Numerical Solutions of the Hamilton-Jacobi Equations Arising in Nonlinear H[infinity] and Optimal Control PDF Download
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![Numerical Solutions of the Hamilton-Jacobi Equations Arising in Nonlinear H[infinity] and Optimal Control PDF](https://books.google.com/books/content/images/frontcover/4E4gOAAACAAJ?fife=w150-h200)
Author: Jerry Markman
Publisher:
ISBN:
Category :
Languages : en
Pages : 228
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![Numerical Solutions of the Hamilton-Jacobi Equations Arising in Nonlinear H[infinity] and Optimal Control PDF](https://books.google.com/books/content/images/frontcover/4E4gOAAACAAJ?fife=w80-h100)
Author: Jerry Markman
Publisher:
ISBN:
Category :
Languages : en
Pages : 228
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Author: Rajnish Sharma
Publisher:
ISBN:
Category :
Languages : en
Pages :
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The Bolza-form of the finite-time constrained optimal control problem leads to the Hamilton-Jacobi-Bellman (HJB) equation with terminal boundary conditions and tobe- determined parameters. In general, it is a formidable task to obtain analytical and/or numerical solutions to the HJB equation. This dissertation presents two novel polynomial expansion methodologies for solving optimal feedback control problems for a class of polynomial nonlinear dynamical systems with terminal constraints. The first approach uses the concept of higher-order series expansion methods. Specifically, the Series Solution Method (SSM) utilizes a polynomial series expansion of the cost-to-go function with time-dependent coefficient gains that operate on the state variables and constraint Lagrange multipliers. A significant accomplishment of the dissertation is that the new approach allows for a systematic procedure to generate optimal feedback control laws that exactly satisfy various types of nonlinear terminal constraints. The second approach, based on modified Galerkin techniques for the solution of terminally constrained optimal control problems, is also developed in this dissertation. Depending on the time-interval, nonlinearity of the system, and the terminal constraints, the accuracy and the domain of convergence of the algorithm can be related to the order of truncation of the functional form of the optimal cost function. In order to limit the order of the expansion and still retain improved midcourse performance, a waypoint scheme is developed. The waypoint scheme has the dual advantages of reducing computational efforts and gain-storage requirements. This is especially true for autonomous systems. To illustrate the theoretical developments, several aerospace application-oriented examples are presented, including a minimum-fuel orbit transfer problem. Finally, the series solution method is applied to the solution of a class of partial differential equations that arise in robust control and differential games. Generally, these problems lead to the Hamilton-Jacobi-Isaacs (HJI) equation. A method is presented that allows this partial differential equation to be solved using the structured series solution approach. A detailed investigation, with several numerical examples, is presented on the Nash and Pareto-optimal nonlinear feedback solutions with a general terminal payoff. Other significant applications are also discussed for one-dimensional problems with control inequality constraints and parametric optimization.
Author: Yves Achdou
Publisher: Springer
ISBN: 3642364330
Category : Mathematics
Languages : en
Pages : 316
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These Lecture Notes contain the material relative to the courses given at the CIME summer school held in Cetraro, Italy from August 29 to September 3, 2011. The topic was "Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications". The courses dealt mostly with the following subjects: first order and second order Hamilton-Jacobi-Bellman equations, properties of viscosity solutions, asymptotic behaviors, mean field games, approximation and numerical methods, idempotent analysis. The content of the courses ranged from an introduction to viscosity solutions to quite advanced topics, at the cutting edge of research in the field. We believe that they opened perspectives on new and delicate issues. These lecture notes contain four contributions by Yves Achdou (Finite Difference Methods for Mean Field Games), Guy Barles (An Introduction to the Theory of Viscosity Solutions for First-order Hamilton-Jacobi Equations and Applications), Hitoshi Ishii (A Short Introduction to Viscosity Solutions and the Large Time Behavior of Solutions of Hamilton-Jacobi Equations) and Grigory Litvinov (Idempotent/Tropical Analysis, the Hamilton-Jacobi and Bellman Equations).
![Approximation of Hamilton-Jacobi Equations Arising in Nonlinear H [infinity] Control Problems PDF](https://books.google.com/books/content/images/frontcover/If_ZtgAACAAJ?fife=w80-h100)
Author: Fabio Camilli
Publisher:
ISBN:
Category : H infinity symbol control
Languages : en
Pages : 15
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![On the Hamilton-Jacobi Equation of Nonlinear H[infinity] Optimal Control PDF](https://books.google.com/books/content/images/frontcover/dCJgHAAACAAJ?fife=w80-h100)
Author: A. J. van der Schaft
Publisher:
ISBN:
Category :
Languages : en
Pages : 20
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Author: Adriano Festa
Publisher: LAP Lambert Academic Publishing
ISBN: 9783659140532
Category :
Languages : en
Pages : 128
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Book Description
This book deals with the development and the analysis of numerical methods for the resolution of first order nonlinear differential equations of Hamilton-Jacobi type on irregular data. These equations arises for example in the study of front propagation via the level set methods, the Shape-from-Shading problem and, in general, in Control theory. Our contribution to the numerical approximation of Hamilton-Jacobi equations consists in the proposal of some semiLagrangian schemes for different kind of discontinuous Hamiltonian and in an analysis of their convergence and a comparison of the results on some test problems. In particular we will approach with an eikonal equation with discontinuous coefficients in a well posed case of existence of Lipschitz continuous solutions. Furthermore, we propose a semiLagrangian scheme also for a Hamilton-Jacobi equation of a eikonal type on a ramified space, for example a graph. This is a not classical domain and only in last years there are developed a systematic theory about this. We present, also, some applications of our results on several problems arise from applied sciences.
Author: Piermarco Cannarsa
Publisher:
ISBN:
Category :
Languages : en
Pages : 23
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Author: Murad Abu-Khalaf
Publisher: Springer Science & Business Media
ISBN: 1846283507
Category : Technology & Engineering
Languages : en
Pages : 218
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Book Description
This book provides techniques to produce robust, stable and useable solutions to problems of H-infinity and H2 control in high-performance, non-linear systems for the first time. The book is of importance to control designers working in a variety of industrial systems. Case studies are given and the design of nonlinear control systems of the same caliber as those obtained in recent years using linear optimal and bounded-norm designs is explained.
Author: Hamood Amur Hamood Alwardi
Publisher:
ISBN:
Category : Hamilton-Jacobi equations
Languages : en
Pages : 96
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Book Description
This thesis addresses the construction of some algorithms for numerically solving optimal feedback control problems. Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. More precisely, optimal control problems involve a dynamic system with input quantities, called controls, and some quantity, called cost, to be minimized. An optimal control is a set of differential equations describing the paths of the control variables that optimise the cost. Finding solutions to problems of this nature involves a significantly high degree of difficulty in terms of cost and power compared with the related task of solving optimal open-loop control problems. Moreover, stability is a major problem in the feedback control problem, which may tend to overcorrect errors that can cause oscillations of constant or changing amplitude. A feedback control problem essentially depends on both state and time variables, and so its determination by numerical schemes has one serious drawback, it is the so called curse of dimensionality. Therefore, efficient numerical methods are needed for the accurate determination of optimal feedback controls. There are essentially two equivalent ways in widespread use today to solve optimal feedback control problems. In the first approach, often referred to as the direct approach, the optimal feedback control problem is approximated by considering the optimisation of an objective functional with respect to the control function. This optimisation is subject to the system dynamics and numerous constraints on the state and control variables. In the second approach, the optimal feedback control problem is transformed into a first order terminal value problem by formulating the problem as a nonlinear hyperbolic partial differential equation, known as the Hamilton-Jacobi-Bellman (HJB) equation. In this thesis we consider some numerical algorithms for solving the HJB equation, based on Radial Basis Functions (RBFs). We present a new adaptive least-squares collocation RBFs method for solving a HJB equation. The method involves the use of the least squares method using a set of RBFs in space variables, combined with the implicit backward Euler finite difference method in time, to create an unconditionally stable solution scheme. We also present some of the more theoretical aspects related to the solution of the HJB equation using the adaptive least-squares collocation RBFs method, especially, the relevant existence, uniqueness and stability results. We demonstrate the accuracy and effectiveness of this method by performing numerical experiments on test problems with up to three states and two control variables. Furthermore, we construct another numerical method based on a domain decomposition method using a matrix inversion technique for solving HJB equation. In this method, we propose a new formula for inverting nonsymmetric and full dense coefficient matrix faster than the classical matrix inversion techniques. We also investigate the accuracy of the numerical solution, condition numbers of the system matrix, and the computational time when increasing the number of subdomains. We perform some numerical experiments to illustrate the usefulness and accuracy of the method.
Author: Yangang Chen
Publisher:
ISBN:
Category : Differential equations, Nonlinear
Languages : en
Pages : 197
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Book Description
Hamilton-Jacobi-Bellman (HJB) equations are nonlinear controlled partial differential equations (PDEs). In this thesis, we propose various numerical methods for HJB equations arising from three specific applications. First, we study numerical methods for the HJB equation coupled with a Kolmogorov-Fokker-Planck (KFP) equation arising from mean field games. In order to solve the nonlinear discretized systems efficiently, we propose a multigrid method. The main novelty of our approach is that we subtract artificial viscosity from the direct discretization coarse grid operators, such that the coarse grid error estimations are more accurate. The convergence rate of the proposed multigrid method is mesh-independent and faster than the existing methods in the literature. Next, we investigate numerical methods for the HJB formulation that arises from the mass transport image registration model. We convert the PDE of the model (a Monge-Ampère equation) to an equivalent HJB equation, propose a monotone mixed discretization, and prove that it is guaranteed to converge to the viscosity solution. Then we propose multigrid methods for the mixed discretization, where we set wide stencil points as coarse grid points, use injection at wide stencil points as the restriction, and achieve a mesh-independent convergence rate. Moreover, we propose a novel periodic boundary condition for the image registration PDE, such that when two images are related by a combination of a translation and a non-rigid deformation, the numerical scheme recovers the underlying transformation correctly. Finally, we propose a deep neural network framework for the HJB equations emerging from the study of American options in high dimensions. We convert the HJB equation to an equivalent Backward Stochastic Differential Equation (BSDE), introduce the least squares residual of the BSDE as the loss function, and propose a new neural network architecture that utilizes the domain knowledge of American options. Our proposed framework yields American option prices and deltas on the entire spacetime, not only at a given point. The computational cost of the proposed approach is quadratic in dimension, which addresses the curse of dimensionality issue that state-of-the-art approaches suffer.
