Two-point boundary value problems for quasi-monotone dynamical systems

Decentralized optimal control for large-scale nonlinear systems with respect to local performance criteria is considered. A decoupled linear Two-Point Boundary Value (TPBV) problem sequence is constructed to approximate the nonlinear coupling large-scale TPBV problem, which is the necessary condition of the nonlinear optimal control problem. We prove that the constructed linear TPBV problem sequence uniformly converges to the nonlinear TPBV problem. By iteratively solving this linear TPBV problem sequence, a decentralized linear feedback optimal control law is obtained. An algorithm is presented to design some qualified approximate optimal control law by using a finite iteration result. An illustrative example shows the validity of the algorithm.

This paper deals with an optimal control problem for a class of singularly perturbed time-delay large-scale systems. The optimal control laws for the order-reduced slow subsystem with time-delay and fast subsystem are designed, respectively. For the slow subsystem, the sensitivity approach is proposed to solve the coupled two-point boundary value (TPBV) problem with both time delay and advance terms. The TPBV problem is transformed into a decoupled sequence of inhomogeneous linear TPBV problems without time delay and advance terms. By solving the sequence of TPBV problems and Riccati equations, the approximate optimal control law of the original system is obtained. The control law consists of analytic state feedback and a time-delay compensation term which is a series sum of adjoint vectors. The compensation term can be approximately obtained by a recursion formula of adjoint vectors. Numerical examples show that the proposed method is valid.

State Transition Tensors for Continuous-Thrust Control of Three-Body Relative Motion

In this paper, we establish close relationships between the stability constants on one hand and the global behaviour of fundamental matrices on the other hand to the two-point boundary value problems on time-scale dynamical systems. We introduce the concept of conditioning number k and show that co nditioning number is the right criteria in estimating the global error due to small perturbations of two point boundary value problems on time scale dynamical systems. Further, the moderate stability constants imply a dichotomy with moderate k-bound will be developed. Further, the exponential behaviour of solutions of the Green’s matrix will be investigated. We also investigate the conditions under which strong dichotomy exists for two-point boundary value problems when the boundary conditions are separable.

<p>One of a powerful application in the age of cloud computing is the outsourcing of scientific computations to cloud computing which makes cloud computing a very powerful computing paradigm, where the customers with limited computing resource and storage devices can outsource the sophisticated computation workloads into powerful service providers. One of scientific computations problem is Two-Point Boundary Value Problems(BVP) is a basic engineering and scientific problem, which has application in various domains. In this paper, we propose a privacy-preserving, verifiable and efficient algorithm for Two-Point Boundary Value Problems in outsourcing paradigm. We implement the proposed schema on the customer side laptop and using AWS compute domain elastic compute cloud (EC2) for the cloud side.</p>

A finite difference method for obtaining sixth order accurate approximation to the solution of the two-point linear boundary value problem is given. The convergence of the method is proved. Numerical results for a typical problem are tabulated and in each case the observed error is compared with its theoretical estimate. The numerical results are compared with those obtained from an earlier method of the author and the method of Noumerov.

We will begin this chapter with a discussion of two-point boundary value ODEs and then move on to two-point eigenvalue problems. The Blasius equation in Example 1.6.4 is one of the most celebrated two-point boundary value problems in fluid dynamics. Two-point boundary value problems form a challenging class of numerical computations with a wide literature. For an exposition of the theoretical ideas on two-point boundary value problems see Press et al. (1986), Chapter 16, or Keller (1968). Broadly speaking there are two classes of methods for such problems, shooting methods and relaxation methods. Shooting methods are related to integration schemes for initial value problems, while relaxation methods are based on discrete approximations to derivative operators. Thus, we are ready to tackle both these methods. In a shooting method you start from one end as in an initial value calculation using the actual boundary conditions specified at that boundary supplemented with a few other assumed (or guessed) ones which replace the actual boundary conditions of the ODE specified at the other end, and integrate (or “shoot”) towards the other boundary. In general when you eventually reach the other boundary, the boundary conditions there will not be satisfied (in other words you “miss” the target). Hence, you modify your guessed boundary conditions at the starting boundary (“aim” again) and integrate forward (“shoot” again). In this way you have in effect generated a mapping of initially guessed boundary conditions onto errors in matching the actual boundary conditions to be enforced at the other end point. Iterative procedures can now be invoked to converge to the desired value of the boundary conditions at the other end.

The ultimate purpose of this article is to introduce and describe a combined approach, based on asymptotic boundary conditions (ABCs) and a fourth order cubic B-spline collocation, for the numerical solution of a general class of two-point linear boundary-value problems (BVPs) over a semi-infinite interval that arises in various engineering applications. The scheme will be extended and then implemented to handle a system of BVPs. The idea of the proposed strategy is to first reduce the condition at infinity to an asymptotic boundary condition that approaches the specified value at infinity over a large finite interval. Then, the problem complimented with the resulting ABC is solved using a fourth–order spline collocation approach constructed over uniform meshes. The scheme is numerically verified to have a fourth order rate of convergence. The work is illustrated by considering a number of test examples that confirm the accuracy, efficient treatment of the boundary condition at infinity, and applicability of the approach. The computational results show that the scheme is reliable, converges fast, and compares very well with the existing analytic solutions.

Bidirectional path and motion planning approaches decrease planning time, on average, compared to their unidirectional counterparts. In single-query feasible motion planning, using bidirectional search to find a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">continuous</i> motion plan requires an edge connection between the forward search tree and the reverse search tree. Such a tree–tree connection requires solving a two-point boundary value problem (BVP). However, obtaining a closed-form two-point BVP solution can be difficult or impossible for many systems. While numerical methods can provide a reasonable solution in many cases, they are often computationally expensive, numerically unstable, or sensitive (to an initial guess) for the purposes of single-query sampling-based motion planning. To overcome this challenge, we present a novel bidirectional search strategy that does not require solving the two-point BVP. Instead of connecting the forward and reverse trees directly, the reverse tree’s cost information is used as a guiding heuristic for forward search. This enables the forward search to quickly grow down the reverse tree—converging to a fully feasible solution <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">without</i> a direct tree–tree connection and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">without</i> the solution to a two-point BVP. In this article, we propose two algorithms that use this strategy for single-query feasible motion planning for various dynamical systems, performing experiments in both simulation and hardware testbeds. We find that these algorithms perform better than or comparable to the existing state-of-the-art methods with respect to quickly finding an initial feasible solution.

A new superconvergent method based on a sextic spline is described and analysed for the solution of systems of nonlinear singular two-point boundary value problems (BVPs). It is well known that the optimal orders of convergence could not be achieved using standard formulation of a sextic spline for the solution of BVPs. Based on the method used in our earlier research papers [J. Rashidinia and M. Ghasemi, B-spline collocation for solution of two-point boundary value problems, J. Comput. Appl. Math. 235 (2011), pp. 2325–2342; J. Rashidinia, M. Ghasemi, and R. Jalilian, An o(h 6) numerical solution of general nonlinear fifth-order two point boundary value problems, Numer. Algorithms 55(4) (2010), pp. 403–428], we construct a new O(h 8) locally superconvergent method for the solution of general nonlinear two-point BVPs up to order 6. The error bounds and the convergence properties of the method have been proved theoretically. Then, the method is extended to solve the system of nonlinear two-point BVPs. Some test problems are given to demonstrate the applicability and the superconvergent properties of the proposed method numerically. It is shown that the method is very efficient and applicable for stiff BVPs too.

Multipoint approach to the two-point boundary value problem

This research aims to use the shooting method (SM) to find numerical solutions to the boundary value problems of ordinary differential equations (ODEs). Applied mathematics, theoretical physics, engineering, control, and optimization theory all have two-point boundary value problems. If the two-point boundary value problem cannot be solved analytically, numerical approaches must be used. The scenario in the two-point boundary value issue for a single second-order differential equation with prescribed initial and final values of the solution gives rise to shooting method. Firstly, the method is discussed, and some boundary value problems of ODEs are solved by using the proposed method. Obtained results are compared with the exact solution for the validation of the proposed method and represented both in graphical and tabular form. It has been found that the convergence rate of the shooting method to the exact solution is so high. As a finding of this research, it has been determined that the shooting method produces the best-fit numerical results of boundary value problems.

Optimal sliding mode control for linear time-delay systems with sinusoidal disturbances

This paper presents the approximate optimal vibration control methodology for discrete nonlinear vehicle active suspension subject to persistent road disturbances. Based on a dynamic model of nonlinear vehicle active suspension and a linear exogenous system model of the persistent road disturbances, a nonlinear two-point boundary value (TPBV) problem is introduced. By introducing a sensitive parameter, the original TPBV problem is reformed as a series of TPBV problems without nonlinear items. A discrete approximate optimal vibration controller (DAOVC) can be obtained by solving a Riccati equation, Stein equation, and nonlinear compensation item, respectively. An iteration algorithm is designed to realize the computational realizability of DAOVC based on the control performance. It is demonstrated that the control performance of ride comfort, road holding ability, and suspension deflection under the DAOVC are much smaller than the ones under the classical feedforward and feedback optimal vibration controller (FFOVC) and open-loop vehicle suspension system.

The sliding-mode control for linear systems with time-delay Is concerned. By treating some state variables as virtual control, a quadratic performance index is given. According to the necessary conditions for the optimality, the two-point boundary value (TPBV) problems with both time-delay and time-advance terms are obtained. By developing the successive approximation approach (SAA) of differential equation into infinite horizon, the original TPBV problem which is derived from the optimal switching manifold design is transformed into a sequence of linear TPBV problems without delay and advance terms. The solution sequence of the linear TPBV problems uniformly converges to the solution of the original problem.. By using a finite term of the adjoint vector sequence, a suboptimal switching manifold is obtained. The switching manifold designed assures that the state trajectories of the closed-loop system converge to zero as fast as possible on ideal sliding surface. The convergence velocity of every state trajectory on the ideal sliding surface can be adjusted through choosing quadratic performance index. Numerical simulation is given to show the effectiveness of the proposed design approach.