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This paper presents a Lie-group shooting method for the numerical solutions of multi-dimensional nonlinear boundary-value problems, which may exhibit multiple solutions. The Lie-group shooting method is a powerful technique to search unknown initial conditions through a single parameter, which is determined by matching the multiple targets through a minimum of an appropriately defined measure of the mis-matching error to target equations. Several numerical examples are examined to show that the novel approach is highly efficient and accurate. The number of solutions can be identified in advance, and all possible solutions can be numerically integrated by using the fourth-order Runge–Kutta method. We also apply the Lie-group shooting method to a numerical solution of an optimal control problem of the Duffing oscillator.

We investigate variants of Goddard’s problems for nonvertical trajectories. The control is the thrust force, and the objective is to maximize a certain final cost, typically, the final mass. In this article, performing an analysis based on the Pontryagin maximum principle, we prove that optimal trajectories may involve singular arcs (along which the norm of the thrust is neither zero nor maximal), that are computed and characterized. Numerical simulations are carried out, both with direct and indirect methods, demonstrating the relevance of taking into account singular arcs in the control strategy. The indirect method that we use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method. The homotopic approach leads to a quadratic regularization of the problem and is a way to tackle the problem of nonsmoothness of the optimal control.

In this paper, we consider a one-dimensional dam-river system, described by a diffusive-wave equation and often used in hydraulic engineering to model the dynamic behavior of the unsteady flow in a river for shallow water when the flow variations are not important. We propose an integral boundary control which leads to a nondissipative closed-loop system with noncollocated actuators and sensors; hence, two main difficulties arise: first, how to show the C0-semigroup generation and sec- ond, how to achieve the stability of the system. To overcome this situation, the Riesz basis methodology is adopted to show that the closed-loop system generates an ana- lytic semigroup. Concerning the stability, the shooting method is applied to assign the spectrum of the system in the open left-half plane and ensure its exponential stability as well as the output regulation. Numerical simulations are presented for a family of system parameters.

A method for the numerical solution of state-constrained optimal control problems subject to higher-index differential-algebraic equation (DAE) systems is introduced. For a broad and important class of DAE systems (semiexplicit systems with algebraic variables of different index), a direct multiple shooting method is developed. The multiple shooting method is based on the discretization of the optimal control problem and its transformation into a finite-dimensional nonlinear programming problem (NLP). Special attention is turned to the mandatory calculation of consistent initial values at the multiple shooting nodes within the iterative solution process of (NLP). Two different methods are proposed. The projection method guarantees consistency within each iteration, whereas the relaxation method achieves consistency only at an optimal solution. An illustrative example completes this article.

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.

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.

To solve the multipoint boundary-value problem (MPBVP) associated with a constrained optimal control problem, one needs a good guess not only for the state but also for the costate variables. A direct multiple shooting method is described, which yields approximations of the optimal state and control histories. The Kuhn–Tucker conditions for the optimal parametric control are rewritten using adjoint variables. From this representation, estimates for the adjoint variables at the multiple shooting nodes are derived. The estimates are proved to be consistent, in the sense that they converge toward the MPBVP solution if the parametrization is refined. An optimal aircraft maneuver demonstrates the transition from the direct to the indirect method.

Various methods have been proposed for the numerical solution of optimal control problems with bounded state variables. In this paper, a new method is put forward and compared with two other methods, one of which makes use of adjoint variables whereas the other does not. Some conclusions are drawn on the usefulness of the three methods involved.

Complex pursuit-evasion games with state variable inequality constraints are investigated. Necessary conditions of the first and the second order for optimal trajectories are developed, which enable the calculation of optimal open-loop strategies. The necessary conditions on singular surfaces induced by state constraints and non-smooth data are discussed in detail. These conditions lead to multi-point boundary-value problems which can be solved very efficiently and very accurately by the multiple shooting method. A realistically modelled pursuit-evasion problem for one air-to-air missile versus one high performance aircraft in a vertical plane serves as an example. For this pursuit-evasion game, the barrier surface is investigated, which determines the firing range of the missile. The numerical method for solving this problem and extensive numerical results will be presented and discussed in Part 2 of this paper; see Ref. 1.

In Part 1 of the paper (Ref. 2), we have shown that the necessary conditions for the optimal control problem of the abort landing of a passenger aircraft in the presence of windshear result in a multipoint boundary-value problem. This boundary-value problem is especially well suited for numerical treatment by the multiple shooting method. Since this method is basically a Newton iteration, initial guesses of all variables are needed and assumptions about the switching structure have to be made. These are big obstacles, but both can be overcome by a so-called homotopy strategy where the problem is imbedded into a one-parameter family of subproblems in such a way that (at least) the first problem is simple to solve. The solution data to the first problem may serve as an initial guess for the next problem, thus resulting in a whole chain of problems. This process is to be continued until the objective problem is reached. Techniques are presented here on how to handle the various changes of the switching structure during the homotopy run. The windshear problem, of great interest for safety in aviation, also serves as an excellent benchmark problem: Nearly all features that can arise in optimal control appear when solving this problem. For example, the candidate for an optimal trajectory of the minimax optimal control problem shows subarcs with both bang-bang and singular control functions, boundary arcs and touch points of two state constraints, one being of first order and the other being of third order, etc. Therefore, the results of this paper may also serve as some sort of user's guide for the solution of complicated real-life optimal control problems by multiple shooting. The candidate found for an optimal trajectory is discussed and compared with an approximate solution already known (Refs. 3–4). Besides the known necessary conditions, additional sharp necessary conditions based on sign conditions of certain multipliers are also checked. This is not possible when using direct methods.

The landing of a passenger aircraft in the presence of windshear is a threat to aviation safety. The present paper is concerned with the abort landing of an aircraft in such a serious situation. Mathematically, the flight maneuver can be described by a minimax optimal control problem. By transforming this minimax problem into an optimal control problem of standard form, a state constraint has to be taken into account which is of order three. Moreover, two additional constraints, a first-order state constraint and a control variable constraint, are imposed upon the model. Since the only control variable appears linearly, the Hamiltonian is not regular. Thus, well-known existence theorems about the occurrence of boundary arcs and boundary points cannot be applied. Numerically, this optimal control problem is solved by means of the multiple shooting method in connection with an appropriate homotopy strategy. The solution obtained here satisfies all the sharp necessary conditions including those depending on the sign of certain multipliers. The trajectory consists of bang-bang and singular subarcs, as well as boundary subarcs induced by the two state constraints. The occurrence of boundary arcs is known to be impossible for regular Hamiltonians and odd-ordered state constraints if the order exceeds two. Additionally, a boundary point also occurs where the third-order state constraint is active. Such a situation is known to be the only possibility for odd-ordered state constraints to be active if the order exceeds two and if the Hamiltonian is regular. Because of the complexity of the optimal control, this single problem combines many of the features that make this kind of optimal control problems extremely hard to solve. Moreover, the problem contains nonsmooth data arising from the approximations of the aerodynamic forces and the distribution of the wind velocity components. Therefore, the paper can serve as some sort of user's guide to solve inequality constrained real-life optimal control problems by multiple shooting.

The paper proposes a special iterative method for a nonlinear TPBVP of the form\(\dot x\)(t)=f(t, x(t),p(t)),\(\dot p\)(t)=g(t, x(t),p(t)), subject toh(x(0),p(0))=0,e(x(T),p(T))=0. Certain stability properties of the above differential equations are taken into consideration in the method, so that the integration directions associated with these equations respectively are opposite to each other, in contrast with the conventional shooting methods. Via an embedding and a Riccati-type transformation, the TPBVP is reduced to consecutive initial-value problems of ordinary differential equations. A preliminary numerical test is given by a simple example originating in an optimal control problem.

A method based on matching a zero of the right-hand side of the differential equations, in a two-point boundary-value problem, to the boundary conditions is suggested. Effectiveness of the procedure is tested on three nonlinear, two-point boundary-value problems.

An initial-value technique, which is simple to use and easy to implement, is presented for a class of nonlinear, singularly perturbed two-point boundary-value problems with a boundary layer on the left end of the underlying interval. It is distinguished by the following fact: The original second-order problem is replaced by an asymptotically equivalent first-order problem and is solved as an initial-value problem. Numerical experience with several examples is described.

In the parameter variation method, a scalar parameterk, ke[0, 1], is introduced into the differential equations. The parameterk is inserted in such a way that, whenk=0, the solution of the boundary-value problem is known or readily calculated and, whenk=1, the problem is identical with the original problem. Thus, bydeforming the solution step-by-step throughk-space fromk=0 tok=1, the original problem may be solved. These solutions then provide good starting values for any convergent, iterative scheme such as the Newton-Raphson method.

In a recent paper, Sukhanov derived a new method for transforming a nonlinear two-point boundary-value problem into an initial-value problem. Sukhanov's equations involve only the solution of ordinary differential equations and not partial differential equations. An earlier paper by the authors presented their interpretation of Sukhanov's method. An alternative method is presented in this paper. Numerical results are given.

A new method is presented to obtain a state feedback form solution to an optimal control problem with nonlinear dynamics and a quadratic performance index. The method is based on solving an integral equation equivalent to the two-point boundary-value problem related to the optimization problem by applying an inverse theorem concerning analytic nonlinear operators. Compared with the previous methods, this one is straightforward, more generally applicable, and gives important additional knowledge about the solution. An example is presented to illustrate the use of the method.

We show that, like the method of adjoints, the method of complementary functions can be effectively used to solve nonlinear boundary-value problems.

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.

We consider the solution of the following two-point boundary-value problem: $$\begin{gathered} \dot x(t) = f(t,x(t),p(t)), \dot p(t) = g(t,x(t),p(t)), t \in [0,T], \hfill \\ h(x(0),p(0)) = 0, p(T) = q. \hfill \\ \end{gathered} $$ We propose a combination technique consisting of the interval length continuation method and the back-and-forth shooting method. Certain alternative ways of employing continuation are discussed, and some of them are well suited for the problem under consideration. As a test for the method, a numerical example of a problem originating in optimal control is given.

The back-and-forth shooting method of Orava and Lautala (Ref. 1) is considered. The method transforms a given boundary-value problem to a sequence of initial-value problems. The present paper studies the convergence properties of this sequence. A local convergence theorem is given, and the rate of convergence is found to be quadratic in sufficiently smooth cases. The necessary tools for this analysis concerning the Frechet differentiability of certain mappings are given in the Appendix.

The direct shooting method for the solution of boundary-value problems for ordinary, nonlinear differential equations is analyzed from the point of view of linearization. Some relations between this method and perturbation methods are established. The relations between the direct shooting algorithm and Whittaker's algorithm are also established, considering the problem from the point of view of solving nonlinear algebraic equations.

Algorithms for calculating the junction points between optimal nonsingular and singular subarcs of singular control problems are developed. The algorithms consist in formulating appropriate initialvalue and boundary-value problems; the boundary-value problems are solved with the method of multiple shooting. Two examples are detailed to illustrate the proposed numerical methods.

A new iterative method is developed to solve the boundary-value problems for ordinary nonlinear differential equations. The method requires that only the original system of differential equations is solved once in each iteration. The initial conditions for a new iteration are evaluated directly from the given boundary conditions and the initial and boundary conditions obtained in the previous iteration, thus avoiding the necessity of solving a system of algebraic equations. The convergence proofs of the method are given. Examples of the application of the method are presented and discussed.

Several numerical methods for solving the nonlinear two-point boundary value problem associated with an optimum spacecraft trajectory are considered. A comparative evaluation of the methods is made to determine the relative merits of each method. Particular attention is given to such characteristics as the simplicity of formulation and implementation, the convergence sensitivity, the computing time required, and the computer storage requirements. The methods considered are the perturbation method, the quasilinearization method, and the gradient method. The numerical comparison is made by considering a two-dimensional, low-thrust, minimum-time, Earth-Mars trajectory.

Much of the theory of invariant imbedding is devoted to transforming two-point boundary-value problems into Cauchy problems. Numerous references are given. In this paper, a proof is given that the solution of the Cauchy problem does indeed satisfy the boundary-value problem. The discussion is self-contained and general enough to cover many applications in optimal control and radiative transfer.