SPLICING OF TIME OPTIMAL CONTROLS

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1 Dynamic Sysems and Applicaions 21 (212) SPLICING OF TIME OPTIMAL CONTROLS H. O. FATTORINI Deparmen of Mahemaics, Universiy of California Los Angeles, California , USA ABSTRACT. Two conrols u 1 () and u 2 () are spliced (in symbols (u 1 u 2 )()) if u 2 () is applied afer u 1 (); boh are defined in finie inervals. We seek condiions under which ime opimaliy of u 1, u 2 imply ime opimaliy of u 1 u 2, and eend he resuls o a finie or infinie number of conrols. The heorems (or raher eamples, since hey are resriced o he righ ranslaion semigroup in L 2 (, )) are used o consruc ime opimal conrols ha do no saisfy Ponryagin s maimum principle. AMS (MOS) Subjec Classificaion. 93E2, 93E25 1. PRELIMINARIES The sysem is (1.1) y () = Ay() + u(), y() = ζ wih conrols u( ) L (, T; E). The operaor A generaes a srongly coninuous semigroup S() in a Banach space E, called he sae space of (1.1). The soluion of (1.1) is he coninuous funcion (1.2) y() = y(, ζ, u) = S()ζ + A conrol is admissible if S( σ)u(σ)dσ. (1.3) u( ) L (,T;E) 1. The conrol is ime opimal if i drives he iniial poin ζ o a poin arge (1.4) y(t, ζ, u) = ȳ in opimal ime T, ha is, no oher admissible conrol does he drive in less ime. Under resricions on ζ, ȳ, necessary and sufficien condiions for opimaliy can be given in erms of Ponryagin s maimum principle (1.6) below. The principle requires mulipliers [4], [6] Secion 2.3 whose consrucion we ouline. When A has a bounded inverse, E 1 is he compleion of he dual E in he norm y E 1 = (A 1 ) y E. Received November 15, $15. c Dynamic Publishers, Inc.

2 17 H. O. FATTORINI The adjoin semigroup S() can be eended o an operaor S() : E 1 E 1. If S() is srongly coninuous, Z(T) E 1 consiss of all z E 1 such ha S() z E ( > ) and (1.5) z Z(T) = T S() z E d <. The norm (1.5) makes Z(T) a Banach space. All Z(T) coincide and all norms Z(T) are equivalen for T >. Z(T) is an eample of a muliplier space, a linear space Z E o which S() can be eended in such a way ha S() Z E for > and ha he semigroup equaion S(s + ) ζ = S(s) S() ζ is valid for ζ Z and s, >. When A does no have a bounded inverse, E 1 is he compleion of E in any of he equivalen norms y E 1,λ = ((λi A) 1 ) y E (λ large enough). The definiion of Z(T) (and of muliplier spaces) doesn change. An admissible conrol ū( ) L (, T; E) saisfies Ponryagin s maimum principle if (1.6) S(T ) η, ū() = ma u 1 S(T ) η, u a. e. in < T,, he dualiy of he spaces E and E, η in some muliplier space Z. We call η he muliplier and z() = S(T ) η E he cosae corresponding o (or associaed wih) he conrol ū(). We assume ha (1.6) is nonrivial, ha is, ha S(T ) z is no idenically zero in he inerval < T, alhough i may be zero in par of he inerval (where (1.6) gives no informaion on ū()). When E is a Hilber space (1.6) reduces o (1.7) ū() = S(T ) η S(T ) η (T δ < T), where (T δ, T] is he maimal inerval where S(T ) η ; if δ = T he inerval is < T. We say ha ū() is regular if i saisfies (1.6) wih η = z Z(T). Theorem 1.1. Assume ū() drives ζ E o ȳ = y(t, ζ, ū) ime opimally in he inerval T and ha (1.8) ȳ S(T)ζ D(A). Then ū() is regular. Theorem 1.2. Le ū() be a regular conrol wih (a) ȳ = y(t, ζ, ū) D(A), Aȳ < 1,or (b) ζ D(A), Aζ < 1, S(T ) z ( < T). Then ū() drives ζ E o ȳ = y(t, ζ, ū) ime opimally in he inerval T.

3 SPLICING OF TIME OPTIMAL CONTROLS 171 Theorem 1.1 is [6] Theorem I does no guaranee ha S() z in he whole inerval (see Remark 4.4). Theorem 1.2 is [6] Theorem We only use i here when ζ = (which fis ino (b), wih no assumpions on ȳ). We say ha ȳ E is reachable if here eiss an admissible conrol driving o ȳ. Theorem 1.3 (eisence of opimal conrols). Le E be a Hilber space. If ȳ is reachable hen here eiss an opimal conrol, ha is, an admissible conrol ū( ) driving o ȳ in opimal ime. Alhough sufficien for his paper, he resul is rue in spaces much more general han Hilber; see [6] Secion 3.1. The following resul is [6] Theorem and is oally independen of he maimum principle (1.6). Theorem 1.4. Le ū( ) be ime opimal. 1 Then (1.9) ū() = 1. Theorem 1.4 implies uniqueness of opimal conrols in a Hilber space. In fac, if wo conrols ū 1 () and ū 2 () drive o ȳ in opimal ime T so does ū() = (ū 1 () + ū 2 ())/2; since (1.9) is mandaory, ū 1 () = ū 2 (). The resul is valid in sricly conve spaces; see [6] Theorem We don know wheher sric conveiy is necessary for uniqueness, alhough nonuniqueness eamples in spaces no sricly conve eis (see [6], Secions 1.2 and 5.5). Theorem 1.1 says nohing abou ime opimal conrols driving ζ o ȳ no saisfying (1.8), in paricular abou conrols driving o ȳ / D(A). In fac, he main purpose of his paper is o consruc ime opimal conrols ha do no saisfy he maimum principle (1.6). For hese, y(t,, ū) / D(A). 2. SPLICING Le u 1 ( ) L (, T 1 ; E), u 2 ( ) L (, T 2 ; E). The splice u 1 u 2 of u 1 and u 2 is u 1 () ( < T 1 ) (2.1) u() = (u 1 u 2 )() = u 2 ( T 1 ) (T 1 < T 1 + T 2 ) and belongs o L (, T 1 +T 2 ; E). Splicing of n conrols u j ( ) L (, T j ; E) is defined direcly, or inducively by (2.2) u = u 1 u 2 u n = (u 1 u 2 u n 1 ) u n 1 Saemens like (1.9) should be qualified by almos everywhere. We don boher wih his.

4 172 H. O. FATTORINI and u( ) L (, T 1 + T T n ; E). The definiion eends o infinie sequences u n ( ) L (, T n ; E); n = 1, 2,... }. If u n ( ) L (,T n;e) C hen he splice ( (2.3) u() = u n )() n=1 belongs o L (, T; E), T = n=1 T n. We only use his case when T <. Obviously, he finie or infinie splice of admissible conrols is admissible. Splicing is associaive bu no commuaive. In general, splicing opimal conrols does no preserve opimaliy; for he simples one-dimensional sysem y () = u() he only ime opimal conrols in [, T] are u 1 () = 1 and u 1 () = 1 and no nonrivial splicing is opimal. One can only hope for eamples where splicing preserves opimaliy, and he resuls are only for he conrol sysem (2.4) y(, ) y(, ) = + u(, ), y(, ) = ζ(), y(, ) = (, < ). This sysem can be wrien as (1.1) in he space E = L 2 (, ) wih (2.5) Ay() = y (), domain D(A) = all y( ) L 2 (, ) wih y ( ) in L 2 (, ) and y() = }. This operaor generaes he (isomeric) righ ranslaion semigroup y( ) ( ) (2.6) S()y() = ( < ). We show below ha here eiss classes SP(T) of opimal conrols 2 such ha if u 2 ( ) SP(T 2 ) and u 1 ( ) is an arbirary opimal conrol in [, T 1 ] hen he splice (u 1 u 2 )() is opimal (Theorem 4.1). The resul is easily eended o finie splicings (Theorem 4.2). Infinie splicings are no as simple; all we can show (Theorem 5.3) is ha here eis inducively defined sequences of opimal conrols ū j ( )} (ū 1 ( ) ime opimal, u j ( ) SP(T j ) for j = 2, 3,...) whose splice is ime opimal. Since he resuls are resriced o he sysem (2.4) splicing seems lile more han a curiosiy. However, i can be used o produce ime opimal conrols no saisfying Ponryagin s maimum principle (1.6). The resuls are generalizaions of he resuls in [5] and [6] Secion 2.7 (where splicing was used in paricular cases bu no eplicily defined) and we are able o produce ime opimal conrols rougher han he ones herein. 2 Each conrol u() is defined in is own inerval [, T], possibly differen for differen conrols. Tha u() is opimal in [, T] means i drives ime opimally o he arge ȳ = y(t,, ū), ha is, he arge is no fied beforehand. Opimaliy has he same meaning: no oher admissible conrol u( ) can drive o y(t,, ū) in less ime. Someimes we abbreviae opimal in [, T] o simply opimal.

5 SPLICING OF TIME OPTIMAL CONTROLS 173 Splicing (alhough no of opimal conrols) is known in conrol heory; i has been especially used in conrollabiliy of sysems in Euclidean spaces and Lie groups. For references see [6], Secion THE RIGHT TRANSLATION SEMIGROUP The adjoin of (2.6) is he lef ranslaion (and chop-off) semigroup (3.1) S() y( + ) ( ) y() = ( < ). We have (3.2) S() S() = I, S()S() y() = χ ()y(), χ () he characerisic funcion of [, ). Formula (1.2) for he conrol u()() = u(, ) is (3.3) y(,, ζ, u) = y(, ζ, u)() = = ζ( ) + ( ) S()ζ + S( σ)u(σ)dσ () u(σ, ( σ))dσ, hus he conribuion of he conrol u(σ, ) o y(,, ζ, u) is he inegral of u(σ, ) over he inersecion of he posiive quadran, wih he line (σ, ( σ)) joining (, ) wih (, ) as shown in Figure 1. The space Z of all mulipliers consiss of all measurable z() defined in > and such ha κ(, z) = S() z( ) = z( + ) 2 d = z() 2 d <, for >, while Z(T) consiss of all z( ) Z saisfying he inegrabiliy condiion (1.5), T S(σ) z( ) dσ = T κ(σ, z)dσ <. Since we are in a Hilber space, (1.7) applies, and any conrol ha saisfies (1.6) is given by (3.4) ū(, ) = S(T ) z() S(T ) z( ) z( + (T )) = χ () κ(t, z) (T δ < T),

6 174 H. O. FATTORINI Figure 1. Inegraion lines in formula (3.3). T T s(ȳ) R s(ȳ) K T R s(ȳ) K Figure 2. Opimal ime and opimal conrol when s(ȳ) <. where (T δ, T] is he maimal inerval where S(T ) z ; if δ = T, he inerval in (3.4) is < T. Using he second equaliy (3.2) (3.5) S(T σ)ū(σ, ) = S(T σ)s(t σ) z() S(T σ) z( ) = χ T σ()z() κ(t σ, z) (T δ < T). Consequenly, if κ(t σ) in < σ T, y(t,,, u) is he inegral in σ T of he righ side of (3.5). This leads o a simple eplici formula for y(t,,, u) in erms of z() ([6] formula (2.6.25)) which we don need o use here. Le ϕ() be measurable in >. Define s(ϕ( )) as he minimum s such ha suppor ϕ( ) [, s] (suppor defined modulo null ses). If he suppor of ϕ( ) is no conained in any inerval [, s] we se s(ϕ( )) =. Lemma 3.1. Assume ha ȳ( ) L 2 (, ) is reachable and s(ȳ( )) <. Then (a) if T is he opimal driving ime from o ȳ() we have T s(ȳ), (b) if ū(, ) is he opimal conrol doing he drive, he suppor of ū(, ) is conained in he region R in Figure 2 righ.

7 SPLICING OF TIME OPTIMAL CONTROLS 175 Proof. Assume T > s(ȳ) so ha we have he configuraion in Figure 2 lef. If he ime opimal conrol ū(, ) is no zero in K we may modify is definiion o ū(, ) = here. This doesn affec he arge and, if anyhing, improves each norm ū(, ) L 2 (, ). The modified conrol is admissible, hus ime opimal; by uniqueness i is equal o ū(, ). If T > s(ȳ) hen ū(, ) L 2 (, ) = in T s(ȳ). However, Theorem 1.4 says ha a ime opimal conrol mus saisfy u() = 1, hus we have a conradicion. (b) If he opimal conrol ū(, ) is no zero in K we modify is definiion o ū(, ) = here and argue as in (a). Lemma 3.2. Assume ȳ() is reachable wih a conrol of he form (3.4). Then (3.6) s(z( )) = s(ȳ( )). Proof. Equaliy (3.6) follows from he far more precise saemen (3.7) z() = ȳ() =. To see ha (3.7) holds we look a formula (3.5). The arge ȳ() is obained inegraing in σ T he funcion on he righ side, hus z() = ȳ() =. On he oher hand, he funcion χ T σ () (as a funcion of ) is nonzero for T σ hus ȳ() = z() = and we are done. Lemma 3.1 and Lemma 3.2 have oally differen scopes. In Lemma 3.1 ȳ() is jus reachable by an admissible conrol (hus by an opimal conrol in view of Theorem 1.3). In Lemma 3.2, ȳ() is reachable by a conrol of he form (3.4), his conrol being auomaically opimal by Theorem 1.2. The aim of his paper is o produce opimal conrols no of he form (3.4). 4. FINITE SPLICING A conrol u(, ) in L (, T; L 2 (, )) belongs o he class SP(T) if and only if i is admissible and drives in opimal ime T o a arge ȳ() wih s(ȳ( )) = T. Eamples of conrols in SP(T) are easy o produce wih he help of Theorem 1.2 and (3.4): if z( ) Z(T) wih s(z( )) = T hen he conrol ū(, ) in (3.4) drives o a arge ȳ() = y(t,,, ū) in opimal ime T and hus is in SP(T). Since Z(T) L 2 (, ) we may ake z( ) L 2 (, ) for he purposes of his paper. Any conrol in SP(T) mus be zero in he region K in Figure 3; if no, we can modify is definiion o ū(, ) = here and argue as in he proof of Lemma 3.1 The condiion s(z( )) = s(ȳ( )) = T alone does no guaranee ha T is he opimal ime o reach ȳ( ); if s(z( )) = T and T < T he conrol ū(, ) = S(T ) z() S(T ) z( ) = χ () z( + (T )) κ(t, z) drives o a arge ȳ( ) wih s(ȳ( )) = T in ime T < T. ( < T )

8 176 H. O. FATTORINI T s(ȳ) = T K Figure 3. Conrols in he class SP(T). T 2 ȳ 2 () T 2 ȳ 2 () T 2 ȳ 1 ( T 2 ) T 1 + T 2 R ū 2 ( T 1, ) K R ū 2 (, ) K T 1 ȳ 1 () ū 1 (, ) Figure 4. Splicing of ū 1 (, ) and ū 2 (, ). Theorem 4.1. Le he admissible conrol ū 1 (, ) L (, T 1 ; L 2 (, )) be opimal in [, T 1 ] and le u 2 (, ) SP(T 2 ). Then he splice (u 1 u 2 )(, ) L (, T 1 + T 2 ; L 2 (, )) is opimal in [, T 1 + T 2 ]. Proof. ū 2 (, ) is zero in he complemen K of he riangle R in Figure 4 lef, hus he splice ū(, ) = (ū 1 ū 2 )(, ) is zero in he corresponding region K of Figure 4 righ. If ū 2 (, ) drives o he arge ȳ 2 () in ime T 2 and ū 1 (, ) drives o he arge ȳ 1 () in ime T 1, formula (3.3) shows ha ū(, ) drives o he arge ȳ 2 () ( T 2 ) (4.1) ȳ() = ȳ 1 ( T 2 ) (T 2 < ) in ime T = T 1 + T 2. I remains o show ha his drive is opimal. Since ū(, ) is admissible he arge (4.1) is reachable and Theorem 1.3 says ha a ime opimal conrol ũ(, ) eiss. Le T be he opimal driving ime, and assume firs ha T < T 2. The conrol ũ(, ) (, ) u(, ) = R (, ) K

9 SPLICING OF TIME OPTIMAL CONTROLS 177 ȳ 2 () T 2 ȳ 1 ( T 2 ) T 1 + T 2 T ȳ 2 () T 2 R K R K T 1 δ Figure 5. Time opimaliy of he splice. ( R, K he wo regions in Figure 5 lef) is admissible and drives o ȳ2 () in ime T < T 2, which conradics he opimaliy of ū 2 (, ). Accordingly, T T2. If T < T 1 + T 2, le δ = T 1 + T 2 T. The admissible conrol does he drive The conrol u δ (, ) = ũ( δ, ) (δ T 1 + T 2 ) y(δ,,, u δ ) =, y(t 1 + T 2,,, u δ ) = ȳ(). v(, ) = u δ (, ) (, ) R (, ) K (T 1 T 1 + T 2 ) (R, K he regions in Figure 5 righ) is admissible and does he drive y(t 1,,, v) =, y(t 1 + T 2,,, v) = ȳ 2 () while ū 2 ( T 1, ) does he same drive opimally; i follows ha v(, ) is also opimal and, by uniqueness of opimal conrols, 3 (4.2) v(, ) = u δ (, ) = ū 2 ( T 1, ) (T 1 T 1 + T 2 ) which shows ha u δ (, ) = in K. By Formula (3.3) (Figure 1) we mus have (4.3) y(t 1,,, u δ ) = ȳ 1 (). Accordingly, u δ ( + δ, ) = ũ(, ) drives o ȳ 1 () in ime T 1 δ, which conradics he opimaliy of ū 1 (, ). Theorem 4.2. Le ū 1 (, ) L (, T 1 ; L 2 (, )) be opimal in [, T 1 ] and le ū j (, ) SP(T j ), j = 2,..., n. Then he splice v n (, ) = (ū 1 ū 2 ū n )(, ) L (, T 1 + T T n ; L 2 (, )) is opimal in [, T 1 + T T n ]. 3 Time opimaliy can be defined for an arbirary inerval [a, b]. Resuls are he same.

10 178 H. O. FATTORINI Proof. Theorem 4.1 and inducion: use formula (2.2). Remark 4.3. We can figure ou he arge hi by v n (, ) arguing as in Theorem 4.1. Le ȳ j () = y(t j,,, ū j ) be he arge hi by he conrol ū j (, ) in opimal ime T j, and define S j = T T j (j = 1,...n 1), T = T 1 + T T n. Then (4.4) y(t,,, v n ) ȳ 1 ( (T S 1 )) (T S 1 < ) = ȳ j ( (T S j )) (T S j < T S j 1, j = 2,...,n 1) ȳ n () ( < T n ). This formula reads from righ o lef as j increases. We can pu i in able form: Inerval Targe ȳ() [, T n ) ȳ n () [T n, T n + T n 1 ) ȳ n 1 ( T n ) [T n + T n 1, T n + T n 1 + T n 2 ) ȳ n 2 ( T n T n 1 )..... T n + + T 3, T n + + T 2 ) ȳ 2 ( T n T 3 ) [T n + + T 2, ) ȳ 1 ( T n T 2 ) Remark 4.4. The finie splices of opimal conrols in Theorem 4.2 have a seemingly conflicual relaionship wih Theorem 1.1. Assume he opimal conrols ū j (, ) in he splice are produced from formula (3.4) and le z j () be he muliplier for each conrol. I is proved in [6] Secion 2.6 ha z j () can be chosen in such a way ha ȳ j () = (T j,,, ū j ) is coninuously differeniable in [, T j ] wih ȳ j () = ȳ j() = ȳ j (T j ) = = ȳ j (T j) =. In view of he definiion (2.5) of he infiniesimal generaor A and of formula (4.4) he arge ȳ(t,,, v n ) belongs o D(A). In fac, he differen pieces of he arge are smooh and mach (ogeher wih heir derivaives) a he divisory poins of (4.4) and he arge is zero for =. Theorem 1.1 applies and v(, ) saisfies he maimum principle (1.6) in is Hilber space version (1.7), which seems o clash wih he disconinuous naure of v n (, ). The eplanaion lies in he fac ha Theorem 1.1 does no guaranee ha he cosae z() = S(T ) z in (1.6) is nonzero in he enire conrol inerval [, T], jus in an inerval (T δ, T]. The muliplier provided by Theorem 1.1 for he conrol v(, ) is he muliplier z n ( ) (eended = for T n ) for he las piece ū n (, ) of he splice v n (, ), and he cosae z() = S(T ) z n vanishes ouside of he inerval (T T n, T]. In fac, he conrol v n (, ) swiches cosaes every ime crosses one of he divisory poins S j = T T j in he inerval [, T]. For more deails on his see [6] Secion 2.7.

11 SPLICING OF TIME OPTIMAL CONTROLS INFINITE SPLICING If y E = sae space of (1.1) we denoe by T (y) he opimal driving ime from o y. If y is reachable and E is a Hilber space, Theorem 1.3 says ha T (y) eiss and is finie. If y is no reachable, ha is, if canno be driven o y by any admissible conrol in any ime we define T (y) =. The firs resul below is in [4], [6] Lemma 2.7.3; he second resul is specific o he sysem (2.4). Lemma 5.1. Le E be a Hilber space and le y n } E be a sequence wih y n y E. Then (5.1) T (y) lim inf n T (y n). Lemma 5.2. Assume y 1 () y 2 (). Then (5.2) T (y 1 ( )) T (y 2 ( )). Proof. We may assume ha T (y 2 ( )) <. If ū 2 (, ) is he conrol driving o y 2 () opimally we define a new conrol u 1 (, ) muliplying ū 2 (, ) by he quoien y 1 ()/y 2 () over each inegraion line (σ, (T σ)) in Formula (3.3) (Figure 1); precisely u 1 (σ, (T σ)) = φ()ū 2 (σ, (T σ)) ( <, σ T, + (T σ) ) where φ() = if y 2 () =, φ() = y 1 ()/y 2 () if y 2 (). We have φ() 1, hus u 1 (σ, ) ū 2 (σ, ) and i follows ha u 1 (σ, ) is admissible. Formula (3.3) shows ha y(t,,, u 1 ) = y 1 (), so ha y 1 () is reachable in ime T = T (y 2 ( )). The opimal driving ime is T, hus inequaliy (5.2) holds and we are done. The infinie sequence o be spliced is ū n (, )}, where he firs conrol ū 1 (, ) L (, T 1 ; L 2 (, )) is ime opimal in [, T 1 ] and u n (, ) SP(T n ) (n = 2, 3,...). The sequence is far from arbirary; in fac i will be consruced inducively, each ū n (, ) (or, raher, is suppor) depending on he choice of he preceding erms ū 1 (, ),...,ū n 1 (, ). The procedure implies T = n=1 T n <, hus he splice is defined in a finie inerval [, T]. The primary objecs are no he T n bu wo sequences S n ; n =, 1, 2,... } and S n ; n =, 1, 2,... } such ha (5.3) < S < S 1 < S 2 <... S > S 1 > S 2 >... S n < S n (n =, 1, 2,...). In addiion, he sequences S n } and S n } saisfy (5.4) S n+1 S n α n, S n S n α n (n =, 1, 2,...),

12 18 H. O. FATTORINI S n+1 y(τ,,, w n ) y(s n,,, v n ) S n S n 1 Figure 6. Choice of S n+1 from v n (, ). where he summable posiive series α n ; n =, 1,... } is given in advance and oally arbirary. The T n are obained from he S n by (5.5) T n = S n S n 1 (n = 1, 2,...), hence (5.6) T = T n = (S n S n 1 ) n=1 n=1 α n 1 = α n = α. n=1 n= The sequences begin wih S = and S > saisfying he second condiion (5.4) for n =, ha is, S S α. For he inducive sep, we assume ha he S j, he S j and he ū j (, ) have been consruced for j n. Define v n (, ) as he finie splice (5.7) v n (, ) = (ū 1 ū 2 ū n )(, ). By Theorem 4.2 i drives in opimal ime S n o he arge y(s n,,, v n ). If we define (5.8) w n (, ) = v n (, ) ( S n ) (S n < ) we have (Figure 6) (5.9) y(τ,,, w n ) = ( < τ S n ) y(s n, (τ S n )),, v n ) (τ S n ) for τ > S n, so ha y(τ,,, w n ) y(s n,,, v n ) L 2 (, ) as τ S n. Accordingly, Lemma 5.1 can be applied and we can selec S n+1 > S n wih (5.1) T (y(τ,,, w n )) T (y(s n,,, v n )) 1 n (S n τ S n+1 ).

13 SPLICING OF TIME OPTIMAL CONTROLS 181 In fac, if his choice is no possible here eiss a sequence τ k }, τ k S n such ha T (y(s n,,, v n )) lim inf k T (y(τ k,,, w n )) + 1 n conradicing (5.1). Since T (y(s n,,, v n )) = S n, (5.1) becomes (5.11) T (y(τ,,, w n )) S n 1 n (S n τ S n+1 ). Moving S n+1 o he lef if necessary we may assume ha (5.12) S n+1 S n α n+1 and hen selec S n+1 wih (5.13) S n < S n+1 < S n+1 which, in view of (5.12) gives he second condiion (5.4). The conrol ū n+1 (, ) added a his sep o he splice is an arbirary conrol in SP(T n+1 ), where T n+1 = S n+1 S n. This conrol eiss, since formula (3.4) applied o z( ) wih s(z( )) = T (T arbirary) produces a conrol in SP(T). The sequences S n } and S n } consruced by repeaed applicaion of his inducive sep saisfy he hree condiions (5.3) and he wo condiions (5.4). We hen define and splice he chosen sequence ū n (, ), T = sup S n = lim S n = lim S n+1 = inf n n n n Sn+1 ū = n=1 obaining an admissible conrol ū(, ) L (, T; L 2 (, )). This conrol drives o he arge ū n (5.14) ȳ() = y(t,,, ū) ȳ 1 ( (T S 1 )) (T S 1 < ) = ȳ j ( (T S j )) (T S j < T S j 1, j = 2, 3,...) where ȳ j () = y(t j,,, ū j ) has suppor in [, T j ] for j 2; he suppor of ȳ 1 () is unresriced. (This formula is he generalizaion of (4.4) o infinie splices.) Consider now a second splice where each ū j (, ) (j n + 1) is replaced by ξ j (, ) = in he same inerval [, T j ], ( n ) ( w n = ū n j=1 j=n+1 ξ j ). This conrol is he same defined in (5.8). I drives o a arge y(t,,, w n ) which can be described by (5.14) wih he only difference ha ȳ j () = for j n+1;

14 182 H. O. FATTORINI precisely (5.15) ȳ() = y(t,,, ū) ȳ 1 ( (T S 1 )) (T S 1 ) = ȳ j ( (T S j )) (T S j < T S j 1, j = 2, 3,..., n) ( < T S n ) which can be wrien in he form (5.16) y(t,,, w n ) = χ()y(t,,, ū) where χ() is he characerisic funcion of he inerval [, T S n ]. I follows from (5.16) ha y(t,,, w n ) y(t,,, ū) hence, using Lemma 5.2 in he firs inequaliy and aking (5.11) ino accoun in he second, (5.17) T (ȳ( )) T (y(t,,, w n )) S n 1 n T as n. Taking limis, we deduce ha T (ȳ( )) T. Since ū(, ) drives o ȳ() in ime T, i follows ha T (ȳ( )) = T and ha ū(, ) is opimal. We have proved Theorem 5.3. The infinie splice (2.3) of he conrols ū n (, ); n = 1, 2,... } is ime opimal. Remark 5.4. The funcion T (y) is a version of he value funcion used in he Hamilon-Jacobi approach o opimal conrol, alhough he usual value funcion is defined as he ime needed o drive a arge y E o (see [1], [2], [3]). Lemma 5.1 is relaed o eising resuls on smoohness of he value funcion. 6. APPLICATIONS: STRONGLY SINGULAR TIME OPTIMAL CONTROLS A conrol is called singular if does no saisfy he maimum principle (1.6) wih any muliplier z Z(T). If a singular conrol saisfies (1.6) wih a muliplier η Z i is called weakly singular; if i does no saisfy (1.6) wih any muliplier η, i is called srongly singular. The conrol ū(, ) consruced in Theorem 5.3 using conrols ū n (, ) SP(T n ) of he form (3.4) as building blocks is srongly singular, ha is, i does no saisfy he maimum principle (1.6) wih any muliplier η. In fac, if i did, i would have iself he form (3.4) wih z( ) Z = space of all mulipliers (Secion 3) in some inerval [T δ, T]. This is impossible; conrols of he form (3.4) are eiher = or on lines + (T ) = c, a qualiy ha ū(, ) conspicuously lacks. However,

15 SPLICING OF TIME OPTIMAL CONTROLS 183 ū(, ) is ime opimal in any subinerval 4 [, T ], T < T and is regular in each of hese subinervals. This is he class of conrols consruced in [5] [6] Theorem We can apply he same idea o consruc opimal conrols ha are more singular han he ones in [5], [6]. We prove firs an addendum o Theorem 5.3 having o do wih he assumpions on ū 1 (, ). In Theorem 5.3 ū 1 (, ) is jus assumed o be an opimal conrol in [, T 1 ] (he oher conrols ū n (, ) are in SP(T n )). We assume now ha u 1 (, ) belongs o he class SP(T 1 ) as well. Theorem 6.1. Assume ha all conrols ū n (, ); n = 1, 2,... } in Theorem 5.3 belong o SP(T n ). Then he infinie splice ū(, ) belongs o SP(T). Proof. We know from Theorem 5.3 ha ū(, ) drives in opimal ime T o a arge ȳ() described by formula (5.14); all we have o show is ha (6.1) s(ȳ( )) = T. By he assumpion on ū 1 (, ) we have s(ȳ 1 ( )) = s(y(t 1,,, ū 1 )) = T 1, in paricular, suppor ȳ 1 ( ) [, T 1 ]. This means he suppor of y 1 ( (T S 1 )) in he firs line of (5.14) is conained in [T S 1, T] = [T T 1, T], so ha suppor ȳ( ) [, T]. If suppor ȳ( ) [, T ǫ] for ǫ > we reverse his implicaion and deduce ha s(ȳ 1 ( ))) T ǫ, a conradicion. We call SP (T) SP(T) he se of all conrols of he form (3.4) wih s(z( )) = T. The se SP 1 (T) SP(T) consiss of all infinie splices produced under he auspices of Theorem 6.1 wih building blocks ū n (, ) SP (T n ). Lemma 6.2. SP 1 (T) for all T >. Proof. A resul like his is needed because he opimal ime T in Theorems 5.3 and 6.1 is no chosen in advance; i is a byproduc of he heorem, precisely, i depends on he choice of he S n. Given T > arbirary we consruc firs a conrol v 1 (, ) SP 1 (T ) for some T T. This is possible since he series α n } and is sum α in (5.6) are compleely arbirary; we may ake α = T. If T = T we are done. Oherwise, we ake a conrol v (, ) SP (T) wih opimal ime T T and define ū = v v 1, ha is, we add he building block v a he beginning of he splice. By Theorem 6.1 he conrol ū belongs o SP 1 ((T T ) + T ) = SP 1 (T). 4 The opimaliy principle says ha if ū( ) is ime opimal in [, T] hen i is ime opimal in any subinerval [a, b] [, T]. The proof is immediae; if we can improve he driving ime in [a, b] hen (via a ime ranslaion) we can improve he driving ime in [, T]. The ime ranslaion works because he sysem (1.1) does no depend eplicily on ime.

16 184 H. O. FATTORINI Theorem 6.3. The conrols ū n (, ) making up he infinie splice in Theorem 6.1 can be aken in SP 1 (T n ). Proof. We go over he proof of Theorem 5.3 as modified in Theorem 6.1, especially over he inducive choice of ū n+1 (, ) in he paragraph following (5.11). The only qualificaion on ū n+1 (, ) is an arbirary conrol in SP(T n+1 ) wih T n+1 = S n+1 S n where S n+1 has been chosen on he basis of (5.11). By Lemma 6.2 we can selec ū(, ) in SP 1 (T n+1 ) for arbirary T n+1. The res of he proof of Theorem 5.3 remains he same. The se of all splices produced by Theorem 6.1 wih building blocks ū n (, ) SP 1 (T n ) is SP 2 (T) SP(T). Conrols in his class are srongly singular no only in he whole inerval [, T] bu in each subinerval [, S n, ] S n } he sequence in Theorem 5.3 wih S n T. Eploiing he same idea we prove firs ha SP 2 (T) for all T > and hen use Theorem 6.1 o consruc conrols whose building blocks are in SP 2 (T n ). This is he class SP 3 (T); he inervals where he conrols are srongly singular are [, S mn ] where S mn S n as m and S n T. The classes SP n (T) are inducively defined in a similar way for all n and conain opimal conrols which are srongly singular in more and more subinervals of [, T]. A ime opimal conrol ū( ) for (1.1) is called hypersingular in [, T] if i is srongly singular in any inerval [a, b] [, T] (for an eample, see [7]). Obviously, he conrols in any class SP n (T) fall shor of hypersingular. To produce a hypersingular conrol aking ū n (, ) SP n (T) and hen aking limis as n won work since he conrols in he sequence would have suppor in a shrinking band T, ǫ n. We don know if he consrucion in his paper can be modified o give hypersingular conrols, in fac we don know if hypersingular conrols for (2.4) eis a all. The oy conrol sysem (2.4) (one of he simples involving parial differenial equaions) is he carrier of a number of ineresing conrol heory eamples and counereamples; for informaion on his see [6], [9], [1]. For a recen survey on regular, singular and srongly singular conrols for he sysem (1.1) and relaed opics see [1]. REFERENCES [1] P. Cannarsa and O. Cǎrjâ, On he Bellman equaion for he minimum ime problem in infinie dimensions, SIAM J. Conrol & Opimizaion, 43: , 24. [2] O. Cǎrjâ, On he minimum ime funcion and he minimum energy problem; a nonlinear case. Sysems & Conrol Leers, 55: , 26. [3] O. Cǎrjâ and A. Lazu, On he minimal ime null conrollabiliy of he hea equaion. Discree Conin. Dyn. Sys, Proc. 7h AIMS Conference on Dynamical Sysems, Differenial Equaions and Applicaions, , 29.

17 SPLICING OF TIME OPTIMAL CONTROLS 185 [4] H. O. Faorini, Eisence of singular eremals and singular funcionals in reachable spaces, Jour. Evoluion Equaions, 1: , 21. [5] H. O. Faorini, Vanishing of he cosae in Ponryagin s maimum principle and singular ime opimal conrols, Jour. Evoluion Equaions, 4:99 123, 24. [6] H. O. Faorini, Infinie Dimensional Linear Conrol Sysems, Norh-Holland Mahemaical Sudies 21, Elsevier, Amserdam 25. [7] H. O. Faorini, Linear conrol sysems in sequence spaces, Funcional Analysis and Evoluion Equaions. The Güner Lumer Volume , 27. [8] H. O. Faorini, Time and norm opimaliy of weakly singular conrols, Progress in Nonlinear Differenial Equaions and Their Applicaions 8: Springer Basel AG [9] H. O. Faorini, Srongly regular ime and norm opimal conrols, o appear in Dynamics of Coninuous, Discree and Impulsive Sysems Series B: Applicaions and Algorihms 18: , 211. [1] H. O. Faorini, Time and norm opimal conrols: a survey of recen resuls and open problems, Aca Mahemaica Scienia 31B: , 211.