## Optimization of Dynamic SystemsThis textbook deals with optimization of dynamic systems. The motivation for undertaking this task is as follows: There is an ever increasing need to produce more efficient, accurate, and lightweight mechanical and electromechanical de vices. Thus, the typical graduating B.S. and M.S. candidate is required to have some familiarity with techniques for improving the performance of dynamic systems. Unfortunately, existing texts dealing with system improvement via optimization remain inaccessible to many of these students and practicing en gineers. It is our goal to alleviate this difficulty by presenting to seniors and beginning graduate students practical efficient techniques for solving engineer ing system optimization problems. The text has been used in optimal control and dynamic system optimization courses at the University of Deleware, the University of Washington and Ohio University over the past four years. The text covers the following material in a straightforward detailed manner: • Static Optimization: The problem of optimizing a function that depends on static variables (i.e., parameters) is considered. Problems with equality and inequality constraints are addressed. • Numerical Methods: Static Optimization: Numerical algorithms for the solution of static optimization problems are presented here. The methods presented can accommodate both the unconstrained and constrained static optimization problems. • Calculus of Variation: The necessary and sufficient conditions for the ex tremum of functionals are presented. Both the fixed final time and free final time problems are considered. |

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### Contents

STATIC OPTIMIZATION | 19 |

CALCULUS OF VARIATIONS | 41 |

BASIC RESULTS | 61 |

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### Common terms and phrases

approximation axquop Suox boundary conditions boundary value boundary value problems boundary-value problem BVP-DAE Calculus of Variations chapter computed conditions for optimality constrained problem control inputs control u(t control variables controller gain cost functional costate equations degree-of-freedom denotes derivatives described dfun differential equations dynamic optimization dynamic systems end points equality constraints Example extended penalty function extrema extremum Fabien finding the control first-order necessary conditions given gradient Hessian matrix higher order terms higher-order increment initial guess input[n iteration linear system long double minimum multiple shooting method Newton's method nodes nonlinear numerical solution obtained optimal control problems optimality equations optimization problem order necessary conditions penalty function penalty parameter positive definite procedure quadratic Quasi-Newton Method Riccati equation robot satisfy scalar shooting method slack variables solve sufficient conditions trajectories unconstrained value problem variable inequality constraint vector of dimension x(tf Xj(t zero