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1 Mah-Ne.Ru All Russian mahemaical poral Aleksei S. Rodin, On he srucure of singular se of a piecewise smooh minimax soluion of Hamilon-Jacobi-Bellman equaion, Ural Mah. J., 2016, Volume 2, Issue 1, DOI: hps://doi.org/ /umj Use of he all-russian mahemaical poral Mah-Ne.Ru implies ha you have read and agreed o hese erms of use hp:// Download deails: IP: Augus 17, 2018, 08:12:19

2 URAL MATHEMATICAL JOURNAL, Vol. 2, No. 1, 2016 ON THE STRUCTURE OF SINGULAR SET OF A PIECEWISE SMOOTH MINIMAX SOLUTION OF HAMILTON JACOBI BELLMAN EQUATION 1 Aleksei S. Rodin Krasovskii Insiue of Mahemaics and Mechanics, Ural Branch of he Russian Academy of Sciences; Ural Federal Universiy, Ekaerinburg, Russia alexey.rodin.ekb@gmail.com Absrac: The properies of a minimax piecewise smooh soluion of he Hamilon Jacobi Bellman equaion are sudied. We ge a generalizaion of he nesessary and sufficien condiions for he poins of nondiffereniabiliy (singulariy) of he minimax soluion and he Rankine Hugonio condiion. We describe he dimensions of smooh manifolds conaining in he singular se of he piecewise smooh soluion in erms of sae characerisics crossing on his singular se. New srucural properies of he singular se are obained for he case of he Hamilonian depending only on he impulse variable. Key words: Hamilon Jacobi Bellman equaion, Minimax soluion, Singular se, Piecewise smooh soluion, Tangen subspace, Rankine Hugonio condiion. Inroducion As is known [1 3], he firs-order parial differenial equaions of he Hamilon Jacobi Bellman ype are associaed wih problems of opimal conrol heory. In he presen paper, we sudy he properies of he generalized (minimax) soluion of he Hamilon Jacobi Bellman equaion (HJBE) proposed by A.I. Subboin. Necessary and sufficien condiions for a poin belonging o he singular se of he minimax soluion, i. e., o he se of poins of nondiffereniabiliy, were obained by E.A. Kolpakova [4, 5]. These resuls are developed in he presen paper. We sudy properies of he singular se of a minimax piecewise smooh soluion and esablish connecions beween he dimension of singular submanifolds and he sae characerisics ha come o hese submanifolds. We also obain he connecion beween he srucure of he Hamilonian and he srucure of he singular se in he case when he Hamilonian depends only on he impulse variable. One of he close works in his area is he book by A.A. Melikyan [6], where parial differenial equaions of he firs order. Where considered in he class of coninuously differeniable and piecewise smooh inpu daa. His work is mainly devoed o he issue of developmen of he mehod of characerisics and is furher applicaion o consrucions of soluions in he following cases: a) he generalized viscosiy soluion is no smooh and hen he Hamilonian is smooh or non-smooh funcion; b) he soluion is smooh, bu he Hamilonian is non smooh. The Poisson brackes are he main ools in he sudy [6]. Anoher work deserves special aenion. I is he monograph P. Cannarsa and C. Sinesari [7]. The auhors sudy soluions of he Hamilon Jacobi Bellman equaion in he class of semiconvex or semiconcave funcions. This soluions have he bounded second derivaives along any direcion. Singular se has a simple srucure in he case when a soluion is in he class of semiconvex or semiconcave funcions. The class of semiconvex or semiconcave funcions is more narrow hen he class of piecewise smooh funcions in he presening paper. 1 This work was suppored by he Russian Foundaion for Basic Research (projec no ) and by he Program of he Presidium of he Russian Academy of Sciences Mahemaical Problems of Conemporary Conrol Theory (projec no ).

3 Srucure of Singular Se Piecewise smooh soluion of Hamilon Jacobi Bellman equaion and is singular se 1.1. Problem saemen Consider he Cauchy boundary value problem for he Hamilon Jacobi Bellman equaion ϕ(, x) where [0, T ], x R n, and D x ϕ(, x) = + H(, x, D x ϕ(, x)) = 0, ϕ(t, x) = σ(x), (1.1) ( ϕ(, x) ϕ(, x) ϕ(, x) ),,..., = s. x 1 x 2 x n Define Π T = {(, x) : [0, T ], x R n }, he symbol in Π T denoe he inerior of he se Π T. We invesigae problem (1.1) under he following assumpions: (A1) he funcion H(, x, s) is coninuously differeniable wih respec o he variables, x, s and is concave wih respec o he variable s; (A2) he funcions D s H(, x, s), D x H(, x, s) are Lipschiz coninuous on he variables x and s, here exis consans L 1 > 0, L 2 > 0 such ha: D s H(, x, s ) D s H(, x, s ) L 1 ( x x + s s ), D x H(, x, s ) D x H(, x, s ) L 2 ( x x + s s ) for any (, x ), (, x ) Π T and for any s, s R n ; (A3) he funcion σ(x) is coninuously differeniable; (A4) here exis α > 0 and β > 0 such ha D x H(, x, s) α(1 + x + s ), D s H(, x, s) β(1 + x + s ) for any poin (, x, s) Π T R n. Here, he symbol denoes he Euclidean norm in R n. The aim of his paper is o sudy he srucure of a soluion ϕ( ) o problem (1.1) Generalized soluion o problem (1.1) Under he above assumpions, a classical soluion ϕ( ) o problem (1.1) may exis only locally in a neighborhood of he boundary manifold C T = {(, x, z): = T, x = ξ, z = σ(ξ); ξ R n }. This soluion ϕ( ) can be consruced using he Cauchy mehod of characerisics [8]. Le us wrie he characerisic sysem wih he boundary condiions a = T for problem (1.1): x = D s H(, x, s), s = D x H(, x, s), z = s, D s H(, x, s) H(, x, s), (1.2) x(t, ξ) = ξ, s(t, ξ) = D x σ(ξ), z(t, ξ) = σ(ξ) ξ R n. (1.3) The symbol, denoes he inner produc. The soluions x, s, and z are called, respecively, he sae, impulse, and cos of characerisics of he Hamilon Jacobi Bellman equaion (1.1). We noe ha, under condiions (A1) (A4), for any ξ R n a soluion of he characerisic sysem exiss, is unique, and can be exended o he inerval [0, T ]. According o he Cauchy mehod [8], we have he formulas x = x(, ξ), ϕ(, x) = z(, ξ) and D x ϕ(, x) = s(, ξ), [0, T ], ξ R n x(, ξ), if he Jacobian is no equal o zero. (, ξ)

4 60 Aleksei S. Rodin In wha follows, we consider nonclassical, nonsmooh soluions o problem (1.1). We apply he following generalizaion of he noion of differeniabiliy of a funcion [9]; his generalizaion is a useful ool of nonsmooh analysis. Definiion 1. The superdifferenial D + ϕ( 0, x 0 ) of a funcion ϕ( ): Π T R a a poin ( 0, x 0 ) is defined as he following se { D + ϕ( 0, x 0 ) = co (α, s) R n+1 : ϕ( 0 +, x 0 + x) ϕ( 0, x 0 ) (α, s), (, x) } lim sup 0. 0, x 0 + x The superdifferenial of a funcion ϕ( ) a he poins of is differeniabiliy consiss of he unique elemen equal o he gradien of his funcion. We recall (see [4, 5]) he definiion of a generalized soluion o problem (1.1). Definiion 2. A generalized soluion o problem (1.1) is a locally Lipschiz superdiffereniable funcion Π T (, x) ϕ(, x) R such ha, for any poin ( 0, x 0 ) Π T, here exis ξ 0 R n and soluions x(, ξ 0 ), s(, ξ 0 ), and z(, ξ 0 ) of sysem (1.2), (1.3) saisfying he condiion x( 0, ξ 0 ) = x 0, z( 0, ξ 0 ) = ϕ( 0, x 0 ), z(, ξ 0 ) = ϕ(, x(, ξ 0 )) [ 0, T ]. A superdifferenial funcion is he funcion ϕ( ) : Π T R such ha D + ϕ(, x) for any (, x) in Π T. The following asserion on he connecion of Definiion 2 wih he definiions of he minimax soluion and he viscosiy soluion is a consequence of resuls in [4, 5, 10 12]. Proposiion 1. If condiions (A1) (A4) are saisfied o problem (1.1), hen here exiss a unique generalized soluion o problem (1.1) in he sense of Definiion 2. In addiion, Definiion 2 is equivalen o he definiions of minimax soluion and viscosiy soluion o problem (1.1) Singular se Le us recall he definiion of he singular se of a generalized soluion ϕ( ) o problem (1.1). Definiion 3. The singular se Q of a generalized soluion ϕ( ) o problem (1.1) is he se of poins (, x) Π T where he funcion ϕ is no differeniable. According o [4, 5], he following asserions hold. Proposiion 2. Le condiions (A1) (A4) be saisfied for problem (1.1). Then (, x) Q if and only if here exis ξ 1, ξ 2 R n, ξ 1 ξ 2, such ha x(, ξ 1 ) = x(, ξ 2 ) = x, z(, ξ 1 ) = z(, ξ 2 ) = ϕ(, x), s(, ξ 1 ) s(, ξ 2 ), where x(, ξ i ), s(, ξ i ), and z(, ξ i ), i = 1, 2, are soluions of he characerisic sysem (1.2), (1.3). Proposiion 3. If he singular se Q conains he curve given by a coninuously differeniable funcion x(), 0 < 0 < T, hen s(, ξ 1 ) s(, ξ 2 ), dx() = H(, x(), s(, ξ 1 )) H(, x(), s(, ξ 2 )) ( 0, T ]. d This relaion generalizes he known Rankine Hugonio condiion o he case of he n-dimensional sae variable x.

5 1.4. Class of piecewise smooh funcions Srucure of Singular Se 61 In he presen paper, we consider generalized soluions ϕ( ) o problem (1.1) in he class of piecewise smooh funcions (see, for insance, [10]). Definiion 4. A funcion ϕ( ) : Π T R is called piecewise smooh in Π T if (1) The domain of his funcion Π T has he following srucure: in Π T = i I M i, M i M j = for i, j I, i j, where I = {1, 2,..., N}, M i are differeniable submanifolds in Π T. (2) The resricion of a piecewise smooh funcion ϕ( ) o M j, j J, is a coninuously differeniable funcion, where J := {i I : M i is an (n + 1)-dimensional manifold}, M j is he closure of he se M j. (3) For any i I, ( 1, x 1 ), ( 2, x 2 ) M i, he condiion J( 1, x 1 ) = J( 2, x 2 ) holds, where J(, x) := {j J : x M j }. Le us explain Definiion 4. The manifolds M j, j J I, of dimension n + 1 are open, and j J M j = Π T. All he remaining manifolds M i, i I \ J, of dimension less hen n + 1 belong o he boundary of he closure of (n + 1)-dimensional manifolds. In addiion, he following propery holds: J( 1, x 1 ) = J( 2, x 2 ) for any poins ( 1, x 1 ) and ( 2, x 2 ) belonging o he same manifold. Therefore, for any poin (, x) M i, i I, we have (, x) M j1... M jk for j 1,..., j k J(, x) and (, x) / M i for i J \ J(, x). 2. Characerisics and dimension of a singular manifold 2.1. Srucure of a singular manifold Le us consider a minimax soluion ϕ( ) o problem (1.1) in he class of piecewise smooh funcions. We fix a manifold M i, i I, of dimension n + 1 k, where k 1, n, and denoe i by M [k] o simplify he furher presenaion. Le L [k] (, x) be he angen subspace o he manifold M [k] a he poin (, x). We call he se of vecors orhogonal o vecors from he angen subspace a poin (, x) as he normal subspace a poin (, x). The projecion of he superdifferenial of he funcion ϕ( ) o he normal subspace a he poin (, x), is denoed as S + [k] (, x) := { q + R n+1 : p L [k] (, x), p + q + D + ϕ(, x), q +, (1, f) = 0 (1, f) L [k] (, x) }. The normal subspace a poin (, x) is he subspace of he minimal dimenion conaining he se S + [k](, x). Fix a poin (, x) Q. The symbol Index (, x) denoes he se consising wo or more parameers ξ R n, such ha for any pairs ξ, ξ Index (, x) he following condiions are valid: x(, ξ ) = x(, ξ ) = x, z(, ξ ) = z(, ξ ) = ϕ(, x), s(, ξ ) s(, ξ ), ξ ξ. (2.1) According o Asserion 2, his se is nonempy for all (, x) Q. Le us fix a poin (, x) M [k] Q.

6 62 Aleksei S. Rodin Lemma 1. If he superdifferenial D + ϕ(, x) of a piecewise smooh minimax soluion ϕ( ) o problem (1.1) a a poin (, x) M [k] Π T consiss of more hen one elemen, hen he difference of wo elemens d and d of his superdifferenial belongs o he normal subspace o manifold a he poin (, x). If he dimension of he normal subspace is k, where 1 k n, hen here exis k linearly independen vecors of he form d d, where d, d D + ϕ(, x). P r o o f. The proof follows from he properies of a piecewise smooh minimax soluion ϕ( ) o problem (1.1) [10]. Any elemen d of he superdifferenial D + ϕ(, x) can be represened as he sum p + q +, where p belongs o he angen subspace o he singular se a he poin (, x) and q + belongs o he normal subspace a he same poin. I was shown in [10] ha he projecion of he superdifferenial D + ϕ(, x) o he angen subspace is a singleon. As is known (see [4]), he superdifferenial of a locally Lipschiz minimax soluion ϕ( ) o problem (1.1) is a closed bounded se and has he form D + ϕ(, x) := co{d(ξ ) R n+1 : ξ Index (, x)}, d(ξ ) = ( H(, x(, ξ ), s(, ξ )), s(, ξ )). (2.2) Le he angen subspace o he singular se a a poin (, x) have dimension n + 1 k. We consider he vecors d(ξ ) d(ξ1 ), where ξ, ξ1 Index (, x) and d(ξ ) d(ξ1 ) = q+ (ξ ) q + (ξ1 ). The symbol q + (ξ ) denoes he projecion of d(ξ ) o he normal subspace a he poin (, x). Since q + (ξ ), q + (ξ1 ) S+ [k] (, x), ξ, ξ1 Index (, x), we find ha he vecors q+ (ξ ) q + (ξ1 ), ξ, ξ1 Index (, x), also lie in he normal subspace of dimension k. We will show ha here exis no less han k linearly independen differences following form q + (ξ i ) q + (ξ 1), ξ i, ξ 1 Index (, x), i 2, k + 1. We show ha here is no elemen q 0 saisfing he following condiion: q is orogonal o he se S + [k](, x) and q belongs o he normal subspace. We prove he fac by reducio ad absurdum. Le he convex se S + [k](, x) has dimenion k l, 0 < l k [9]. Then here is an elemen q such ha q is orogonal o he se S + [k](, x) and q belongs o he normal subspace. The equaliy p, q = 0 follows from p L [k] (, x), q belongs o he normal subspace, q + (ξ ), q = 0, ξ Index (, x), q + (ξ ) S + [k](, x), q is orogonal o he se S + [k] (, x). This implies ha p + q + (ξ ), q = 0, for any p L [k] (, x), ξ Index (, x), q + (ξ ) S + [k](, x), as any supergradien may be represened as he sum p + q + (ξ ) [10], han q is orogonal o any supergradien. Therefore q belongs in inersecion of hyperplanes whose normals are he supergradiens. Consecuenly, q belongs o angen subspace, bu q belongs o normal subspace and q 0. This is a conradicion. Theorem 1. Le condiions (A1) (A4) be saisfied for problem (1.1), and le (, x) Q. Then (, x) M [k], where dim M [k] = n + 1 k, k 1, n, if and only if here exis soluions x(, ξi ), s(, ξi ), and z(, ξ i ), of sysem (1.2), (1.3), ξ i Index (, x), i 1, k + 1, such ha properies (2.1) hold and he k (n + 1)-marix (H 2 H 1 ) s 1 2 s1 1 s 2 2 s s n 2 sn 1 D = (H 3 H 1 ) s 1 3 s1 1 s 2 3 s s n 3 sn (2.3) (H k+1 H 1 ) s 1 k+1 s1 1 s 2 k+1 s s n k+1 sn 1 has he rank equal o k. Here (s 1 i, s2 i,..., sn i ) = s(, ξ i) and H i = H(, x(, ξ i ), s(, ξ i )). If one adds any row of he form ( (H k+2 H 1 ) s 1 k+2 s1 1 s 2 k+2 s s n k+2 sn 1 )

7 Srucure of Singular Se 63 where (s 1 k+1, s2 k+1,..., sn k+1 ) = s(, ξ k+1), H k+1 = H(, x(, ξ k+1 ), s(, ξ k+1 )), ξ k+1 Index (, x), o he marix of D, hen he rank of he received (k + 1) (n + 1)-marix equal o k. P r o o f. Necessiy. Le (, x) M [k] Q. We noe ha he dimension of he angen subspace L [k] (, x) coincides wih he dimension of he manifold M [k]. Since dim L [k] (, x) = n+1 k, we conclude ha he dimension of he normal subspace S + [k](, x) is n + 1 (n + 1 k) = k. I follows from Lemma 1 ha here exis elemens q + (ξi ) S+ [k] (, x), ξ i Index (, x) i 1, k + 1, such ha he vecors q + (ξi ) q+ (ξ1 ), ξ i, ξ 1 Index (, x) i 2, k + 1, are linearly independen. We denoe by Basic [k] (, x) he following se {{ξ 1, ξ 2,..., ξ k+1 } : ξ i Index (, x), i 1, k + 1, q + (ξ i ) q + (, ξ 1) linearly independen}. Consequenly, he marix D consising of rows of he form q + (ξi ) q+ (ξ1 ), ξ i, ξ 1 Index (, x), i 2, k + 1, has he rank k. If we add a row of he form q + (ξ ) q + (ξ1 ), ξ, ξ1 Index (, x), o he marix consising of rows of he form q + (ξi ) q+ (ξ 1 ), {ξ1, ξ 2,..., ξ k+1 } Basic [k](, x), hen he rank of he marix remains he same. Sufficiency. Assume ha he rank of he marix (H 2 H 1 ) s 1 2 s1 1 s 2 2 s s n 2 sn 1 D = (H 3 H 1 ) s 1 3 s1 1 s 2 3 s s n 3 sn (H k+2 H 1 ) s 1 k+2 s1 1 s 2 k+2 s s n i+1 sn 1 is k, for any parameer ξk+2 Index (, x). The rows of his marix are elemens d(ξ i ) d(ξ 1 ), where d(ξi ), d(ξ 1 ) D+ ϕ(, x), i 2, k + 1. In addiion, hey can be considered as normals o hyperplanes of dimension n. Moreover, k of hese normals are linearly independen. From Lemma 1, he vecors d(ξi ) d(ξ 1 ) = q+ (ξi ) q+ (ξ1 ), {ξ 1, ξ 2,..., ξ k+1 } Basic [k](, x), belong o he normal subspace a poin (, x). The vecors d(ξi ) d(ξ 1 ), {ξ 1, ξ 2,..., ξ k+1 } Basic [k] (, x) form a basic of he normal subspace. I implies ha he dimension of he normal subspace a he poin (, x) is k and he dimension of he angen subspace is n + 1 k. Hence, (, x) M [k]. Remark 1. According o [5], he inclusions (, x) Q and d(ξ i ) D+ ϕ(, x), {ξ 1, ξ 2,..., ξ k+1 } Basic [k](, x), imply ha he following Rankine Hugonio condiion holds for a curve x( ) lying on M [k] : s(, ξ i ) s(, ξ 1), dx() d {ξ 1, ξ 2,..., ξ k+1 } Basic [k](, x). = H(, x(), s(, ξi )) H(, x(), s(, ξ1)), (2.4) We rewrie his condiion in he form ( ( H(, x(), s(, ξi )) H(, x(), s(, ξ1)) ) ), s(, ξi ) s(, ξ1), (1, ẋ()) = q + (ξ i ) q + (ξ 1), (1, ẋ()) = 0, (2.5) {ξ1, ξ 2,..., ξ k+1 } Basic [k](, x). We see from condiion (2.5) ha he vecor (1, ẋ()) is orhogonal o all he vecors q + (ξi ) q+ (ξ1 ), ξ i, ξ 1 Basic [k](, x), i 2, k + 1, ha form a basis in he normal subspace. Hence, we conclude ha he vecor (1, ẋ()) belongs o he angen subspace L [k] (, x). The Theorem 1 is proven.

8 64 Aleksei S. Rodin 2.2. Properies of he superdifferenial Theorem 2. Le condiions (A1) (A4) be saisfied for problem (1.1). Le (, x) Q and (, x) M [k], where dim M [k] = n + 1 k and 1 k n. Assume ha he Hamilonian H = H(s) is concave in variable s. For any characerisics x(, ξ i ), s(, ξ i ), z(, ξ i ), {ξ1, ξ 2,..., ξ k+1 } Basic [k] (, x) such ha he k (n + 1)-marix D of he form (2.3) has he rank k, here is no characerisic x(, ξ k+2 ), s(, ξ k+2 ), z(, ξ k+2 ), ξ k+2 Index (, x) saysfing he condiion s(, ξ k+2 ) = α i s(, ξ i ), α i 0, α i = 1. P r o o f. Le us inroduce he convenien noaion ( q i,j (, x) = q + (ξ i ) q + (ξ j ) = d(ξ i ) d(ξ j ) = ( H( s(, ξ i )) H( s(, ξ j )) ) ), s(, ξ i ) s(, ξ j ), (2.6) ξ i, ξ j Index (, x). Assume ha he saemen of Theorem 2 does no hold and here exiss a characerisic x(, ξ k+2 ), z(, ξ k+2 ), s(, ξ k+2 ), ξ k+2 Index (, x), saisfying condiion (2.1) and such ha s(, ξ k+2 ) = α i s(, ξ i ), α i 0, α i = 1. (2.7) In view of Theorem 1 and he inclusion (, x) M [k], he rank of he (k + 1) (n + 1)-marix D obained by adding o he marix D he row q k+2,1 (, x) is equal o k. The added row q k+2,1 (, x) is a linear combinaion of rows of he marix D, i.e. exis b i R, i 1, k, such ha q k+2,1 (, x) = b i q i+1,1 (, x). (2.8) Relaions (2.6) and (2.8) imply ha d(ξ k+2 ) = ( 1 b i )d(ξ 1 ) + b i d(ξ i+1 ). (2.9) Since equaliy (2.9) is applicable o all he componens of he vecor d(ξ k+2 ), we rewrie equaliy (2.9) in he form of he following wo equaliies: H( s(, ξ k+2 )) = s(, ξ k+2 ) = ( 1 b i )H( s(, ξ 1 )) + ( 1 I follows from (2.10) and (2.11) ha (( H 1 b i ) s(, ξ 1 ) + b i ) s(, ξ 1 ) + ) ( b i s(, ξ i+1 ) = 1 b i H( s(, ξ i+1 )), (2.10) b i s(, ξ i+1 ). (2.11) b i )H( s(, ξ 1 )) + b i H( s(, ξ i+1 )). We also noe ha he sum of he coefficiens a H( s(, ξ i )) and s(, ξ i ) is equal o 1, i 1, k + 1. There exis wo represenaions for s(, ξ k+2 ): formula (2.7) from he above assumpion condiion of he problem ha s(, ξ k+2 ) is a convex combinaion of s(, ξ i ), i 1, k + 1, and formula (2.11) obained from he linear of he row q k+2,1 (, x) releive rows of he marix D.

9 Subracing (2.7) from (2.11), we ge 0 = (( 1 We subrac and add he erm Srucure of Singular Se 65 b i ) α 1 ) s(, ξ 1 ) + (b i α i+1 ) s(, ξ i+1 ). (2.12) (b i α i+1 ) s(, ξ 1 ) o he righ-hand side of (2.12). Then, grouping he erms, we ge 0 = ( 1 α 1 ) (b i b i + α i+1 ) s(, ξ 1 ) + (b i α i+1 ) ( s(, ξ i+1 ) s(, ξ 1 ) ). Hence, aking ino accoun ha he coefficien a s(, ξ 1 ) is equal o zero, we obain 0 = (b i α i+1 ) ( s(, ξ i+1 ) s(, ξ 1 ) ). (2.13) If he differences s(, ξ i+1 ) s(, ξ 1 ), i 1, k, are linearly independen, hen equaliy (2.13) is equivalen o he equaliies b i = α i+1, i 1, k. In his case, equaliy (2.13) and he condiion k+1 α i = 1 imply α 1 = (1 k b i). Le b 1 α 2 and vecor s(, ξ 2 ) s(, ξ 1 ) be a linear combinaion of he differences s(, ξ i+1 ) s(, ξ 1 ), i 2, k. Le muliply scalarly boh sides of ideniy (2.13) by ẋ 0, where (1, ẋ) L [k] (, x). Taking ino accoun he Rankine Hugonio condiion (2.4), we ge 0 = (b i α i+1 ) ( H( s(, ξ 1 )) H( s(, ξ i+1 )) ). This implies ha he difference H(, ξ 1 ) H(, ξ 2 ) is a linear combinaion of he differences H( s(, ξ 1 )) H( s(, ξ i+1 )), i 2, k, and he rank of he marix D is equal k 1. However, his is no possible, because of he assumpion of he Theorem 2, ha he rank of he k (n + 1)-marix D is equal k. The Rankine Hugonio condiion (2.4) for ẋ = 0 imply, ha H(, ξ 1 ) H(, ξ i+1 ) = 0, i 1, k. Therefore, he differences s(, ξ i+1 ) s(, ξ 1 ), i 1, k, are linearly independen. Then, as menioned above, b i = α i+1, i 1, k, and α 1 = 1 k b i. I follows from (2.10) and (2.11), ha H( s(, ξ k+2 )) = α i H( s(, ξ i )), s(, ξ k+2 ) = α i s(, ξ i ), (2.14) α i = 1, α i 0, i 1, k + 1. Le s consider he simplex S of dimension of k spanned by he poins ( H( s(, ξ i )), s(, ξ i )), i 1, k + 1. Convexiy of funcion s H(s) and (2.14) imply ha he simplex S lies on he graph of funcion s H(s). Smoohness of funcion of H(s) in variable s (condiion A1), implies ha he supporing hyperplane o he hypograph of H( ) a any poin ( H( s(, ξ i )), s(, ξ i )) S, i 1, k + 1,

10 66 Aleksei S. Rodin is unique and conains he simplex S. We will denoe he normal o he supporing hyperplane conaining simplex S by ( 1, N) R n+1, where D s H( s(, ξ i )) = N, i 1, k + 2. (2.15) Since H = H(s), we can obained from (1.2) and (1.3) he relaions s(, ξ i ) = D x H( s(, ξ i )) = 0, ξ i Index(, x), i 1, k + 2, T. Therefore, s(, ξ i ) D x σ(ξ i ). I follows from (2.1) and (2.15), ha T x = x() = ξ i H T T s (D xσ(ξ i ))dτ = ξ i Ndτ = ξ j Ndτ, (2.16) ξ i, x j Index (, x), i, j 1, k + 2. Relaion (2.16) implies ha ξ i = ξ j where i, j 1, k + 2 wha he conradics condiion (2.1). The Theorem 2 is proven. Remark 2. I follows from he saemen of Theorem 2 in case H=H(s) ha he poins of he form ( H( s(ξ )), s(ξ )), ξ Index (, x), (, x) Q, (2.17) are corner poins of he convex se D + ϕ(, x). Really, le (, x) M [k] Q. I follows from (2.2) and Theorem 1, ha he superdifferenial is convex, closed se and i lies in he subspace of dimension k. According o he Caraheodory heorem [9] any elemen of D + u(, x) can be presened as a convex combinaion of no more hen (k + 1) supergradiens of he form (2.17) and such ha he differences d(ξ i ) d(ξ 1 ) = ( H( s(ξ i )) + H( s(ξ 1 )), s(ξ i ) s(ξ 1 )), where ξ i, ξ 1 Index (, x), i 2, k + 1, are linearly independen. If a supergradien ( H( s(ξ )), s(ξ )) D + u(, x) would be presened in he form ( H( s(ξ )), s(ξ )) = α i ( H( s(ξ i )), s(ξ i )) ξ i Index (, x), i 1, k + 1, where α i 0, k+1 α i = 1, hen his componen s(ξ ) should be also presened as he convex combinaion s(ξ i ) ξ i Index (, x), i 1, k + 1. Tha conradics Theorem 2. Therefore, he saemen of remark 2 is rue. Corollary 1. If condiions (A1) (A4) are saisfied, (, x) Q and H = H(s), hen he relaion s(ξ ) s(ξ ), D s H( s(ξ )) = H( s(ξ )) H( s(ξ )), is valid for any ξ, ξ Index (, x) and ξ ξ. We will prove he corollary by reducio ad absurdum. Le here exis ξ, ξ Index (, x), ξ ξ, saisfied he relaions The equaliy can be rewrien in he form s(ξ ) s(ξ ), D s H( s(ξ )) = H( s(ξ )) H( s(ξ )). ( s(ξ ) s(ξ ), H( s(ξ )) H( s(ξ ))), (D s H( s(ξ )), 1) = 0. (2.18)

11 Srucure of Singular Se 67 Noe ha he vecor (D s H( s(ξ )), 1) is he normal o he supporing hyperplane Γ o he hypograph of funcion s H(s) a poin ( s(ξ ), H( s(ξ ))). The difference ( s(ξ ) s(ξ ), H( s(ξ )) H( s(ξ ))) lies in he supporing hyperplane Γ following condiion (2.18). The poin ( s(ξ ), H( s(ξ ))) belongs o he graph of he concave funcion s H(s) and lies in he hyperplane Γ. I follows ha Γ is he supporing hyperplane o he hypograph of H( ) a his poin. As H( ) is coninuously differeniable, hen Γ is he angen hyperplane o he hypograph of H( ) a poin ( s(ξ ), H( s(ξ ))). Consequenly, (D s H( s(ξ )), 1) = (D s H( s(ξ )), 1). We remain ha D s H( s(ξ )) = D s H( s(ξ )) = N. As he Hamilonian has he form H = H(s), hen s = D x H( s) = 0 (1.2), (1.3). I implies ha s(, ξ ) D x σ(ξ ), s(, ξ ) D x σ(ξ ). Using condiion (2.1), we ge ha saes of characerisics x(, ξ) wih parameers ξ and ξ saisfy he relaion T x = x() = ξ H T s (D xσ(ξ ))dτ = ξ This equaliy can be rewrien in he following form T T x = x() = ξ Ndτ = ξ Ndτ. This equaliy imply ha ξ = ξ. I conradics he assumpion ξ ξ. 3. Conclusion H s (D xσ(ξ ))dτ. In his paper resuls presened in [13] are modified and developed. New properies of superdifferenials of a piece-smooh minimax soluion of he HJBE and characerisics of he HJBE on he singular se are obained and discussed. REFERENCES 1. Bellman R. Dynamic programming. Princeon: Princeon Univ. Press, p. 2. Ponryagin L.S., Bolyanskii V.G., Gamkrelidze R.V. and Mishchenko E.F. The mahemaical heory of opimal processes. New York: Wiley, Krasovskii N.N. Theory of moion conrol. Moscow: Nauka, p. [in Russian]. 4. Subboina N.N., Kolpakova E.A., Tokmansev T.B. and Shagalova L.G. The mehod of characerisics for Hamilon Jacobi Bellman equaions. Ekaerinburg: RIO UrO RAN, p. [in Russian]. 5. Kolpakova E.A. The generalized mehod of characerisics in he heory of Hamilon Jacobi equaions and conservaion laws // Trudy Ins. Ma. Mekh. UrO RAN Vol. 16, no. 5, P [in Russian] 6. Melikyan A.A. Generalized characerisics of firs order PDEs: applicaions in opimal conrol and differenial games. Boson: Birkhäuser, Cannarsa P., Sinesrari C. Semiconcave funcions, Hamilon Jacobi equaions and opimal conrol, p. 8. Perovskii I.G. Lecures on he heory of ordinary differenial equaions. Moscow: Mosk. Gos. Univ., p. [in Russian]. 9. Rockafellar R. Convex analysis. Princeon: Princeon Univ. Press, p. 10. Subboin A.I. Generalized soluions of firs order PDEs. The dynamical opimizaion perspecive. New York: Birkhäuser, p. DOI: / Crandall M.G. and Lions P.L. Viscosiy soluions of Hamilon Jacobi equaions // Trans. Amer. Mah. Soc Vol. 277, no. 1. P

12 68 Aleksei S. Rodin 12. Subboina N.N. and Kolpakova E.A. On he srucure of locally Lipschiz minimax soluions of he Hamilon Jacobi Bellman equaion in erms of classical characerisics // Proc. Seklov Ins. Mah Vol. 268, Suppl. 1, P. S222 S Rodin A.S. On he Srucure of he Singular Se of a Piecewise Smooh Minimax Soluion of he Hamilon Jacobi Bellman Equaion // Trudy Ins. Ma. Mekh. UrO RAN Vol. 21, no. 2, P [in Russian]