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## Über dieses Buch

This book provides an introductory yet rigorous treatment of Pontryagin’s Maximum Principle and its application to optimal control problems when simple and complex constraints act on state and control variables, the two classes of variable in such problems. The achievements resulting from first-order variational methods are illustrated with reference to a large number of problems that, almost universally, relate to a particular second-order, linear and time-invariant dynamical system, referred to as the double integrator. The book is ideal for students who have some knowledge of the basics of system and control theory and possess the calculus background typically taught in undergraduate curricula in engineering.

Optimal control theory, of which the Maximum Principle must be considered a cornerstone, has been very popular ever since the late 1950s. However, the possibly excessive initial enthusiasm engendered by its perceived capability to solve any kind of problem gave way to its equally unjustified rejection when it came to be considered as a purely abstract concept with no real utility. In recent years it has been recognized that the truth lies somewhere between these two extremes, and optimal control has found its (appropriate yet limited) place within any curriculum in which system and control theory plays a significant role.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Since its appearance in the early sixties, the Pontryagin’s Maximum Principle is a fundamental part of Control Theory. Indeed it is the basic tool when we have to cope with Optimal Control problems, in particular those stated in terms of finite-dimensional, continuous-time dynamical systems.
Arturo Locatelli

### Chapter 2. The Maximum Principle

We present here the most significant results which are customarily grouped under the name of Maximum Principle. It supplies a set of necessary conditions of optimality for a wide class of optimal control problems. Firstly, we give a formal, yet synthetic, description of these problems is given. Then we state the theorems which, in principle, allow one to find a solution of them which satisfies the necessary conditions of the Maximum Principle.
Arturo Locatelli

### Chapter 3. Simple Constraints:

The necessary conditions of the Maximum Principle are successfully applied to the problems of this chapter. Their main features are: (1) the initial time is zero; (2) the initial state is given; (3) the performance index is of a purely integral type; (4) only simple constraints act on the system, namely we only require that (a) the final state is free or belongs to a regular variety, (b) the final time is either free or given, (c) the control variable is free or takes on values in a closed subset of the real numbers. Accordingly, we consider problems where (i) both the final time and state are given; (ii) the final time is free and the final state is given; (iii) the final time is given and the final state is not given; (iv) the final time is free and the final state is not given.
Arturo Locatelli

### Chapter 4. Simple Constraints:

The main features of the optimal control problems considered here are: (1) the initial state is not given; (2) the performance index is of a purely integral type; (3) only simple constraints act on the system, namely we only require that (a) the initial and final states are free or belong to a regular variety, (b) the final time is either free or given, (c) the control variable is free or takes on values in a closed subset of the real numbers. Accordingly, we consider problems where (i) the final state and both the initial and final times are given; (ii) the final state is not given and both the initial and final times are given; (iii) the final state is not given and both the initial and final times are free; (iv) the initial time is always zero but in the last section.
Arturo Locatelli

### Chapter 5. Simple Constraints:

The main features of the optimal control problems considered here are: (1) the initial time is zero; (2) the performance index is not of a purely integral type, that is it always includes a function of the final event; (3) the initial state is given; (4) only simple constraints act on the system, namely we only require that (a) the final state is free or belongs to a regular variety, (b) the final time is either free or given, (c) the control variable is free or takes on values in a closed subset of the real numbers. In the considered problems we have the final time given or free.
Arturo Locatelli

### Chapter 6. Nonstandard Constraints on the Final State

In this chapter we consider control problems where the final state of the controlled system is not constrained to belong to a set with the properties of a regular variety but rather to a set which is a generalization of it. In the last problem of this chapter the final time is not given but must be smaller than an assigned value.
Arturo Locatelli

### Chapter 7. Minimum Time Problems

The performance index of the seven problems considered in this chapter is simply the time required to transfer the state of the system from a regular variety to a set which is always a regular variety but in the fifth problem. The amplitude of the control variable is constrained.
Arturo Locatelli

### Chapter 8. Integral Constraints

In this chapter we tackle problems where the controlled system must satisfy integral-type constraints. They amount to requiring that the integral of some functions of time as well as of the state and control variables take on values which either are equal or less than given quantities. We first consider equality constraints and then inequality ones.
Arturo Locatelli

### Chapter 9. Punctual and Isolated Constrains

In this chapter we consider punctual and isolated equality constraints which concern the values taken on by the state variables at isolated time instants inside the control interval. These instants may or may not be specified. First we present seven problems where the initial and final state are given. Furthermore: (1) the constraints are either functions of the state only or also functions of time; (2) the final time is given or free; (3) different scenarios are discussed according to whether the time of constraints satisfaction is given or free; (4) occasionally the comparison with the constraint-free situation is presented. We end this chapter by considering a problem which leads to a seemingly contradictory conclusion because the necessary conditions can not be satisfied, even if an optimal solution is shown to exist.
Arturo Locatelli

### Chapter 10. Punctual and Global Constraints

In this chapter we consider punctual and global constraints which concern the values taken on by the state and control variables at each time instants inside the control interval. First we consider punctual and global equality constraints and assume that the controlled system possesses more than one control variable. This assumption is needed because, otherwise, the problems will be either trivial or unsolvable. Then we tackle punctual and global inequality constraints. The main features of the relevant problems are: (1) the initial state is always given; (2) the final state is either given or partially specified or constrained to belong to a set which is not a regular variety; (3) the final time is given in one problem and is free in all the remaining ones; (4) other complex constraints (punctual and isolated equality constraints, integral inequality constraints) are also occasionally present.
Arturo Locatelli

### Chapter 11. Singular Arcs

Here we focus the attention on the issues raised by the presence of the so-called singular arcs. Accordingly, we always have a linear hamiltonian function and constraints on the absolute value of the control variable. Only simple constraints are present in all considered problems but in the second one where also a punctual and isolated constraint on the state variables is included.
Arturo Locatelli

### Chapter 12. Local Sufficient Conditions

We briefly mention an important topic of optimal control theory which is strictly connected to the necessary conditions framework and is a natural complement of it, namely the sufficient conditions issue. As in the (at least conceptually) related field of minimization of functions, we can distinguish between global and local conditions. Some of the most significant results for global conditions stem from the Hamilton–Jacoby theory: however they are not mentioned in the present treatment as they are rooted in a somehow unrelated context. On the contrary, we focus the attention on local conditions resulting from a variational approach closely related to the one underlying the achievements of the Maximum Principle. Indeed if we require that, once the first variation of the performance index is zero, its second variation is positive, then we end up with local sufficient conditions. We illustrate them with reference to four significant scenarios.
Arturo Locatelli

### Backmatter

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