Appendix II Existence of Time Optimal Controls and the Bang-Bang Principle

Appendix I1

Existence of Time Optimal Controls and the Bang-Bang

1. LINEAR CONTROL SYSTEMS We shall now suppose that we are given a control system of the form X(t) = A(t)x(t)

+ b(t)u(t),

~ ( 0=) x O ,

(1)

where x ( t ) E R”, u ( t ) E R, and A and b are n x n and n x 1 matrices of functions that are integrable on any finite interval. The controls u will be restricted to the set U = {u : u is Lebesgue measurable and lu(z)l I1 a.e.},

(2)

where U is commonly called the set of admissible controls; note that U c Lm(R). A “target” point z E R” is given, and we shall consider the control problem of hitting z in minimum time by trajectories of (l),using only controls u E U . Given a control u, denote the solution of (1) at time t by x ( t : u). In fact, by variation of parameters,

+ X ( t ) Ji X-’(z)b(z)u(z)dz,

x ( t : u) = X(t)xO

(3)

where X ( t ) is the principal matrix solution of the homogeneous system X ( t ) = A(t)X(t)with X ( 0 ) = I. 169

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The attainable set at time t [namely, the set of all points (in R") reachable by solutions of (1) in time t using all admissible controls] is just d ( t ) ={ x ( t : U ) : u E U } c R " .

It will also be of use to consider the set B(t)=

{Ji X-'(z)b(z)u(z)dz : u

Note that by (3) d ( t )= X ( t ) [ x ,

I

U c R".

E

+ a(t)]= { X ( t ) x , + X(t)y :y E B(t)l

and B(t)= X - ' ( t ) d ( t ) - xo,

so that z E d ( t l )if and only if ( X - ' ( t , ) z - x o ) E B(t,). Finally, we define the bang-bang controls to be elements of ubb

=

= ( u : [ u ( z ) [ 1 a.e. z},

and we denote the set of all points attainable by bang-bang controls by dbb(f)=

{ x ( f : u) : f.4 E

ubb},

t > 0.

2. PROPERTIES OF THE ATTAINABLE SET In this section we derive the properties of d ( t )that will be central to the study of the control problem. Define a mapping I : L"([O, T I )-+ R", T > 0, by

I(u) =

[X-'(z)b(z)u(z)dz,

u E L"([O, TI).

Lemma 1 I is a continuous linear mapping between U c L"( [0, T I ) with the weak-star topology and R" with the usual topology. Proof The linearity follows directly from the additivity of integration. For the continuity, recall that on bounded subsets of L"([O, T I ) the weak-star topology is metrizable, and u, 5u if and only if JOT

y(r)u,(z)dz -+ JOT y(z)u(z) dz as r + co

for all y E L'([o, TI).

Existence of Time Optimal Controls and the Bang-Bang Principle

Suppose that (u,) E U and u,

1 71

2u. Let X-'(z)b(z) = (yi(z)). Then

{ yi>E L'([O, T I ) by assumption, and so

for all i = 1,2,. . . ,n. In other words Z(ur)-+ I(#).

As an immediate consequence of this lemma, we have the following theorem. Theorem 1 d ( T )is a compact, convex subset of R" for any T > 0. Proof The set of admissible controls is just the unit ball in Lm([0, T I ) , and so is w*-compact and convex (Banach-Alaoglu theorem). Consequently,

9 ( T )= Z(u) is the continuous, linear image of a w*-compact, convex set, so must be compact and convex. However, d ( T )= X(TKx0 + 9 w I

is just an affine translate of W(T)in R", and so itself must be compact andconvex.

A much deeper result about the structure of the attainable is contained in the following theorem, usually called the bang-bang principle. It says essentially that any point reachable by some admissible control in time T is reachable by a bang-bang control in the same time. Theorem 2 (Bang-Bang Principle) For any T > 0, d(T ) = dbb( T). Proof Note that d ( T )is just an affine translate of B ( T ) .It will then be sufficient to show that for every point x E W(T ) ,x = Z(u*) for some bang-bang control u*. Let B = I - ' ( { x ) ) n U = { u : I ( u ) = x ) n U.

( B is just the set of all admissible controls in U hitting x in time T.) By Lemma 1, B is a weak-star compact, convex subset of L"([O, T I ) , and so by Krein-Milman theorem it has an extreme point, say u*. If we can show Iu*l = 1 a.e., then we shall have found our bang-bang control and be finished.

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AppeodixII

Suppose not. Then there must exist a set E c [0, T ] of positive Lebesgue measure such that

Iu*(z)I < 1

for z E E.

In fact we can do a little better. Namely, let

m = 1,2,. . . . Em = { z E E : lu*(z)l < 1 - l/m}, Then Em = E , and since E has positive measure, at least one Em must also have positive measure. So there must exist an E > 0 and a nonnull set F c [0, TI with

(u*(z)[ < 1 - E

for z E F.

(2)

Since F is nonnull, the vector space L"(F) (again with respect to Lebesgue measure) must be infinite dimensional, and as a consequence, the integration mapping Z,:L"(F) -,R", Z,(U) =

J, X-'(z)b(z)u(z)dz,

cannot be 1-1. ( I F maps an infinite-dimensional vector space into a so we finite-dimensional vector space.) Consequently, Ker(Z,) # can choose a bounded measurable function u f 0 on F with I&) = 0. We now set u = 0 on [0,T ] - F so that

a,

Z(U) = 0, (3) and by dividing through by a large-enough constant we can certainly suppose Iu( _< 1 a.e. on [O, T I . Then by (2), Iu* +_ EUI I1,so that u* & EU E U, and by (3) Z(u* & EU) = I(u*) & d ( u ) = Z(u*) = x, i.e., u* & EU E B. Since clearly, u* = f(u* + E U ) + &* - E U )

and u f 0, u* cannot be an extreme point of B, a contradiction. 3. EXISTENCE OF TIME OPTIMAL CONTROLS

We return to the time optimal control considered in Section 1 of this Appendix. We shall assume throughout the rest of this section that the target point z is hit in some time by an admissible control, that is: (*) There exists a tl > 0 and u E U such that x(tl : u) = z.

Existence of Time Optimal Controls and the Bang-Bang Principle

173

Assumptions of the type (*) are called controllability assumptions, and, needless to say, are essential for the time-optimal control problem to be well posed. If we set t* = infit : 3u E U with x(t : u) = z} (1) [this set is non empty by (*) and bounded below by 0, so we can talk about its infimum], then t* would be the natural candidate for the minimum time, and it remains to construct an optimal control u* E U such that x(t* : u*) = z.

Theorem 1 If (*) holds, then there exists an optimal control u* such that x(t* :u*) = z in minimum time t*.

E

U

Proof Define t* as in (1).By this definition we can choose a sequence of times t, 1t* and of controls {u,) c U such that x ( t , : u,) = z, n = 1 , 2 , . . . . Then 12 - x(t* :u,)J =

Jx(t,: u,) - x(t* :u,)f.

However, from the fundamental solution,

Jr

x(tn : Un) - x(t* : u,) = X(t,)Xo - X(t*)Xo

- X(t*) Consequently,

X(tJ

J:

(2)

X-'(T)b(T)u,(T)dT

X-'(z)b(z)u,(z)dz.

Ix(t, : u,) - x(t* :u,)] IIX(t,)xo - X(t*)xoJ

jsa'

+ Ix(t*)- x(t,)l

x-'(z)b(.r)u,,(z)dz

The first term on the right-hand side of (3) clearly tends to zero as n .+ 00. [X(-)is continuous.] The second term can be bounded above by

l:

IJx(tn)ll

lx-'(T)b(T)l

6:"

lun(z)l d~ IIIx(tn)lJ IX-'(T)~(T)I dz

since Iu,I I1. Consequently, as n .+ 00 this term also tends to zero. Finally, the third term tends to zero, again by the continuity of X(.). Plugging (3) back into (2), we get x(t*:u,) .+ z as n .+ 00,

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i.e., z E d ? t * ) . However, by Theorem 2.1, d ( t * ) is compact, and so z E d(t*). Consequently, 2 = x(t*

for some u*

E

: u*)

U.

Note that once we have shown that an optimal control exists, the bang-bang principle guarantees that an optimal bang-bang control exists. The above proof also carries over with only cosmetic changes to continuously moving targets z(t), 0 It < T , in fact to “continuously moving” compact target sets and to the general state equation

+

x = A(t)x(t) B(t)u(t),

where B(t) is IZ x m and u(t) = (ul(t),. . . ,u,(t))’.