While optimality conditions for optimal control problems with state constraints have been extensively investigated in the literature the results pertaining to numerical methods are relatively scarce. This book fills the gap by providing a family of new methods. Among others, a novel convergence analysis of optimal control algorithms is introduced. The analysis refers to the topology of relaxed controls only to a limited degree and makes little use of Lagrange multipliers corresponding to state constraints. This approach enables the author to provide global convergence analysis of first order and superlinearly convergent second order methods. Further, the implementation aspects of the methods developed in the book are presented and discussed. The results concerning ordinary differential equations are then extended to control problems described by differential-algebraic equations in a comprehensive way for the first time in the literature.

"Optimal Control" reports on new theoretical and practical advances essential for analysing and synthesizing optimal controls of dynamical systems governed by partial and ordinary differential equations. New necessary and sufficient conditions for optimality are given. Recent advances in numerical methods are discussed. These have been achieved through new techniques for solving large-sized nonlinear programs with sparse Hessians, and through a combination of direct and indirect methods for solving the multipoint boundary value problem. The book also focuses on the construction of feedback controls for nonlinear systems and highlights advances in the theory of problems with uncertainty. Decomposition methods of nonlinear systems and new techniques for constructing feedback controls for state- and control constrained linear quadratic systems are presented. The book offers solutions to many complex practical optimal control problems.

In this thesis we consider numerical methods for solving state-constrained optimal control problems. There are two main focii in the research, i.e. state- constrained optimal open-loop and feedback control problems. For all cases, we reformulate the constrained optimal control problem to the unconstrained problem through a penalty method. The state-constraints which we discuss here are only in the form of inequalities but for both purely state-constraint and control-state constraint types. For solving state-constrained optimal open-loop control problems, we establish a power penalty method and analyze its convergence. This method is then implemented in MISER 3.3 to do some numerical tests. The results con rm that the method work very well. Furthermore, we use the power penalty method to discuss a sensitivity analysis. On the other hand, for solving state-constrained optimal feedback control problems we construct a new numerical algorithm. The algorithm based on upwind nite di erence scheme is iterated in order to increase the accuracy and speed of computation. In particular to address the curse of dimensionality, a special method for generating grid points in the domain is developed. Numerical experiment shows that the computational speed increases significantly with this modi ed method. Moreover, for further improvement in the accuracy the algorithm can be combined with Richardson Extrapolation Method.

A focused presentation of how sparse optimization methods can be used to solve optimal control and estimation problems.

This work presents recent mathematical methods in the area of optimal control with a particular emphasis on the computational aspects and applications. Optimal control theory concerns the determination of control strategies for complex dynamical systems, in order to optimize some measure of their performance. Started in the 60's under the pressure of the "space race" between the US and the former USSR, the field now has a far wider scope, and embraces a variety of areas ranging from process control to traffic flow optimization, renewable resources exploitation and management of financial markets. These emerging applications require more and more efficient numerical methods for their solution, a very difficult task due the huge number of variables. The chapters of this volume give an up-to-date presentation of several recent methods in this area including fast dynamic programming algorithms, model predictive control and max-plus techniques. This book is addressed to researchers, graduate students and applied scientists working in the area of control problems, differential games and their applications.