Authors:

Martens, Björn

Document type:

Dissertation / Thesis

Title:

Necessary Conditions, Sufficient Conditions, and Convergence Analysis for Optimal Control Problems with Differential-Algebraic Equations

Advisor:

Gerdts, Matthias, Prof. Dr.

Referee:

Gerdts, Matthias, Prof. Dr.; Pickenhain, Sabine, Prof. Dr.

Submission date:

11.03.2019

Date oral examination:

12.07.2019

Publication date:

11.11.2019

Year:

2019

Pages (Book):

173

Language:

Englisch

Subject:

Optimale Kontrolle ; Differential-algebraisches Gleichungssystem ; Optimierungsproblem ; Riccati-Differentialgleichung ; Karush-Kuhn-Tucker-Bedingungen ; Hochschulschrift

Keywords:

optimal control, necessary conditions, sufficient conditions, convergence analysis, differential-algebraic equations, Hessenberg DAE, generalized equations, Hamilton Jacobi inequality, Euler discretization, uniform strong regularity

Abstract:

In this thesis, we study optimal control problems subject to differential-algebraic equations in Hessenberg form and mixed control-state constraints. Specifically, we derive necessary and sufficient conditions for problems with Hessenberg
differential-algebraic equations of arbitrary order, and then examine convergence properties of an approximated optimal control problem for the index two case. The first part of this thesis is dedicated to proving vital lemmas and theorems for later chapters. Herein, we consider linear operators, bilinear forms, generalized equations, parametric optimization problems, and linear differential-algebraic equations. In the second part, we derive a local minimum principle for optimal control problems with index one differential-algebraic equations and mixed control-state constraints by writing the problem as an infinite optimization problem, for which non-trivial Lagrange multipliers exist. Then, an explicit representation is derived for these multipliers, which yields a local minimum principle. The results are then applied to an optimal control problem with Hessenberg differential-algebraic equations of arbitrary order by reducing the index to one. The third part of this thesis examines second-order sufficient conditions for optimal control problems subject to index one differential-algebraic equations and mixed control-state constraints. Herein, a Riccati equation is used to construct a quadratic function, which satisfies a Hamilton Jacobi inequality. The main task of the verification is to prove second-order sufficient conditions for a parametric optimization problems with the assumptions at hand, which is done in the first part. Analog to the second part, the results are applied to problems with Hessenberg differential-algebraic equations of arbitrary order by reducing the index. In the last part of this thesis, we consider the implicit Euler discretization for an optimal control problem subject to an index two differential-algebraic equation in semi-explicit form and mixed control-state constraints. Typically, convergence is proven by comparing the respective Karush-Kuhn-Tucker conditions. However, there is a discrepancy between the continuous and discrete necessary conditions of optimal control problems with differential-algebraic equations of index two or higher. Hence, standard techniques fail. This was overcome by equivalently reformulating the discrete optimization problem, which has suitable Karush-Kuhn-Tucker conditions. The respective necessary conditions are then rewritten as generalized equations and a fitting convergence theorem is applied, which results in a linear convergence rate of the solution and multipliers in the essential supremum norm. «

In this thesis, we study optimal control problems subject to differential-algebraic equations in Hessenberg form and mixed control-state constraints. Specifically, we derive necessary and sufficient conditions for problems with Hessenberg
differential-algebraic equations of arbitrary order, and then examine convergence properties of an approximated optimal control problem for the index two case. The first part of this thesis is dedicated to proving vital lemmas and theorems for later chapters... »

DDC notation:

515.642

Department:

Fakultät für Luft- und Raumfahrttechnik

Institute:

LRT 1 - Institut für Angewandte Mathematik und Wissenschaftliches Rechnen

Chair:

Gerdts, Matthias

Open Access yes or no?:

Ja / Yes

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