Shooting methods for numerical solutions of control problems constrained by linear and nonlinear hyperbolic partial differential equations

국문 초록 (Abstract)

본 논문에는 뉴튼방법을 기본으로한 슈팅방법을 통해 제어 가능한 문제(controllability problem)를 최적의 제어(optimal control)문제로 전환하여 해결하는 수치적 방법에 대해 연구되어졌다. 최적제어 문제는 선형(linear)과 반선형(semilinear) 미분방정식을 조건으로 갖는 문제로 제어 목표는 원하는 마지막 상태를 추적하는 것이고 제어방식은 분포된 형태(distributed type)과 경계 형태(boundary type)을 모두 다루었다. 일반적으로 최적의 문제의 해를 수치적으로 해결하는데 Gradient 알고리듬을 이용하는 것이 보편적인데, 이는 초기치의 설정에 따라 그 계산량이 크게 변하는 경향을 갖고 있어 이를 해결하기 위해 많은 연구들이 이루어지고 있다. 이 논문에서는 주어진 문제를 효과적이면서 빠르게 수치해를 구해 낼 수 있는 슈팅방법을 소개함과 동시에 최적의 수치해가 제어 가능한 문제의 해로 접근함을 증명하였다. 최적제어 문제의 해를 구하기 위해서 Lagrange의 법칙을 써서 최적화 문제를 Optimality 시스템으로 전환한다. 이 시스템은 초기치 문제와 최종치 문제를 병행하고 있어 시간이 증가함에 따라 동시에 문제를 해결할 수 없는 특징을 갖고 있다. 그리하여 슈팅방법을 통해 이 시스템을 두 개의 초기치 문제로 전환하여 시간의 증가에 따라 동시에 수치해를 구할 수 있게 해 준다. 또한 이 방법을 통해 1차원 그리고 2차원의 예를 통해 계산된 수치 결과를 제시하였다.

다국어 초록 (Multilingual Abstract)

We consider shooting methods for computing approximate solutions of control problems constrained by linear or nonlinear hyperbolic partial differential equations. Optimal control problems and exact controllability problems are both studied, with the latter being approximated by the former with appropriate choices of parameters in the cost functional. The types of equations include linear wave equations, semilinear wave equations, and first order linear hyperbolic equations. The controls considered are either distributed in part of the time-space domain or of the Dirichlet type on the boundary. Each optimal control problem is reformulated as a system of equations that consists of an initial value problem (IVP) for the state equations and a terminal value problem for the adjoint equations. The opimality systems are regarded as a system of an IVP for the state equation and an IVP for the adjoint equations with unknown initial conditions. Then the optimality system is solved by shooting methods, i.e. we attempt to find adjoint initial values such that the adjoint terminal conditions are met. The shooting methods are implemented iteratively and Newton's method is employed is employed to update the adjoint initial values. The convergence of the algorithms are theoretically discussed and numerically verified. Computational experiments are performed extensively for a variety of settings : different types of constraint equations in 1-D or 2-D, distributed or boundary controls, optimal control or exact controllability.

목차 (Table of Contents)

• TABLE OF CONTENTS = ⅲ
• ABSTRACT = ⅴ
• 1 INTRODUCTION = 1
• 2 Shooting Methods for Numerical Solutions of Distributed Optimal Control Problems Constrained by Linear and Nonlinear Wave Equations = 6
• 2.1 Distributed optimal control problems for the wave equations = 6
• 2.2 The solution of the exact controllability problem as the limit of optimal control solutions = 8
• 2.3 Shooting methods for 1D control problems = 13
• 2.3.1 Algorithm for 1D control problems = 15
• 2.3.2 1D Computational results = 19
• 2.4 Shooting methods in 2D control problems = 26
• 2.4.1 Algorithm for 2D control problems = 26
• 2.4.2 2D computational results = 31
• 3 Shooting Methods for Numerical Solutions of Exact Controllability Problems Constrained by Linear and Nonlinear Wave Equations with Local Distributed Controls = 34
• 3.1 Exact controllability problems for the wave equations = 34
• 3.2 The solution of the exact local controllability problem as the limit of optimal control solutions = 36
• 3.3 Computational results = 42
• 3.3.1 Exact controllability problems with linear cases = 42
• 3.3.2 Exact controllability problems with nonlinear cases = 56
• 4 Shooting Methods for Numerical Solutions of Exact Boundary Controllability Problems for the 1-D Wave Equation = 96
• 4.1 Introduction = 96
• 4.2 The solution of the exact controllability problem as the limit of optimal control solutions = 100
• 4.3 An optimality system of equations and a continuous shooting method = 105
• 4.4 The discrete shooting method = 107
• 4.5 Computational experiments for controllability of the linear wave equation = 112
• 4.5.1 An example with known smooth exact solution = 112
• 4.5.2 Generic examples with minimum L^(2)(∑)-norm boundary control = 115
• 4.6 Computational experiments for controllability of semilinear wave equations = 117
• 5 Shooting Methods for Numerical Solutions of Distributed Optimal Control Problems Constrained by First Order Linear Hyperbolic Equation = 131
• 5.1 Distributed optimal control problems for first order linear hyperbolic equation = 131
• 5.2 An optimality system of equations = 136
• 5.3 Computational results = 138
• 6 CONCLUSION = 141
• BIBLIOGRAPHY = 144
• ACKNOWLEDGMENTS = 148