This paper is concerned with the numerical solution of optimal control problems for distributed parameter systems by means of nonlinear programming methods. Specifically, we will consider optimal control problems for systems described by reaction diffusion convection equations subject to control and state inequality constraints. We assume that for these problems there exists a unique solution and the dynamic systems have an equilibrium state stable (in the sense of Lyapunov) and are completely controllable and observable. We transcribe these problems into large finite dimensional nonlinear programming problems by introducing suitable finite difference discretization schemes. A basic point of this transcription is a consistency condition for which the condition of optimality of the discretized problems (Karush–Kuhn–Tucker conditions) reflect the optimality conditions of the original continuous problems (Pontryagin Maximum Principle). This relationship between the discrete and the continuous necessary optimality conditions suggests to discretize the reaction diffusion convection equation with a finite difference scheme of type FTCS (Forward Time, Centered Space) which is second order accurate in space and first order accurate in time and is numerically stable in the sense of Von Neumann under suited conditions. The numerical solutions of the above nonlinear programming problems are determined by solving the constrained system of nonlinear equations obtained by the Karush–Kuhn–Tucker (KKT) optimality conditions. For solving these KKT system we use a Modified Inexact Newton (MIN) method. The convergence properties of the MIN method are stated under standard assumptions on the KKT systems. These assumptions are the same for which, at each iteration of the MIN method, the perturbed Newton equation has a unique solution, which can be determined by solving an equality constrained quadratic programming problem with the Hestenes’ method of multipliers. Numerical studies show that the MIN method leads to satisfactory results if we take care of choosing the starting point of the MIN iterative procedure and the perturbation parameter in such a way that the damping parameter is sufficiently large for all iterations.
GALLIGANI, Emanuele. "Application of inexact Newton method to the optimization of systems with distributed parameters" Working paper, Dipartimento di Ingegneria Enzo Ferrari - Università di Modena e Reggio Emilia, 2011. https://doi.org/10.25431/11380_1060323
|Titolo:||Application of inexact Newton method to the optimization of systems with distributed parameters|
|Data di pubblicazione:||2011|
|Mese di pubblicazione:||Gennaio|
|Digital Object Identifier (DOI):||10.25431/11380_1060323|
|Citazione:||GALLIGANI, Emanuele. "Application of inexact Newton method to the optimization of systems with distributed parameters" Working paper, Dipartimento di Ingegneria Enzo Ferrari - Università di Modena e Reggio Emilia, 2011. https://doi.org/10.25431/11380_1060323|
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