The brachistochrone motion of a mechanical system with non-holonomic, non-linear and non-stationary constraints

Further to previous studies /1, 2/ of the brachistochrone motion of non-holonomic mechanical systems with linear homogeneous constraints, consideration is given here to non-holonomic, non-linear and non-stationary mechanical systems. The problem is to formulate the differential equations of the brachistochrone motion of non-holonomic, non-linear and non-stationary mechanical systems and to determine the additional forces which must be introduced in order to implement motion of this type.     

Nonclassical pseudospectral method for the solution of brachistochrone problem

In this paper, nonclassical pseudospectral method is proposed for solving the classic brachistochrone problem. The brachistochrone problem is first formulated as a nonlinear optimal control problem. Properties of nonclassical pseudospectral method are presented, these properties are then utilized to reduce the computation of brachistochrone problem to the solution of algebraic equations. Using this method, the solution to the brachistochrone problem is compared with those in the literature.     

A note on the classical brachistochrone

Summary It is shown that the classical Bernoulli's brachistochrone problem and the brachistochrone problem in a central force field may be solved by the maximum principle of Pontryagin. According to the optimum control theory these problems are singular.

---

Zusammenfassung In der Arbeit ist gezeigt, daß das klassische Bernoullische brachistochrone Problem und Brachistochronen im zentralen Schwerefeld durch das Pontryagin Maximum Prinzip gelöst werden können. Gemäss Theorie der optimalen Regelung sind diese Probleme singulär.

     

A note on the classical brachistochrone

It is shown that the classical Bernoulli's brachistochrone problem and the brachistochrone problem in a central force field may be solved by the maximum principle of Pontryagin. According to the optimum control theory these problems are singular.     

A new look at the brachistochrone problem

Summary The classic brachistochrone problem is solved as an optimal control problem in conjunction with the use of Chebyshev series.

---

Résumé Le problème classique du brachistochrone est résolu comme un problème de contrôle optimale à l'aide des séries de Chebyshev.

     

Efficient Sequential Quadratic Programming Implementations for Equality-Constrained Discrete-Time Optimal Control

Efficient sequential quadratic programming (SQP) implementations are presented for equality-constrained, discrete-time, optimal control problems. The algorithm developed calculates the search direction for the equality-based variant of SQP and is applicable to problems with either fixed or free final time. Problem solutions are obtained by solving iteratively a series of constrained quadratic programs. The number of mathematical operations required for each iteration is proportional to the number of discrete times N. This is contrasted by conventional methods in which this number is proportional to N ³. The algorithm results in quadratic convergence of the iterates under the same conditions as those for SQP and simplifies to an existing dynamic programming approach when there are no constraints and the final time is fixed. A simple test problem and two application problems are presented. The application examples include a satellite dynamics problem and a set of brachistochrone problems involving viscous friction.     

Convergence Acceleration of Direct Trajectory Optimization Using Novel Hessian Calculation Methods

Sparse sequential quadratic programming (SQP) has offered fast and robust convergence of trajectory optimization based on direct collocation. However, the conventional approach of calculating the Hessian of the Lagrangian is sometimes inefficient in view of the computational time. Therefore, this paper proposes two novel Hessian calculation methods that exploit the doubly-bordered block diagonal structure of the Hessian. Through applications to the constrained brachistochrone problem and the space shuttle reentry problem, the proposed methods were demonstrated to show faster convergence speeds as compared with the conventional methods. This work was supported by a Grant-In-Aid from the Japan Society for the Promotion of Science.     

Computational results for extensions in quasilinearization techniques for optimal control

In a companion paper (Ref. 1), several extensions in quasilinearization were presented. These results are studied computationally in this paper for two problems: the brachistochrone problem and the reentry vehicle problem. For the brachistochrone free-time problem, it is shown that much more rapid convergence is obtained than that presented in previous literature (Ref. 2). It is also shown that, if the techniques of the companion paper are used, the normal region of convergence is extended significantly. Similar results are obtained for the reentry vehicle problem.The research presented in this paper was supported in part by the Air Force Office of Scientific Research, Grant No. AF-AFOSR-699-67.     

The arrival time brachistochrones in general relativity

We consider a general relativistic version of the classical brachistochrone problem, whose solutions are causal curves, parameterized by a constant multiple of their proper time and with 4-acceleration perpendicular to a given observer field, that extremize the arrival time measured by an observer at the final endpoint. This kind of brachistochrones presents characteristics different from the travel time brachistochrones, that were studied in [8, 9, 10]. In this article we formulate the variational problem in a general context; moreover, in the case of a stationary metric, we prove two variational principles and we determine the second order differential equation satisfied by the arrival time brachistochrone. Using these variational principles and techniques from Critical Point Theory we establish some results concerning the existence and the multiplicity of travel time brachistochrones with a given energy between an event and an observer.     

An extended quasilinearization algorithm

The quasilinearization algorithm for the solution of two-point boundary-value problems is extended to handle a general class of multipoint boundary value problems involving multiple subarcs, state and/or control variable inequality constraints, and discontinuous state and/or adjoint variables. The corner and final times are unspecified since they are implicitly defined by the satisfaction of subarc stopping conditions. The inequality constraints are handled directly without the use of penalty functions. The extended algorithm is applied to a discontinuous version of the brachistochrone problem, and good convergence properties are obtained.This research was supported in part by AFOSR Grant No. 72–2166.     

Dynamic Programming Method for Constrained Discrete-Time Optimal Control

A dynamic programming method is presented for solving constrained, discrete-time, optimal control problems. The method is based on an efficient algorithm for solving the subproblems of sequential quadratic programming. By using an interior-point method to accommodate inequality constraints, a modification of an existing algorithm for equality constrained problems can be used iteratively to solve the subproblems. Two test problems and two application problems are presented. The application examples include a rest-to-rest maneuver of a flexible structure and a constrained brachistochrone problem.     

A problem on brachistochrone as invariant variational problem

Based on the modified theory of invariant variational problems developed by the author, we describe a theoretical group approach to the problem of brachistochrone.     

The brachistochrone with acceleration: A running track

The equations for the brachistochrone of a self-accelerating particle are found by solving a problem in the calculus of variations with a constraint and a free end condition then applied to find the shape of a running track.     

Variational approach to a non-local identification problem related to non-linear ion transport

A variational approach to a non-linear non-local identification problem related to the non-linear transport equation is studied. Introducing a similarity transformation, the problem is formulated as an identification problem for a non-linear differential equation of second order with an additional non-local condition. For the solution of the forward problem stability in H¹-norm with respect to the identification parameter is obtained. Using this result the existence of a solution to the identification problem is proved. Some results of computational experiments are given. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Multi-model predictive control of Hammerstein-Wiener systems based on balanced multi-model partition

ABSTRACT

Model analysis of Hammerstein-Wiener systems has been made, and it is found that the included angle is applicable to such systems to measure the non-linearity. Then, a dichotomy gridding algorithm is proposed based on the included angle. Supporting by the gridding algorithm, a balanced multi-model partition method is put forward to partition a Hammerstein-Wiener system into a set of local linear models. For each linear model, a linear model predictive controller (MPC) is designed. After that, a multi-MPC is composed of the linear MPCs via soft switching. Thus, a complex non-linear control problem is transformed into a set of linear control problems, which simplifies the original control problem and improves the control performance. Two non-linear systems are built into Hammerstein-Wiener models and investigated using the proposed methods. Simulations demonstrate that the proposed gridding and partition methods are effective, and the resulted multi-MPC controller has satisfactory performance in both set-point tracking and disturbance rejection control.     

Maximum a Posteriori Sequence Estimation Using Monte Carlo Particle Filters

We develop methods for performing maximum a posteriori (MAP) sequence estimation in non-linear non-Gaussian dynamic models. The methods rely on a particle cloud representation of the filtering distribution which evolves through time using importance sampling and resampling ideas. MAP sequence estimation is then performed using a classical dynamic programming technique applied to the discretised version of the state space. In contrast with standard approaches to the problem which essentially compare only the trajectories generated directly during the filtering stage, our method efficiently computes the optimal trajectory over all combinations of the filtered states. A particular strength of the method is that MAP sequence estimation is performed sequentially in one single forwards pass through the data without the requirement of an additional backward sweep. An application to estimation of a non-linear time series model and to spectral estimation for time-varying autoregressions is described.     

A posteriori error analysis for linearization of nonlinear elliptic problems and their discretizations

The paper is devoted to a posteriori quantitative analysis for errors caused by linearization of non-linear elliptic boundary value problems and their finite element realizations. We employ duality theory in convex analysis to derive computable bounds on the difference between the solution of a non-linear problem and the solution of the linearized problem, by using the solution of the linearized problem only. We also derive computable bounds on differences between finite element solutions of the nonlinear problem and finite element solutions of the linearized problem, by using finite element solutions of the linearized problem only. Numerical experiments show that our a posteriori error bounds are efficient.     

A novel algorithm for area traffic capacity control with elastic travel demands

A non-linear area traffic control system with limited capacity is considered in this paper. Optimal signal settings and link capacity expansions can be determined while trip distribution and network flow are in equilibrium. This problem can be formulated as a non-linear mathematical program with equilibrium constraints. For the objective function a non-linear constrained optimization program for signal settings and link capacity expansion is determined. For the constraint set the elastic user equilibrium traffic assignment obeying Wardrop’s first principle can be formulated as a variational inequality. Since the constrained optimization problem is non-convex, only local optima can be obtained. In this paper, a novel algorithm using a non-smooth trust region approach is proposed. Numerical tests are performed using a real data city network and various example test networks in which the effectiveness and robustness of the proposed method are confirmed as compared to other well-known solution methods.     

The game problem on the dolichobrachistochrone : PMM vol.40, n 6, 1976, pp. 1003–1013

The capture and evasion sets, the players' optimal strategies and the game value determined for the game problem on the dolichobrachistochrone, analysed within the framework of a position formalism similar to [1]. Singularities inherent in the game of the minimax-maximin time to contact [1, 2] become apparent; they are determined in the given problem by the specific behavior of the optimal paths close to the target set. Isaacs [4] examined the game problem on the dolichobrachistochrone, being the game analog of the classical variational problem on the brachistochrone [3]. However, as was shown in [5], the solution proposed by Isaacs contains erroneous statements.     

On the initial value problem for the Vlasov-Poisson-Fokker-Planck system with initial data in Lp spaces

In this paper the global existence of weak solutions for the Vlasov-Poisson-Fokker-Planck equations in three dimensions is proved with an L¹ ∩ L^p initial data. Also, the global existence of weak solutions in four dimensions with small initial data is studied. A convergence of the solutions is obtained to those built by E. Horst and R. Hunze when the Fokker-Planck term vanishes. In order to obtain the a priori necessary estimates a sequence of approximate problems is introduced. This sequence is obtained starting from a non-linear regulation of the problem together with a linearization via a time retarded mollification of the non-linear term. The a priori bounds are reached by means of the control of the kinetic energy in the approximate sequence of problems. Then, the proof is completed obtaining the equicontinuity properties which allow to pass to the limit.