5. Controllability of Constrained Linear Autonomous Systems

5.

CONTROLLABILITY OF CONSTRAINED LINEAR AUTONOMOUS SYSTEMS

5.1

Introduction

I n optimal c o n t r o l and g e n e r a l l y i n t h e c o n t r o l o f dynamic systems t h e r e a r e c e r t a i n system-theoretic concepts which play v i t a l r o l e s . One o f these i s s t a b i l i t y , but perhaps the most important i s c o n t r o l l a b i l i t y . a dynamic system iC = f(x,u)

Loosely speaking we c a l l

c o n t r o l l a b l e i f t h e r e e x i s t s an

admissible c o n t r o l f u n c t i o n u( . ) which t r a n s f e r s t h e s t a t e of t h e system from an a r b i t r a r y i n i t i a l value xo t o t h e o r i g i n o f t h e s t a t e space i n f i n i t e time.

I n t u i t i v e l y t h e concept

o f c o n t r o l l a b i l i t y i s basic t o t h e problem o f designing minimal time c o n t r o l l e r s , v i z . c o n t r o l l e r s which s t e e r xo t o the o r i g i n o f t h e s t a t e space i n minimum time. This chapter i s devoted t o a study o f t h e c o n t r o l l a b i l i t y p r o p e r t i e s of t h e 1i n e a r autonomous dynamic system iC=AxtBu where A u

E

E

nxn R ,B

E

(5.1.1)

R n x m a r e constant matrices and x

E

Rn,

Rm and where t h e c o n t r o l v a r i a b l e u i s required t o s a t i s f y

the constraint u(t)

E

R

= Rm,

t

E

lo,-).

(5.1.2)

The f o l l o w i n g sections review c e r t a i n well-known conditions f o r c o n t r o l l a b i l i t y o f (5.1.1) t o the i n t e r i o r o f R (0 (5.1.1)

E

when 52 = Rm and when 0 belongs

Int(R)).

The c o n t r o l l a b i l i t y o f

when 0 L I n t ( R ) i s then i n v e s t i g a t e d i n d e t a i l and

new r e s u l t s a r e presented p e r t a i n i n g t o t h e ' a r b i t r a r y 151

152

LINEAR-QUADRATIC EXTENSIONS

i n t e r v a l n u l l - c o n t r o l l a b i l i t y ' o f (5.1.1) 5.1.1

subject t o (5.1.2).

Unconstrained Case

When s2 E Rm, c o n d i t i o n s f o r t h e c o n t r o l l a b i l i t y o f (5.1.1) a r e much s i m p l i f i e d .

F i r s t we d e f i n e p r e c i s e l y t h e concept

o f c o n t r o l 1a b i 1ity. DEFZNZTZON 5 . 1 . 1

The l i n e a r autonomous system (5.1.1)

c o n t r o l l a b l e i f f o r each xo

E

is

Rn t h e r e e x i s t s a bounded

measurable c o n t r o l f u n c t i o n which steers xo t o t h e o r i g i n of t h e s t a t e space i n f i n i t e time. Necessary and s u f f i c i e n t c o n d i t i o n s f o r c o n t r o l l a b i l i t y were supplied by R.E. f o l 1ows

.

Kalman d u r i n g t h e e a r l y 1960s and a r e as

A necessary and s u f f i c i e n t c o n d i t i o n f o r i s that t o be c o n t r o l l a b l e when Q 5

THEOREM 5 . 1 . 1 (5.1.1)

Rank(Q) = n where

A Q = [B,AB,.

. .,A

(5.1.3)

n-1

Bl

.

(5.1.4)

Equivalently, W(0,t)

t

4 J@(t,.r)BB

T T

cp (t,-r)d.r

0

for a l l t

>0

>0

(positive-definite) (5.1.5)

where cp(t,.r) s a t i s f i e s

aT @(t,.r)

= A@(t,.r),

@(T,T) =

and i s t h e t r a n s i t i o n m a t r i x associated w i t h

I

(5.1.6) = Ax.

CONTROLLABILITY CONDITIONS

153

Inequality (5.1.5) actually allows one to compute a continuous control function which steers xo to 0. THEOREM 5.7.2

u(t)

=

The control function

-BT@ T(T,t)W-l(O,T)@(T,O)xo,

t

E

[O,Tl

(5.1.7)

steers xo to the origin in time T. PROOF

The solution of (5.1.1) at time T is given by

x(T)

=

@(T,O)xo

T

.

+ .f@(T,T)Bu(T)dT 0

(5.1.8)

Substituting (5.1.7) into (5.1.8) and using (5.1.5) yields

x(T)

=

@(T,O)x0

- @(T,O)x0

=

0

.

(5.1.9)

Note that the rank condition (5.1.3) is easy to check, especially when n is small. On the other hand (5.1.3) is not the correct condition when A and B are time-varying, but (5.1.5) remains valid. Note also that controllability here implies not only that there is a control function u( .) which steers xo to 0 in finite time but also that there is a control function u(.), given by (5.1.7) which performs this function in an arbitrary time interval [O,Tl. Thus when a z Rm controllability actually implies 'arbitrary-interval controllability', and the converse is trivially also true. 5.1.2

Zero Interior to

0

We assume here that R 5 Rm is arbitrary but that

154

L INEAR-QUADRATIC EXTENS IONS

0

E

Int(R).

(5.1 .lo)

In general, whether or not (5.1.10) is satisfied, it would be too much to expect that (5.1.1) is controllable subject to the restriction (5.1.2). Therefore we introduce the following definition. D E F I N I T I O N 5.1.2 The dynamic system (5.1.1) is nullcontrollable if there exists an open set V in Rn which contains the origin and for which any xo E V can be controlled to the origin in a finite time by a bounded, measurable control function.

Clearly null-controllability is really controllability in a sphere which surrounds the origin and so controllability implies null-controllability but the converse is not usually true. We can now state the relevant null-controllability theorem. THEOREM 5.1.3

System (5.1.1) is null-controllable if (5.1.10) holds, if and only if Rank(Q)

=

n.

(5.1.11)

PROOF

If (5.1.1) is null-controllable it is easy to show that (5.1.1) is controllable if the restraint (5.1.2) is removed: condition (5.1.11) then follows. On the other hand, (5.1.11) implies that (5.1.7) is well defined and so for xo E CxlIlxII < E ~ Iit follows that Ilu(t)ll < c2(c1) for t E [O,Tl where E ~ ( E ~-+) 0 as cl 0. As 0 E Int(R) it is clear that for E~ sufficiently small, u(t) E R , t E [O,Tl. -+

Note that as T is arbitrary in the above construction there

CONTROLLABILITY CONDITIONS

155

exists for each T > 0 a s e t V(T) as specified in Definition (5.1.2). Therefore when 0 E I n t ( R ) null-controllability i s , actually, equivalent t o 'arbitrary-interval nu1 1-controllability'

.

5.1.3

Reachable Sets

The notion of a reachable s e t and certain of i t s properties play important roles in controllability theory, as we shall see i n the following pages. We now define the reachable sets See Section 5.A, the Appendix, for R R ( t ) , R C H ( n ) ( t ) and R,. certain set-theoretic definitions. DEFTNITION 5. I . 3 The reachable s e t Rn( t )( RCH(n) ( t ) )a t time t i s defined t o be the s e t of a l l points i n Rn t o which the origin can be steered a t time t by a bounded measurable control function u ( - ) which s a t i s f i e s U ( T ) E 0 ( u ( T ) E Convhull(f2)) for a l l T E [ O , t l . The reachable s e t R, i s the u n i o n over positive t of the s e t s R R ( t ) .

[ll

Consider the linear process (5.1.1) w i t h constraint s e t Rc, Rm. The reachable s e t R R ( t ) a t time t i s convex. If in addition R i s compact then R R ( t ) i s convex, compact, and varies continuously with t on t 2 0 and

TffEOREM 5 . 7 . 4

RCH(Q) t > = R ~ ( t ) *

The f i r s t part of the theorem and the compactness of R Q ( t ) f Q i s compact follow d i r e c t from Lemma 4A [l, p. 1631 on the range of a vector measure. The equivalence of R R ( t ) , RCH(Q) ( t ) i s proved in Theorem 1 A [l, p. 1641 by n o t i n g t h a t R R ( t ) E R C H ( n ) ( t ) and t h a t these s e t s are convex and compact if R i s compact and by proving t h a t R n ( t ) i s dense in PROOF

RCH( Q ) ( t,

L I NEAR-QUADRATIC EXTENSIONS

156

The n e x t theorem shows t h a t t h e n a t u r e o f t h e reachable s e t s o f (5.1.1)

s u b j e c t t o (5.1.2)

i s d i r e c t l y related t o the

c o n t r o l l a b i l i t y of t h e n e g a t i v e o f system (5.1.1).

TtlEOREM 5 . 7 . 5 t o (5.1.2)

o f system (5.1.1)

The reachable s e t ,R

subject

c o n t a i n s a neighbourhood o f t h e o r i g i n i f and o n l y

i f t h e system

iC s u b j e c t t o (5.1.2)

=

-

AX

-

BU

(5.1.12)

i s null-controllable.

PROOF

Suppose t h a t u ( t ) , t E [O,Tl s t e e r s (5.1.12) t o 0 a t t = T. Then i t i s easy t o show t h a t u ( T - t ) ,

from 0 t o xo a t t = T.

s t e e r s (5.1.1) eas i1y v e r i f i e d

.

from xo t

[O,Tl

E

The converse i s a l s o

The n e x t theorem i s o f fundamental importance i n s t u d i e s of null-controllability.

THEOREM 5.7.6

[2]

such t h a t Bu = 0. reachable s e t ,R zero v

E

Suppose t h a t t h e r e e x i s t s a v e c t o r u

E

R

Then, t h e o r i g i n i s i n t e r i o r t o t h e o f (5.1.1)

i f and o n l y i f t h e r e i s no non-

Rn such t h a t T At v e Bu G O f o r a l l t

>0

and f o r a l l u

E

R

. (5.1.13)

PROOF

By Theorem 5.1.4,

RR(t) i s convex f o r a l l t

Furthermore t h e f a c t t h a t t h e r e i s a u i m p l i e s t h a t RR(tl) all t

> 0.

E

R such t h a t Bu = 0

5 R R ( t 2 ) f o r tl S t 2 and 0

T h e r e f o r e i t f o l l o w s t h a t ,R

> 0.

E

RR(t) for

i s t h e u n i o n of

nested i n c r e a s i n g convex s e t s which c o n t a i n t h e o r i g i n , and

CONTROLLAB I LITY CONDITIONS

157

is thus a convex set which contains the origin. If in fact the origin is interior to ,R there exist n + 1 points in ,R whose convex hull contains the origin as an interior point. It follows that these n + 1 points must be contained in RO(t) for some t > 0 since these sets are increasing. Therefore it is clear that the origin is interior to ,R if and only if the origin is interior to RO(t) for some t > 0. Consequently if the origin is not interior to R, there exists a v E Rn such that

v Tx(t) g o

(5.1.14)

for all t > 0 and all admissible controls u(*). (5.1.14) is just t

vT JeA(t-T)Bu(T)dr

Inequality

(5.1.15)

G0

0

and it follows by continuity and a special choice of u(.) that

v TeAt Bu G O for all t > O and for all u

E

R.

(5.1.16) On the other hand, if the origin is interior to ,R there cannot exist a vector v E Rn which satisfies (5.1.16), as the existence of such a vector would imply inequality (5.1.14). 5.1.4

Zero Interior to Convhull(0)

The last theorem and Theorems 5.1.4 and 5.1.5 may now be combined to yield the following result which is similar to Theorem 5.1.3.

158

LINEAR-QUADRATIC EXTENSIONS

THEOREM 5 . 1 . 7 0

E

Suppose t h a t f2 i s compact and t h a t

Int(Convhull(f2)).

Then, system (5.1.1)

s u b j e c t t o (5.1.2)

i s n u l l - c o n t r o l l a b l e i f and o n l y i f Rank(Q) = n.

PROOF

(5.1.17)

i s compact, Theorem 5.1.4 i m p l i e s t h a t f2 can be replaced by Convhull(f2) as f a r as reachable s e t s a r e concerned. As z e r o belongs t o Convhull(f2) we have t h a t t h e r e e x i s t s a v e c t o r u E Convhull(f2) such t h a t Bu = 0. Theorem 5.1.6 t h e n i m p l i e s t h a t z e r o i s i n t e r i o r t o ,R i f and o n l y i f t h e r e i s no non-zero v E Rn such t h a t As

$2

T At v e Bu G O f o r a l l t

> O and f o r a l l u

E

Convhull(f2). 18) ( 5 .l.

We now prove t h a t t h i s i s t r u e if and o n l y i f (5.1.17) holds. Suppose f i r s t t h a t Q does n o t have r a n k n. Then, t h e r e e x i s t s a v e c t o r v E Rn such t h a t T vQ=O.

(5.1.19)

Now i t i s w e l l known t h a t t h i s i m p l i e s t h a t T At v e B = O

(5.1.20)

so t h a t T At v e Bu = 0 f o r a l l t > O and a l l u

E

Convhull(0) (5.1.21)

which c o n t r a d i c t s t h e statement t h a t z e r o i s i n t e r i o r t o . R,

CONTROLLABILITY CONDITIONS

159

Suppose now that Q has rank n but that vTeAtBu = 0 for all t and all u E Convhull(R). Then by setting t = 0 we obtain

v TBu

=

0 for all u

E

Convhull(S2)

(5.1.22)

and successive differentiations of vTeAtBu and evaluations at t = 0 yield

vTAi Bu

=

0 for all u

E

Convhull(R),

i=l ,... ,n-1. ( 5.1.23)

As zero is interior to Convhull(R), Equations (5.1.22) and (5.1.23) imply that Q has rank less than n, which i s a contradiction. Therefore (5.1.17) implies that there exists u E Convhull(R) such that vTeAtBu is not identically zero for all t. Now either v TeAtBu changes sign as a function of t for this fixed u E Convhull(R) or it is of one sign. Suppose in fact that vTeAt Bu G O (not identically zero, as we have proved). As zero is interior to Convhull(S2) we can switch the sign of vTeAtBu to be positive (whenever it is non-zero) by reversing the sign of u so that (5.1.18) does not hold for any v E Rn. This completes the proof that zero is interior to R, if and only if (5.1.17) holds. The observation that Rank[B,AB, ...,A n-1Bl

=

Rank[-B,AB,-A 2B,...,(-l) nAn-1B l (5.1.24)

then yields the theorem.

160 5.1.5

LINEAR-QUADRATIC EXTENSIONS Global and Non-linear Results

As remarked prior to Definition 5.1.2 controllability (as distinct from null-controllabil ity) cannot in general be expected of ( 5 . 1 . 1 ) when it is subjected to the constraint ( 5 . 1 . 2 ) . However, the following theorems provide conditions which guarantee that controllability (null-controllability in the large) can be achieved by (5.1.1) subject to ( 5 . 1 . 2 ) . THEOREM 5 . 1 . 8

[l]

Consider the system ( 5 . 1 . 1 ) subject to the control constraint ( 5 . 1 . 2 ) . Suppose that zero belongs to the interior of R , that Rank(Q) = n, and that A is stable (Re()\) < 0). Then ( 5 . 1 . 1 ) is controllable. The stability of i = Ax ensures that any initial point xo E Rn can be steered by u(-) E 0 until x(t) approaches 0 and therefore enters the domain of null-controllability of ( 5 . 1 . 1 ) . But then x(t) can be steered to the origin in a finite time. PROOF

When the control function is scalar the strict inequality on Re()\) can be relaxed, but then compactness of R is assumed. THEOREM 5.1.9

[l]

Suppose that m = 1 and that R is a compact set which contains zero in its interior. Then (5.1.1) is controllable if and only if Rank(Q) = n and every eigenvalue X of A satisfies Re(A) GO. When m > 1 we have the following theorem.

THEOREM 5.1.10

[ll

Suppose that R is a compact set which contains zero in its interior. Assume that no two Jordan canonical blocks of A contain equal eigenvalues of A. Then (5.1.1) is controllable if and only if Rank(Q1 = n and every

161

CONTROLLABILITY CONDITIONS eigenvalue A of A satisfies Re(A) G O .

Finally, we remark that null-controllability of a non-linear system can sometimes be deduced from the null-controllabil ity of its linearization. This is made precise in the following theorem [ l l .

THEOREM 5.7. I 7

Consider the non-1 inear autonomous system i

=

(5.1.25)

f(x,u)

where f:Rn+' Rn is once continuously differentiable in x and u. Suppose that zero is interior to the constraint set R c _ Rm, and assume that -f

f(0,O)

=

( 5 , l . 26)

0

Rank[B,AB, ...,A n-1 Bl

=

where A

=

-(O,O), af ax

B

=

-(O,O) af

au

(5.1.27)

n

.

(5.1.28)

Then (5.1.25) is null-controllable. 5.2

Zero Not Interior to Constraint Set or its Convex Hull

5.2.1

Oscillatory Systems

In this section we devote attention to recent work on the controllability of (5.1.1) subject to (5.1.2) when 0 k Int(R). Initially research on this problem was motivated by the question [31: can the motion of a simple pendulum be brought to rest in a finite time by the application of a unit force acting only in one direction? In mathematical terms one asks

162

LINEAR-QUADRATIC EXTENSIONS

whether t h e system Xl

=

x2

x2

=

- x1 + u

(5.2.1)

is ( n u l l - ) c o n t r o l l a b l e when u(t)

E

[OYlI

y

t

E

[O,..).

(5.2.2)

Clearly (5.2.1) i s c o n t r o l l a b l e when u ( t ) E R 1 and i s not r e s t r i c t e d according t o (5.2.2). Furthermore i f u ( t ) E [ - ~ , 1 1 E > 0 Theorem 5.1.3 guarantees t h a t (5.2.1) i s null-controllable. Saperstone and Yorke were t h e f i r s t t o prove t h e following r e s u l t when R = [0,11 . Suppose t h a t m = 1 and R = [0,11. The system (5.1.1) is null-controllable i f and only i f a l l the eigenvalues o f A have non-zero imaginary p a r t s and Rank(()) = n .

THEOREM 5.2.7

T h i s interesting theorem implies t h a t i f n i s odd (implying a t l e a s t one real eigenvalue) (5.1.1) i s not null-controllable. Note t h a t the conditions of the theorem a r e s a t i s f i e d by (5.2.1) and (5.2.2) so t h a t (5.2.1) i s indeed null-controllable subject t o the c o n s t r a i n t t h a t u ( t ) E [0,11. Note a l s o t h a t Theorem 5.1.4 allows one t o replace the interval [ O y l l i n Theorem 5.2.1 by the s e t ( 0 , l ) consisting o f t h e two end points of the compact i n t e r v a l . I t turns out t h a t Theorem 5.2.1 i s not t h e best t h a t can be obtained when 0 k Int(R) and 0 k Int(Convhull(R)). The

CONTROLLABILITY CONDITIONS

163

a p p r o p r i a t e g e n e r a l i z a t i o n o f Theorem 5.2.1 which we i n v e s t i g a t e f u l l y i n t h e n e x t s e c t i o n i s due t o Bramner [2]. Brammer ' s Theorem

5.2.2

The f o l l o w i n g theorem i s a major g e n e r a l i z a t i o n o f Theorem 5.2.1 though i t i s simple i n form. However, Brammer's proof i s exceedingly l e n g t h y and i n t r i c a t e so t h a t we p r e f e r t o p r e s e n t H a j e k ' s s h o r t p r o o f [4].

THEOREM 5 . 2 . 2

[21

Suppose t h a t

there i s a u

E

Sl such t h a t Bu = 0

(5.2.3)

and t h a t t h e s e t Convhull(C2) has non-empty i n t e r i o r i n Rm. (5.2.4) Then (5.1.1)

s u b j e c t t o (5.1.2)

i s n u l l - c o n t r o l l a b l e i f and

only i f Rank(Q1 = n

(5.2.5)

and t h e r e i s no r e a l e i g e n v e c t o r v o f A T v Bu G O f o r a l l u

E

Sl.

T satisfying (5.2.6)

C l e a r l y (5.2.6) i s t r i v i a l l y s a t i s f i e d i f A has no r e a l eigenvalues so t h a t Theorem 5.2.2 reduces t o Theorem 5.2.1 when m = 1 and Sl = [0,11

.

HSjek's p r o o f o f t h e above theorem r e q u i r e s t h e f o l l o w i n g lemmas, which we o n l y s t a t e .

L INEAR-QUADRATIC EXTENSIONS

164

LEMMA 5.2.7 Suppose t h a t A E R n x and V i s a closed convex cone i n Rn such t h a t v and -v belong t o V only i f v = 0. If At V i s invariant under eAt, v i z . e V 5 V f o r a l l t > 0 then V contains an eigenvector of A. At The l i n e a r hull of {e B u l t > 0, u E Q) i s Rn LEMMA 5 . 2 . 2 i f and only i f Rank(Q) = n (provided t h a t (5.2.3), (5.2.4) hold)

.

The system A = Ax t Bu s a t i s f i e s (5.2.5), LEMMA 5 . 2 . 3 (5.2.6) i f and only i f the negative system d = - Ax - Bu does. PROOF OF THEOREM 5 . 2 . 2 As a consequence of Theorems 5.1.5, 5.1.6 and Lemma 5.2.3 we have t h a t (5.1.1) i s null-controllable subject t o conditions (5.2.3) and (5.2.4) i f and only i f there is no non-zero vector v E Rn such t h a t

v T eA t Bu G O f o r a l l t > O and f o r a l l u

E

Q. (5.2.7)

If (5.2.5) f a i l s i t is easy t o show via Lemma 5.2.2 t h a t there is a non-zero vector v E Rn such t h a t

v T eA t Bu = 0 f o r a l l t and a l l u

E

If (5.2.6) is s a t i s f i e d by a real eigenvector v we have t h a t

0. E

T At T At v e Bu = v Bu.e where A i s the real eigenvalue associated w i t h v . from (5.2.6) and (5.2.9) t h a t

(5.2.8)

T Rn of A (5.2.9) I t follows

CONTROLLABILITY CONDITIONS At vTe Bu GO f o r a l l t > O and a l l u E fl.

165 (5.2.10)

We conclude from t h e above t h a t t h e c o n d i t i o n s (5.2.5) (5.2.6)

and

a r e necessary.

Next we assume t h a t (5.1.1) t h e c o n d i t i o n s (5.2.5),

i s not null-controllable but that

(5.2.6)

hold;

t h i s leads t o a contra-

diction. L e t W be t h e s e t o f a l l l i n e a r combinations w i t h non-negative c o e f f i c i e n t s o f p o i n t s eAtBu f o r t > 0, u E fl. Then from (5.2.7) we have t h a t i f (5.1.1) t h e r e i s a v E Rn such t h a t

i s not null-controllable

T v w G 0 for all w

E

W.

(5.2.11)

Now W i s a closed convex cone i n Rn which by Lemma 5.2.2 and c o n d i t i o n (5.2.5)

satisfies

W

+

(-W)

and i s o b v i o u s l y i n v a r i a n t under eAt, eAtWs

w

(5.2.12)

= Rn

viz.

f o r a l l t 2 0.

(5.2.13)

Next we note t h a t t h e s e t V c o n s i s t i n g o f those vectors v

E

Rn which s a t i s f y

T v w

<0

f o r each w

E

W

(5.2.14)

i s a l s o a closed convex cone i n Rn. Furthermore (5.2.12) guarantees t h a t v and -v both belong t o V o n l y i f v = 0.

We

166

L INEAR-QUADRATIC EXTENS IONS

see then that the invariance of W under eAt implies invariance

T

.'

of V under eA Lemma 5.2.1 then implies that V contains an T eigenvector v of A But then (5.2.6) is satisfied by an eigenvector of AT, which contradicts our assumption, and the sufficiency of the conditions of the theorem is proved.

.

We note that Heymann and Stern [5] have also offered a geometric proof o f the above theorem. However, Hajek's approach seems to us to be the simplest. 5.2.3

Global Results

The following theorem shows that if s2 is suitably unbounded, the reachable set ,R of (5.1.1) is all of Rn.

THEOREM 5 . 2 . 3 [21 If s2 is a cone with vertex at the origin and has non-empty interior in Rm then (5.2.5), (5.2.6) are necessary and sufficient for the reachable set ,R of (5.1.1) to be all of Rn. PROOF

The necessity of (5.2.5) and (5.2.6) is already proved. For sufficiency we first note that as 0 is a cone with vertex Furthermore at the origin so also is the reachable set .,R the reachable set i s convex and contains the origin in its interior if (5.2.51, (5.2.6) are satisfied. Combining these two properties of ,R we obtain the desired result.

COROLLARY 5 . 2 . 1 If s2 is a cone with vertex at the origin and has non-empty interior in Rm then (5.2.5), (5.2.6) are necessary and sufficient for (5.1.1) to be control 1 able. PROOF

Theorem 5.2.3 implies via the proof of Theorem 5.1.5 that (5.1.12) is controllable if and only if ,R is all of Rn. It then follows under the stated assumptions that (5.1.12) is

CONTROLLABILITY CONDITIONS c o n t r o l l a b l e i f and o n l y i f (5.2.5) However (5.1.1) (5.1.12)

s a t i s f i e s (5.2.5),

167

and (5.2.6)

are satisfied.

if and o n l y i f

(5.2.6)

does, and t h e c o r o l l a r y i s proved. i n d i c a t e s i t i s n o t always necessary t h a t R

As Theorem 5.1.8

be unbounded f o r ,R controllable.

t o be a l l o f Rn o r f o r (5.1.1)

t o be

The f o l l o w i n g theorem i s t h e a p p r o p r i a t e

g e n e r a l i z a t i o n o f Theorem 5.1.8

and t h e proof i s t h e same.

Suppose t h a t t h e c o n d i t i o n s o f Theorem 5.2.2

THEOREM 5.2.4

a r e s a t i s f i e d and t h a t A i s s t a b l e (Re(X)
Then (5.1.1)

i s controllable. 5.3

Arbitrary-Interval Null-Controllability

As noted i n S e c t i o n 5.1.1,

t h e necessary and s u f f i c i e n t con-

d i t i o n s f o r c o n t r o l l a b i l i t y o f (5.1.1) when R = Rm a r e such t h a t if (5.1.1) p o i n t xo

E

s u b j e c t t o (5.1.2) i s c o n t r o l l a b l e then any

Rn can be t r a n s f e r r e d t o t h e o r i g i n o f t h e s t a t e

space i n t i m e

T > 0, where T i s a r b i t r a r y . T h i s i s e a s i l y

seen t o be t r u e as a consequence o f (5.1.7) which e x p l i c i t l y constructs a control function u ( t ) , 0


which t r a n s f e r s

xo a t t i m e 0 t o 0 a t t i m e T. When R c R m and 0

E

I n t ( R ) we saw t h a t (5.1.1)

c o n t r o l l a b l e i f and o n l y i f Rank(Q) = n. o f Theorem 5.1.3 when 0

E

shows t h a t i f (5.1.1)

i s null-

I n fact, the proof

i s null-controllable

Int(f2) then t h e r e i s a neighbourhood V(T) o f t h e

o r i g i n which can be t r a n s f e r r e d t o t h e o r i g i n i n t i m e T where

> 0,

T i s arbitrary.

On t h e o t h e r hand i f 0 i I n t ( R ) we a r e n o t always a b l e t o t r a n s f e r a neighbourhood o f t h e o r i g i n , whose s i z e depends

LINEAR-QUADRATIC EXTENSIONS

168

upon T, t o the o r i g i n i n time T > 0. The example defined by (5.2.1) and (5.2.2) i s a case i n point. Indeed if x2(0) > 0 application of t h e non-negative control u ( t ) E [0,11 only increases the value of x 2 ( t ) away from i t s s t a r t i n g value of x2(0). More precisely, we have t h a t

xl(t)

= x,(O)cos(t)

t

+ x 2 ( 0 ) s i n ( t ) t ./

sin(t-T)u(T)dT

0

(5.3.1) and x2(t) =

-

t x , ( O ) s i n ( t ) + x,(o)cos(t) + ./ cos(t-T)u(T)dT 0

(5.3.2)

so t h a t when x l ( 0 ) x2(t)

= 0 we have

= x2(0)cos(t)

t

+ ./ cos(t-T)u(T)dT 0

(5.3.3)

which i s positive f o r x2(0) > O and t
The above observations therefore lead us t o a study o f ' a r b i t r a r y - i nterval nu1 1-control l a b i l i t y ' which we define precisely i n the next section. A number of the r e s u l t s presented i n the following sections were developed i n collaboration w i t h M. Pachter 161.

CONTROLLABILITY CONDITIONS 5.3.1

169

Preliminary Results

DEFlNlTlON 5.3.7 We call (5.1.1) arbitrary-interval nullcontrollable if for each T > 0 there exists an open set V(T) in Rn which contains the origin and for which any xo E V(T) can be controlled to the origin at time T by a bounded measurable control function. The next theorem relates the properties of the reachable sets of (5.1.1) to arbitrary-interval null-controllability. The system (5.1.1) subject to (5.1.2) is arbitrary-interval nu1 1-control lable if and only if for each T > 0 the reachable set RR(T) of (5.1.1) contains the origin in its interior. THEOREM 5.3.7

PROOF

For fixed but arbitrary T we have

RR(T)

=

T

U eA(T-T)BU(r)drlu(T) 0

E

R , O < T GT)

(5.3.4)

If (5.1.1) is arbitrary-interval null-controllable we have for each T > 0 that -r

V(T)

=

;eA(T-T)Bu(r)dTlu(r) 0

E

R, 0

< T < T). (5.3.5)

It follows from (5.3.4), (5.3.5) and the invertibi ity of eAT that RR(T) contains the origin in its interior because V(T) does. The converse, namely that V(T) exists f RQ(T) contains the origin in its interior, follows in the same way via the invertibility of eAT . Note that the relationship between arbitrary-intervul nu1 1-

LINEAR-QUADRATIC EXTENSIONS

170

controllability and reachabil ity is simpler than that between null-controllabil ity and reachabil ity (cf. Theorems 5.1.5, 5.1.7).

The following theorem exploits the relationship.

THEOREM 5.3.2 A necessary condition for arbitrary-interval null-controllability o f (5.1.1) subject to (5.1.2) is that there should exist for each non-zero v E Rn and for each T > 0 a time T , 0 < T < T and a u E R such that vTeATBu > 0.

(5.3.6)

Suppose that (5.1.1) subject to (5.1.2) i s arbitraryinterval null-controllable. Theorem 5.3.1 then implies that for each T > 0, Rn(T) contains the origin in its interior. This implies that there i s no non-zero v E: Rn such that PROOF

T

vTJeA(T-T)Bu(-r)d-r
for all u( - ) such that

U(T)

E R , O<

0

T

GT.

(5.3.7)

Now if (5.3.6) were not satisfied for each non-zero v E Rn for some T, O < T < T, there would exist a non-zero v E Rn such that

v TeAT Bu

GO

for all

T,

0<

T

< T and all u

E

R (5.3.8)

but this would imply that vT;eA(T-')Bu(r)d.r 0

< 0 for all u(-) such that

U(T)

E

R, O< TGT (5.3.9)

CONTROLLABILITY CONDITIONS

171

which contradicts the fact that RQ(T) contains the origin in its interior. Therefore the theorem is proved. We can now deduce an important necessary condition when bounded.

52

is

Assume that 52 is bounded. Then a necessary condition for arbitrary-interval null-controllability of (5.1.1) subject to (5.1.2) is that

THEOREM 5 . 3 . 3

0

E

Cl(Convhull(B52)).

(5.3.10)

If 52 is compact, (5.3.10) becomes 0

E

BConvhull(52).

(5.3.11)

Clearly if the system (5.1.1) subject to (5.1.2) is arbitrary-interval null-controllable , so also is the system PROOF

k = A x t i i

(5.3.12)

where G(t)

E

Cl(Convhull(B52)).

(5.3.13)

Now assume that (5.3.10) does not hold so that there exists a hyperplane strictly separating 0 and C1 (Convhull(B52)), viz. there is a non-zero v E Rn such that

vTz < O for all z

E

Cl(Convhull(B52)).

(5.3.14)

However from Theorem 5.3.2 arbitrary-interval null-controllability implies that given a sequence Tny Tn > 0, Tn -+ 0, there is a sequence T,,, 0 G .cn G Tn and a sequence

172

LINEAR-QUADRATIC EXTENSIONS AT

zn E Cl(Convhull(BR)) such that vTe 'zn > 0. Now because R is bounded Cl(Convhull(Bfi)) is compact so that there is a sub-sequence (2,) of {z,,) which converges to .? E Cl(Convhull(BS2)) as k 03 and 'rk +. 0. Hence Lim v T eATk zk = vT? 2 0, which contradicts (5.3.14). Now R -+

k-+m

compact implies that Convhull(R) is compact and (5.3.11) follows from (5.3.10). The following necessary and sufficient condition is the counterpart of Theorem 5.1.6.

THEOREM 5.3.4 Suppose that there exists a vector u E R such that Bu = 0. Then, (5.1.1) subject to (5.1.2) is arbitrary-interval nu1 1-control lable if and only if there exists for each non-zero v E Rn and for each T > 0 a time T , 0 G T G T and a u E R such that v T eAT Bu > 0. Equivalently, there should exist no v E Rn such that for some T > 0, T AT v e Bu GO, 0 G T G T for all u E R . The assumption that there is a u E R such that Bu = 0 implies that Rn(tl) C Rn(t2), tl < t2 d T. Furthermore Rn(t) is convex (Theorem 5.1.4) and 0 E RR(t), t E (0,Tl , T > 0 and arbitrary.

PROOF

We already showed (Theorem 5.3.2) that the condition of the theorem is necessary for arbitrary-interval nu1 1 -controll ability. To prove sufficiency we show that if the system is not arbitrary-interval null-controllable then the condition of the theorem is violated. If the system is n o t arbitraryinterval null-controllable then 0 tf Int(RR(T)) for some T > 0 (Theorem 5.3.1). As a consequence of the convexity and

CONTROLLABILITY CONDITIONS

173

t h e n e s t i n g o f t h e reachable sets we see t h a t t h i s i m p l i e s t h e existence o f a non-zero v

E

t vTleA(t-T)Bu(T)d.r

Rn such t h a t

< 0,

0

< t < T,

0

f o r a l l u ( - ) such t h a t

U(T)

E

.

R

(5.3.15)

For a p a r t i c u l a r choice o f u ( - ) t h i s i m p l i e s t h a t

T At v e Bu

< 0,

0

< t
for all u

E

R

(5.3.16)

which e s t a b l i s h e s t h e desired c o n t r a d i c t i o n . 5.3.2

Necessary and S u f f i c i e n t Conditions

I n t h e f o l l o w i n g pages we s h a l l o f t e n r e f e r t o (5.1.1) t o (5.1.2)

subject

B, R ) .

as t h e system (A,

We s h a l l assume throughout t h a t there i s a u

E

R such t h a t Bu = 0.

(5.3.17)

The next theorem, due t o M. Pachter, r e l a t e s t h e a r b i t r a r y i n t e r v a l n u l l - c o n t r o l l a b i l i t y o f (A, B, R ) t o t h a t of (A, B, C1 (Conichull (Convhull (L?)))). i s r e f e r r e d t o Section 5.A,

The u n i n i t i a t e d reader

t h e Appendix, f o r t h e d e f i n i t i o n s

o f Conichull , Convhull and o t h e r s e t - t h e o r e t i c concepts.

THEOREM 5.3.5

The system (A, B, R ) i s a r b i t r a r y - i n t e r v a l

n u l l - c o n t r o l l a b l e i f and o n l y i f t h e system (A, B, Cl(Conichul1 (Convhull nu11- c o n t r o l l a b 1e.

( a ) ) ) )i s a r b i t r a r y - i n t e r v a l

174

LINEAR-QUADRATIC EXTENSIONS

Theorem 5.3.4 states that (A, By a ) is arbitraryinterval null-controllable if and only if for each non-zero v E Rn and T > 0 there is a T~ 0 < T < T and a u E R such that vTeAT Bu > 0. NOW, trivially, vTeATBu = uTBTeA ' T ~so PROOF

T

that the preceding statement is equivalent to the statement that for each non-zero v E Rn and T > 0 there is a T~ 0 G T < T such that BTeA ~v TL R', where (see Section 5.A) Q' is the polar of R. Now Q' is a closed, convex cone so that

AT^

the separating hyperplane theorem implies that BTe v f R' is equivalent to the existence of a non-zero a E Rm such that T

aTw ' < 0 for all w ' E R' and aTBTeA v' > 0. It follows from this that (A, By R) is arbitrary-interval nu1 l-controllable if and only if for each non-zero v E Rn and T > 0 there is a

T

0 < T < T and a E R" such that aTBTeA v' = vTeAT Ba > 0 where n'' A= (R')'. Thus (A, By R) is arbitrary-interval nullcontrollable if and only if (A, By ,") is arbitrary-interval null-controllable. T,

Now it is clear from the definition of the polar of a set that 5 R" and Q" is a closed, convex cone. Therefore it follows that Convhull(R) C- Q " , Conichull(Convhull(C2)) E and Cl(Conichull(Convhull(Q))) ~ n " .Hence il c C1 (Conichull(Convhull(s2))) E so that arbitrary-interval null-controllability with the constraint set C1 (Conichull(Convhull ( a ) ) ) imp1 ies the same with R" and hence with R, and the theorem is proved.

,"

,I'

The above theorem and its proof indicate that one' may work with the 'nicer' closed, convex, cone C1 (Conichull (Convhull(0)))

175

CONTROLLABILITY CONDITIONS which is equivalent to Cl(Convhull(Conichull(i2))) R S Rm.

for any

The following theorem shows that if (A, B, Q ) is arbitraryinterval null-controllable so is (A, B, 6) where 6 is a suitable, bounded, constraint set. THEOREM 5.3.6

Let K

= {z

E

RmIIlzll G 1).

Then , (i) the system ( A , B, Q ) is arbitrary-interval nullcontrollable if and only if the system (A, B, Conichull(Q)n K) is arbitrary-interval null-control lable; (ii) if 0 E Convhull(Q) then the system (A, B, Q ) is arbitrary-interval null-controllable if and only if the system (A, B, Convhull(0) n K) is arbitrary-interval nullcontrol 1 able. PROOF

(i)

If we can show that

Conichull (Q) = Conichull (Con chu

i(n) n

then the theorem follows from Theorem 5.3 5. would allow us to write Cl(Conichul1 (Convhull (Conichull( R )

n

K)

(5.3.17)

Indeed (5.3.17)

K)))

=

Cl(Convhull(Conichull(Conichull(R) n K ) ) )

=

C1 (Convhull (Conichull (0)) ) = C1 (Conichull (Convhull(52))) (5.3.18)

176

L INEAR-QUADRATIC

EXTENS IONS

and an application of Theorem 5.3.5 would yie d the desired result We thus proceed now to prove that ( 5 3.17) holds. First, it is obvious that Conichull ( a ) fl K so

E

Conichull (f2)

that Conichull(Conichull(f2) fl K)

Next, suppose that z a

E

E

(5.3.19)

Conichull(i2).

Conichull(Q), z Z 0. Then

1

y = z E Conichull(i2) so that y E Conichull(R) fl K. It follows then that IlzlLy E Conichull(Conichull(f2) fl K) so that z E Conichull(Conichull(i2) fl K), and

Conichull(S2) c Conichull(Conichull(i2) fl K ) . (5.3.20) The inclusions (5.3.18) is proved.

and (5.3 19) yield (5.3.17) and (i)

(ii) Here Theorem 5.3.5 wou d be again applicable if one could show that

Conichull (Convhull(i2))

=

Conichull(Convhull(f2) fl K ) . (5.3.21)

Clearly we have that Convhull(f2) 2 Convhull(i2) fl K so that Conichull(Convhull(i2) fl K) c Conichull(Convhull(S2)) ( 5 3.22)

Now let z E Conichull(Convhull(R)), z Z O . Then there s a A > 0 such that Az E Convhull(i2) and we have that if IIAz 11 1 then Az E Convhull (f2) fl K so that z E Conichull(Convhull(i2) fl K). If it turns out that IIAz II > 1 then since 0 < 1 and 0 E Convhull(i2) we have


CONTROLLABILITY CONDITIONS

1

177

m

Az E Convhull(i2) r l K. Convhull (a) and Hence Az always belongs t o Conichull(Convhull(i2) rl K) so t h a t

that

Xz

E

Conichull(Convhull(i2)) 5 Conichull(Convhull(i2) rl K ) . (5.3.23) Inclusions (5.3.22) and (5.3.23) yield (5.3.21) and ( i i ) i s proved. 5.3.3

Necessary Conditions

In this section we provide a necessary condition f o r arbitraryinterval n u l l - c o n t r o l l a b i l i t y which has an appealing geometric character. First we need the following lemma.

Let C be a closed cone i n Rn and l e t v

LEMMA 5.3. I

E

Rn be

such t h a t

IlvII

=

T 1, v c < O , f o r a l l c

E

C , c f 0.

(5.3.24)

n I V Tx = -1, x

E

c}

(5.3.25)

Then the set A A =

(X

E

R

is compact. As A is the intersection of two closed sets i t i s closed. W e now prove t h a t A is bounded. We have t h a t x E A implies t h a t x = - v + N(v T ), where N ( . ) denotes ' t h e null space o f ' . Let us assume t h a t A i s unbounded, which implies T t h a t there i s a sequence xk E N(v ) such t h a t - v + X k E C and II X k II 03 a s k + 03 Clearly the sequence

PROOF

+

.

178

LINEAR-QUADRATIC EXTENSIONS

T A Xk ' k = mK c - N(v ) fl (xlIlxII G 1 ) which is a compact set, so that yk -+ j E N(V T ), viz. vTi = 0. NOW since c is a cone we k' have that zk = E C and as C is closed it follows

,f$-,, + T i p

that Lim zk = j E C, so that vTj < 0, a contradiction. k-+a A necessary condition for arbitrary-interval THEOREM 5 . 3 . 7 null-controllability of (A, B, L?) is that the cone C 4 Cl(Conichull(Convhull(BR))) is not pointed. If the cone C is pointed there exists a v E Rn such that the hypothesis of Lemma 5 . 3 . 1 holds, which implies that the set A is compact. Clearly A is also convex and non-empty and is such that 0 A. Theorem 5 . 3 . 3 then implies that the system (A, I, A) is not arbitrary-interval null-controllable. Noting that Conichull(A) = C we see that Theorem 5.3.5 implies that the system (A, I, C) is not arbitrary-interval nullcontrollable. A further application of Theorem 5 . 3 . 5 yields the fact that (A, B, L?) is not arbitrary-interval nullcontrollable. PROOF

The above theorem has the following implications. If u E R E R' and 0 f Int(Convhull(R)) then COROLLARY 5 . 3 . 1 the system (A, B, L?) is not arbitrary-interval null-controllable. Note, however, that such a system can be null-controllable, cf. Section 5.2.1. COROLLARY 5 . 3 . 2 If B has full rank then the cone C is not Cl(Conichull(Convhull(R))) pointed if and only if the cone is not pointed. Note also that regardless of the rank of B we always have that

CONTROLLABILITY CONDITIONS

179

Conichull (Convhull(BQ)) = Conichull (B(Convhul1 (Q))) = BConichull(Convhull(Q)). If B has full rank then BC1 (Conichull(Convhull(Q))) = C1 (Conichull (Convhull (BQ))). Finally, Theorem 5.3.7 and Corollary 5.3.2 imply the following result.

If B has full rank then a necessary condiCOROLLARY 5 . 3 . 3 tion for arbitrary-interval null-controllability of the system (A, B, Q ) is that the cone t = Cl(Conichull(Convhull(Q))) is not pointed. 5.3.4

Necessary and Sufficient Conditions Generated Cones

-

Finitely

When c = Cl(Conichull(Convhul1 ( a ) ) ) is a finitely generated cone we can deduce a necessary and sufficient condition for arbitrary-interval null-controllabil ity of the system (A, B, 0 ) . We sha 1 present an example which illustrates the use of the theorem and another which illustrates that in general c" cannot be replaced by R in the theorem. A sufficient condition in the spirit of Theorem 5.1.3 is also presented. TffEOREM 5.3.8

Suppose that = Cl(Conichull(Convhull(Q))) is a finitely generated cone. Then the system (A, B, Q ) i s arbi trary-interval nu1 1 -control1 able if and only if there exists for each non-zero vector v E R n (in fact, for each v E (Bt)', where is the polar o f ( . ) ) an integer i(v), 0 < i(v) < n-1 and a u E A(v,i(v)) such that v ~ A ~ ( ~ ) B>u0, where ( - ) I

A(v,i) A= (u

E

tlvTA jBu=O, j=O ,..., i-1)

(5.3.26)

180

LINEAR-QUADRATIC EXTENSIONS A(v,O)

PROOF

6

c.

( 5.3.27)

Sufficiency follows easily upon noting that

T AT i(v)Bu i(v> v e Bu=vA T(-qT + higher-order terms in

T

(5.3.28)

for all u E A(v,i(v)). Indeed this equation and satisfaction of the conditions of the theorem imply that for each v there is a t(v) > 0 sufficiently small and a u E t such that vTeAT Bu > 0 for all T, 0 < T < t(v). Theorems 5.3.4 and 5.3.5 then imply that the system (A, B y a ) is arbitraryinterval null-controllable. Suppose now that the system (A, By a ) is arbitrary-interval null-controllable. Then for a given v E Rn either there is a u E such that v TBu> 0, in which case the conditions of the theorem are satisfied with i(v) = 0, or v TBu < 0 for all u E C. If in addition to this last-mentioned case A(v,l) = (0) or v TABu < 0 for all u E A(v,l) we have by the continuity in T of vTeATBu that there exists for each generating vector uk of t, k=l, ...,p, a time tk(v) such that v TeAT Buk < 0 for all T, 0 < T < tk(v). Then, since each vector u E t is a non-negative combination of the generating vectors uk, k=l, ...,p we have vTeATBu S 0 for all u E and 0 G T St(v), where t(v) = min tk(v). Hence Theorems 5.3.4 k and 5.3.5 imply that the system is not arbitrary-interval null-control lable, a contradiction. This then imp1 ies that either v TABu > 0 for some u E A(v,l) in which case i(v) = 1, or i(v) > 1. A similar argument may be used to investigate Clearly i(v) < n-1 as the larger possible values of i(v).

c

CONTROLLABILITY CONDITIONS

181

otherwise the Cay1 ey-Hami 1 ton theorem would imply that v TAk Bu = 0 for all u E A(v,k) and all k and A(v,k) = A(v,n) for all k > n. This in turn would imply, using a similar argument to that developed above which is based upon the finite generation of the cone t , that there is a time t(v)>O such that v T eAT Bu < 0 for all T, 0 < T < t(v) and all u E t, contraditting the assumed arbitrary-interval null -controll ability of (A, B, 52). The following remarks which relate to Theorem 5.3.8 are important.

(i) When checking the conditions of Theorem 5.3.8 on a particular system (A, B, 52) we need only consider those vectors v E Rn such that v E (BConvhull(52))' where denotes the polar of ( - ) I

( a ) .

(ii) If the cone C = Cl(Conichull(Convhull(B52))) is finitely generated, then Theorem 5.3.8 is applicable because the arbitrary-interval null-controllability of system (A, I, C) is equivalent to that of (A, B, t ) and (A, B, 52) - see Theorem 5.3.5. ( i i i ) If m = 2 or n = 2, Theorem 5.3.8 is always applicable because every cone in R 2 is finitely generated.

(iv) If the control constraint set 52 is a polyhedron, is finitely generated and Theorem 5.3.8 can be used. The following theorem provides us with a set of sufficient conditions for arbitrary-interval null-controllabil ity.

THEOREM 5 . 3 . 9 Suppose there exist * B E R" such that

QS ~ 5 2 ,P E

R"x m and

t

182

LINEAR-QUADRATIC EXTENSIONS = Epus f o r a l l us

BU,

t h e s e t (Puslus

E

E

os

(5.3.29)

;

GSl c o n t a i n s t h e o r i g i n Pus = 0 i n i t s interior;

Rank[i,Ai,

...,A n - l -B]

(5.3.30) = n.

(5.3.31)

Then, t h e system (A, B y R) i s a r b i t r a r y - i n t e r v a l n u l l controllable.

PROOF

Naming t h e v e c t o r Pus as z we see t h a t t h e system

R = AX

t

EZ

(5.3.32) (5.3.33)

s a t i s f i e s t h e c o n d i t i o n s o f Theorem 5.3.8; c o n d i t i o n on t h e p a i r (A,

i) and

i.e.

t h e rank

t h e f a c t t h a t Rz c o n t a i n s

zero i n i t s i n t e r i o r a r e e a s i l y seen t o be s u f f i c i e n t . We remark t h a t a s p e c i a l case o f t h e c o n d i t i o n s o f Theorem 5.3.9

has been used p r e v i o u s l y i n t h e l i t e r a t u r e .

Specifi-

c a l l y i n [7] t h e minimum-time o p t i m a l c o n t r o l problem i s considered and t h e f o l l o w i n g assumptions a r e made, n R c R i s a polyhedron.

(5.3.34)

The s o - c a l l e d P o n t r y a g i n c o n d i t i o n o f general p o s i t i o n i s assumed, namely i f b i s a v e c t o r c o - l i n e a r w i t h an edge o f t h e polyhedron R then [b,Ab,.

. .,A

n-1

bl = n

(5.3.35)

CONTROLLABILITY CONDITIONS 0

183

R and 0 k vertex(f2).

E

(5.3.36)

We now show t h a t these assumptions imply the existence of an as, P and a s specified i n Theorem 5.3.9. Indeed, i f 0 E Int(R) we can s e t Rs = R, P = I , B = I . If 0 I Int(R) e can then then by (5.3.36) i t must belong t o an edge of R . W t o be b (a vector co-linear w i t h define Rs t o be t h a t edge, bT t h a t edge) and P t o be bTb

5.3.5

.

Examples

Our f i r s t example i l l u s t r a t e s t h a t i n Theorem 5.3.8 the s e t e consider the following system: cannot be replaced by R. W

A =

$2 =

[: :] [: 7 ’

(5.3.37)

=

{u

First we note t h a t

-C = Cl(Conichu

1 Convhull(R))) = { u l u l

E

1 R , u2

> 0). (5.3.39)

Now T

v BU = vlul + v2u2 I

(5.3.40)

so t h a t f o r each v E R2 f o r which v1 f 0 o r v2 > 0 there i s T a u E so t h a t v Bu > 0. The only case t h a t remains i s v1 = 0, v2 < 0. Here we note t h a t

184

LINEAR-QUADRATIC EXTENSIONS T v Bu = 0 f o r u2 = 0,

u1 a r b i t r a r y

(5.3.41)

so t h a t A(v,l]

= {ulul

and t h e r e c l e a r l y e x i s t s a u

E

1

, u2

A(v,l)

E

T v ABu =

R

-

= 0)

(5.3.42)

such t h a t

u1 > O .

(5.3.43)

Therefore t h e c o n d i t i o n s o f Theorem 5.3.8 a r e s a t i s f i e d and t h e system ( A , B y $2) i s a r b i t r a r y - i n t e r v a l n u l l - c o n t r o l l a b l e . However, i f we r e p l a c e v2

<0

t h e r e i s no u

E

c

by t h e s e t R we see t h a t when v1 = 0, T R which causes v Bu > 0 and A ( v , l ) i s

j u s t t h e s e t (01, so t h a t t h e c o n d i t i o n s o f t h e theorem cannot be s a t i s f i e d . Our n e x t example i s one which s a t i s f i e s t h e c o n d i t i o n s o f Theorem 5.3.8 b u t n o t those o f Theorem 5.3.9.

The system

equations a r e

A3 =

-

x1

+

x2

-

x3

+ u3 - u4

and R = iulu.

1

so t h a t

> O , i=1,...,41

(5.3.45)

CONTROLLABILITY CONDITIONS

1 A =

-1 -1

y]

-

1 - 1

,

B = [ l

0

o

o

1 -1

185 0

0

0 0 1 -11

.

(5.3.46)

We have t h a t

v TBU = v1(u1-u2) T C l e a r l y v Bu

+

v2U1

+ v 3 ( U 3-U 4 ).

(5.3.47)

>0

i f v3 f 0 and i f v1 f 0 and f o r some u E T I f v2 f 0 then v Bu f a i l s t o be p o s i t i v e f o r some

v2 = 0. u E o n l y i f v1 = v3 = 0, o r v1 = -v2 and v3 = 0, o r v1 + v2 and v1 a r e o f opposite s i g n and v3 = 0. The s e t A(v,l) f o r these t h r e e cases i s , r e s p e c t i v e l y iUIUl

= 0, u2

u3

> 0,

u4 2 01

( 5.3.48)

hlUl

2 0 , u2 = 0, u3

> 0,

u4

> 01

( 5.3.49)

iUIUl

= u2 = 0, u3

> 0,

>o,

u4

> 01.

(5.3.50)

Furthermore,

v TABu = v1(u1-u2)

+

v2(-2ul+u2+u3-u4)

+

v3(u2-u3+u4) (5.3.51)

and as

v2 f 0 and v3

= 0 i n a l l t h e above cases, (5.3.51)

be made p o s i t i v e f o r a u

E

(ulul

=

can

u2 = 0, u3 > 0, u4 > 01.

Thus t h e c o n d i t i o n s o f Theorem 5.3.8

a r e s a t i s f i e d and t h e

system i s a r b i t r a r y - i n t e r v a l n u l l - c o n t r o l l a b l e . Now we note t h a t (5.3.29)

and (5.3.30)

change s i g n as a f u n c t i o n o f u

S

E

as

imply t h a t Bus must where as i s a subset of

186 s2.

L INEAR-QUADRATIC EXTENSIONS

But

I n our example t h i s can happen o n l y i f u1 = u2 = 0.

then (5.3.44)

i s n o t even n u l l - c o n t r o l l a b l e ,

l e t alone

a r b i t r a r y - i n t e r v a l nu1l - c o n t r o l l a b l e , and t h e s u f f i c i e n c y conditions o f Theorem 5.3.9 5.3.6

cannot be s a t i s f i e d .

Minimum Time Function

One o f t h e most important and i n t e n s i v e l y studied problems i n optimal c o n t r o l theory i s t h a t o f s t e e r i n g t h e s t a t e o f a l i n e a r dynamic system t o t h e o r i g i n i n minimum time.

Usually

one places enough r e s t r i c t i o n s and assumptions on t h e dynamic system and t h e c o n s t r a i n t s e t R t o ensure t h a t t h e minimum time f u n c t i o n T(x), i.e.

t h e minimum t i m e t o reach t h e o r i g i n

from s t a t e x, i s continuous i n an open neighbourhood of t h e origin.

I n t h i s s e c t i o n we show t h a t t h e c o n d i t i o n o f

a r b i t r a r y - i n t e r v a l n u l l - c o n t r o l l a b i l i t y o f t h e autonomous l i n e a r dynamic system i s necessary and s u f f i c i e n t f o r t h e c o n t i n u i t y o f t h e minimum t i m e f u n c t i o n . We need t h e f o l l o w i n g theorem which we quote i n s l i g h t l y modified form from [ll

.

THEOREM 5.3.10

Suppose t h a t R i s compact.

a measurable c o n t r o l f u n c t i o n ul(-) steers t h e s t a t e xo

E

w i t h ul(t)

Ift h e r e e x i s t s E s2 which

Rn t o t h e o r i g i n o f t h e s t a t e space i n

time tl then t h e r e e x i s t s a measurable c o n t r o l f u n c t i o n u ( ' ) with u(t)

E

Q which steers xo t o t h e o r i g i n i n minimum time T.

I n c i d e n t a l l y , note t h a t t h e converse o f Theorem 5.3.10

is

t r i v i a l l y true, v i z . i f xo can be steered t o t h e o r i g i n i n minimum t i m e T, then i t can be steered t o t h e o r i g i n i n f i n i t e time tl 2 T!

CONTROLLABILITY CONDITIONS

187

The above theorem implies that the minimum time function T(x) is defined in an open neighbourhood of x = 0 when fl is compact if and only if (5.1.1) is null-controllable. However, nullcontrollability is not sufficient to ensure that T(x) is continuous in an open neighbourhood of x = 0. Suppose that fl is compact. A necessary and sufficient condition for the minimum time function T(x) to be continuous in an open neighbourhood of the origin of the state space is that (5.1.1) subject to (5.1.2) is arbitraryinterval null-control lable.

TffEOREM 5 . 3 . 1 7

PROOF Note that T(0) = 0 and suppose that T(x) is continuous in an open neighbourhood of the origin. Then the definition of continuity implies that for each t > O there is an open neighbourhood of the origin, say V(t), such that all states in V(t) can be steered to the origin in time t, which is arbitrary-interval null-controllability. Suppose now that (5.1.1) subject to (5.1.2) is arbitraryinterval null-controllable. Then by Theorem 5.3.10 and subsequent remarks the minimum time function T(x) is defined in an open neighbourhood o f the origin. Furthermore arbitraryinterval null-controllability implies that for each t > 0 there is an open neighbourhood of the origin, say V(t>, such that all states in V(t) can be steered to the origin in time t. Owing to the continuous dependence of the solutions of the differential equation on the initial values we know that given a point x in the domain of definition of T(-) there is a neighbourhood of x, say i , such that all points y E can be steered in time T(x) to V(t). This then implies that all points y E can be steered in time T(x) + t to the origin.

188

LINEAR-QUADRATIC EXTENSIONS

It follows that for each t > 0 and x in the domain of T(.) there is a neighbourhood v" of x such that T(y) - T(x) < t for all y E which implies that T(x) is upper-semicontinuous in an open neighbourhood of the origin. Now the set of states R(t) which can be steered to the origin in time t > 0 is compact, convex, and varies continuously with t on t > 0, and ,.. R(tl) c R(t2), tl < t p (cf. Theorem 5.1.4, the proof of Theorem 5.1.6 and Theorem 5.3.1). Moreover, if the minimum time to steer x to the origin is T(x) we have that x E ai(T(x)). Also such an x cannot belong to the sets fi(T(x)-t), t > 0 as T(x) is the minimum time required to steer x to the origin. Therefore there exists for each t > 0 a neighbourhood of x such that T(y) 2 T(x)-t for all y E i, or T(x) - T(y) G t for all y E &, which implies that T(x) is lower-semicontinuous in an open neighbourhood of the origin. The fact that T(x) is both upper- and lower-semicontinuous in an open neighbourhood of the origin implies that it is continuous in an open neighbourhood of the origin.

-

The above theorem implies that Pontryagin's conditions (5.3.34) - (5.3.36) which are sufficient for arbitraryinterval null-controllability also guarantee the continuity of the minimum time function T(x). It also follows that our necessary and sufficient conditions for arbitrary-interval null-controllability are the minimal conditions required for the continuity of the minimum time function. 5.3.7

Further Necessary Conditions and Sufficient Conditions

In this section we present necessary conditions and sufficient conditions due to M. Pachter for arbitrary-interval null-

189

CONTROLLABIL ITY CONDITIONS c o n t r o l l a b i l i t y which do n o t r e q u i r e t h e assumption t h a t

C1 ( C o n i c h u l l (Convhull ( a ) ) ) i s f i n i t e l y generated. Furthermore t h e y a r e o f geometric c h a r a c t e r and y i e l d i n s i g h t i n t o t h e geometric aspects o f a r b i t r a r y - i n t e r v a l n u l l - c o n t r o l l a b i l i t y . We r e q u i r e t h e f o l l o w i n g p r e l i m i n a r y lemma and subsequent

observations. LEMMA 5.3.2

Suppose t h a t C i s a convex cone i n Rn and t h a t

S i s a subspace c o n t a i n e d i n C.

c

=

s t

Then

s'n

c.

(5.3.52)

s'

PROOF C l e a r l y S t n C E ( S + s ' ) fl C = C. To prove t h e converse we suppose t h a t c E C and c = c + c 2 where c1 E S 1 and c 2 E Then c 2 = c - cl, and -cl E S s i n c e S i s a subspace. As S E C we have -cl E C and t h e c o n v e x i t y o f C nC i m p l i e s t h a t c 2 E C. Consequently we have t h a t c E S t which i n t u r n i m p l i e s t h a t C 5 S t n C.

8.

s'

From t h i s p o i n t on we suppose t h a t S i s t h e l a r g e s t subspace contained i n C

8

Cl(Conichull(Convhull(Bs2))).

C l e a r l y we

have t h a t

s

=

cn

(-c)

( 5.3.53)

and an a l g o r i t h m f o r computing S i s a v a i l a b l e 181. Note t h a t Theorem 5.3.7 i m p l i e s t h a t (01 i s a s t r i c t subset o f S. Lemma 5.3.2 i m p l i e s t h a t C can be w r i t t e n as (5.3.54)

L I NEAR-QUADRATIC EXTENSIONS

190

fl C is a closed, convex, pointed cone. That s' tl C where is pointed follows from the fact that ($ n C) tl [-(?nC)] =

sl n c n

n (-c) = sl n c n (-c) n

(-.+I

SL n

=

sn sl

{O). Furthermore we note that (BConvhull ( a ) ) ' c s' and (BConvhull ($2)) = (C1 (Conichull (Convhull (BQ)) ) ) I

I

=

.

Before presenting our main theorem we define the following we denote by Xo the interior of notation. Given a set X c the set X relative to the subspace s', plus the set (0). We denote by {AIS) the smallest subspace invariant under A which contains the subspace S. It is well known that {AIS) = S t AS t + A n-1 S .

...

THEOREM 5.3.12 A necessary condition for arbitrary-interval null-controllability of (5.1.1) subject to (5.1.2) is that ((BConvhull(Q))')o

fl {AlSIl = (01.

( 5.3.55)

A sufficient condition for arbitrary-interval null -control 1 ability is that (BConvhull(Q))' fl {A[ S>l = t-0) which is equivalent, as 0 E BConvhull(Q) to the sufficient condition 0

E

-

Int(Convhull(BQ)t{AlS)).

(5.3.56)

cf. Theorem 5.3.4,

(5.3.57)

We first consider necessity. Let v E ((BConvhull(Q))l)o, with IlvII = 1. It follows that vTc < O for all c E C f l SL, c f O so that Lemma 5.3.1 implies that the set

PROOF

CONTROLLABILITY COND I T 1ONS A

A T = ( y ( v y = -1, y

V

E

C

n

191

$1 i s compact and convex and

n s'.

c l e a r t h a t Conichull(Av) = C

it i s

Then i t f o l l o w s because

o f t h e compactness o f Av t h a t t h e r e i s a t > 0 such t h a t v TeAT y Q 0 f o r a l l 'I, 0 Q T Q t and f o r a l l y E Av. Conseq u e n t l y by Theorem 5.3.2

t h e system (A, I,Av) i s n o t

a r b i t r a r y - i n t e r v a l n u l l - c o n t r o l l a b l e and by Theorem 5.3.5

s'),

t h i s i s t r u e a l s o o f t h e system (A, I, C n which i m p l i e s T AT t h a t t h e r e i s a t > 0 such t h a t v e y Q 0 f o r a l l T, 0 Q

'I

Q t and f o r a l l y

t o a subspace of eATTv

E+

for all

E

C

n

s'.

Now assume t h a t v belongs

9 i n v a r i a n t under AT. T

Then

2 0 so t h a t v TeAT s = 0 f o r a l l s

E

S and

>O.

Combining t h e two r e s u l t s w i t h t h e a i d o f Lemma 5.3.2 y i e l d s v TeAT c Q O f o r a l l T , 0 Q T Q t and a l l c E C

all

T

so t h a t Theorem 5.3.2

i m p l i e s t h a t t h e system i s n o t a r b i t r a r y -

i n t e r v a l nu1 l - c o n t r o l l a b l e .

Thus i t i s necessary f o r

arbitrary-interval null-controllabil ity that

T ( (BConvhull(C2))')' n ( l a r g e s t subspace i n v a r i a n t under A contained i n $) = (0). Reference [9] provides. t h e l a s t step, T v i z . t h e l a r g e s t subspace i n v a r i a n t under A c o n t a i n e d i n! 5

i s CA~SI'. Now we go on t o s u f f i c i e n c y . 0

+

(AIS)) which i s e q u i v a l e n t t o (Convhull(BC2) t ( A I S ) ) ' = (0). A s e t - t h e o r e t i c i d e n t i t y then E

Int(Convhull(BQ)

We assume t h a t

implies t h a t (Convhull(BQ))' Next we n o t e t h a t i f (5.3.58)

n

C A l S I l = CO).

( 5.3.58)

holds then S # (0).

Indeed, i f

S = (0) we would have from (5.3.58)

t h a t (Convhull(Bi2))' = (0)

192

LINEAR-QUADRATIC EXTENSIONS

which is equivalent to 0 E Int(Convhull(Bi2)). This then implies that 0 E Int(Convhull(R)) and R(B) = Rn so that S = Rn, which i s a contradiction, which means that (0) is a strict subset of S. As mentioned in the proof of necessity, (AISI1 is the largest subspace invariant under A T contained in $, which implies that for all v E (Convhull(BQ))' there is an i, 0 < i < n-1 and an s E S E C such that v TA j s = 0 for all s E S and j=O,l, ...,i-1 and v TA i -s > O . Also, if v t (Convhull(BQ))' then there is a y E C such that v Ty > 0. In either case, then, we have that for each v there exists a y E C and a t > 0 such that v TeAT y > 0, 0 < T < t which implies that the system ( A , I, C) is arbitrary-interval nullcontrollable. Theorem 5.3.5 then implies that the system (A, I, BQ) is arbitrary-interval null-controllable, which implies that the system (A, B, Q) is arbitrary-interval nullcontrollable. Conditions (5.3.56) and (5.3.57) are equivalent if 0 E Convhull(,BR) because in that case (Convhull(BR) t {AIS))' = (Convhull(BR))' tl {AlSl' - see Section 5.A. The following corollaries follow from the above theorem.

A necessary condition for arbitrary-interval COROLLARY 5.3.4 null-controllability o f (5.1.1) subject to (5.1.2) is that Rank(Q) = n. COROLLARY 5.3.5 Assume that 0 E Int(Convhull(R)). Then a necessary and sufficient condition for arbitrary-interval null-controllability o f (5.1.1) subject to (5.1.2) is that Rank(Q) = n.

Furthermore we note that the sufficient conditions of Theorem 5.3.12 are less stringent than those o f Theorem 5.3.9. Indeed, returning to the example of Section 5.3.5 we see that

CONTROLLABILITY CONDITIONS

S

=

193

linear span(

BConvhull ( Q ) = {y

E

(5.3.59)

3

R Iyl G y2, y2 > O , y3

and { A I S ) = linear span(

[ ;] 0

9

[-;1).

E

R 11 (5.3.60)

(5.3.61)

It follows that

{A~s}' = linear span( and

[

(5.3.62)

does not belong to (BConvhull(i2))'.

Therefore

condition (5.3.56) is satisfied and the system (5.3.44) subject to (5.3.45) is arbitrary-interval null-controllable. We note, however, that there is a gap between the necessary condition and the sufficient conditions o f Theorem 5.3.12.

' 0 0 A = 0 1 0 1 ,o 0

0 1 0 0 0 1 0 0

, B=14

(5.3.63)

and Q = linear span(

11

( 5.3.64)

194

L INEAR-QUADRATIC EXTENSIONS

we find that the sufficient conditions of Theorem 5.3.12 are not satisfied but it can be verified direct using Theorem 5.3.4 that the system (A, B, 0 ) is arbitrary-interval nullcontrol 1 able. 5.3.8

Further Special Cases

THEOREM 5 . 3 . 1 3 Assume that $ E R2. Then a necessary and sufficient condition for arbitrary-interval null-controllability of (5.1.1) subject to (5.1.2) is that (Convhull(Bi2))'

n {A~SI'

=

(01

(5.3.65)

which is equivalent, because of our assumption that Bu for some u E 0 , to 0

PROOF

E

Int(Convhull(BR)+{AlSI).

=

0

(5.3.66)

We need only prove the necessity of the condition, as sufficiency is already proved in Theorem 5.3.12. In fact all we need to prove is that all vectors in the boundary of (Convhull(Bi2))' are not in {AIS) 1 . Thus, suppose v E a(Convhull(Bi2))'. If v E {AIS)1 then since $ E R 2 either v is an eigenvector of AT or $ is invariant under A T . In the former case vTeAtc = eAtv Tc G 0 for all c E C and t > 0 so that we would not have arbitrary-interval nullcontrollability. Assume then that is invariant under A T . Then those vectors v E (Convhull(BR))' such that vTc < 0 for all c E C n 9, c f-0, are also in (AlS}' so that, following the proof of necessity in Theorem 5.3.12, there exists t > 0 such that v TeAT c GO for all c E C and T, 0 G T G t so that

CONTROLLABILITY CONDITIONS

195

we would not have arbitrary-interval null-controllability. Suppose that n = 3, i.e. that x E R3 . Then (5.3.65) and (5.3.66) are necessary and sufficient for arbitrary-interval null-controllability of (5.1.1) subject to COROLLARY 5.3.6

(5.1.2).

PROOF

Again we need only prove necessity. Arbitraryinterval null-controllability implies that C is not pointed so that the dimension of the largest subspace S is greater than or equal to 1. Hence the dimension of 2 is less than or equal to n - 1 = 2 and the corollary follows from Theorem 5.3.13. We remark that Corollary 5.3.6 allows us to always use Theorem 5.3.8 when n < 3. Indeed, as stated above, the dimension of S is not less than one and that of s' is not greater than 2. As every cone in R 2 is finitely generated so is 2 n C and as S is a subspace the cone C is finitely generated. 5.4

Conclusion

In the introductory section of this chapter we reviewed wellknown results on control 1 abi 1 i ty and nu1 1 -control 1 abi 1 i ty of autonomous linear systems and described and discussed the most important properties of reachable sets. Section 2 was devoted to a study of null-controllability when R and its convex hull do not contain zero in their interior. Specifically we proved Brammerls important theorem via Hajek's approach.

In Section 3 we introduced the important notion of arbitrary-

196

LINEAR-QUADRATIC EXTENSIONS

interval null-controllability and proved a number of preliminary necessary and sufficient conditions. Then we proved that in questions of arbitrary-interval nu1 1 -control1ability R can be replaced by the cone C = Cl(Conichull(Convhull(R))) and this led to the development of necessary conditions, such as C should be not pointed, and necessary and sufficient conditions when is a finitely generated cone. We showed by means of a sufficient condition that Pontryagin's condition of general position imp1 ies arbitrary-interval null-control lability. We also presented a number of illustrative examples. Next we proved the important result that the minimum time function is continuous in an open neighbourhood of the origin of the state space if and only if the system (A, By n), n compact, is arbitrary-interval nullcontrollable. This justifies further the introduction of the notion of arbitrary-interval null-controllabil ity. Following this we presented necessary conditions and sufficient conditions of a geometric nature which are due to M. Pachter. These conditions do not require that the cone C be finitely generated . The chapter provides a rather complete treatment of the 'controllability' of constrained, 1 inear, autonomous continuous-time, dynamic systems. 5.5 References

[ll

LEE, E.B. & MARKUS, L. Foundations of Optimal Control Theory. Wiley, New York, 1967.

[2]

BRAMMER, R.F. Controllability in Linear Autonomous Systems with Positive Control 1 ers. SIAM J . Control , 10, 1972, pp. 339-353.

CONTROLLABILITY CONDITIONS

197

SAPERSTONE, S.H. & YORKE, J.A. Controllability of Linear Oscillatory Systems Using Positive Controls. SIAM J. Control, 9, 1971, pp. 253-262. HiJEK, 0. A Short Proof of Brammer's Theorem. Unpublished preprint, 1975. HEYMANN, M. & STERN, R.J. Controllability of Linear Geometric ConsideraSystems with Positive Controls: tions. J. Math. Anal. Appl., to appear. PACHTER, M. & JACOBSON, D.H. Conditions for the Controllability of Constrained Linear Autonomous Systems on Time Intervals of Arbitrary Length. CSIR Special Report WISK 210, July 1976, 23 p. BOLTYANSKII, V.E. Mathematical Methods of Optimal Control, Holt, Rinehart, Winston-, New York, 1971, pp. 119-120. ECKHARDT, V. Theorems on the Dimension of Convex Sets. Linear Algebra and its Applications, 12, 1975, pp. 63-76. WONHAM, W.M. On the Matrix Riccati Equation of Stochastic Control. SIAM J. Control, 6, 1968, pp. 681-697. ROCKAFELLAR, R.T. Press, 1970.

Convex Analysis.

Princeton University

Appendix The following definitions and facts are used in Chapter 5; for further details see [lo]. The set C 5 Rn is a cone if for all c E C and all scalars a > 0 we have CIC E C. A cone C is convex if and only if C + C SC. A cone C is pointed if C l (-C)

=

(01.

A cone C is pointed if and only if there is a v that v Tc < 0 for all c E C, c f 0.

E

Rn such

LINEAR-QUADRATIC EXTENSIONS

198

The convex hull o f a set E R", denoted Convhull(R), i s the smallest convex set which contains 0 . The conic hull o f a s e t 52 c Rm, denoted Conichull (n), i s the smallest cone with vertex a t the origin which contains R. The closure o f a set Sl E Rm, denoted C1 ( a ), i s the smallest closed set containing n. If the set !J i s bounded, so i s Convhull(Q); i f the set n i s closed, so i s Convhull(C2). T h u s n compact implies t h a t Convhull(f2) i s compact. If the set S2 i s convex, so i s the set Conichul l(R).

Let B be a map B:Rm + Rn. W e denote by B(n) or Bn the image under B of the set R . Let T c _ R n Then T i , the p o l a r of T, i s defined as T' = Ix E R n IxTy < 0 for a l l y E TI. The set T' i s a closed 1 and convex cone, and if T i s a subspace then T' = T W e A If T i s convex then note t h a t T E TI' where T" = ( T I ) ' T' = I01 i f and only i f 0 E Int(T).

.

.

.

Let T , V c Rn. Then T' T ' n V' = (TtV)'.

n V' c (T+V)' and

if 0

E

T

n V then