Quadratic order conditions for bang-singular extremals

QUADRA TIC ORDER CONDITI O NS F OR BANG-SINGULAR EXTREMALS M. Soledad Ar onna CONICET CIF ASIS, Argentina INRIA Saclay - CMAP Ecole Polytec hnique Route de Saclay , 91128 Palaisea u, F rance J. Fr ´ ed ´ eric Bonnans INRIA Saclay - CMAP Ecole Polytec hnique Route de Saclay , 91128 Palaisea u, F rance Andrei V. Dmitruk Russian Academy of Sciences - CEMI and Moscow State U niversi ty 47 N akhimo vsky Prospect, 117418 Mosco w, Ru ssia P ablo A. Lot ito CONICET PLADEMA - U niv. Nacional de Centro de la Prov. de Buenos Aires Campus Universitario Pa ra je Arroy o Seco, B7000 T andil, Argentina Abstra ct. This pap er deals with optimal con trol problems for systems aﬃne in t h e control vari able. W e consider nonn egativit y constrain ts on th e control, and ﬁnitely many equality and inequality constraints on the ﬁnal state. First, w e obtain second order necessary optimality conditions. Secondly , w e d eriv e a second order suﬃcient condition for the scalar control case. 1. I n troduction In this article w e obtain second ord er conditions for an op timal cont rol problem aﬃne in th e control. First w e consider a p oint wise nonnegativ- it y constrain t on the con trol, end-p oint state constraints and a ﬁxed time in terv al. Then w e extend the result to b ound constr aints on the con trol, initial-ﬁnal state constraint s an d p roblems inv olving parameters. W e do n ot assume th at the multipliers are unique. W e study wea k and P on try agin minima. There is already an imp ortan t literature on this sub ject. The case with- out con trol constraint s, i.e. when the extremal is total ly singu lar, has b een extensiv ely s tudied since the m id 1960s. Kelle y in [32] treated the scalar 1991 Mathematics Subje ct Classiﬁc ation. Primary: 49K15. Key wor ds and phr ases. optimal control, second order condition, control constraint, singular arc, bang-singular solution. The ﬁrst tw o authors are supp orted by the Europ ean U nion under the 7th F ramework Programme FP7-PEOPLE-2010-ITN Grant agreement number 264735-SADCO. 1 2 M.S. ARONNA, J.F. BON NANS, A.V. DMITRUK AND P .A. LOTITO con trol case and presente d a necessary cond ition in v olving the second ord er deriv ativ e of the switc h ing fu nction. The r esult was extended b y Kopp and Mo y er [34] for higher order deriv ative s, and in [33] it was shown that th e order had to b e ev en. Goh in [27] prop osed a sp ecial c hange of v ariables obtained via a linear ODE and in [26] used this transf orm ation to deriv e a necessary condition for the v ector con trol pr ob lem. An extensiv e surv ey of these articles can b e foun d in Gabaso v and K irillo v a [24]. Jacobson and Sp eye r in [30], and to gether with L ele in [31] obtained necessary conditions b y adding a p enalizatio n term to the cost f unctional. Gabaso v and Kir illo v a [24], Krener [35], Agrac hev and Gamkrelidze [1] obtained a countable series of necessary conditions th at in fact use the idea b ehind the Goh transforma- tion. Milyutin in [43] d isco vered an ab s tract essence of this app roac h and obtained even stron ger necessary conditions. In [2] Agrac hev and Sac hk o v in v estigate d second order optimalit y cond itions of the minim um time prob- lem of a single-input sy s tem. The main feature of this kind of problem, where th e con trol en ters linearly , is that the corresp onding second v ariation do es not con tain the Legendre term, so the metho d s of th e classica l calculus of v ariations are not applicable for obtaining suﬃcien t conditions. This is wh y the literature wa s mostly dev oted to necessary conditions, wh ich are actually a consequence of the non n egativit y of the second v ariation. A su f- ﬁcien t condition for time optimalit y was giv en by Mo y er [45] for a system with a scalar con trol v ariable and ﬁxed endp oin ts. O n the other h and, Goh’s transformation ab o v e-men tioned allo ws one to co nv ert the seco nd v ariation in to another functional that h op efully turns out to b e co erciv e w ith resp ect to the L 2 − norm of some state v ariable. Dmitruk in [12] prov ed that this co ercivit y is a suﬃ cien t co ndition for the wea k optimalit y , and pr esen ted a closely related necessary condition. He used the abstract approac h devel - op ed by Levitin, Milyutin and Osmolo vskii in [38], and considered ﬁn itely man y inequalit y and equalit y constraints on the endp oin ts and the p ossible existence of sev eral multipliers. In [13, 15] he also obtained n ecessary and suﬃcien t conditions for this norm, again closely related, f or P on try agin min- imalit y . More recentl y , Bonnard et al. in [6] provided second ord er su ﬃcien t conditions for the min im um time problem of a s ingle-input system in terms of the existence of a conjugate time. On the other hand, th e case with linear con trol constraints an d a “purely” bang-bang con trol without singular subarcs has b een extensiv ely inv esti- gated ov er the past 15 ye ars. Milyutin and Osmolo vskii in [44] p ro vided necessary and s uﬃcien t co nditions based on the general theory of [38]. Os- molo vskii in [46] completed some of the pro ofs of th e latter article. S aryc hev in [53] ga v e ﬁrs t and second ord er suﬃcient condition for Pon try agin solu- tions. A grac hev, Stefani, Zezza [3] redu ced the pr oblem to a ﬁnite d imen- sional problem w ith the switc hing instant s as v ariables and obtained a suf - ﬁcien t condition for strong optimal it y . The r esu lt wa s recen tly extended by P oggio lini and Spadin i in [47]. O n the other hand, Maurer and Osmolo v s kii QUADRA TIC ORDER CONDITIONS F OR BANG-SINGULAR EXTREMALS 3 in [42, 41] ga ve a second order su ﬃcien t condition that is suitable for prac- tical v eriﬁcations and present ed a numerical p ro cedure that allo ws to v erify the p ositivit y of certain quadratic forms. F elgenhauer in [21, 22, 23] stud- ied b oth second order op timalit y conditions and sensitivit y of the optimal solution. The mixed case, wh ere the con trol is partly b ang-bang, partly singular w as stud ied in [48] b y Po ggiolini and S tefani. T hey obtained a s econd order suﬃcien t condition with an add itional geometrical h yp othesis (which is not needed here) and claimed th at it is not clear whether this h yp othesis is ‘almost necessary’, in the sen se that it is not obtained straigh tforw ard from a necessary condition by strengthening an inequ ality . In [49, 50] they derive d a sec ond order su ﬃcien t condition for the sp ecial case of a time-optimal problem. The main result of the present article is to provide a suﬃcient condition that is ‘almost necessary’ for bang-singular extremals in a general Ma y er problem. On the other hand, the single-input time-optimal problem w as extensiv ely studied by means of synt hesis-lik e metho ds. See, among others, S ussmann [59, 58, 57], S c h¨ attler [54] and S c h¨ attler-Jank o vic [55]. Both bang-bang and bang-singular s tr uctures were analysed in these w orks. The article is organized as follo ws. In the second section we present the p roblem and give basic deﬁnitions. In the third section w e p erform a second order analysis. More p recisely , we obtain the second v ariation of the Lagrangian fun ctions and a necessary condition. Afterw ards, in the four th section, w e present the Goh transformation and a n ew necessary condition in the transformed v ariables. In th e ﬁfth section we sho w a s uﬃcien t condition for scalar con trol. Finally , w e giv e an example w ith a scalar con trol wh er e the second ord er suﬃcient condition can b e veriﬁed. The app endix is dev oted to a series of tec hnical prop erties that are used to p ro v e the main results. 2. St a tem e nt of t h e p roblem and assump t ions 2.1. Stat emen t of the problem. Cons id er the sp aces U := L ∞ (0 , T ; R m ) and X := W 1 ∞ (0 , T ; R n ) as con trol and state spaces, resp ectiv ely . Denote with u and x their elements, r esp ectiv ely . When needed, put w = ( x, u ) for a p oint in W := X × U . In this pap er we in v estigat e the optimal control problem J := ϕ 0 ( x ( T )) → min , (1) ˙ x ( t ) = m X i =0 u i f i ( x ) , x (0) = x 0 , (2) u ( t ) ≥ 0 , a . e . on t ∈ [0 , T ] , (3) ϕ i ( x ( T )) ≤ 0 , for i = 1 , . . . , d ϕ , η j ( x ( T )) = 0 , for j = 1 . . . , d η . (4) where f i : I R n → I R n for i = 0 , . . . , m, ϕ i : I R n → I R for i = 0 , . . . , d ϕ , η j : I R n → I R for j = 1 , . . . , d η and u 0 ≡ 1 . Ass ume that data fu nctions f i 4 M.S. ARONNA, J.F. BON NANS, A.V. DMITRUK AND P .A. LOTITO are twice cont inuously diﬀeren tiable. F u nctions ϕ i and η j are assumed to b e t wice diﬀeren tiable. A tr aje ctory is an elemen t w ∈ W that satisﬁes the state equation (2). If, in addition, constraint s (3) and (4 ) hold, we say that w is a fe asible p oint of the p r oblem (1)-(4). Denote by A the set of fe asible p oints. A fe asible variation for ˆ w ∈ A is an elemen t δw ∈ W suc h that ˆ w + δ w ∈ A . Deﬁnition 2.1. A p air w 0 = ( x 0 , u 0 ) ∈ W is said to b e a we ak minimum of problem (1)-(4) if there exists an ε > 0 suc h that the cost function attains at w 0 its minim um on the set  w = ( x, u ) ∈ A : k x − x 0 k ∞ < ε, k u − u 0 k ∞ < ε  . W e sa y w 0 is a Pontryagin minimum of problem (1)-(4) if, for an y p ositiv e N , there exists an ε N > 0 suc h that w 0 is a minim um p oin t on the set  w = ( x, u ) ∈ A : k x − x 0 k ∞ < ε N , k u − u 0 k ∞ ≤ N , k u − u 0 k 1 < ε N  . Consider λ = ( α, β , ψ ) ∈ I R d ϕ +1 , ∗ × I R d η , ∗ × W 1 ∞ (0 , T ; I R n, ∗ ) , i.e. ψ is a Lips c hitz-co nt in uous function w ith v alues in the n − dimensional space of ro w-v ectors with real comp onent s I R n, ∗ . Deﬁne th e pr e-H amiltonian function H [ λ ]( x, u, t ) := ψ ( t ) m X i =0 u i f i ( x ) , the terminal L agr angian fu nction ℓ [ λ ]( q ) := d ϕ X i =0 α i ϕ i ( q ) + d η X j = 1 β j η j ( q ) , and the L agr angian function (5) Φ[ λ ]( w ) := ℓ [ λ ]( x ( T )) + Z T 0 ψ ( t ) m X i =0 u i ( t ) f i ( x ( t )) − ˙ x ( t ) ! d t. In this a rticle the optimali t y of a g iv en feasible tra jectory ˆ w = ( ˆ x, ˆ u ) is stu died. Whenev er some argument of f i , H , ℓ, Φ or their deriv ativ es is omitted, assume that they are ev aluated o v er this tra jectory . Without loss of generalit y sup p ose that (6) ϕ i ( ˆ x ( T )) = 0 , for all i = 0 , 1 , . . . , d ϕ . 2.2. First order analysis. Deﬁnition 2.2. Denote by Λ ⊂ I R d ϕ +1 , ∗ × I R d η , ∗ × W 1 ∞ (0 , T ; I R n, ∗ ) the set of Pontryagin multipliers asso ciated with ˆ w consisting of the elements λ = ( α, β , ψ ) satisfying the Pontryagin Maximum Principle, i.e. ha ving the follo w ing prop erties: | α | + | β | = 1 , (7) α = ( α 0 , α 1 , . . . , α d ϕ ) ≥ 0 , (8) QUADRA TIC ORDER CONDITIONS F OR BANG-SINGULAR EXTREMALS 5 function ψ is solution of th e c ostate e quation and satisﬁes the tr ansversality c ondition at the end p oin t T , i.e. (9) − ˙ ψ ( t ) = H x [ λ ]( ˆ x ( t ) , ˆ u ( t ) , t ) , ψ ( T ) = ℓ ′ [ λ ]( ˆ x ( T )) , and the follo w ing minimum c ondition holds (10) H [ λ ]( ˆ x ( t ) , ˆ u ( t ) , t ) = min v ≥ 0 H [ λ ]( ˆ x ( t ) , v , t ) , a . e . on [0 , T ] . Remark 1. F or ev ery λ ∈ Λ , the follo wing t wo co nditions hold. (i) H u i [ λ ]( t ) is con tin uous in time, (ii) H u i [ λ ]( t ) ≥ 0 , a.e . on [0 , T ] . Recall th e follo wing w ell known r esult for whic h a pro of can b e found e.g. in Alekseev and T ikhomiro v [4], Kurcyusz and Zo w e [36]. Theorem 2.3. The set Λ is not empty. Remark 2. Since ψ ma y b e expressed as a linear con tin uous mapping of ( α, β ) and since (7) holds, Λ is a ﬁn ite-dimensional compact set. Thus, it can b e identiﬁed with a compact subset of I R s , wh ere s := d ϕ + d η + 1 . The follo wing expression for the d eriv ativ e of the Lagrangian function holds (11) Φ u [ λ ]( ˆ w ) v = Z T 0 H u [ λ ]( ˆ x ( t ) , ˆ u ( t ) , t ) v ( t )d t. Consider v ∈ U and the line arize d state e quation: (12)      ˙ z ( t ) = m X i =0 ˆ u i ( t ) f ′ i ( ˆ x ( t )) z ( t ) + m X i =1 v i ( t ) f i ( ˆ u ( t )) , a . e . on [0 , T ] , z (0) = 0 . Its solution z is called the line arize d state variable. With ea c h index i = 1 , . . . , m, w e asso ciate the s ets (13) I i 0 :=  t ∈ [0 , T ] : max λ ∈ Λ H u i [ λ ]( t ) > 0  , I i + := [0 , T ] \ I i 0 , and the active set (14) ˜ I i 0 := { t ∈ [0 , T ] : ˆ u i ( t ) = 0 } . Notice that I i 0 ⊂ ˜ I i 0 , and that I i 0 is relativ ely op en in [0 , T ] as eac h H u i [ λ ] is con tin uous. Assumption 1. Assu me strict c omplementarity for the c ontr ol c onstr aint, i.e. for ev ery i = 1 , . . . , m, (15) I i 0 = ˜ I i 0 , up to a set of n ull measure . Observe then that for an y ind ex i = 1 , . . . , m, th e con trol ˆ u i ( t ) > 0 a.e. on I i + , and giv en λ ∈ Λ , H u i [ λ ]( t ) = 0 , a . e . on I i + . 6 M.S. ARONNA, J.F. BON NANS, A.V. DMITRUK AND P .A. LOTITO Assumption 2. F or ev ery i = 1 , . . . , m, the activ e set I i 0 is a ﬁn ite union of interv als, i.e. I i 0 = N i [ j = 1 I i j , for I i j subinterv als of [0 , T ] of the form [0 , d ) , ( c, T ]; or ( c, d ) if c 6 = 0 and d 6 = T . Denote by c i 1 < d i 1 < c i 2 < . . . < c i N i < d i N i the end p oin ts of these in terv als. C onsequen tly , I i + is a ﬁnite u nion of in terv als as w ell. Remark 3 (On the multi-dimensional con trol case) . W e w ould lik e to mak e a comment concerning solutions with more than one con trol comp onent b e- ing singular at the same time. In [9, 10], Chitour et al. prov ed that generic systems with three or m ore co nt rol v ariables, or with t wo con trols and drift did not adm it singular optimal tra jectories (b y means of Goh’s necessary condition [26]). Consequent ly , the study of generic prop erties of con tr ol- aﬃne systems is r estricted to problems ha ving either one dimensional control or t w o con trol v ariables and no drift. Neverthele ss, there are motiv ations for in v estigati ng problems with an arbitrary num b er of inputs that w e p oin t out next. In [37], L ed zewicz and S c h¨ attler work ed on a mo del of cancer treat- men t havi ng tw o control v ariables entering linearly in the pr e-Hamilto nian and n onzero d rift. T hey p ro vided necessary optimalit y cond itions for so- lutions with b oth con trols b eing singular at the same time. Eve n if they w ere not able to giv e a pro of of optimalit y they claimed to ha ve strong ex- p ectations that this str ucture is part of the solution. Other examples can b e found in the literature. Maurer in [40] analyzed a resour ce allo cation problem (tak en from Bryson-Ho [8]). Th e mo del h ad t w o controls and drift, and numerical computations yielded a candid ate solution con taining t w o si- m ultaneous singular arcs. F or a system with a similar structure, Ga jardo et al. in [25] discussed the optimalit y of an extremal with t w o singular con trol comp onen ts at the same time. Another motiv ation th at we wo uld lik e to p oint out is the tec hnique used in Ar onna et al. [5] to study the sho oting algorithm for bang-singular solutions. In order to treat this kin d of extremals, they p erform a transform ation that yields a new system and an asso ciated totally singular solution. This n ew s y s tem in v olv es as many con trol v ariables as singular arcs of the original solution. Hence, ev en a one-dimensional problem can lead to a m ulti-dimensional totally singular solution. Th ese facts giv e a motiv ation for th e in v estigation of m ulti-input con trol-aﬃne problems. 2.3. Crit ical cones. Let 1 ≤ p ≤ ∞ , and call U p := L p (0 , T ; I R m ) , U + p := L p (0 , T ; I R m + ) and X p := W 1 p (0 , T ; I R n ) . Recal l that giv en a top ological v ector space E , a su b set D ⊂ E and x ∈ E , a tangent dir e ction to D at x is an elemen t d ∈ E suc h that there exists sequences ( σ k ) ⊂ I R + and ( x k ) ⊂ D with x k − x σ k → d. QUADRA TIC ORDER CONDITIONS F OR BANG-SINGULAR EXTREMALS 7 It is a well kn o wn result, see e.g. [11], that the tangen t cone to U + 2 at ˆ u is { v ∈ U 2 : v i ≥ 0 on I i 0 , for i = 1 , . . . , m } . Giv en v ∈ U p and z the solution of (12), consider the line arization of the c ost and ﬁnal c onstr aints (16) ( ϕ ′ i ( ˆ x ( T )) z ( T ) ≤ 0 , i = 0 , . . . , d ϕ , η ′ j ( ˆ x ( T )) z ( T ) = 0 , j = 1 , . . . , d η . F or p ∈ { 2 , ∞} , deﬁne the L p − critic al c one as C p :=  ( z , v ) ∈ X p × U p : v tangen t to U + p , (12) and (16) hold  . Certain relations of inclusion and d ensit y b et w een some app ro ximate critical cones are n eeded. Given ε ≥ 0 and i = 1 , . . . , m, d eﬁne the ε − active sets, up to a s et of null m easure I i ε := { t ∈ (0 , T ) : ˆ u i ( t ) ≤ ε } , and the sets W p,ε := { ( z , v ) ∈ X p × U p : v i = 0 on I i ε , (12) h olds } . By Assum ption 1, the follo wing explicit expression for C 2 holds (17) C 2 = { ( z , v ) ∈ W 2 , 0 : (16) holds } . Consider the ε − critic al c ones (18) C p,ε := { ( z , v ) ∈ W p,ε : (16 ) holds } . Let ε > 0 . Note that by (17), C 2 ,ε ⊂ C 2 . O n the other hand, giv en ( z , v ) ∈ C ∞ ,ε , it easily follo ws that ˆ u + σ v ∈ U + for sm all p ositiv e σ. Th us v is tangen t to U + at ˆ u, and this y ields C ∞ ,ε ⊂ C ∞ . Recall the follo wing tec h nical result, see Dmitruk [16]. Lemma 2.4 (on d ensit y) . Consider a lo c al ly c onvex top olo gic al sp ac e X , a ﬁnite-fac e d c one C ⊂ X , and a line ar manifold L dense in X . Then the c one C ∩ L is dense in C. Lemma 2.5. G iven ε > 0 the fol lowing pr op erties hold. (a) C ∞ ,ε ⊂ C 2 ,ε with dense inclu si on. (b) S ε> 0 C 2 ,ε ⊂ C 2 with dense inclusion. Pr o of. (a) Th e inclusion is immediate. As U is dense in U 2 , W ∞ ,ε is a dense subspace of W 2 ,ε . By Lemma 2.4, C 2 ,ε ∩ W ∞ ,ε is dense in C 2 ,ε , as desired. (b) The inclusion is immediate. In order to pro v e density , consider the follo w ing dense subs pace of W 2 , 0 : W 2 , S := [ ε> 0 W 2 ,ε , and the ﬁn ite-face d cone in C 2 ⊂ W 2 , 0 . By Lemma 2.4, C 2 ∩ W 2 , S is d ense in C 2 , whic h is what we needed to prov e.  8 M.S. ARONNA, J.F. BON NANS, A.V. DMITRUK AND P .A. LOTITO 3. S econd order anal ys is 3.1. Second v ariation. Consider the follo wing quadratic map p ing on W ; Ω[ λ ]( δ x, δ u ) := 1 2 ℓ ′′ [ λ ]( ˆ x ( T ))( δx ( T )) 2 + 1 2 Z T 0 [( H xx [ λ ] δ x, δ x ) + 2( H ux [ λ ] δ x, δ u )] d t. The next lemma pr o vides a second order expansion for the Lagrangian func- tion in v olving op erator Ω . Recall the follo wing notation: giv en t w o functions h : I R n → I R n h and k : I R n → I R n k , we say that h is a big-O of k around 0 and denote it by h ( x ) = O ( k ( x )) , if there exists p ositiv e constan ts δ and M suc h that | h ( x ) | ≤ M | k ( x ) | for | x | < δ. I t is a smal l-o if M go es to 0 as | x | go es to 0. D enote this by h ( x ) = o ( k ( x )) . Lemma 3.1. L et δ w = ( δ x, δ u ) ∈ W . Then for every multiplier λ ∈ Λ , the function Φ has the fol lowing exp ansion (omitting time ar g uments): Φ[ λ ]( ˆ w + δ w ) = Z T 0 H u [ λ ] δ u d t + Ω[ λ ]( δ x, δ u ) + 1 2 Z T 0 ( H uxx [ λ ] δ x, δ x, δ u )d t + O ( | δ x ( T ) | 3 ) + Z T 0 | ( ˆ u + δ u )( t ) |O ( | δ x ( t ) | 3 ) d t. Pr o of. Omit th e dep en dence on λ for the sak e of simplicit y . Use the T a ylor expansions ℓ ( ˆ x ( T )+ δ x ( T )) = ℓ ( ˆ x ( T ))+ ℓ ′ ( ˆ x ( T )) δx ( T )+ 1 2 ℓ ′′ ( ˆ x ( T ))( δx ( T )) 2 + O ( | δ x ( T ) | 3 ) , f i ( ˆ x ( t ) + δx ( t )) = f i ( ˆ x ( t )) + f ′ i ( ˆ x ( t )) δ x ( t ) + 1 2 f ′′ i ( ˆ x ( t ))( δ x ( t )) 2 + O ( | δ x ( t ) | 3 ) , in th e expression Φ( ˆ w + δ w ) = ℓ ( ˆ x + δx ( T )) + Z T 0 ψ " m X i =0 ( ˆ u i + δ u i ) f i ( ˆ x + δ x ) − ˙ ˆ x − ˙ δ x # d t. Afterw ards, use the iden tit y Z T 0 ψ m X i =0 ˆ u i f ′ i ( ˆ x ) δ x d t = − ℓ ′ ( ˆ x ( T )) δx ( T ) + Z T 0 ψ ˙ δ x d t, obtained by int egration b y parts and equation (2) to get the desired result.  The p revious lemma yields the follo w ing iden tit y for every ( δx, δ u ) ∈ W : Ω[ λ ]( δ x, δ u ) = 1 2 D 2 Φ[ λ ]( ˆ w )( δ x, δ u ) 2 . QUADRA TIC ORDER CONDITIONS F OR BANG-SINGULAR EXTREMALS 9 3.2. Necessary condition. T his section pro vides the follo w ing seco nd or- der necessary condition in terms of Ω and the critical cone C 2 . Theorem 3.2. If ˆ w is a we ak minimum then (19) max λ ∈ Λ Ω[ λ ]( z , v ) ≥ 0 , for all ( z , v ) ∈ C 2 . F or the sak e of simplicit y , deﬁne ¯ ϕ : U → I R d ϕ +1 , and ¯ η : U → I R d η as ¯ ϕ i ( u ) := ϕ i ( x ( T )) , for i = 0 , 1 , . . . , d ϕ , ¯ η j ( u ) := η j ( x ( T )) , for j = 1 , . . . , d η , (20) where x is the solution of (2 ) corresp ond ing to u. Deﬁnition 3.3. W e s a y that the e quality c onstr aints ar e nonde g e ner ate if (21) ¯ η ′ ( ˆ u ) is on to from U to I R d η . If (21) does not hold, w e call them de gener ate . W rite the problem in the follo wing w a y (P) ¯ ϕ 0 ( u ) → min; ¯ ϕ i ( u ) ≤ 0 , i = 1 , . . . , d ϕ , ¯ η ( u ) = 0 , u ∈ U + . Supp ose t hat ˆ u is a lo cal weak solution of (P) . Next we pro v e Theo- rem 3.2. Its pro of is divided in to t w o cases: degenerate and nondegenerate equalit y constrain ts. F or the ﬁrst case the result is immed iate and is tac kled in the next Lemma. I n ord er to sh o w Theorem 3.2 for th e latter case we in tro duce an auxiliary pr oblem parameterized by certain critical dir ections ( z , v ) , denoted b y (QP v ). W e prov e th at v al(QP v ) ≥ 0 and , by a result on dualit y , the desired second order condition will b e deriv ed. Lemma 3.4. If e quality c onstr aints ar e de gener ate, then (19) holds. Pr o of. Notice that there exists β 6 = 0 suc h th at P d η j = 1 β j η ′ j ( ˆ x ( T )) = 0 , since ¯ η ′ ( ˆ u ) is not onto. Consider α = 0 and ψ = 0 . T ak e λ := ( α, β , ψ ) and n otice that both λ and − λ are in Λ . Observe that Ω[ λ ]( z , v ) = 1 2 d η X j = 1 β j η ′′ j ( ˆ x ( T ))( z ( T )) 2 . Th us Ω [ λ ]( z , v ) ≥ 0 either for λ or − λ. The requir ed result follo ws.  T ake ε > 0 , ( z , v ) ∈ C ∞ ,ε , and rewrite (18) usin g the notation in (20) , C ∞ ,ε = { ( z , v ) ∈ X × U : v i ( t ) = 0 on I i ε , i = 1 , . . . , m , (12) holds , ¯ ϕ ′ i ( ˆ u ) v ≤ 0 , i = 0 , . . . , d ϕ , ¯ η ′ ( ˆ u ) v = 0 } . Consider the problem δ ζ → min ¯ ϕ ′ i ( ˆ u ) r + ¯ ϕ ′′ i ( ˆ u )( v , v ) ≤ δ ζ , for i = 0 , . . . , d ϕ , ¯ η ′ ( ˆ u ) r + ¯ η ′′ ( ˆ u )( v , v ) = 0 , − r i ( t ) ≤ δ ζ , on I i 0 , for i = 1 , . . . , m. (QP v ) 10 M.S. ARONNA, J.F. BON NANS, A.V. DMITRUK AND P .A. LOTITO Prop osition 1. L et ( z , v ) ∈ C ∞ ,ε . If the e quality c onstr aints ar e nonde ge n- er ate, pr oblem (QP v ) is fe asible and v al (QP v ) ≥ 0 . Pr o of. Let us ﬁr st pr o v e feasibilit y . As ¯ η ′ ( ˆ u ) is onto , there exists r ∈ U such that the equalit y constrain t in (QP v ) is satisﬁed. T ak e δ ζ := m ax( k r k ∞ , ¯ ϕ ′ i ( ˆ u ) r + ¯ ϕ ′′ ( ˆ u )( v , v )) . Th us th e pair ( r , δ ζ ) is feasible for (QP v ). Let u s n ow pr o v e that v al (QP v ) ≥ 0 . On the contrary supp ose that there exists a f easible solution ( r, δζ ) with δ ζ < 0 . The last constraint in (QP v ) implies k r k ∞ 6 = 0 . Set, for σ > 0 , (22) ˜ u ( σ ) := ˆ u + σ v + 1 2 σ 2 r , ˜ ζ ( σ ) := 1 2 σ 2 δ ζ . The goal is ﬁ nding u ( σ ) feasible for (P) suc h that for sm all σ, u ( σ ) U → ˆ u, and ¯ ϕ 0 ( u ( σ )) < ¯ ϕ 0 ( ˆ u ) , con tradicting the w eak optimalit y of ˆ u. Notice that ˆ u i ( t ) > ε a.e. on [0 , T ] \ I i ε , and then ˜ u ( σ ) i ( t ) > − ˜ ζ ( σ ) for suﬃcien tly small σ. On I i ε , if ˜ u ( σ ) i ( t ) < − ˜ ζ ( σ ) then necessarily ˆ u i ( t ) < 1 2 σ 2 ( k r k ∞ + | δ ζ | ) , as v i ( t ) = 0 . Thus, deﬁning th e set J i σ := { t : 0 < ˆ u i ( t ) < 1 2 σ 2 ( k r k ∞ + | δ ζ | ) } , w e get { t ∈ [0 , T ] : ˜ u ( σ ) i ( t ) < − ˜ ζ ( σ ) } ⊂ J i σ . Ob serv e that on J i σ , the fun ction | ˜ u ( σ ) i ( t ) + ˜ ζ ( σ ) | /σ 2 is d omin ated by k r k ∞ + | δ ζ | . Since meas( J i σ ) goes to 0 b y th e Dominated Con v ergence Theorem, we obtain Z J i σ | ˜ u ( σ ) i ( t ) + ˜ ζ ( σ ) | d t = o ( σ 2 ) . T ake ˜ ˜ u ( σ ) :=  ˜ u ( σ ) o n [0 , T ] \ J i σ , − ˜ ζ ( σ ) on J i σ . Th us, ˜ ˜ u satisﬁes (23) ˜ ˜ u ( σ )( t ) ≥ − ˜ ζ ( σ ) , a . e . on [0 , T ] , k ˜ ˜ u ( σ ) − ˆ u k 1 = o ( σ 2 ) , k ˜ ˜ u ( σ ) − ˆ u k ∞ = O ( σ 2 ) , and the follo w ing estimates h old ¯ ϕ i ( ˜ ˜ u ( σ )) = ¯ ϕ i ( ˆ u ) + σ ¯ ϕ ′ i ( ˆ u ) v + 1 2 σ 2 [ ¯ ϕ ′ i ( ˆ u ) r + ¯ ϕ ′′ i ( ˆ u )( v , v )] + o ( σ 2 ) < ¯ ϕ i ( ˆ u ) + ˜ ζ ( σ ) + o ( σ 2 ) , (24) ¯ η ( ˜ ˜ u ( σ )) = σ ¯ η ′ ( ˆ u ) v + 1 2 σ 2 [ ¯ η ′ ( ˆ u ) r + ¯ η ′′ ( ˆ u )( v , v )] + o ( σ 2 ) = o ( σ 2 ) . As ¯ η ′ ( ˆ u ) is ont o on U w e can ﬁnd a corrected con trol u ( σ ) satisfying the equalit y constrain t and suc h that k u ( σ ) − ˜ ˜ u ( σ ) k ∞ = o ( σ 2 ) . De duce by (23) that u ( σ ) ≥ 0 a.e. on [0 , T ] , and b y (24) that it satisﬁes the terminal QUADRA TIC ORDER CONDITIONS F OR BANG-SINGULAR EXTREMALS 11 inequalit y constraint s. Thus u ( σ ) is feasible for (P) and it satisﬁes (22). This cont radicts the w eak optimalit y of ˆ u.  Recall that a L agr ange multip lier asso ciated w ith ˆ w is a pair ( λ, µ ) in I R d ϕ +1 × I R d η × W 1 ∞ (0 , T ; I R n, ∗ ) × U ∗ with λ = ( α, β , ψ ) satisfying (7), (8), µ ≥ 0 and the stationarity c ond ition Z T 0 H u [ λ ]( t ) v ( t )d t + Z T 0 v ( t )d µ ( t ) = 0 , for every v ∈ U . Here U ∗ denotes the dual space of U . Simple computations sho w that ( λ, µ ) is a Lagrange m ultiplier if and only if λ is a Po n try agin multiplie r and µ = H u [ λ ] . T h us µ ∈ L ∞ (0 , T ; I R m, ∗ ) . Let us come bac k to Theorem 3.2. Pr o of. [of Theorem 3.2] Lemma 3.4 co v ers the degenerate case. Assu me th us that ¯ η ′ ( ˆ u ) is onto . T ak e ε > 0 and ( z , v ) ∈ C ∞ ,ε . App lyin g Prop osition 1, we see th at there cannot exist r and δζ < 0 suc h that ¯ ϕ ′ i ( ˆ u ) r + ¯ ϕ ′′ i ( ˆ u )( v , v ) ≤ δ ζ , i = 0 , . . . , d ϕ , ¯ η ′ ( ˆ u ) r + ¯ η ′′ ( ˆ u )( v , v ) = 0 , − r i ( t ) ≤ δ ζ , on I i 0 , for i = 1 , . . . , m. By the Dub ovit skii-Milyutin Theorem (see [19]) we obtain the existence of ( α, β ) ∈ I R s and µ ∈ U ∗ with supp µ i ⊂ I i 0 , and ( α, β , µ ) 6 = 0 suc h that (25) d ϕ X i =0 α i ¯ ϕ ′ i ( ˆ u ) + d η X i =1 β j ¯ η ′ j ( ˆ u ) − µ = 0 , and denoting λ := ( α, β , ψ ) , with ψ b eing solution of (9), the follo w in g holds: d ϕ X i =0 α i ¯ ϕ ′′ i ( ˆ u )( v , v ) + d η X i =1 β j ¯ η ′′ j ( ˆ u )( v , v ) ≥ 0 . By Lemma 8.2 we obtain (26) Ω[ λ ]( z , v ) ≥ 0 . Observe that (25) implies that λ ∈ Λ . C onsider now ( ¯ z , ¯ v ) ∈ C 2 , and note that Lemma 2.5 guarantee s the existence of a sequence { ( z ε , v ε ) } ⊂ C ∞ ,ε con v erging to ( ¯ z , ¯ v ) in X 2 × U 2 . Recall Remark 2. Let λ ε ∈ Λ b e su ch that (26) h olds for ( λ ε , z ε , v ε ) . S in ce ( λ ε ) is b ounded, it con tains a limit p oin t ¯ λ ∈ Λ . Th us (26) holds for ( ¯ λ, ¯ z , ¯ v ) , as requ ired.  4. Goh Tran sforma tion Consider an arbitrary linear system: (27)  ˙ z ( t ) = A ( t ) z ( t ) + B ( t ) v ( t ) , a . e . on [0 , T ] , z (0) = 0 , 12 M.S. ARONNA, J.F. BON NANS, A.V. DMITRUK AND P .A. LOTITO where A ( t ) ∈ L ( I R n ; I R n ) is an essen tiall y b ound ed fu nction of t, and B ( t ) ∈ L ( I R m ; I R n ) is a Lipschitz-c on tin uous function of t. With eac h v ∈ U asso- ciate the state v ariable z ∈ X solutio n of (12). Let us presen t a transforma- tion of the v ariables ( z , v ) ∈ W , ﬁrst introdu ced b y Go h in [27]. Deﬁne t wo new state v ariables as follo ws: (28)    y ( t ) := Z t 0 v ( s )ds , ξ ( t ) := z ( t ) − B ( t ) y ( t ) . Th us y ∈ Y := W 1 ∞ (0 , T ; I R m ) , y (0) = 0 and ξ is an elemen t of sp ace X . It easily follo ws that ξ is a solution of the linear diﬀeren tial equ ation (29) ˙ ξ ( t ) = A ( t ) ξ ( t ) + B 1 ( t ) y ( t ) , ξ (0) = 0 , where (30) B 1 ( t ) := A ( t ) B ( t ) − ˙ B ( t ) . F or the pu rp oses of this article tak e (31) A ( t ) := m X i =0 ˆ u i f ′ i ( ˆ x ( t )) , and B ( t ) v ( t ) := m X i =1 v i ( t ) f i ( ˆ u ( t )) . Then (27) coincides w ith the linearized equatio n (12). 4.1. T ransformed critical directions. As optimalit y conditions on the v ariables obtai ned b y th e Goh T rans f ormation will b e derive d, a n ew set of critical dir ections is needed. T ak e a p oin t ( z , v ) in C ∞ , and d eﬁne ξ and y by th e transform ation (28 ). Let h := y ( T ) and notice that since (16) is satisﬁed, the follo wing inequalities h old, ϕ ′ i ( ˆ x ( T ))( ξ ( T ) + B ( T ) h ) ≤ 0 , for i = 0 , . . . , d ϕ , η ′ j ( ˆ x ( T ))( ξ ( T ) + B ( T ) h ) = 0 , for j = 1 , . . . , d η . (32) Deﬁne the set of tr ansforme d c ritic al dir e ctions P := ( ( ξ , y , h ) ∈ X × Y × I R m : ˙ y i = 0 o v er I i 0 , y (0) = 0 , h := y ( T ) , (29) and (32) h old ) . Observe that for ev ery ( ξ , y , h ) ∈ P and 1 ≤ i ≤ m, (33) y i is constant ov er eac h connected comp onent of I i 0 , and at the end p oin ts th e follo wing conditions hold y i = 0 on [0 , d i 1 ) , if 0 ∈ I i 0 , and y i = h i on ( c i N i , T ] , if T ∈ I i 0 , (34) where c i 1 and d i 1 w ere in trod u ced in Assumption 2. Deﬁne the set P 2 := { ( ξ , y , h ) ∈ X 2 × U 2 × I R m : (29 ) , (32) , (33) and (3 4) hold } . Lemma 4.1. P i s a dense subset of P 2 in the X 2 × U 2 × I R m − top olo gy. QUADRA TIC ORDER CONDITIONS F OR BANG-SINGULAR EXTREMALS 13 Pr o of. Th e inclusion is immediate. In order to pr o v e the d ensit y , consider the follo wing sets. X := { ( ξ , y , h ) ∈ X 2 × U 2 × I R m : (29 ) , (33) and (34) hold } , L := { ( ξ , y , y ( T )) ∈ X × Y × I R m : y (0) = 0 , (29) and (33) hold } , C := { ( ξ , y , h ) ∈ X : (32) holds } . By Lemma 8.1 , L is a dense subset of X. Th e conclusion follo ws with Lemma 2.4.  4.2. T ransformed second v ariat ion. W e are inte rested in writing Ω in terms of v ariables y and ξ deﬁned in (28). In tro duce the follo wing n otatio n for th e sak e of simplifying the pr esentati on. Deﬁnition 4.2. Consider the follo wing matrices of sizes n × n, m × n and m × n, resp ectiv ely . (35) Q [ λ ] := H xx [ λ ] , C [ λ ] := H ux [ λ ] , M [ λ ] := B ⊤ Q [ λ ] − ˙ C [ λ ] − C [ λ ] A, where A and B w ere deﬁned in (31 ). Notice that M is w ell-deﬁned as C is Lipsc hitz-con tin uous on t. Decomp ose matrix C [ λ ] B into its symmetric an d sk ew-symmetric parts, i.e. c onsider (36) S [ λ ] := 1 2 ( C [ λ ] B + ( C [ λ ] B ) ⊤ ) , V [ λ ] := 1 2 ( C [ λ ] B − ( C [ λ ] B ) ⊤ ) . Remark 4. Observ e th at, since C [ λ ] and B are Lip sc hitz-con tinuous, S [ λ ] and V [ λ ] are Lipsc h itz-co nt inuous as w ell. In fact, simple computations yield (37) S ij [ λ ] = 1 2 ψ ( f ′ i f j + f ′ j f i ) , V ij [ λ ] = 1 2 ψ [ f i , f j ] , for i, j = 1 , . . . , m, where (38) [ f i , f j ] := f ′ i f j − f ′ j f i . With this notation, Ω tak es the form Ω[ λ ]( δ x, v ) = 1 2 ℓ ′′ [ λ ]( ˆ x ( T ))( δx ( T )) 2 + 1 2 Z T 0 [( Q [ λ ] δx , δ x ) + 2( C [ λ ] δ x, v )]d t . Deﬁne the m × m matrix (39) R [ λ ] := B ⊤ Q [ λ ] B − C [ λ ] B 1 − ( C [ λ ] B 1 ) ⊤ − ˙ S [ λ ] , where B 1 w as in trod u ced in equatio n (30). Consider the function g [ λ ] from I R n × I R m to I R deﬁned by: (40) g [ λ ]( ζ , h ) := 1 2 ℓ ′′ [ λ ]( ˆ x ( T ))( ζ + B ( T ) h ) 2 + 1 2 ( C [ λ ]( T )(2 ζ + B ( T ) h ) , h ) . Remark 5. (i) W e use the same notatio n for th e matrices Q [ λ ] , C [ λ ] , M [ λ ] , ℓ ′′ [ λ ]( ˆ x ( T )) and for the bilinear mapping they deﬁne. (ii) Ob serv e that when m = 1 , the function V [ λ ] ≡ 0 sin ce it b ecomes a sk ew-symmetric scalar. 14 M.S. ARONNA, J.F. BON NANS, A.V. DMITRUK AND P .A. LOTITO Deﬁnition 4.3. Deﬁne the mapping ov er X × Y × U give n by Ω P [ λ ]( ξ , y , v ) := g [ λ ]( ξ ( T ) , y ( T )) + Z T 0 { 1 2 ( Q [ λ ] ξ , ξ ) + 2( M [ λ ] ξ , y ) + 1 2 ( R [ λ ] y , y ) + ( V [ λ ] y , v ) } d t, (41) with g [ λ ] , Q [ λ ] , M [ λ ] , R [ λ ] and V [ λ ] deﬁn ed in (35)-(40). The follo wing theorem sho ws that Ω P coincides w ith Ω . See e.g. [15]. Theorem 4.4. L et ( z , v ) ∈ W satisfying (12) and ( ξ , y ) b e deﬁne d by (28) . Then Ω[ λ ]( z , v ) = Ω P [ λ ]( ξ , y , v ) . Pr o of. W e omit the dep end ence on λ for the sak e of simplicit y . R ep lace z b y its expression in (28) and obtain Ω( z , v ) = 1 2 ℓ ′′ ( ˆ x ( T ))( ξ ( T ) + B ( T ) y ( T )) 2 + 1 2 Z T 0 [( Q ( ξ + B y ) , ξ + B y ) + ( C ( ξ + B y ) , v ) + ( C ⊤ v , ξ + B y )]d t. (42) In tegrating b y parts yields (43) Z T 0 ( C ξ , v )d t = [( C ξ , y )] T 0 − Z T 0 ( ˙ C ξ + C ( Aξ + B 1 y ) , y )d t, and Z T 0 ( C B y , v )d t = Z T 0 (( S + V ) y , v )d t = 1 2 [( S y , y )] T 0 + Z T 0 ( − 1 2 ( ˙ S y , y ) + ( V y , v ))d t. (44) Com bining (42), (43) and (4 4) w e get the desired result.  Corollary 1. If V [ λ ] ≡ 0 then Ω do es not involve v explicitly, and it c an b e expr esse d in terms of ( ξ , y , y ( T )) . In view of (37), th e previous corolla ry holds in particular if [ f i , f j ] = 0 on the referen ce tra jectory for eac h pair 1 ≤ i < j ≤ m. Corollary 2. If ˆ w is a we ak minimum, then max λ ∈ Λ Ω P [ λ ]( ξ , y , v ) ≥ 0 , for every ( z , v ) ∈ C 2 and ( ξ , y ) deﬁne d by (28) . 4.3. New second order condition. In this section we present a n ecessary condition in v olving the v ariable ( ξ , y , h ) in P 2 . T o achiev e this w e remo v e the explicit dep endence on v from the second v ariat ion, for certain su bset of multipliers. Recall that we consider λ = ( α, β ) as elements of I R s . Deﬁnition 4.5. Giv en M ⊂ I R s , deﬁ n e G ( M ) := { λ ∈ M : V ij [ λ ]( t ) = 0 on I i + ∩ I j + , for an y pair 1 ≤ i < j ≤ m } . QUADRA TIC ORDER CONDITIONS F OR BANG-SINGULAR EXTREMALS 15 Theorem 4.6. L et M ⊂ I R s b e c onvex and c omp act, and assume that (45) max λ ∈ M Ω P [ λ ]( ξ , y , ˙ y ) ≥ 0 , for all ( ξ , y , h ) ∈ P . Then max λ ∈ G ( M ) Ω P [ λ ]( ξ , y , ˙ y ) ≥ 0 , for all ( ξ , y , h ) ∈ P . The p ro of is based on some tec h niques in tro duced in Dmitruk [12, 15] for the pr o of of similar theorems. Let 1 ≤ i < j ≤ m and t ∗ ∈ int I i + ∩ I j + . T ak e y ∈ Y satisfying (46) y (0) = y ( T ) = 0 , y k = 0 , for k 6 = i, k 6 = j. Suc h fu nctions deﬁn e a linear con tin uous mapping r : I R s, ∗ → I R b y (47) λ 7→ r [ λ ] := Z T 0 ( V [ λ ]( t ∗ ) y , ˙ y )d t. By condition (46), and since V [ λ ] is sk ew-symmetric, Z T 0 ( V [ λ ]( t ∗ ) y , ˙ y )d t = V ij [ λ ]( t ∗ ) Z T 0 ( y i ˙ y j − y j ˙ y i )d t. Eac h r is an element of the dual space of I R s, ∗ , and it can thus b e iden tiﬁed with an elemen t of I R s . Consequ en tly , the su bset of I R s deﬁned by R ij ( t ∗ ) := { r ∈ I R s : y ∈ Y satisﬁes (46) , r is d eﬁned by (47) } , is a linear subspace of I R s . No w, consider al l the ﬁ nite co llections Θ ij := n θ = { t 1 < · · · < t N θ } : t k ∈ int I i + ∩ I j + for k = 1 , . . . , N θ o . Deﬁne R := X i 0 } . By Assum ption 2, I ij can b e expressed as a ﬁnite union of in terv als, i.e. I ij = K ij [ k =1 I k ij , wh ere I k ij := ( c k ij , d k ij ) . Let ( z , v ) ∈ C ∞ , i 6 = j, and y b e deﬁned b y (28). Notice that y i is constan t on eac h ( c k ij , d k ij ) . De note with y k i,j its v alue on this interv al. Prop osition 2. L et ( z , v ) ∈ C ∞ , y b e deﬁne d by (28) and λ ∈ G (Λ) . Then Z T 0 ( V [ λ ] y , v )d t = m X i 6 = j i,j =1 K ij X k =1 y k i,j ( [ V ij [ λ ] y j ] d k ij c k ij − Z d k ij c k ij ˙ V ij [ λ ] y j d t ) . 18 M.S. ARONNA, J.F. BON NANS, A.V. DMITRUK AND P .A. LOTITO Pr o of. Ob serv e that (56) Z T 0 ( V [ λ ] y , v )d t = m X i 6 = j i,j =1 Z T 0 V ij [ λ ] y i v j d t, since V ii [ λ ] ≡ 0 . Fix i 6 = j, and r ecall th at that V ij [ λ ] is diﬀerent iable in time (see expr ession (37)). Since ( z , v ) ∈ C ∞ and λ ∈ G (Λ) , Z T 0 V ij [ λ ] y i v j d t = Z I ij V ij [ λ ] y i v j d t = K ij X k =1 Z d k ij c k ij V ij [ λ ] y i v j d t = K ij X k =1 y k i,j ( [ V ij [ λ ] y j ] d k ij c k ij − Z d k ij c k ij ˙ V ij [ λ ] y j d t ) , (57) where the last equalit y wa s obtained by inte grating by p arts and knowing that y i is constan t on I ij . The desired result follo ws from (56) and (57).  Giv en a real function h and c ∈ I R, d eﬁne h ( c +) := lim t → c + h ( t ) , and h ( c − ) := lim t → c − h ( t ) . Deﬁnition 4.8. Let ( ξ , y , h ) ∈ P 2 and λ ∈ G (Λ) . Deﬁne Ξ[ λ ]( ξ , y , h ) := 2 m X i 6 = j i,j =1 K ij X k =1 c k ij 6 =0 y k i,j ( V ij [ λ ]( d k ij ) y j ( d k ij +) − V ij [ λ ]( c k ij ) y j ( c k ij − ) − Z d k ij c k ij ˙ V ij [ λ ] y j d t ) , where the ab o v e expression is in terpreted as follo w s: (i) y j ( d k ij +) := h j , if d k ij = T , (ii) V ij [ λ ]( c k ij ) y j ( c k ij − ) := 0 , if ˆ u i > 0 and ˆ u j > 0 for t < c k ij , (iii) V ij [ λ ]( d k ij ) y j ( d k ij +) := 0 , if ˆ u i > 0 and ˆ u j > 0 for t > d k ij . Prop osition 3. The fol lowing pr op erties for Ξ hold. (i) Ξ [ λ ]( ξ , y , h ) is wel l-deﬁne d for e ach ( ξ , y , h ) ∈ P 2 , and λ ∈ G (Λ) . (ii) If { ( ξ ν , y ν , y ν ( T )) } ⊂ P c onver ges in the X 2 × U 2 × I R m − top olo gy to ( ξ , y , h ) ∈ P 2 , then Z T 0 ( V [ λ ] y ν , ˙ y ν )d t ν → ∞ − → Ξ[ λ ]( ξ , y , h ) . Pr o of. (i) T ake ( ξ , y , h ) ∈ P 2 . First observ e that y i ≡ y k i,j o v er ( c k ij , d k ij ) . As c k ij 6 = 0 , t w o p ossible situations can arise, (a) for t < c k ij : ˆ u j = 0 , thus y j is constan t, and consequen tly y j ( c k ij − ) is w ell-deﬁned, (b) for t < c k ij : ˆ u i > 0 and ˆ u j > 0 , th us V ij [ λ ]( c k ij ) = 0 since λ ∈ G (Λ) . QUADRA TIC ORDER CONDITIONS F OR BANG-SINGULAR EXTREMALS 19 The same analysis can b e done for t > d k ij when d k ij 6 = T . W e conclude that Ξ is correctly deﬁned. (ii) Observe that since y ν con v erges to y in the U 2 − top ology and since y ν i is constant o ver I ij , then y i is constan t as w ell, and y ν i go es to y i p oin t wise on I ij . Th us, y ν i ( c k ij ) − → y k i,j , and y ν i ( d k ij ) − → y k i,j . No w, for the terms on y j , the same analysis can b e made, whic h yields either y ν j ( c k ij ) − → y j ( c k ij − ) or V ij [ λ ]( c k ij ) = 0; and, either y ν j ( d k ij ) − → y j ( d k ij +) or V ij [ λ ]( d k ij ) = 0 , when d k ij < T . F or d k ij = T , y ν j ( T ) − → h j holds.  Deﬁnition 4.9. F or ( ξ , y , h ) ∈ P 2 and λ ∈ G (Λ) deﬁne Ω P 2 [ λ ]( ξ , y , h ) := g [ λ ]( ξ ( T ) , h ) + Ξ[ λ ]( ξ , y , h ) + Z T 0 (( Q [ λ ] ξ , ξ ) + 2( M [ λ ] ξ , y ) + ( R [ λ ] y , y ))d t. Remark 6. Observe that when m = 1 , the mapping Ξ ≡ 0 since V ≡ 0 . Th us, in this case, Ω P 2 can b e deﬁ n ed f or an y element ( ξ , y , h ) ∈ X 2 × U 2 × I R and any λ ∈ Λ . If we tak e ( z , v ) ∈ W satisfying (12), and deﬁne ( ξ , y ) by (28), th en Ω[ λ ]( z , v ) = Ω P [ λ ]( ξ , y , ˙ y ) = Ω P 2 [ λ ]( ξ , y , y ( T )) . F or m > 1 , the pr evious equalit y h olds for ( z , v ) ∈ C ∞ . Lemma 4.10. L e t { ( ξ ν , y ν , y ν ( T ) } ⊂ P b e a se quenc e c onver ging to ( ξ , y , h ) ∈ P 2 in the X 2 × U 2 × I R m − top olo gy. Then lim ν → ∞ Ω P [ λ ]( ξ ν , y ν , ˙ y ν ) = Ω P 2 [ λ ]( ξ , y , h ) . Denote with co Λ the conv ex hull of Λ . Theorem 4.11. L et ˆ w b e a we ak minimum, then (58) max λ ∈ G (co Λ) Ω P 2 [ λ ]( ξ , y , h ) ≥ 0 , for all ( ξ , y , h ) ∈ P 2 . Pr o of. Corollary 2 together with Th eorem 4.6 applied to M := co Λ yield max λ ∈ G (co Λ) Ω P [ λ ]( ξ , y , ˙ y ) ≥ 0 , for all ( ξ , y , y ( T )) ∈ P . The result follo ws from Lemma 4.1 and Lemma 4.10.  Remark 7. Notice that in case (21 ) is not satisﬁed, condition (58) do es not provide an y useful information as 0 ∈ co Λ . O n the other hand , if (21) holds, every λ = ( α, β , ψ ) ∈ Λ necessarily h as α 6 = 0 , and th us 0 / ∈ co Λ . 5. S ufficient c ondition Consider the p roblem for a scalar con trol, i.e . let m = 1 . This section pro vides a suﬃcient condition for Po n try agin optimalit y . 20 M.S. ARONNA, J.F. BON NANS, A.V. DMITRUK AND P .A. LOTITO Deﬁnition 5.1. Giv en ( y , h ) ∈ U 2 × I R , let γ ( y , h ) := Z T 0 y ( t ) 2 d t + | h | 2 . Deﬁnition 5.2. A sequence { v k } ⊂ U c onver ges to 0 in the Pontr yagin sense if k v k k 1 → 0 and there exists N su c h that k v k k ∞ < N . Deﬁnition 5.3. W e sa y that ˆ w s atisﬁes γ − quadr atic gr owth c ondition in the Pontryagin sense if there exists ρ > 0 suc h that, for ev ery sequen ce of feasible v ariations { ( δ x k , v k ) } with { v k } con v erging to 0 in the P on try agin sense, (59) J ( ˆ u + v k ) − J ( ˆ u ) ≥ ργ ( y k , y k ( T )) , holds for a large enough k , where y k is deﬁned by (28). Equiv alent ly , for all N > 0 , there exists ε > 0 suc h that if k v k ∞ < N and k v k 1 < ε, then (59) holds. Deﬁnition 5.4. W e s a y that ˆ w is nor mal if α 0 > 0 for ev ery λ ∈ Λ . Theorem 5.5. Supp ose that ther e exists ρ > 0 such that (60) max λ ∈ Λ Ω P 2 [ λ ]( ξ , y , h ) ≥ ργ ( y , h ) , for all ( ξ , y , h ) ∈ P 2 . Then ˆ w is a P ontryagin minimum satisfying γ − quadr atic gr owth. F urther- mor e, if ˆ w i s normal, the c onverse holds. Remark 8. In case the bang arcs are a bsent, i.e. the con trol is totally singular, th is theorem reduces to one pr o v ed in Dmitruk [13, 15]. Recall that Φ is deﬁned in (5). W e w ill use the f ollo win g tec hnical result. Lemma 5.6. Consider { v k } ⊂ U c onver ging to 0 in the Pontryagin sense. L et u k := ˆ u + v k and let x k b e the c orr esp onding so lution of e quation (2 ) . Then for every λ ∈ Λ , (61) Φ[ λ ]( x k , u k ) = Φ [ λ ]( ˆ x, ˆ u ) + Z T 0 H u [ λ ]( t ) v k ( t )d t + Ω[ λ ]( z k , v k ) + o ( γ k ) , wher e z k is deﬁne d by (12) , γ k := γ ( y k , y k ( T )) , and y k is deﬁne d by (28) . Pr o of. By Lemma 3.1 w e can write Φ[ λ ]( x k , u k ) = Φ [ λ ]( ˆ x, ˆ u ) + Z T 0 H u [ λ ]( t ) v k ( t )d t + Ω[ λ ]( z k , v k ) + R k , where, in view of Lemma 8.4 , (62) R k := ∆ k Ω[ λ ] + Z T 0 ( H uxx [ λ ]( t ) δx k ( t ) , δ x k ( t ) , v k ( t ))d t + o ( γ k ) , with δ x k := x k − ˆ x, and (63) ∆ k Ω[ λ ] := Ω[ λ ]( δ x k , v k ) − Ω[ λ ]( z k , v k ) . QUADRA TIC ORDER CONDITIONS F OR BANG-SINGULAR EXTREMALS 21 Next, we pr ov e that (64) R k = o ( γ k ) . Note that Q ( a, a ) − Q ( b, b ) = Q ( a + b, a − b ) , for any bilinear mappin g Q , and any pair a, b. Put η k := δ x k − z k . Hence, from (63), we get ∆ k Ω[ λ ] = 1 2 ℓ ′′ [ λ ]( ˆ x ( T ))( δx k ( T ) + z k ( T ) , η k ( T )) + 1 2 Z T 0 ( H xx [ λ ]( δx k + z k ) , η k )d t + Z T 0 ( H ux [ λ ] η k , v k )d t. By Lemmas 8.4 and 8.12 in the App en dix, the ﬁrst and the second terms are of order o ( γ k ) . In teg rate b y p arts the last term to obtain Z T 0 ( H ux [ λ ] η k , v k )d t (65) = [( H ux [ λ ] η k , y k )] T 0 − Z T 0 { ( ˙ H ux [ λ ] η k , y k ) + ( H ux [ λ ] ˙ η k , y k ) } d t. (66) Th us, by Lemma 8.12 w e deduce that the ﬁrst tw o terms in (66 ) are of ord er o ( γ k ) . It remains to deal with last term in the in tegral. Replac e ˙ η k b y its expression in equation (12 8) of L emm a 8.12: Z T 0 ( H ux [ λ ] ˙ η k , y k )d t = Z T 0 ( H ux [ λ ] 1 X i =0 ˆ u i f ′ i ( ˆ x ) η k + v k f ′ 1 ( ˆ x ) δ x k + ζ k ! , y k )d t = o ( γ k ) + Z T 0 d d t  y 2 k 2  H ux [ λ ] f ′ 1 ( ˆ x ) δ x k d t, (67) where the second equalit y f ollo ws from Lemmas 8.4 and 8.12. In tegrating the last term b y parts, w e obtain Z T 0 d d t  y 2 k 2  H ux [ λ ] f ′ 1 ( ˆ x ) δ x k d t =  y 2 k 2 H ux [ λ ] f ′ 1 ( ˆ x ) δ x k  T 0 − Z T 0 y 2 k 2 d d t  H ux [ λ ] f ′ 1 ( ˆ x )  δ x k d t − Z T 0 y 2 k 2 H ux [ λ ] f ′ 1 ( ˆ x ) ˙ δ x k d t = o ( γ k ) − Z T 0 d d t  y 3 k 6  H ux [ λ ] f ′ 1 ( ˆ x ) f 1 ( ˆ x )d t = o ( γ k ) −  y 3 k 6 H ux [ λ ] f ′ 1 ( ˆ x ) f 1 ( ˆ x )  T 0 + Z T 0 y 3 k 6 d d t  H ux [ λ ] f ′ 1 ( ˆ x ) f 1 ( ˆ x )  d t = o ( γ k ) , (68) where we used L emma 8.12 and , in particular, equation (129). F rom (67) and (68), it follo ws that the term in (65) is of order o ( γ k ) . Th us, (69) ∆ k Ω[ λ ] ≤ o ( γ k ) . 22 M.S. ARONNA, J.F. BON NANS, A.V. DMITRUK AND P .A. LOTITO Consider n o w the third order term in (62): Z T 0 ( H uxx [ λ ] δ x k , δ x k , v k )d t = [ y k δ x ⊤ k H uxx [ λ ] δ x k ] T 0 − Z T 0 y k δ x ⊤ k ˙ H uxx [ λ ] δ x k d t − 2 Z T 0 y k δ x ⊤ k H uxx [ λ ] ˙ δ x k d t = o ( γ k ) − Z T 0 d d t ( y 2 k ) δ x ⊤ k H uxx [ λ ] f 1 ( ˆ x )d t = o ( γ k ) − h y 2 k δ x ⊤ k H uxx [ λ ] f 1 ( ˆ x ) i T 0 − Z T 0 y 2 k v k f 1 ( ˆ x ) ⊤ H uxx [ λ ] f 1 ( ˆ x )d t = o ( γ k ) , (70) b y Lemmas 8.4 and 8.12 . The last inequalit y follo w s f rom in tegrating by parts one more time as it was d one in (68) . Consider expression (62). By inequalit y (69) and equation (70), equalit y (64) is obtained and thus, the desired resu lt follo ws.  Pr o of. [of Theorem 5.5] P art 1. First w e pro v e that if ˆ w is a normal Po n try a- gin minim um satisfying the γ − quadratic gro wth condition in the P on try agin sense then (6 0) h olds for some ρ > 0 . Here the necessary condition of The- orem 3.2 is used. D eﬁne ˆ y ( t ) := R t 0 ˆ u ( s )d s, and note that ( ˆ w , ˆ y ) is, for some ρ ′ > 0 , a P ontry agi n minimum of ˜ J := J − ρ ′ γ ( y − ˆ y , y ( T ) − ˆ y ( T )) → min , (2)-(4), ˙ y = u, y (0) = 0 . (71) Observe that the critical cone ˜ C 2 for (71) co nsists of the p oints ( z , v , δ y ) in X 2 × U 2 × W 1 2 (0 , T ; I R ) verifying ( z , v ) ∈ C 2 , ˙ δ y = v and δ y (0) = 0 . Since the pre-Hamiltonian at p oint ( ˆ w , ˆ y ) coincides w ith the original pr e-Hamilto nian, the set of m u ltipliers for (71) consists of the p oints ( λ, ψ y ) with λ ∈ Λ . Applying the second ord er necessary condition of Theorem 3.2 at the p oin t ( ˆ w , ˆ y ) we see that, for ev ery ( z , v ) ∈ C 2 and δ y ( t ) := R t 0 v ( s )d s, there exists λ ∈ Λ suc h th at (72) Ω[ λ ]( z , v ) − α 0 ρ ′ ( k δy k 2 2 + δ y 2 ( T )) ≥ 0 , where α 0 > 0 since ˆ w is normal. T ak e ρ := min λ ∈ Λ α 0 ρ ′ > 0 . App lyin g the Goh transf orm ation in (72), condition (60) for the constan t ρ follo ws. Part 2. W e shall prov e that if (60) holds for some ρ > 0 , th en ˆ w satisﬁes γ − quadratic gro wth in the Pon tryagin sense. On the contrary , assum e that the quadratic gro wth condition (59) is not v alid. Cons equ en tly , th ere exists a sequence { v k } ⊂ U con v erging to 0 in the P ontry agin sense such that, denoting u k := ˆ u + v k , (73) J ( ˆ u + v k ) ≤ J ( ˆ u ) + o ( γ k ) , QUADRA TIC ORDER CONDITIONS F OR BANG-SINGULAR EXTREMALS 23 where y k ( t ) := R t 0 v k ( s ) ds and γ k := γ ( y k , y k ( T )) . Denote b y x k the solution of equation (2) co rresp onding to u k , deﬁne w k := ( x k , u k ) and let z k b e the solution of (12) asso ciated with v k . T ake any λ ∈ Λ . Mu ltiply inequalit y (73) by α 0 , add the n onp ositiv e term P d ϕ i =0 α i ϕ i ( x k ( T )) + P d η j = 1 β j η j ( x k ( T )) to its left-hand side, and obtain the inequalit y (74) Φ [ λ ]( x k , u k ) ≤ Φ [ λ ]( ˆ x, ˆ u ) + o ( γ k ) . Recall expansion (61 ). L et ( ¯ y k , ¯ h k ) := ( y k , y k ( T )) / √ γ k . Note that the ele- men ts of this sequence ha v e u nit norm in U 2 × R . By the Banac h-Alaoglu Theorem, extracting if necessary a sequence, w e ma y assume th at there exists ( ¯ y , ¯ h ) ∈ U 2 × I R suc h that (75) ¯ y k ⇀ ¯ y , and ¯ h k → ¯ h, where the ﬁ rst limit is ta k en in the we ak top ology of U 2 . The remainder of the pr o of is split into t w o parts. (a) Using equations (61) and (74) w e pro v e that ( ¯ ξ , ¯ y , ¯ h ) ∈ P 2 , where ¯ ξ is a solution of (29). (b) W e pro v e that ( ¯ y , ¯ h ) = 0 and that it is the limit of { ( ¯ y k , ¯ h k ) } in the strong sense. This leads to a con tradiction since eac h ( ¯ y k , ¯ h k ) has unit norm. (a) W e shall pro v e that ( ¯ ξ , ¯ y , ¯ h ) ∈ P 2 . F rom (61 ) and (74) it follo ws that 0 ≤ Z T 0 H u [ λ ]( t ) v k ( t )d t ≤ − Ω P 2 [ λ ]( ξ k , y k , h k ) + o ( γ k ) , where ξ k is solution of (29) corresp ondin g to y k . The ﬁrst inequalit y holds as H u [ λ ] v k ≥ 0 almost ev erywhere on [0 , T ] and we replace d Ω P b y Ω P 2 in view of Remark 6. By the con tin uit y of mapping Ω P 2 [ λ ] o v er X 2 × U 2 × I R deduce that 0 ≤ Z T 0 H u [ λ ]( t ) v k ( t )d t ≤ O ( γ k ) , and thus, for eac h comp osing in terv al ( c, d ) of I 0 , (76) lim k →∞ Z d c H u [ λ ]( t ) ϕ ( t ) v k ( t ) √ γ k dt = 0 , for every nonn egativ e Lip sc hitz contin u ou s fun ction ϕ with supp ϕ ⊂ ( c, d ) . The latter expression means th at the su pp ort of ϕ is included in ( c, d ) . In tegrating b y parts in (76) and by (75) w e obtain 0 = lim k →∞ Z d c d d t ( H u [ λ ]( t ) ϕ ( t )) ¯ y k ( t )d t = Z d c d d t ( H u [ λ ]( t ) ϕ ( t )) ¯ y ( t )d t. By Lemma 8.5, ¯ y is n ondecreasing ov er ( c, d ) . Hence, in view of Lemm a 8.7, w e can in tegrate b y parts in the previous equatio n to get (77) Z d c H u [ λ ]( t ) ϕ ( t )d ¯ y ( t ) = 0 . 24 M.S. ARONNA, J.F. BON NANS, A.V. DMITRUK AND P .A. LOTITO T ake t 0 ∈ ( c, d ) . By the strict complemen tary in Assumption 1, there exists λ 0 ∈ Λ su ch that H u [ λ 0 ]( t 0 ) > 0 . Hence, in view of the con tin uit y of H u [ λ 0 ], there exists ε > 0 such that H u [ λ 0 ] > 0 on ( t 0 − 2 ε, t 0 + 2 ε ) ⊂ ( c, d ) . Cho ose ϕ suc h that supp ϕ ⊂ ( t 0 − 2 ε, t 0 + 2 ε ) , and H u [ λ 0 ]( t ) ϕ ( t ) = 1 on ( t 0 − ε, t 0 + ε ) . Since d ¯ y ≥ 0 , equatio n (77) y ields 0 = Z d c H u [ λ ]( t ) ϕ ( t )d ¯ y ( t ) ≥ Z t 0 + ε t 0 − ε H u [ λ ]( t ) ϕ ( t )d ¯ y ( t ) = Z t 0 + ε t 0 − ε d ¯ y ( t ) = ¯ y ( t 0 + ε ) − ¯ y ( t 0 − ε ) . As ε and t 0 ∈ ( c, d ) are arbitrary w e ﬁ nd that (78) d ¯ y ( t ) = 0 , on I 0 , and th us (33) holds. Let us pro v e condition (34) for ( ¯ ξ , ¯ y , ¯ h ) . Supp ose that 0 ∈ I 0 . T ake ε > 0 , and notice that by Assump tion 1 there exists λ ′ ∈ Λ and δ > 0 suc h that H u [ λ ′ ]( t ) > δ for t ∈ [0 , d 1 − ε ] , and th us b y (76) we obtain R d 1 − ε 0 v k ( t ) / √ γ k d t → 0 , as v k ≥ 0 . Then for all s ∈ [0 , d 1 ) , w e ha v e ¯ y k ( s ) → 0 , and thus (79) ¯ y = 0 , on [0 , d 1 ) , if 0 ∈ I 0 . Supp ose th at T ∈ I 0 . Th en, w e can d eriv e R T a N + ε ¯ v k ( t )d t → 0 b y an analog ous argumen t. Thus, the p oin t wise conv ergence ¯ h k − ¯ y k ( s ) → 0 , holds for ev ery s ∈ ( a N , T ] , and then, (80) ¯ y = ¯ h, on ( a N , T ] , if T ∈ I 0 . It remains to chec k the ﬁnal cond itions (32) for ¯ h. L et 0 ≤ i ≤ d ϕ , ϕ ′ i ( ˆ x ( T ))( ¯ ξ ( T ) + B ( T ) ¯ h ) = lim k →∞ ϕ ′ i ( ˆ x ( T ))  ξ k ( T ) + B ( T ) h k √ γ k  = lim k →∞ ϕ ′ i ( ˆ x ( T )) z k ( T ) √ γ k . (81) A ﬁr st order T ayl or expansion of the function ϕ i around ˆ x ( T ) giv es ϕ i ( x k ( T )) = ϕ i ( ˆ x ( T )) + ϕ ′ i ( ˆ x ( T )) δx k ( T ) + O ( | δ x k ( T ) | 2 ) . By Lemmas 8.4 and 8.12 in the Ap p endix, we can write ϕ i ( x k ( T )) = ϕ i ( ˆ x ( T )) + ϕ ′ i ( ˆ x ( T )) z k ( T ) + o ( √ γ k ) . Th us (82) ϕ ′ i ( ˆ x ( T )) z k ( T ) √ γ k = ϕ i ( x k ( T )) − ϕ i ( ˆ x ( T )) √ γ k + o (1) . QUADRA TIC ORDER CONDITIONS F OR BANG-SINGULAR EXTREMALS 25 Since x k satisﬁes (4), equations (81) and (82) yield, for 1 ≤ i ≤ d ϕ : ϕ ′ i ( ˆ x ( T ))( ¯ ξ ( T ) + B ( T ) ¯ h ) ≤ 0 . F or i = 0 use inequalit y (73). Analogously , η ′ j ( ˆ x ( T ))( ¯ ξ ( T ) + B ( T ) ¯ h ) = 0 , for j = 1 , . . . , d η . Th us ( ¯ ξ , ¯ y , ¯ h ) satisﬁes (32), and b y (78), (79) and (80), w e obtain ( ¯ ξ , ¯ y , ¯ h ) ∈ P 2 . (b) Return to th e exp ansion (61). Equatio n (74) an d H u [ λ ] ≥ 0 imply Ω P 2 [ λ ]( ξ k , y k , y k ( T )) = Φ[ λ ]( x k , u k ) − Φ[ λ ]( ˆ x, ˆ u ) − Z T 0 H u [ λ ] v k d t − o ( γ k ) ≤ o ( γ k ) . Th us (83) lim inf k →∞ Ω P 2 [ λ ]( ¯ ξ k , ¯ y k , ¯ h k ) ≤ lim su p k →∞ Ω P 2 [ λ ]( ¯ ξ k , ¯ y k , ¯ h k ) ≤ 0 . Split Ω P 2 as follo ws, Ω P 2 ,w [ λ ]( ξ , y , h ) := Z T 0 { ( Q [ λ ] ξ , ξ ) + ( M [ λ ] ξ , y ) } d t + g [ λ ]( ξ ( T ) , h ) , Ω P 2 , 0 [ λ ]( y ) := Z I 0 ( R [ λ ] y , y )d t, and Ω P 2 , + [ λ ]( y ) := Z I + ( R [ λ ] y , y )d t. Notice that Ω P 2 ,w [ λ ] is we akly conti nuous in the space X 2 × U 2 × I R . C onsider no w the subsp ace Γ 2 := { ( ξ , y , h ) ∈ X 2 × U 2 × I R : (29) , (33) and (34) hold } . Notice th at Γ 2 is itself a Hilb ert space. Let ρ > 0 b e the constant in the p ositivit y co ndition (60) and deﬁne Λ ρ := { λ ∈ co Λ : Ω P 2 [ λ ] − ργ is w eakly l . s . c . on Γ 2 } . Equation (60 ) and Lemma 8.11 in the App end ix imply that (84) max λ ∈ Λ ρ Ω P 2 [ λ ]( ¯ ξ , ¯ y , ¯ h ) ≥ ργ ( ¯ y , ¯ h ) . Denote b y ¯ λ the elemen t in Λ ρ that reac hes the maxim u m in (84). Next w e sho w th at R [ ¯ λ ]( t ) ≥ ρ on I + . Observe th at Ω P 2 , 0 [ ¯ λ ] − ρ R I 0 | y ( t ) | 2 d t is we akly con tin uous in the space Γ 2 . In fact, consider a sequence { ( ˜ ξ k , ˜ y k , ˜ h k ) } ⊂ Γ 2 con v erging weakl y to some ( ˜ ξ , ˜ y , ˜ h ) ∈ Γ 2 . Since ˜ y k and ˜ y are constan t on I 0 , necessarily ˜ y k → ˜ y uniformly in ev ery compact sub set of I 0 . Easily follo w s that (85) lim k →∞ Ω P 2 , 0 [ ¯ λ ]( ˜ y k ) − ρ Z I 0 | ˜ y k ( t ) | 2 d t = Ω P 2 , 0 [ ¯ λ ]( ˜ y ) − ρ Z I 0 | ˜ y ( t ) | 2 d t, 26 M.S. ARONNA, J.F. BON NANS, A.V. DMITRUK AND P .A. LOTITO and therefore, the w eak cont in uit y of Ω P 2 , 0 [ ¯ λ ] − ρ R I 0 | y ( t ) | 2 d t in Γ 2 holds. Since Ω P 2 [ ¯ λ ] − ργ is w eakly l.s.c. in Γ 2 , we get that the (r emainder) qu ad r atic mapping (86) y 7→ Ω P 2 , + [ ¯ λ ]( y ) − ρ Z I + | y ( t ) | 2 d t, is weakly l.s.c. o n Γ 2 . In particular, it is w eakly l.s.c. in the subspace of Γ 2 consisting of the elemen ts for which y = 0 on I 0 . Hence, in view of Lemma 8.10 in the Ap p endix, we get (87) R [ ¯ λ ]( t ) ≥ ρ, on I + . The fol lo wing step is p ro ving th e strong co nv ergence of ¯ y k to ¯ y . With this aim we mak e use of the un iform con v ergence on compact sub sets of I 0 , whic h is p oint ed out in Lemma 8.6. Recall no w Assumption 2, and let N b e the num b er of connected comp o- nen ts of I 0 . Set ε > 0 , and for eac h comp osing in terv al ( c, d ) of I 0 , consider a smaller inte rv al of the form ( c + ε/ 2 N , d − ε/ 2 N ) . Denote their union as I ε 0 . Notice that I 0 \ I ε 0 is of measur e ε. Pu t I ε + := [0 , T ] \ I ε 0 . By the Lemma 8.8 in the Ap p endix, R [ ¯ λ ]( t ) is a con tin uous fun ction of time, and thus from (87 ) we can assure that R [ ¯ λ ]( t ) ≥ ρ/ 2 on I ε + for ε su ﬃcien tly s mall. Consequent ly , Ω ε P 2 , + [ ¯ λ ]( y ) := Z I ε + ( R [ ¯ λ ] y , y )d t, is a Legendre form on L 2 ( I ε + ) , and th us the follo wing inequalit y holds f or the app r o ximating directions ¯ y k , (88) Ω ε P 2 , + [ ¯ λ ]( ¯ y ) ≤ lim inf k →∞ Ω ε P 2 , + [ ¯ λ ]( ¯ y k ) . Since the sequen ce ¯ y k con v erges uniformly to ¯ y on ev ery compact subset of I 0 , deﬁning Ω ε P 2 , 0 [ ¯ λ ]( y ) := Z I ε 0 ( R [ ¯ λ ] y , y )d t, w e get (89) lim k →∞ Ω ε P 2 , 0 [ ¯ λ ]( ¯ ξ k , ¯ y k , ¯ h k ) = Ω ε P 2 , 0 [ ¯ λ ]( ¯ ξ , ¯ y , ¯ h ) . Notice that the weak contin u it y of Ω ε P 2 , 0 [ ¯ λ ] in Γ 2 cannot be applied sin ce ( ¯ ξ k , ¯ y k , ¯ h k ) / ∈ Γ 2 . F rom p ositivit y cond ition (60), equations (88), (89 ), and the weak con tin uit y of Ω P 2 ,w [ ¯ λ ] (in X 2 × U 2 × I R ) we get ργ ( ¯ y , ¯ h ) ≤ Ω P 2 [ ¯ λ ]( ¯ ξ , ¯ y , ¯ h ) ≤ lim k →∞ Ω P 2 ,w [ ¯ λ ]( ¯ ξ k , ¯ y k , ¯ h k ) + lim k →∞ Ω ε P 2 , 0 [ ¯ λ ]( ¯ y k ) + lim inf k →∞ Ω ε P 2 , + [ ¯ λ ]( ¯ y k ) = lim inf k →∞ Ω P 2 [ ¯ λ ]( ¯ ξ k , ¯ y k , ¯ h k ) . On the other hand, inequalit y (83) implies th at the right- hand sid e of the last exp r ession is nonp ositiv e. T herefore, ( ¯ y , ¯ h ) = 0 , and lim k →∞ Ω P 2 [ ¯ λ ]( ¯ ξ k , ¯ y k , ¯ h k ) = 0 . QUADRA TIC ORDER CONDITIONS F OR BANG-SINGULAR EXTREMALS 27 Equation (89 ) yields lim k →∞ Ω ε P 2 , 0 [ ¯ λ ]( ¯ ξ k , ¯ y k , ¯ h k ) = 0 and thus (90) lim k →∞ Ω ε P 2 , + [ ¯ λ ]( ¯ y k ) = 0 . W e hav e: Ω ε P 2 , + [ ¯ λ ] is a Legendre form on L 2 ( I ε + ) and ¯ y k ⇀ 0 on I ε + . Thus, b y (90), ¯ y k → 0 , on L 2 ( I ε + ) . As we a lready noticed, { ¯ y k } con v erges un iformly on I ε 0 , th us the strong con v ergence h olds on [0 , T ] . Th erefore (91) ( ¯ y k , ¯ h k ) − → (0 , 0) , on U 2 × I R . This lea ds to a con tradiction since ( ¯ y k , ¯ h k ) has un it norm f or ev ery k ∈ I N . Th us, ˆ w is a P on try agin minim um sati sfying quadratic gro wth.  6. Ext e nsions a nd an ex ample 6.1. Including para meters. Cons ider the follo w ing optimal control p rob- lem where the in itial state is not determined, some parameters are included and a more ge neral con trol constraint is considered. J := ϕ 0 ( x (0) , x ( T ) , r (0)) → min , (92) ˙ x ( t ) = m X i =0 u i ( t ) f i ( x ( t ) , r ( t )) , (93) ˙ r ( t ) = 0 , (94) a i ≤ u i ( t ) ≤ b i , for a . a . t ∈ (0 , T ) , i = 1 , . . . , m (95) ϕ i ( x (0) , x ( T ) , r (0)) ≤ 0 , for i = 1 , . . . , d ϕ , (96) η j ( x (0) , x ( T ) , r (0)) = 0 , for j = 1 . . . , d η , (97) where u ∈ U , x ∈ X , r ∈ I R n r is a parameter considered as a state v ariable with zero-dynamics, a, b ∈ I R m , fu nctions f i : I R n + n r → I R n , ϕ i : I R 2 n + n r → I R , and η : I R 2 n + n r → I R d η are t wice con tin uously d iﬀeren tiable. As r has zero d ynamics, the costate v ariable ψ r corresp onding to equation (94) do es not app ear in the pr e-Hamilto nian. Denote with ψ th e costate v ariable asso ciated with (93). T he pre-Hamiltonian function for problem (92)-(97) is give n b y H [ λ ]( x, r , u, t ) = ψ ( t ) m X i =0 u i f i ( x, r ) . Let ( ˆ x, ˆ r , ˆ u ) b e a feasible solution for (93)-(97 ). Sin ce ˆ r ( · ) is constan t, w e can denote it by ˆ r . Assume that ϕ i ( ˆ x (0) , ˆ x ( T ) , ˆ r ) = 0 , for i = 0 , . . . , d ϕ . 28 M.S. ARONNA, J.F. BON NANS, A.V. DMITRUK AND P .A. LOTITO An elemen t λ = ( α, β , ψ x , ψ r ) ∈ I R d ϕ + d η +1 × W 1 ∞ (0 , T ; I R n, ∗ ) × W 1 ∞ (0 , T ; I R n r , ∗ ) is a Pon tryagin m ultiplier for ( ˆ x, ˆ r , ˆ u ) if it satisﬁes (7), (8), the costate equa- tion for ψ      − ˙ ψ x ( t ) = H x [ λ ]( ˆ x ( t ) , ˆ r , ˆ u ( t ) , t ) , a . e . on [0 , T ] ψ x (0) = − ℓ x 0 [ λ ]( ˆ x (0) , ˆ x ( T ) , ˆ r ) , ψ x ( T ) = ℓ x T [ λ ]( ˆ x (0) , ˆ x ( T ) , ˆ r ) , and for ψ r (98) ( − ˙ ψ r ( t ) = H r [ λ ]( ˆ x ( t ) , ˆ r , ˆ u ( t ) , t ) , a . e . on [0 , T ] ψ r (0) = − ℓ r [ λ ]( ˆ x (0) , ˆ x ( T ) , ˆ r ) , ψ r ( T ) = 0 . Observe that (98) implies the stationarit y condition ℓ r ( ˆ x (0) , ˆ x ( T ) , ˆ r ) + Z T 0 H r [ λ ]( t )d t = 0 . T ake v ∈ U and consider the linearized s tate equation (99)        ˙ z ( t ) = m X i =0 ˆ u i ( t )[ f i,x ( ˆ x ( t ) , ˆ r ) z ( t ) + f i,r ( ˆ x ( t ) , ˆ r ) δr ( t )] + m X i =1 v i ( t ) f i ( ˆ x ( t ) , ˆ r ) , ˙ δ r ( t ) = 0 , where we can see that δ r ( · ) is constan t and thus we denote it b y δ r . Let the linearized in itial-ﬁnal co nstraints b e ϕ ′ i ( ˆ x (0) , ˆ x ( T ) , ˆ r )( z (0) , z ( T ) , δr ) ≤ 0 , f or i = 1 , . . . , d ϕ , η ′ j ( ˆ x (0) , ˆ x ( T ) , ˆ r )( z (0) , z ( T ) , δr ) = 0 , f or j = 1 , . . . , d η . (100) Deﬁne for eac h i = 1 , . . . , m the sets I i a := { t ∈ [0 , T ] : max λ ∈ Λ H u i [ λ ]( t ) > 0 } , I i b := { t ∈ [0 , T ] : max λ ∈ Λ H u i [ λ ]( t ) < 0 } , I i sing := [0 , T ] \ ( I i a ∪ I i b ) . Assumption 3. Consider the natural extension of Assumption 2, i.e. for eac h i = 1 , . . . , m, the sets I i a and I i b are ﬁnite unions of inte rv als, i.e. I i a = N i a [ j = 1 I i j,a , I i b = N i b [ j = 1 I i j,b , for I i j,a and I i j,b b eing sub in terv als of [0 , T ] of the form [0 , c ) , ( d, T ]; or ( c, d ) if c 6 = 0 and d 6 = T . Notice th at I i a ∩ I i b = ∅ . Call c i 1 ,a < d i 1 ,a < c i 2 ,a < . . . < c i N i a ,a < d i N i a ,a the endp oin ts of these interv als corresp ond ing to b ound a, and deﬁne them analogously for b. Consequentl y , I i sing is a ﬁnite u nion of in terv als as w ell. Assum e that a concatenation of a bang arc follo wed by another b an g arc is forbidd en. QUADRA TIC ORDER CONDITIONS F OR BANG-SINGULAR EXTREMALS 29 Assumption 4. Strict complemen tarit y assu mption for con trol constrain ts: ( I i a = { t ∈ [0 , T ] : ˆ u i ( t ) = a i } , u p to a set of null measure , I i b = { t ∈ [0 , T ] : ˆ u i ( t ) = b i } , u p to a set of null measure . Consider C 2 := ( ( z , δ r, v ) ∈ X 2 × I R n r × U 2 : (99)-(100) hold , v i = 0 on I i a ∪ I i b , for i = 1 , . . . , m ) . The Goh transformation allo ws us to obtain v ariables ( ξ , y ) deﬁned b y y ( t ) := Z t 0 v ( s )d s, ξ := z − m X i =1 y i f i . Notice that ξ satisﬁes the equation ˙ ξ = A x ξ + A r δ r + B x 1 y , ξ (0) = z (0) , (101) where, d enoting [ f i , f j ] x := f i,x f j − f j,x f i , A x := m X i =0 ˆ u i f i,x , A r := m X i =0 ˆ u i f i,r , B x 1 y := m X j = 1 y j m X i =0 ˆ u i [ f i , f j ] x . Consider the transformed v ersion of (100), ϕ ′ i ( ˆ x (0) , ˆ x ( T ) , ˆ r )( ξ (0) , ξ ( T ) + B ( T ) h, δ r ) ≤ 0 , i = 1 , . . . , d ϕ , η ′ j ( ˆ x (0) , ˆ x ( T ) , ˆ r )( ξ (0) , ξ ( T ) + B ( T ) h, δ r ) = 0 , j = 1 , . . . , d η , (102) and let the cone P b e giv en by P := ( ( ξ , δ r , y , h ) ∈ X × I R n r × Y × I R m : y (0) = 0 , h = y ( T ) , (101) and (102) hold , y ′ i = 0 on I i a ∪ I i b , for i = 1 , . . . , m ) . Observe that eac h ( ξ , δ r, y , h ) ∈ P satisﬁes (103) y i constan t o ver eac h comp osing in terv al of I i a ∪ I i b , and at the end p oin ts, (104)  y i = 0 on [0 , d ] , if 0 ∈ I i a ∪ I i b , and , y i = h i on [ c, T ] , if T ∈ I i a ∪ I i b , where [0 , d ) is the ﬁrst maximal comp osing in terv al of I i a ∪ I d b when 0 ∈ I d a ∪ I d b , and ( c, T ] is its last comp osing interv al when T ∈ I i a ∪ I i b . Deﬁne P 2 := ( ( ξ , δ r , y , h ) ∈ X 2 × I R n r × U 2 × I R m : (101) , (102) , (1 03) and (104) hold for i = 1 , . . . , m ) . Recall deﬁ n itions in equations (35), (36), (39), (40), (41 ). Minor simpliﬁ- cations app ear in the compu tations of these functions as the dynamics of r are null and δr is constan t. W e outline these calc ulations in an example. 30 M.S. ARONNA, J.F. BON NANS, A.V. DMITRUK AND P .A. LOTITO Consider M ⊂ I R s and the subset of M ⊂ I R s deﬁned by G ( M ) := { λ ∈ M : V ij [ λ ] = 0 on I i sing ∩ I j sing , for ev ery pair 1 < i 6 = j ≤ m } . Using the same tec hniqu es, we obtain the equiv alen t of Theorem 4.11 : Corollary 3. Su pp ose that ( ˆ x, ˆ r, ˆ u ) is a we ak minimum for pr oblem (92) - (97) . Th en max λ ∈ G (co Λ) Ω P 2 [ λ ]( ξ , δ r, y , h ) ≥ 0 , for all ( ξ , δ r , y , h ) ∈ P 2 . By a simple adaptation of the pro of of T h eorem 5.5 we get the equiv alen t result. Corollary 4. L et m = 1 . Su pp ose that ther e exists ρ > 0 suc h that (105) max λ ∈ Λ Ω P 2 [ λ ]( ξ , δ r, y , h ) ≥ ργ ( y , h ) , for all ( ξ , δ r, y , h ) ∈ P 2 . Then ( ˆ x, ˆ r , ˆ u ) is a Pontryagin minimum that satisﬁes γ − quadr atic gr owth. 6.2. Application to minim um-time problems. Consid er the problem J := T → min , s . t . (9 3) − (97) . Observe that by the c hange of v ariables: (106) x ( s ) ← x ( T s ) , u ( s ) ← u ( T s ) , w e can transform the problem in to the follo w ing form ulation. J := T (0) → min , ˙ x ( s ) = T ( s ) m X i =0 u i ( s ) f i ( x ( s ) , r ( s )) , a . e . on [0 , 1] , ˙ r ( s ) = 0 , a . e . on [0 , 1] , ˙ T ( s ) = 0 , a . e . on [0 , 1] , a i ≤ u i ( s ) ≤ b i , a . e . on [0 , 1] , i = 1 , . . . , m, ϕ i ( x (0) , x (1) , r (0)) ≤ 0 , for i = 1 , . . . , d ϕ , η j ( x (0) , x ( T ) , r (0)) = 0 , for j = 1 . . . , d η . W e can apply Corollaries 3 and 4 to the problem written in this form. W e outline the calculatio ns in the follo wing example. 6.2.1. Example: Markov-D u bins pr oblem. Consider a problem o v er the in- terv al [0 , T ] with free ﬁn al time T : J := T → min , ˙ x 1 = − sin x 3 , x 1 (0) = 0 , x 1 ( T ) = b 1 , ˙ x 2 = cos x 3 , x 2 (0) = 0 , x 2 ( T ) = b 2 , ˙ x 3 = u, x 3 (0) = 0 , x 3 ( T ) = θ , − 1 ≤ u ≤ 1 , (107) QUADRA TIC ORDER CONDITIONS F OR BANG-SINGULAR EXTREMALS 31 with 0 < θ < π , b 1 and b 2 ﬁxed. This problem was originally int ro du ced by Mark o v in [39 ] and studied b y Dubins in [18 ]. More r ecen tly , th e problem w as inv estiga ted b y Su ssmann and T ang [60 ], Soueres and Laumond [56], Boscain and Piccoli [7], among others. Here w e will study the optimalit y of the extremal (108) ˆ u ( t ) :=  1 on [0 , θ ] , 0 on ( θ , ˆ T ] . Observe that by the c hange of v ariables (106) w e can transform (107) in to the follo wing problem on the in terv al [0 , 1] . J := T (0) → m in , ˙ x 1 ( s ) = − T ( s ) sin x 3 ( s ) , x 1 (0) = 0 , x 1 (1) = b 1 , ˙ x 2 ( s ) = T ( s ) cos x 3 ( s ) , x 2 (0) = 0 , x 2 (1) = b 2 , ˙ x 3 ( s ) = T ( s ) u ( s ) , x 3 (0) = 0 , x 3 (1) = θ , ˙ T ( s ) = 0 , − 1 ≤ u ( s ) ≤ 1 . (109) W e obtain for state v ariables: (110) ˆ x 3 ( s ) =  ˆ T s on [0 , θ / ˆ T ] , θ on ( θ / ˆ T , 1] , ˆ x 1 ( s ) =  cos( ˆ T s ) − 1 on [0 , θ / ˆ T ] , ˆ T sin θ ( θ / ˆ T − s ) + cos θ − 1 on ( θ / ˆ T , 1] , ˆ x 2 ( s ) =  sin ˆ T s on [0 , θ / ˆ T ] , ˆ T cos θ ( s − θ / ˆ T ) + sin θ on ( θ , ˆ T ] . Since the terminal v alues for x 1 and x 2 are ﬁxed, the ﬁnal time ˆ T is deter- mined by the p revious equalities. The pre-Hamiltonian for p roblem (109 ) is (111) H [ λ ]( s ) := T ( s )( − ψ 1 ( s ) sin x 3 ( s ) + ψ 2 ( s ) cos x 3 ( s ) + ψ 3 ( s ) u ( s )) . The ﬁn al Lagrangian is ℓ := α 0 T (1) + 3 X j = 1 ( β j x j (0) + β j x j (1)) . As ˙ ψ 1 ≡ 0 , and ˙ ψ 2 ≡ 0 , w e get ψ 1 ≡ β 1 , ψ 2 ≡ β 2 , on [0 , 1] . Since the candidate cont rol ˆ u is singular on [ θ / ˆ T , 1] , w e ha v e H u [ λ ] ≡ 0 . By (111), we obtain (112) ψ 3 ( s ) = 0 , on [ θ / ˆ T , 1] . 32 M.S. ARONNA, J.F. BON NANS, A.V. DMITRUK AND P .A. LOTITO Th us β 3 = 0 . In addition, as the costate equation for ψ 3 is − ˙ ψ 3 = ˆ T ( − β 1 cos ˆ x 3 − β 2 sin ˆ x 3 ) , b y (110) and (112), we get (113) β 1 cos θ + β 2 sin θ = 0 . F rom (110) and (112) an d since H is constant and equal to − α 0 , we get (114) H = ˆ T ( − β 1 sin θ + β 2 cos θ ) ≡ − α 0 . Prop osition 4. The fol lowing pr op erties hold (i) α 0 > 0 , (ii) H u [ λ ]( s ) < 0 on [0 , θ / ˆ T ) for al l λ ∈ Λ . Pr o of. Ite m ( i) Supp ose th at α 0 = 0 . By (113) and (114), we obtain β 1 cos θ + β 2 sin θ = 0 , and − β 1 sin θ + β 2 cos θ = 0 . Supp ose, w.l.g., that cos θ 6 = 0 . Then β 1 = − β 2 sin θ cos θ and thus β 2 sin 2 θ cos θ + β 2 cos θ = 0 . W e conclude that β 2 = 0 as w ell. This imp lies ( α 0 , β 1 , β 2 , β 3 ) = 0 , whic h con tradicts the non-trivialit y condition (7) . So, α 0 > 0 , as requ ir ed. Item (ii) O bserv e that H u [ λ ]( s ) ≤ 0 , on [0 , θ / ˆ T ) , and H u [ λ ] = ψ 3 . Let us pro v e that ψ 3 is nev er 0 on [0 , θ / ˆ T ) . S upp ose th er e exists s 1 ∈ [0 , θ / ˆ T ) suc h that ψ 3 ( s 1 ) = 0 . T h us, since ψ 3 ( θ / ˆ T ) = 0 as indicated in (112), there exists s 2 ∈ ( s 1 , θ / ˆ T ) such that ˙ ψ 3 ( s 2 ) = 0 , i.e. (115) β 1 cos( ˆ T s 2 ) + β 2 sin( ˆ T s 2 ) = 0 . Equations (113) and (115) imply that tan( θ / ˆ T ) = tan( s 2 / ˆ T ) . Th is con tra- dicts θ < π . Th us ψ 3 ( s ) 6 = 0 for ev ery s ∈ [0 , θ / ˆ T ) , and consequen tly , H u [ λ ]( s ) < 0 , f or s ∈ [0 , θ / ˆ T ) .  Since α 0 > 0 , then δ T = 0 for eac h element of the critical cone, where δ T is the linearized state v ariable T . Observ e that as ˆ u = 1 on [0 , θ / ˆ T ] , th en y = 0 and ξ = 0 , on [0 , θ / ˆ T ] , f or all ( ξ , δ T , y , h ) ∈ P 2 . W e look for the second v ariation in the in terv al [ θ / ˆ T , 1] . Th e Goh transfor- mation gives ξ 3 = z 3 − ˆ T y , QUADRA TIC ORDER CONDITIONS F OR BANG-SINGULAR EXTREMALS 33 and since ˙ z 3 = ˆ T v, we ge t z 3 = ˆ T y and thus ξ 3 = 0 . Then , as H ux = 0 and ℓ ′′ = 0 , w e get Ω[ λ ] = Z 1 θ / ˆ T ( β 1 sin θ − β 2 cos θ ) y 2 dt = α 0 Z 1 0 y 2 dt. Notice that if ( ξ , δ T , y , h ) ∈ P 2 , th en h satisﬁes ξ 3 ( T ) + ˆ T h = 0 , and, as ξ 3 ( T ) = 0 , w e get h = 0 . Th us Ω[ λ ]( ξ , y , h ) = α 0 Z T 0 y 2 dt = α 0 γ ( y , h ) , on P 2 . Since Assumptions 3 and 4 hold, we conclude b y Corollary 4 th at ( ˆ x, ˆ T , ˆ u ) is a P on try agin minim um s atisfying quadratic growth. 7. Conc lusion W e pro vided a set of necessary and suﬃcien t conditions for a bang-singular extremal. The suﬃ cien t condition is restricted to the scalar con trol case. These n ecessary and suﬃcien t conditions are close in the sense that, to pass from one to the other, one has to strengthen a n on-negativit y inequalit y transforming it in to a co ercivit y condition. This is the ﬁr st time that a suﬃcient condition that is ‘almost necessary’ is established f or a b ang-singular extremal for the general May er problem. In some cases th e condition can b e easily c hec ked as it can b e seen in the example. 8. Ap pendix Lemma 8.1. L et X := { ( ξ , y , h ) ∈ X 2 × U 2 × I R m : (29) , (33) - (34) hold } , L := { ( ξ , y , y ( T )) ∈ X × Y × I R m : y (0) = 0 , (29) and (33) } . Then L is a dense subset of X in the X 2 × U 2 × I R m − top olo gy. Pr o of. (See Lemma 6 in [17].) Let us prov e the result for m = 1 . The general case is a tr ivial extension. Let ( ¯ ξ , ¯ y , ¯ h ) ∈ X and ε, δ > 0 . C onsider φ ∈ Y suc h that k ¯ y − φ k 2 < ε/ 2 . In order to satisfy condition (34) tak e  y δ ( t ) := 0 , for t ∈ [0 , d 1 ] , if c 1 = 0 , y δ ( t ) := h, f or t ∈ [ c N , T ] , if d N = T , where c j , d j w ere int ro du ced in Assump tion 2. S ince ¯ y is constant on eac h I j , deﬁne y δ constan t o v er th ese inte rv als with the same constan t v alue as ¯ y . It remains to d eﬁne y δ o v er I + . Over eac h maximal comp osing interv al ( a, b ) of I + , deﬁne y δ as describ ed b elow. T ak e c := ¯ y ( a − ) if a > 0 , or c := 0 if a = 0; and let d := ¯ y ( b +) if b < T , or d := h w h en b = T . Deﬁne t w o aﬃne functions ℓ 1 ,δ and ℓ 2 ,δ satisfying ℓ 1 ,δ ( a ) = c, ℓ 1 ,δ ( a + δ ) = φ ( a + δ ) , ℓ 2 ,δ ( b ) = d, ℓ 2 ,δ ( b − δ ) = φ ( b − δ ) . (116) 34 M.S. ARONNA, J.F. BON NANS, A.V. DMITRUK AND P .A. LOTITO T ake (117) y δ ( t ) :=      ℓ 1 ,δ ( t ) , for t ∈ [ a, a + δ ] , φ ( t ) , for t ∈ ( a + δ, b − δ ) , ℓ 2 ,δ ( t ) , for t ∈ [ b − δ, b ] , and notice that k φ − y δ k 2 , [ a,b ] ≤ 1 k max ( | c | , | d | , M ) , where M := su p t ∈ [ a,b ] | φ ( t ) | . Finally , observ e that y δ ( T ) = h, and, for suﬃcient ly small δ, k ¯ y − y δ k 2 ≤ k ¯ y − φ k 2 + k φ − y δ k 2 < ε. Th us, the result follo ws.  Lemma 8.2. L et λ ∈ Λ and ( z , v ) ∈ C 2 . Then (118) d ϕ X i =0 α i ¯ ϕ ′′ i ( ˆ u )( v , v ) + d η X i =1 β j ¯ η ′′ j ( ˆ u )( v , v ) = Ω[ λ ]( z , v ) . Pr o of. Let us compute the left-hand side of (118). Notice that (119) d ϕ X i =0 α i ¯ ϕ i ( ˆ u ) + d η X i =1 β j ¯ η j ( ˆ u ) = ℓ [ λ ]( ˆ x ( T )) . Let u s lo ok for a second order expansion for ℓ. Consider ﬁrst a second order expansion of the stat e v ariable: x = ˆ x + z + 1 2 z vv + o ( k v k 2 ∞ ) , where z vv satisﬁes (120) ˙ z vv = Az vv + D 2 ( x,u ) 2 F ( ˆ x, ˆ u )( z , v ) 2 , z vv (0) = 0 , with F ( x, u ) := P m i =0 u i f i ( x ) . C onsider the second order expansion for ℓ : ℓ [ λ ]( x ( T )) = ℓ [ λ ](( ˆ x + z + 1 2 z vv )( T )) + o ( k v k 2 1 ) = ℓ [ λ ]( ˆ x ( T )) + ℓ ′ [ λ ]( ˆ x ( T ))( z ( T ) + 1 2 z vv ( T )) + 1 2 ℓ ′′ [ λ ]( ˆ x ( T ))( z ( T ) + 1 2 z vv ( T )) 2 + o ( k v k 2 1 ) . (121) Step 1. Comp ute ℓ ′ [ λ ]( ˆ x ( T )) z vv ( T ) = ψ ( T ) z vv ( T ) − ψ (0) z vv (0) = Z T 0 [ ˙ ψ z vv + ψ ˙ z vv ]d t = Z T 0 {− ψ Az vv + ψ ( Az vv + D 2 F ( x,u ) 2 ( z , v ) 2 ) } d t = Z T 0 D 2 H [ λ ]( z , v ) 2 d t. Step 2. C ompute ℓ ′′ [ λ ]( ˆ x ( T ))( z ( T ) , z vv ( T )) . Applyin g Gron w all’s Lemma, w e obtain k z k ∞ = O ( k v k 1 ) , and k z vv k ∞ = O ( k v 2 k 1 ) . Th us | ( z ( T ) , z vv ( T )) | = O ( k v k 3 1 ) , and we conclude that | ℓ ′′ [ λ ]( ˆ x ( T ))( z ( T ) , z vv ( T )) | = O ( k v k 3 1 ) . QUADRA TIC ORDER CONDITIONS F OR BANG-SINGULAR EXTREMALS 35 Step 3. S ee that ℓ ′′ [ λ ]( ˆ x ( T ))( z vv ( T )) 2 = O ( k v k 4 1 ) . Then b y (121) we get, ℓ [ λ ]( x ( T )) = ℓ [ λ ]( ˆ x ( T )) + ℓ ′ [ λ ]( ˆ x ( T )) z ( T ) + 1 2 ℓ ′′ [ λ ]( ˆ x ( T )) z 2 ( T ) + 1 2 Z T 0 D 2 ( x,u ) 2 H [ λ ]( z , v ) 2 d t + o ( k v k 2 1 ) = ℓ [ λ ]( ˆ x ( T )) + ℓ ′ [ λ ]( ˆ x ( T )) z ( T ) + Ω[ λ ]( z , v ) + o ( k v k 2 1 ) . The conclusion follo w s b y (11 9 ).  Lemma 8.3. Given ( z, v ) ∈ W satisfying (12) , the fol lowing estimation holds for some ρ > 0 : k z k 2 2 + | z ( T ) | 2 ≤ ργ ( y , y ( T )) , wher e y is deﬁne d by (28 ) . Remark 9. ρ dep end s on ˆ w , i.e. it do es not v ary with ( z , v ) . Pr o of. Every time we mentio n ρ i w e are referrin g to a constant dep ending on k A k ∞ , k B k ∞ or b oth. Cons ider ξ , the solution of equation (2 9 ) cor- resp ondin g to y . Gron w all’s Lemma and the Cauc hy-Sc hw artz inequalit y imply (122) k ξ k ∞ ≤ ρ 1 k y k 2 . This last inequalit y , together with expr ession (28), implies (123) k z k 2 ≤ k ξ k 2 + k B k ∞ k y k 2 ≤ ρ 2 k y k 2 . On the other h and, equations (28) and (122) lead to | z ( T ) | ≤ | ξ ( T ) | + k B k ∞ | y ( T ) | ≤ ρ 1 k y k 2 + k B k ∞ | y ( T ) | . Then, by the inequalit y ab ≤ a 2 + b 2 2 , we get (124) | z ( T ) | 2 ≤ ρ 3 ( k y k 2 2 + | y ( T ) | 2 ) . The conclusion follo w s from equ ations (123) and (124).  The n ext lemma is a generalizatio n of the pr evious result to the n onlinear case. See Lemma 6.1 in Dmitruk [15]. Lemma 8.4. L et w = ( x, u ) b e the solution of (2) with k u k 2 ≤ c for some c onstant c. P ut ( δ x, v ) := w − ˆ w . Then | δ x ( T ) | 2 + k δ x k 2 2 ≤ ργ ( y , y ( T )) , wher e y is deﬁne d by (28 ) and ρ dep ends on c. Lemma 8.5. L et { y k } ⊂ L 2 ( a, b ) b e a se que nc e of c ontinuous non-de cr e asing functions that c onver ges we akly to y ∈ L 2 ( a, b ) . Then y is non-de cr e asing. 36 M.S. ARONNA, J.F. BON NANS, A.V. DMITRUK AND P .A. LOTITO Pr o of. Let s, t ∈ ( a, b ) b e such that s < t, and ε 0 > 0 suc h that s + ε 0 < t − ε 0 . F or ev ery k ∈ I N , and every 0 < ε < ε 0 , the follo wing inequalit y h olds Z s + ε s − ε y k ( ν ) dν ≤ Z t + ε t − ε y k ( ν ) dν. T aking the limit as k goes to inﬁnity and multiplying by 1 2 ε , we deduce that 1 2 ε Z s + ε s − ε y ( ν ) dν ≤ 1 2 ε Z t + ε t − ε y ( ν ) dν . As ( a, b ) is a ﬁ nite m easure space, y is a function of L 1 ( a, b ) and almo st all p oin ts in ( a, b ) are L eb esgue p oin ts (see Rudin [52, Theorem 7.7]). Thus, by taking ε to 0, it follo ws from the pr evious inequalit y that y ( s ) ≤ y ( t ) , whic h is what we wan ted to pro v e.  Lemma 8.6. Consider a se quenc e { y k } of non-de cr e asing c ontinuous func- tions i n a c omp act r e al interval I and assume that { y k } c onver ges we akly to 0 in L 2 ( I ) . Then it c onver ges uniformly to 0 on any interval ( a, b ) ⊂ I . Pr o of. T ak e an arb itrary in terv al ( a, b ) ⊂ I . First pro v e the p oint wise con- v ergence of { y k } to 0. On the con tr ary , supp ose that ther e exists c ∈ ( a, b ) suc h that { y k ( c ) } do es not con v erge to 0. Th us ther e e xist ε > 0 and a subsequence { y k j } su c h that y k j ( c ) > ε for eac h j ∈ I N , or y k j ( c ) < − ε for eac h j ∈ I N . S upp ose, without loss of generalit y , that the ﬁrst statemen t is true. Thus (125) 0 < ε ( b − c ) < y k j ( c )( b − c ) ≤ Z b c y k j ( t )d t, where the last inequalit y holds sin ce y k j is nondecreasing. But the right -hand side of (125) go es to 0 as j goes to inﬁ nit y . This con tradicts the h yp oth- esis and th us the p oin t wise conv ergence of { y k } to 0 f ollo ws. The uniform con v ergence is a direct consequence of the monotonicit y of the functions y k .  Lemma 8.7. [20, Theorem 22, P age 154 - V olume I] L e t a and b b e two functions of b ounde d variation in [0 , T ] . Supp ose that one is c ontinuous and the other is right-c ontinuous. Th en Z T 0 a ( t ) db ( t ) + Z T 0 b ( t ) da ( t ) = [ ab ] T + 0 − . Lemma 8.8. L et m = 1 , i.e. c onsider a sc alar c ontr ol variable. Then, f or any λ ∈ Λ , the function R [ λ ]( t ) deﬁne d in (39) is c ontinuous in t. Pr o of. Consid er deﬁnition (36). Condition V [ λ ] ≡ 0 yields S [ λ ] = C [ λ ] B , and since R [ λ ] is scalar, w e ca n write R [ λ ] = B ⊤ Q [ λ ] B − 2 C [ λ ] B 1 − ˙ C [ λ ] B − C [ λ ] ˙ B . QUADRA TIC ORDER CONDITIONS F OR BANG-SINGULAR EXTREMALS 37 Note that B = f 1 , B 1 = [ f 0 , f 1 ] , C [ λ ] = − ψ f ′ 1 , and Q [ λ ] = − ψ ( f ′′ 0 + ˆ uf ′′ 1 . Th us R [ λ ] = ψ ( f ′′ 0 + ˆ uf ′′ 1 )( f 1 , f 1 ) − 2 ψf ′ 1 ( f ′ 0 f 1 − f ′ 1 f 0 ) + ψ ( f ′ 0 + ˆ uf ′ 1 ) f ′ 1 f 1 − ψ f ′′ 1 ( f 0 + ˆ uf 1 ) f 1 − ψ f ′ 1 f ′ 1 ( f 0 + ˆ uf 1 ) = ψ [ f 1 , [ f 1 , f 0 ]] . Since f 0 and f 1 are t wice con tin uously diﬀeren tiable, w e conclude that R [ λ ] is conti nuous in time.  Lemma 8.9. [28] Consider a quadr atic form Q = Q 1 + Q 2 wher e Q 1 is a L e gendr e form and Q 2 is we akly c ontinuous over some Hilb ert sp ac e. Then Q is a L e gendr e form. Lemma 8.10. [28, Theorem 3.2] Consider a r e al interval I and a quadr atic form Q over the H ilb ert sp ac e L 2 ( I ) , given by Q ( y ) := Z I y ⊤ ( t ) R ( t ) y ( t )d t. Then Q is we akly l.s.c. o ver L 2 ( I ) iﬀ (126) R ( t )  0 , a . e . on I . Lemma 8.11. [14, Th eorem 5] Given a Hilb ert sp ac e H , and a 1 , a 2 , . . . , a p ∈ H , set K := { x ∈ H : ( a i , x ) ≤ 0 , f or i = 1 , . . . , p } . L et M b e a c onvex and c omp act subset of I R s , and let { Q ψ : ψ ∈ M } b e a family of c ontinuous quadr atic forms over H with the mapping ψ → Q ψ b eing aﬃne. Set M # := { ψ ∈ M : Q ψ is wea kly l . s . c . } and assume that max ψ ∈ M Q ψ ( x ) ≥ 0 , for all x ∈ K. Then max ψ ∈ M # Q ψ ( x ) ≥ 0 , for all x ∈ K. The follo wing result is an adaptation of Lemma 6.5 in [1 5]. Lemma 8.12. Consider a se quenc e { v k } ⊂ U and { y k } their primitives deﬁne d b y (28) . Cal l u k := ˆ u + v k , x k its c orr esp onding solution of (2) , and let z k denote the line arize d state c orr e sp onding to v k , i.e . the solution of (12) . Deﬁne, for e ach k ∈ I N , (127) δ x k := x k − ˆ x, η k := δ x k − z k , γ k := γ ( y k , y k ( T )) . Supp ose that { v k } c onver ges to 0 in the Pontryagin sense. Then (i) (128) ˙ η k = m X i =0 ˆ u i f ′ i ( ˆ x ) η k + m X i =1 v i,k f ′ i ( ˆ x ) δ x k + ζ k , 38 M.S. ARONNA, J.F. BON NANS, A.V. DMITRUK AND P .A. LOTITO (129) ˙ δ x k = m X i =0 u i,k f ′ i ( ˆ x ) δ x k + m X i =1 v i,k f i ( ˆ x ) + ζ k , wher e k ζ k k 2 ≤ o ( √ γ k ) and k ζ k k ∞ → 0 , (ii) k η k k ∞ ≤ o ( √ γ k ) . Pr o of. (i,ii) Consider the second ord er T a ylor expan s ions of f i , f i ( x k ) = f i ( ˆ x ) + f ′ i ( ˆ x ) δ x k + 1 2 f ′′ i ( ˆ x )( δ x k , δ x k ) + o ( | δ x k ( t ) | 2 ) . W e can w r ite (130) ˙ δ x k = m X i =0 u i,k f ′ i ( ˆ x ) δ x k + m X i =1 v i,k f i ( ˆ x ) + ζ k , with (131) ζ k := 1 2 m X i =0 u i,k f ′′ i ( ˆ x )( δ x k , δ x k ) + o ( | δx k ( t ) | 2 ) m X i =0 u i,k . As { u k } is b oun ded in L ∞ and k δx k k ∞ → 0 , w e get k ζ k k ∞ → 0 and the follo w ing L 2 − norm b ound: k ζ k k 2 ≤ cons t. m X i =0 k u i,k ( δ x k , δ x k ) k 2 + o ( γ k ) k m X i =0 u i,k k 1 ≤ cons t. k u k k ∞ k δ x k k 2 2 = O ( γ k ) ≤ o ( √ γ k ) . (132) Let us lo ok for the diﬀeren tial equation of η k deﬁned in (127). By (130), and addin g and substracting the term P m i =1 ˆ u i f ′ i ( ˆ x ) δ x k w e obtai n ˙ η k = m X i =0 ˆ u i f ′ i ( ˆ x ) η k + m X i =1 v i,k f ′ i ( ˆ x ) δ x k + ζ k . Th us w e obtain ( i). Applying Gronw all’s Lemma to this last diﬀerenti al equation we get (133) k η k k ∞ ≤ k m X i =1 v i,k f ′ i ( ˆ x ) δ x k + ζ k k 1 . Since k v k k ∞ < N and k v k k 1 → 0 , w e also ﬁ nd that k v k k 2 → 0 . Applying the Cauch y-Sc hw artz inequalit y to (133), from (132) w e get (ii).  A cknowledgments The authors thank Professor Helm ut Maurer for his useful remarks . QUADRA TIC ORDER CONDITIONS F OR BANG-SINGULAR EXTREMALS 39 Referen ces [1] A. A. Agrachev and R. V. Gamkrelidze, Se c ond or der optimality principle f or a time- optimal pr oblem , Math. U SSR, Sb ornik, 100 , (1976). [2] A. A. Agrachev and Y. L. Sachk o v, “Control theory from the geometric viewp oint,” vol ume 87 of Encyclopaedia of Mathematical S ciences, S p ringer-V erlag, Berlin, 2004. [3] A. A. 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Quadratic order conditions for bang-singular extremals

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