Sensitivity analysis for parametric control problems with control-state constraints

Parametric nonlinear control problems subject to vector-valued mixed control-state constraints are investigated. The model perturbations are implemented by a parameter p of a Banach-space P. We prove solution differentiability in the sense that the optimal solution and the associated adjoint multiplier function are differentiable functions of the parameter. The main assumptions for solution differentiability are composed by regularity conditions and recently developed second-order sufficient conditions (SSC). The analysis generalizes the approach in [16, 20] and establishes a link between (1) shooting techniques for solving the associated boundary value problem (BVP) and (2) SSC. We shall make use of sensitivity results from finite-dimensional parametric programming and exploit the relationships between the variational system associated to BVP and its corresponding Riccati equation.Solution differentiability is the theoretical backbone for any numerical sensitivity analysis. A numerical example with a vector-valued control is presented that illustrates sensitivity analysis in detail.     

Abort landing in the presence of windshear as a minimax optimal control problem,part 2: Multiple shooting and homotopy

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.An extended abstract of this paper was presented at the 8th IFAC Workshop on Control Applications of Nonlinear Programming and Optimization, Paris, France, 1989 (see Ref. 1).This paper is dedicated to Professor Hans J. Stetter on the occasion of his 60th birthday.     

The Lie-Group Shooting Method for Solving Multi-dimensional Nonlinear Boundary Value Problems

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.     

Numerical solution of singular control problems using multiple shooting techniques

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.The author would like to thank Professor Dr. R. Bulirsch, who stimulated and encouraged this work, which is part of the author's dissertation.     

Shooting Methods for Numerical Solution of Nonlinear Stochastic Boundary-Value Problems

Abstract

In the present investigation, shooting methods are described for numerically solving nonlinear stochastic boundary-value problems. These stochastic shooting methods are analogous to standard shooting methods for numerical solution of ordinary deterministic boundary-value problems. It is shown that the shooting methods provide accurate approximations. An error analysis is performed and computational simulations are described.     

Cubic spline method for a class of nonlinear singularly-perturbed boundary-value problems

In this paper, we present a numerical method for solving a class of nonlinear, singularly perturbed two-point boundary-value problems with a boundary layer on the left end of the underlying interval. The original second-order problem is reduced to an asymptotically equivalent first-order problem and is solved by a numerical method using a fourth-order cubic spline in the inner region. The method has been analyzed for convergence and is shown to yield anO(h ⁴) approximation to the solution. Some test examples have been solved to demonstrate the efficiency of the method.The authors thank the referee for his helpful comments.     

On Smoothing l1 Exact Penalty Function for Constrained Optimization Problems

This article introduces a smoothing technique to the l₁ exact penalty function. An application of the technique yields a twice continuously differentiable penalty function and a smoothed penalty problem. Under some mild conditions, the optimal solution to the smoothed penalty problem becomes an approximate optimal solution to the original constrained optimization problem. Based on the smoothed penalty problem, we propose an algorithm to solve the constrained optimization problem. Every limit point of the sequence generated by the algorithm is an optimal solution. Several numerical examples are presented to illustrate the performance of the proposed algorithm.     

Shooting Methods for Numerical Solution of Stochastic Boundary-Value Problems

Abstract

In the present investigation, numerical methods are developed for approximate solution of stochastic boundary-value problems. In particular, shooting methods are examined for numerically solving systems of Stratonovich boundary-value problems. It is proved that these methods accurately approximate the solutions of stochastic boundary-value problems. An error analysis of these methods is performed. Computational simulations are given.     

A discrete eigenfunctions method for computing mixed hyperbolic problems based on an implicit difference scheme

In this paper numerical solutions of mixed hyperbolic problems are computed using a discrete eigenfunctions method combined with an implicit difference scheme. This new numerical technique preserves the qualitative properties of the analytic solution due to the Sturm-Liouville structure of the underlying discrete linear boundary-value problem and has computational stability advantages vs other methods. Illustrative examples are included.     

Multipoint solution of two-point boundary-value problems

The purpose of this paper is to report on the application of multipoint methods to the solution of two-point boundary-value problems with special reference to the continuation technique of Roberts and Shipman. The power of the multipoint approach to solve sensitive two-point boundary-value problems with linear and nonlinear ordinary differential equations is exhibited. Practical numerical experience with the method is given.Since employment of the multipoint method requires some judgment on the part of the user, several important questions are raised and resolved. These include the questions of how many multipoints to select, where to specify the multipoints in the interval, and how to assign initial values to the multipoints.Three sensitive numerical examples, which cannot be solved by conventional shooting methods, are solved by the multipoint method and continuation. The examples include (1) a system of two linear, ordinary differential equations with a boundary condition at infinity, (2) a system of five nonlinear ordinary differential equations, and (3) a system of four linear ordinary equations, which isstiff.The principal results are that multipoint methods applied to two-point boundary-value problems (a) permit continuation to be used over a larger interval than the two-point boundary-value technique, (b) permit continuation to be made with larger interval extensions, (c) converge in fewer iterations than the two-point boundary-value methods, and (d) solve problems that two-point boundary-value methods cannot solve.     

Complex differential games of pursuit-evasion type with state constraints,part 1: Necessary conditions for optimal open-loop strategies

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.This paper is dedicated to the memory of Professor John V. Breakwell.The authors would like to express their sincere and grateful appreciation to Professors R. Bulirsch and K. H. Well for their encouraging interest in this work.     

Adjoint Estimation from a Direct Multiple Shooting 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.     

Asymptotic Initial-Value Method for Singularly-Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations

A computational method is presented to solve a class of nonturning-point singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative, subject to Dirichlet-type boundary conditions. In this method, first we construct a zeroth order asymptotic expansion for the solution of the given boundary-value problem. Then, this problem is integrated to get an equivalent initial-value problem for first-order ordinary differential equations. This initial-value problem is solved by either a classical method or a fitted operator method after approximating some of the terms in the differential equations by using the zeroth order asymptotic expansion. This method is effective and easy to implement. An error estimate is derived for the numerical solution. Examples are given to illustrate the method.     

Helly’s Principle and Its Application to an Infinite-Horizon Optimal Control Problem

In this paper, a numerical method is presented to solve singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a discontinuous source term. First, an asymptotic expansion approximation of the solution of the boundary-value problem is constructed using the basic ideas of the well-known WKB perturbation method. Then, some initial-value problems and terminal-value problems are constructed such that their solutions are the terms of this asymptotic expansion. These initial-value problems and terminal-value problems are singularly-perturbed problems and therefore fitted mesh method (Shishkin mesh) are used to solve these problems. Necessary error estimates are derived and examples are provided to illustrate the method.     

Numerical solution of singular perturbation problems by a terminal boundary-value technique

In this paper, we present a new approach for numerically solving linear singularly perturbed two-point boundary-value problems in ordinary differential equations with a boundary layer on the left end of the interval. The original problem is divided into outer and inner region problems. A terminal boundary condition in implicit form is introduced. Then, the outer region problem is solved as a two-point boundary-value problem (TPBVP), and an explicit terminal boundary condition is obtained. In turn, the inner region problem is modified and solved as a TPBVP using the explicit terminal boundary condition. The proposed method is iterative on the terminal point of the inner region. Three numerical examples have been solved to demonstrate the applicability of the method.     

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.     

An <Emphasis Type="BoldItalic">O</Emphasis>(<Emphasis Type="BoldItalic">h</Emphasis><Superscript><Emphasis Type="Bold">6</Emphasis></Superscript>) numerical solution of general nonlinear fifth-order two point boundary value problems

A sixth-order numerical scheme is developed for general nonlinear fifth-order two point boundary-value problems. The standard sextic spline for the solution of fifth order two point boundary-value problems gives only O(h ²) accuracy and leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. O(h ⁶) global error estimates obtained for these problems. The convergence properties of the method is studied. This scheme has been applied to the system of nonlinear fifth order two-point boundary value problem too. Numerical results are given to illustrate the efficiency of the proposed method computationally. Results from the numerical experiments, verify the theoretical behavior of the orders of convergence.     

A data assimilation process for linear ill‐posed problems

In this paper, an iteration process is considered to solve linear ill‐posed problems. Based on the randomness of the involved variables, this kind of problems is regarded as simulation problems of the posterior distribution of the unknown variable given the noise data. We construct a new ensemble Kalman filter‐based method to seek the posterior target distribution. Despite the ensemble Kalman filter method having widespread applications, there has been little analysis of its theoretical properties, especially in the field of inverse problems. This paper analyzes the propagation of the error with the iteration step for the proposed algorithm. The theoretical analysis shows that the proposed algorithm is convergence. We compare the numerical effect with the Bayesian inversion approach by two numerical examples: backward heat conduction problem and the first kind of integral equation. The numerical tests show that the proposed algorithm is effective and competitive with the Bayesian method. Copyright © 2017 John Wiley & Sons, Ltd.     

Spline Techniques for Solving Singularly-Perturbed Nonlinear Problems on Nonuniform Grids

A numerical method based on cubic splines with nonuniform grid is given for singularly-perturbed nonlinear two-point boundary-value problems. The original nonlinear equation is linearized using quasilinearization. Difference schemes are derived for the linear case using a variable-mesh cubic spline and are used to solve each linear equation obtained via quasilinearization. Second-order uniform convergence is achieved. Numerical examples are given in support of the theoretical results.     

On the solution of the generalized steiner problem by the subgradient method

This paper is concerned with the problem of constructing a minimal cost weighted tree connecting a set ofn given terminal vertices on an Euclidean plane. Both theoretical and numerical aspect of the problem are considered. As regards the first ones, the convexity of the objective function is studied and the necessary and sufficient optimality conditions are deduced. As regards the numerical aspects, a subgradient type algorithm is presented.