical engine controlled via the thrust direction and thrust setting: (a) trajectories of compromise between transfer time and propellant mass, (b) trajectories of minimum time, (c) trajectories of minimum propellant mass, and (d) relations with the Hohmann transfer trajectory. The resulting fundamental properties are as follows:

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.     

Optimal trajectories for aeroassisted,coplanar orbital transfer

This paper considers both classical and minimax problems of optimal control which arise in the study of aeroassisted, coplanar orbital transfer. The maneuver considered involves the coplanar transfer from a high planetary orbit to a low planetary orbit. An example is the HEO-to-LEO transfer of a spacecraft, where HEO denotes high Earth orbit and LEO denotes low Earth orbit. In particular, HEO can be GEO, a geosynchronous Earth orbit.The basic idea is to employ the hybrid combination of propulsive maneuvers in space and aerodynamic maneuvers in the sensible atmosphere. Hence, this type of flight is also called synergetic space flight. With reference to the atmospheric part of the maneuver, trajectory control is achieved by means of lift modulation. The presence of upper and lower bounds on the lift coefficient is considered.Within the framework of classical optimal control, the following problems are studied: (P1) minimize the energy required for orbital transfer; (P2) minimize the time integral of the heating rate; (P3) minimize the time of flight during the atmospheric portion of the trajectory; (P4) maximize the time of flight during the atmospheric portion of the trajectory; (P5) minimize the time integral of the square of the path inclination; and (P6) minimize the sum of the squares of the entry and exit path inclinations. Problems (P1) through (P6) are Bolza problems of optimal control.Within the framework of minimax optimal control, the following problems are studied: (Q1) minimize the peak heating rate; (Q2) minimize the peak dynamic pressure; and (Q3) minimize the peak altitude drop. Problems (Q1) through (Q3) are Chebyshev problems of optimal control, which can be converted into Bolza problems by suitable transformations.Numerical solutions for Problems (P1)–(P6) and Problems (Q1)–(Q3) are obtained by means of the sequential gradient-restoration algorithm for optimal control problems. The engineering implications of these solutions are discussed. In particular, the merits of nearly-grazing trajectories are considered.This research was supported by the Jet Propulsion Laboratory, Contract No. 956415. The authors are indebted to Dr. K. D. Mease, Jet Propulsion Laboratory, for helpful discussions. This paper is a condensation of the investigation reported in Ref. 1.     

Optimal Ascent Trajectories and Feasibility of Next-Generation Orbital Spacecraft

This paper deals with the optimization of the ascent trajectories for single-stage-to-orbit (SSTO) and two-stage-to-orbit (TSTO) rocket-powered spacecraft. The maximum payload weight problem is studied for various combinations of initial thrust-to-weight ratio, engine specific impulse, and spacecraft structural factor. For TSTO rocket-powered spacecraft, two cases are studied: uniform structural factor and nonuniform structural factor between stages.The main conclusions are that: the design of SSTO configurations might be comfortably feasible, marginally feasible, or unfeasible, depending on the parameter values assumed; the design of TSTO configurations is not only feasible, but the payload appears to be considerably larger than that of SSTO configurations; for the case of a nonuniform structural factor, the most attactive TSTO design appears to be a first-stage structure made of only tanks and a second-stage structure made of engines, tanks, electronics, and so on.Improvements in engine specific impulse and spacecraft structural factor are desirable and crucial for SSTO feasibility; indeed, aerodynamic improvements do not yield significant improvements in payload weight.For SSTO configurations, the maximum payload weight behaves almost linearly with respect to the engine specific impulse and the spacecraft structural factor. The same property holds for TSTO configurations as long as the ratio of the structural factors of Stage 2 and Stage 1 is held constant. With reference to the specific impulse/structural factor domain, this property leads to the construction of a zero-payload line separating the feasibility region (positive payload) from the unfeasibility region (negative payload).     

On the numerical computation of minimum-fuel,Earth-Mars transfer

The minimum-fuel space travel problem for the planar Earth-Mars transfer by a low-thrust ion rocket is considered. The investigation shows that the use of the classical switching function and the natural boundary condition of the Hamiltonian leads to numerical difficulties, caused by the nondifferentiability of the Hamiltonian. It is shown, however, that these difficulties can be overcome by substitution of the switching function and the Hamiltonian condition. The numerical solution by multiple shooting techniques is presented.     

Piecewise-Linear Pathways to the Optimal Solution Set in Linear Programming

An optimal control problem with four linear controls describing a sophisticated concern model is investigated. The numerical solution of this problem by combination of a direct collocation and an indirect multiple shooting method is presented and discussed. The approximation provided by the direct method is used to estimate the switching structure caused by the four controls occurring linearly. The optimal controls have bang-bang subarcs as well as constrained and singular subarcs. The derivation of necessary conditions from optimal control theory is aimed at the subsequent application of an indirect multiple shooting method but is also interesting from a mathematical point of view. Due to the linear occurrence of the controls, the minimum principle leads to a linear programming problem. Therefore, the Karush–Kuhn–Tucker conditions can be used for an optimality check of the solution obtained by the indirect method.     

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.     

A Multiple-Shooting Technique for Optimal Control

A new method for solving optimal control problems, here called multiple NOC shooting, is presented. It is developed from NOC shooting. It has some advantages over its parent and over multiple shooting, which are both successful, high-accuracy methods for optimal control. A comparison of the three methods is given, incorporating two examples.     

Highly Accurate Numerical Schemes for Stochastic Optimal Control via FBSDEs

This work is concerned with numerical schemes for stochastic optimal control problems (SOCPs) by means of forward backward stochastic differential equations (FBSDEs). We first convert the stochastic optimal control problem into an equivalent stochastic optimality system of FBSDEs. Then we design an efficient second order FBSDE solver and an quasi-Newton type optimization solver for the resulting system. It is noticed that our approach admits the second order rate of convergence even when the state equation is approximated by the Euler scheme. Several numerical examples are presented to illustrate the effectiveness and the accuracy of the proposed numerical schemes.     

Decomposition technique and optimal trajectories for the aeroassisted flight experiment

The aeroassisted flight experiment (AFE) refers to a spacecraft to be launched and then recovered by the space shuttle in 1994. It simulates a transfer from a geosynchronous Earth orbit (GEO) to a low Earth orbit (LEO). Specifically, the AFE spacecraft is released from the space shuttle and is accelerated by means of a solid rocket motor toward Earth, so as to achieve atmospheric entry conditions close to those of a spacecraft returning from GEO. Following the atmospheric pass, the AFE spacecraft ascends to the specified LEO via an intermediate parking Earth orbit (PEO). The final maneuver includes the rendezvous with and the capture by the space shuttle. The entry and exit orbital planes of the AFE spacecraft are identical with the orbital plane of the space shuttle. In this paper, with reference to the AFE spacecraft, an actual GEO-to-LEO transfer is considered and optimal trajectories are determined by minimizing the total characteristic velocity. The optimization is performed with respect to the time history of the controls (angle of attack and angle of bank), the entry path inclination and the flight time being free. Two transfer maneuvers are considered: (DA) direct ascent to LEO; (IA) indirect ascent to LEO via PEO. While the motion of the AFE spacecraft in a 3D-space is described by a system of six ODEs, substantial simplifications are possible if one exploits these facts: (i) the instantaneous orbital plane is nearly identical with the initial orbital plane; (ii) the bank angle is small; and (iii) the Earth's angular velocity is relatively small. Under these assumptions, the complete system can be decoupled into two subsystems, one describing the longitudinal motion and one describing the lateral motion. The angle of attack history, the entry path inclination, and the flight time are determined via the longitudinal motion subsystem; in this subsystem, the total characteristic velocity is minimized subject to the specified LEO requirement. The angle of bank history is determined via the lateral motion subsystem; in this subsystem, the difference between the instantaneous bank angle and a constant bank angle is minimized in the least square sense subject to the specified orbital inclination requirement. It is shown that both the angle of attack and the angle of bank are constant. This result has considerable importance in the design of nominal trajectories to be used in the guidance of AFE and AOT vehicles.     

Numerical Schemes for Discontinuous Value Functions of Optimal Control

In this paper we explain that various (possibly discontinuous) value functions for optimal control problem under state-constraints can be approached by a sequence of value functions for suitable discretized systems. The key-point of this approach is the characterization of epigraphs of the value functions as suitable viability kernels. We provide new results for estimation of the convergence rate of numerical schemes and discuss conditions for the convergence of discrete optimal controls to the optimal control for the initial problem.     

Numerical solution of minimax optimal control problems by multiple shooting technique

This paper presents the application of the multiple shooting technique to minimax optimal control problems (optimal control problems with Chebyshev performance index). A standard transformation is used to convert the minimax problem into an equivalent optimal control problem with state variable inequality constraints. Using this technique, the highly developed theory on the necessary conditions for state-restricted optimal control problems can be applied advantageously. It is shown that, in general, these necessary conditions lead to a boundary-value problem with switching conditions, which can be treated numerically by a special version of the multiple shooting algorithm. The method is tested on the problem of the optimal heating and cooling of a house. This application shows some typical difficulties arising with minimax optimal control problems, i.e., the estimation of the switching structure which is dependent on the parameters of the problem. This difficulty can be overcome by a careful application of a continuity method. Numerical solutions for the example are presented which demonstrate the efficiency of the method proposed.     

具有年龄结构的种群线性动力系统的最优收获控制问题

§ 1　Introduction and setting of the problemThe optimal control of age-dependent population dynamics has been intensivelystudied in the last two decades and there is now a vast stock of literature on the topic ofoptimal control problems ofage-structured population dynamics.(see [1 -9] ) .To the bestof our knowledge,the works of Brokate[3,4] are the firstto deal with this topic.Since then,many authors devote to the optimal harvesting problem.In this aspect,we refere to thefundamental papers o…     

Time-optimal extension and retraction of robots: Numerical analysis of the switching structure

The problem of the time-optimal control of robot manipulators is of importance because of its potential for increasing the productivity of assembly lines. This work is part of a series of papers by the authors on this topic using direct and indirect methods of optimization. A cylindrical robot or a spherical polar robot constrained to the horizontal plane is considered, and optimal solutions for radial maneuvers are generated. Indirect methods are employed in order to establish the switching structure of the solutions. The results show that even such apparently simple maneuvers as extension or retraction of a robot with a prismatic joint can produce very complex optimal solutions. Time-optimal retraction can exhibit ten different switching structures with eight switching points and two singular arcs.This research was supported by the Deutsche Forschungsgemeinschaft. The results reported in this paper first appeared in the thesis of the first author and were presented at the 1989 AIAA Guidance, Navigation and Control Conference (Refs. 1 and 2).     

不确定情况下两部门经济模型最优投资率的确定

本依据索罗的经济增长模型，推导出两部门经济模型的最优投资率，并在此基础上讨论了生产函数为柯布—道格拉斯函数时，最优投资率满足的条件。     

Existence of Neighboring Feasible Trajectories: Applications to Dynamic Programming for State-Constrained Optimal Control Problems

In this paper, the value function for an optimal control problem with endpoint and state constraints is characterized as the unique lower semicontinuous generalized solution of the Hamilton-Jacobi equation. This is achieved under a constraint qualification (CQ) concerning the interaction of the state and dynamic constraints. The novelty of the results reported here is partly the nature of (CQ) and partly the proof techniques employed, which are based on new estimates of the distance of the set of state trajectories satisfying a state constraint from a given trajectory which violates the constraint.     

Robust Optimal Onboard Reentry Guidance of a Space Shuttle: Dynamic Game Approach and Guidance Synthesis via Neural Networks

Robust optimal control problems for dynamic systems must be solved ifmodeling inaccuracies cannot be avoided and/or unpredictable andunmeasurable influences are present. Here, the return of a future Europeanspace shuttle to Earth is considered. Four path constraints have to beobeyed to limit heating, dynamic pressure, load factor, and flight pathangle at high velocities. For the air density associated with theaerodynamic forces and the constraints, only an altitude-dependent rangecan be predicted. The worst-case air density is analyzed via an antagonisticnoncooperative two-person dynamic game. A closed-form solution of the gameprovides a robust optimal guidance scheme against all possible air densityfluctuations. The value function solves the Isaacs nonlinear first-orderpartial differential equation with suitable interior and boundaryconditions. The equation is solved with the method of characteristics in therelevant parts of the state space. A bundle of neighboring characteristictrajectories yields a large input/output data set and enables a guidancescheme synthesis with three-layer perceptrons. The difficult andcomputationally expensive perceptron training is done efficiently with thenew SQP-training method FAUN. Simulations show the real-time capability androbustness of the reentry guidance scheme finally chosen.     

具有年龄结构的捕食种群系统的最优收获策略

分析了一类基于年龄结构的食饵-捕食者系统的最优收获问题．证明了系统非负解的存在唯一性、解对控制变量的连续依赖性．讨论了最优策略的存在性,利用法锥和Dubovitskii-Milyutin理论导出了最优性条件．     

New general guidance method in constrained optimal control,part 1: Numerical method

A very fast numerical method is developed for the computation of neighboring optimum feedback controls. This method is applicable to a general class of optimal control problems (for example, problems including inequality constraints and discontinuities) and needs no on-line computation, except for one matrix-vector multiplication. The method is based on the so-called accessory minimum problem. The necessary conditions for this auxiliary optimal control problem form a linear multipoint boundary-value problem with linear jump conditions, which is especially well suited for numerical treatment. In the second part of this paper, the performance of the guidance scheme is shown for the heating-constrained cross-range maximization problem of a space-shuttle-orbiter-type vehicle.This research was supported in part by the Deutsche Forschungsgemeinschaft under the Schwerpunktprogramm Anwendungsbezogene Optimierung und Steuerung.The authors wish to express their sincere and grateful appreciation to Professor Roland Bulirsch who encouraged this work.