The report deals with theoretical and computational aspects or optimal control problems with state variable inequality constraints. The computational methods developed here solve bounded state variable problems by searching the constraint surface (or boundary) for the optimal junction point between the subarc on the boundary and the unconstrained subarc. The results obtained for the bounded state variables problem are extended to problems with discontinuous state variable and/or discontinuous differential equations. (Author).
The topic for this thesis is the state-variable inequality constrainted optimal control problem. The problem is formulated as a standard optimal control problem with one additional constraint of the form S(x(t)) or = 0. This constraint is assumed to be of p-th order where p is an integer and p or = 1. In particular, the p-th time derivative of the constraint is the first to contain the control variable explicity.
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.
This self-contained book presents in a unified, systematic way the basic principles of optimal control governed by ODEs. Using a variational perspective, the author incorporates important restrictions like constraints for control and state, as well as the state system itself, into the equivalent variational reformulation of the problem. The fundamental issues of existence of optimal solutions, optimality conditions, and numerical approximation are then examined from this variational viewpoint. Inside, readers will find a unified approach to all the basic issues of optimal control, academic and real-world examples testing the book’s variational approach, and a rigorous treatment stressing ideas and arguments rather than the underlying mathematical formalism. A Variational Approach to Optimal Control of ODEs is mainly for applied analysts, applied mathematicians, and control engineers, but will also be helpful to other scientists and engineers who want to understand the basic principles of optimal control governed by ODEs. It requires no prerequisites in variational problems or expertise in numerical approximation. It can be used for a first course in optimal control.
Computational Optimal Control: Tools and Practice provides a detailed guide to informed use of computational optimal control in advanced engineering practice, addressing the need for a better understanding of the practical application of optimal control using computational techniques. Throughout the text the authors employ an advanced aeronautical case study to provide a practical, real-life setting for optimal control theory. This case study focuses on an advanced, real-world problem known as the “terminal bunt manoeuvre” or special trajectory shaping of a cruise missile. Representing the many problems involved in flight dynamics, practical control and flight path constraints, this case study offers an excellent illustration of advanced engineering practice using optimal solutions. The book describes in practical detail the real and tested optimal control software, examining the advantages and limitations of the technology. Featuring tutorial insights into computational optimal formulations and an advanced case-study approach to the topic, Computational Optimal Control: Tools and Practice provides an essential handbook for practising engineers and academics interested in practical optimal solutions in engineering. Focuses on an advanced, real-world aeronautical case study examining optimisation of the bunt manoeuvre Covers DIRCOL, NUDOCCCS, PROMIS and SOCS (under the GESOP environment), and BNDSCO Explains how to configure and optimize software to solve complex real-world computational optimal control problems Presents a tutorial three-stage hybrid approach to solving optimal control problem formulations
This book focuses on how to implement optimal control problems via the variational method. It studies how to implement the extrema of functional by applying the variational method and covers the extrema of functional with different boundary conditions, involving multiple functions and with certain constraints etc. It gives the necessary and sufficient condition for the (continuous-time) optimal control solution via the variational method, solves the optimal control problems with different boundary conditions, analyzes the linear quadratic regulator & tracking problems respectively in detail, and provides the solution of optimal control problems with state constraints by applying the Pontryagin’s minimum principle which is developed based upon the calculus of variations. And the developed results are applied to implement several classes of popular optimal control problems and say minimum-time, minimum-fuel and minimum-energy problems and so on. As another key branch of optimal control methods, it also presents how to solve the optimal control problems via dynamic programming and discusses the relationship between the variational method and dynamic programming for comparison. Concerning the system involving individual agents, it is also worth to study how to implement the decentralized solution for the underlying optimal control problems in the framework of differential games. The equilibrium is implemented by applying both Pontryagin’s minimum principle and dynamic programming. The book also analyzes the discrete-time version for all the above materials as well since the discrete-time optimal control problems are very popular in many fields.
This paper briefly reviews the literature on necessary optimality conditions for optimal control problems with state-variable inequality constraints. Then, it attempts to unify the treatment of linear optimal control problems with state variable inequality constraints in the framework of continuous linear programming. The duality theory in this framework makes it possible to relate the adjoint variables arising in different formulations of a problem; these relationships are illustrated by the use of a simple example. This framework also allows more general problems and admits a simplex-like algorithm to solve these problems.
Necessary conditions of optimality for state-variable inequality constrained problems are derived by examining the limiting behavior of the Kelley penalty function technique. The conditions so obtained differ from those presently known, with regard to the behavior of the adjoint variables at junctions of interior and boundary arcs. A second, rigorous, derivation is given; this confirms the necessary conditions obtained by the limiting argument. These conditions are related to those known earlier; in particular, it is shown that the earlier conditions over-specify the behavior of the adjoint variables at the junctions. An example is used to demonstrate that the earlier conditions may yield non-stationary trajectories. For the regular case, it is shown that, under certain conditions, only boundary points, as opposed to boundary arcs, are possible. An analytic example illustrates this behavior. (Author).
