doqqa :: Dynamics and Control of Trajectory Tubes

6.1 The Problem of Impulse Controls 255 due to system ( 6.3 ), under boundary conditions .0/ .1/ 0/ D x ;x.ª C 0/ D x : (6.6) x.t 0 Our main interest in this book lies in closed-loop control, but prior to that we start with open-loop solutions presenting results in terms of the above. Such type of problems in terms of the moment problem was indicated in Sect. 1.3 and addressed in detail in [ 120 , 219 ]. 6.1.1 Open-Loop Impulse Control: The Value Function Define 0 .0/ ;x / D min f J. u . // j x.t 0/ D x g V.t 0 0 0 ;x / D as the value function for Problem 6.1.1 . We shall also use notation V.t 0 V.t ;x j ª C 0; ®. //; emphasizing the dependence on f ª; ®. / g : 0 0 0 .1/ Let us start by minimizing V .t ;x / D V.t ;x j ª; I . j x //: Then we first 1 0 0 find conditions for the solvability of boundary-value problem ( 6.6 ) under constraint Va r U. / ; with given. Since .1/ h l;x.ª C 0/ iDh l;x i ( ) Z ª C 0 0 h l;G.ª;t /x iC max h l; G.ª; Ÿ/B.Ÿ/d U.Ÿ/ ij Va r U. / ; 0 t 0 p and since the conjugate space for vector-valued continuous functions C Œt ;ª is 0 p 0 p V Œt ;ª; then treating functions B .t /G.ª; t /l as elements of C Œt ;ª; we have 0 0 .1/ 0 0 0 ih l;G.ª;t /x iC k B . /G .ª; /l k ; (6.7) h l;x 0 C Œt ;ª 0 n : whatever be l 2 R Here 0 0 0 0 . /G .ª; /l k D max fk B .t /G .ª; t /l kj t 2 Œt ;ª gDk l k ; k B C Œt ;ª 0 V 0 t and k l k in the braces is the Euclidean norm. We further drop the upper index p in p p C ; V while keeping the dimension p. .1/ .1/ .0/ Denote c.t ;x / D x G.ª; t /x . Moving the first term in the right-hand 0 0 0 0 side of ( 6.7 ) to the left, then dividing both sides by k B . /G .ª; /l k , we come C Œt ;ª 0 to the next proposition.

Cover

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Series Info

Title

Book Info

Preface

vi

vii

viii

ix

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Contents

Contents 2 - 4

Contents 5 - 7

Contents 8 - 9.5

Contents 9.6 - 11

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Notations

xviii

1. Linear Control Systems

1.1 The Controlled System

p.3

p.4

1.2 Control and State Constraints

1.2.1 Constraints on Control and State

p.7

1.2.2 Open-Loop and Closed-Loop Control

p.9

1.3 Optimal Control with Norm-Minimal Cost

1.3.1 Minimum Energy Control

1.3.2 Minimum Magnitude Control

p.13

p.14

p.15

1.3.3 Controllability

1.3.4 The Finite-Dimensional Moment Problem

1.4 The Reachability Problem

1.4.1 Reachability

1.4.2 Calculating Controls for the Boundary of Reach Set

p.21

p.22

1.4.3 Time-Optimal Control

1.5 Optimal Open-Loop Control for Linear-Convex Systems

1.5.1 The Necessary Conditions of Optimality

p.26

p.27

p.28

1.5.2 Degenerate Cases

p.30

p.31

1.5.3 Sufficiency Conditions for Optimality

p.33

p.34

p.35

1.5.4 A Geometrical Interpretation

p.37

1.5.5 Strictly Convex Constraints on Control

p.39

1.6 Duality Methods in Problems of Optimal Control

1.6.1 The Primal and Dual Optimization Problems

p.42

p.43

p.44

p.45

1.6.2 General Remark: Feedforward Controls

2. The Dynamic Programming Approach

2.1 The Dynamic Programming Equation

p.49

p.50

p.51

p.52

p.53

2.2 The Linear-Quadratic Problem

p.55

p.56

p.57

2.3 Reachability through the HJB Equation: Hard Bounds

2.3.1 Forward Reachability

p.60

2.3.2 Backward Reachability or Solvability Problem