LIPSCHITZIAN REGULARITY OF MINIMIZERS FOR OPTIMAL CONTROL PROBLEMS WITH CONTROL-AFFINE DYNAMICS

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1 LIPSCHITZIAN REGULARITY OF MINIMIZERS FOR OPTIMAL CONTROL PROBLEMS WITH CONTROL-AFFINE DYNAMICS ANDREI V. SARYCHEV AND DELFIM F. MARADO TORRES Abstrct. We study Lgrnge Problem of Optiml Control with functionl b L (t, x (t, u (t dt nd control ffine dynmics ẋ = f (t, x+ g (t, x u nd ( priori unconstrined control u IR m. We obtin conditions under which the minimizing controls of the problem re bounded the fct which is crucil for pplicbility of mny necessry optimlity conditions, like, for exmple, Pontrygin Mximum Principle. As corollry we obtin conditions for Lipschitzin regulrity of minimizers of the Bsic Problem of the Clculus of Vritions nd of the Problem of the Clculus of Vritions with higher-order derivtives. 1. Introduction Under stndrd hypotheses of the Tonelli existence theory in the Clculus of Vritions, the existence of minimizers is gurnteed in the clss of bsolutely continuous functions possibly with unbounded derivtive. As it is known, in such cses the optimlity conditions like the Euler-Lgrnge eqution my fil. Therefore it is importnt to try to obtin Lipschitzin regulrity conditions under which the minimizers re Lipschitzin. Min prt of the results obtined (strting with those of Leonid Tonelli refer to the Bsic Problem of the Clculus of Vritions (see [1, 2, 6, 8, 9, 10, 14, 17]. Less is known for the problems with high-order derivtives ([11]. For the Lgrnge problem nd for the problems of optiml control, the regulrity results re rrity. We re only wre of progress due to F.H. Clrke & R.B. Vinter ([12] for problems ssocited with liner, utonomous (i.e. time-invrint dynmics (see lso [4]. For this prticulr clss of problems, regulrity results re obtined vi trnsformtion of the initil problem into problem of the Clculus of Vritions with higher order derivtives. In this pper we develop different pproch to estblishing Lipschitzin regulrity (boundedness of minimizing controls for the Lgrnge problems with 1991 Mthemtics Subject Clssifiction. 49J15, 49J30. Key words nd phrses. optiml control, clculus of vritions, Pontrygin mximum principle, boundedness of minimizers, nonliner control ffine systems, lipschitzin regulrity. Prtilly presented t the Interntionl Conference dedicted to the 90th Anniversry of L.S. Pontrygin, Moscow, Russi, September

2 2 ANDREI V. SARYCHEV AND DELFIM F. MARADO TORRES functionl b L(t, x(t, u (t dt nd control-ffine non-utonomous dynmics: ẋ = f (t, x + g (t, x u. This clss of systems ppers in wide rnge of problems relevnt to mechnics, sub-riemnnin geometry, etc. We mke use of n pproch developed by R.V. Gmkrelidze ([13, Chp. 8]: reduction of the Lgrnge problem to n utonomous time-optiml control problem, with subsequent compctifiction of the set of control vlues. If the Pontrygin Mximum Principle is pplicble to the compctified problem, one cn use its formultion to derive conditions for boundedness of minimizers of the originl Lgrnge problem nd to determine the bounds for the mgnitudes of minimizing controls. The min result is Theorem 1 (Section 3. As its corollries we obtin (Section 4 results on Lipschitzin regulrity of minimizers in the Clculus of Vritions (see [18]. The uthors re grteful to A. A. Agrchev for stimulting discussions. 2. Preinries 2.1. Optiml Control Problems with Control-Affine Dynmics nd Unconstrined Controls. We study minim of the problem (P J [x(, u ( ] = b L(t, x(t, u (t dt min ẋ(t = f (t, x(t + g (t, x(t u (t,.e. on [, b], x( = x, x(b = x b, x( AC ([, b]; IR n, u ( L 1 ([, b] ; IR m. Here, b IR, < b; L : IR IR n IR m IR, f : IR IR n IR n nd g : IR IR n IR n m re given functions; m n; x, x b IR n re given; u = (u 1,..., u m IR m, g (t, x = ( g 1 (t, x,..., g m (t, x. The controls u ( re integrble. The bsolute continuous solution x( of the differentil eqution is stte trjectory corresponding to the control u (. We ssume tht f (, nd g (, re C 1 functions in IR 1+n The Pontrygin Mximum Principle for (P. The following 1- st-order necessry optimlity condition for the problem (P, is provided by the Pontrygin Mximum Principle. We use, to denote the usul inner product in IR n. Theorem If ( x(, ũ ( is minimizer of the problem ( (P nd the control ũ ( L m ([, b], then there exists nonzero pir ψ0, ψ (, where ψ 0 0 is constnt nd ψ ( is n bsolutely ( continuous vector-function with domin [, b], such tht the qudruple x(, ψ 0, ψ (, ũ ( stisfies: (i: the Hmiltonin system ẋ = H ψ, ψ = H x, with the Hmiltonin H = ψ 0 L(t, x, u+ ψ, f (t, x + ψ, g (t, x u ;

3 LIPSCHITZIAN REGULARITY FOR OPTIMAL CONTROL PROBLEMS 3 (ii: the mximlity condition ( ( H t, x(t, ψ 0, ψ (t, ũ (t = M t, x(t, ψ 0, ψ (t = = sup u IR m H ( t, x(t, ψ 0, ψ (t, u for lmost ll t [, b]. ( Definition 1. A qudruple x(, ψ 0, ψ (, ũ ( in which ψ 0, ψ ( re s in Theorem 2.2.1, ũ ( is integrble nd stisfies the conditions (i nd (ii of Theorem 2.2.1, is clled extreml for the( problem (P. The control ũ ( is clled n extreml control. An extreml x(, ψ 0, ψ (, ũ ( is clled norml if ψ 0 0 nd bnorml if ψ 0 = 0. We cll ( ũ( bnorml extreml control if it corresponds to n bnorml extreml x(, 0, ψ (, ũ ( The Pontrygin Mximum Principle for Autonomous Time Optiml Control Problems nd Constrined Controls. An utonomous time optiml problem is (2.1 T min subject to ẋ(t = F (x(t, u (t,.e. t [, T], x( AC ([, T] ; IR n, u ( L ([, T] ; U, x( = x, x(t = x T. Here F (, is continuous function on IR n+m, nd hs continuous prtil derivtives with respect to x. The control u (, defined on [, T], tkes its vlues in U IR m, nd is mesurble nd bounded function. The following first-order necessry condition the Pontrygin Mximum Principle holds for ny minimizer of the problem (detils cn be found, for exmple, in [13]. There re plenty of generliztions nd modifictions of the Pontrygin Mximum Principle. For exmple Pontrygin Mximum Principle under less restrictive ssumptions for smoothness of dt, cn be found in [5] or [16]. Theorem Let ( x(, ũ ( be solution of the time-optiml problem (2.1. Then there exists nonzero bsolutely continuous function ψ ( stisfying: the Hmiltonin system ẋ = H (x, ψ, u H (x, ψ, u, ψ =, ψ x with corresponding Hmiltonin H (x, ψ, u = ψ, F (x, u ;

4 4 ANDREI V. SARYCHEV AND DELFIM F. MARADO TORRES the mximlity condition ( ( H x(t, ψ (t, ũ (t = M = sup u U x(t, ψ (t { H for lmost ll t( [, T]; the equlity M x(t, ψ (t const Min result = ( x(t, ψ (t, u Theorem 1. Let L(,, C 1 ([, b] IR n IR m ; IR, f (, C 1 (IR IR n ; IR n, g (, C 1 ( IR IR n ; IR n m ; nd Under the hypotheses: ϕ (t, x, u = f (t, x + g (t, x u. (H1 full rnk condition: g (t, x hs rnk m for ll t [, b] nd x IR n ; (H2 coercivity: there exist function θ : IR IR such tht L(t, x, u θ ( u > ζ, for ll (t, x, u [, b] IR n IR m, nd r r + θ (r = 0; (H3 growth condition: there exist constnts γ, β, η nd µ, with γ > 0, β < 2 nd µ mx {β 2, 2}, such tht for ll t [, b], x IR n nd u IR m there holds ( L t + L x i + L ϕ t L t ϕ + L ϕ x i L x i ϕ u µ γ L β + η, i {1,..., n}; then ll minimizers ũ ( of the Lgrnge problem (P which re not bnorml extreml controls, re essentilly bounded on [, b]. Remrk 1. Recll tht if the dynmics is controllble (which is true for problems of the Clculus of Vritions treted in 4, ll extremls re norml. A priori minimizer ũ ( which is not essentilly bounded, my fil to stisfy the Pontrygin Mximum Principle nd therefore my cese to be n extreml. As fr s for essentilly bounded minimizers the Pontrygin Mximum Principle is vlid, nd unbounded minimizers ũ ( (if there re ny re ccording to the Theorem 1 bnorml extreml controls, then we obtin the following: Corollry 1. Under the conditions of the Theorem 1 ll minimizers of the Lgrnge problem (P stisfy the Pontrygin Mximum Principle. }

5 LIPSCHITZIAN REGULARITY FOR OPTIMAL CONTROL PROBLEMS 5 Remrk 2. We my impose stronger but techniclly simpler forms of ssumption (H3 in Theorem 1. Under these conditions Theorem 1 looses some generlity, but sometimes, for given problem, these conditions re esier to verify. For exmple they cn be: (in incresing order of simplicity nd in decresing order of generlity [L ( ϕ t + ϕ x i + ϕ ( L t + L x i ] u µ γ L β + η; L ( ϕ t + ϕ x i + ϕ ( L t + L x i u µ γ L β + η; ϕ t + ϕ x i + L t + L x i γ L β + η, β < 1. It is esy to see why (H3 follows from ny of these conditions. It is enough to notice tht nd tht 0 mx {β 2, 2}. L ϕ t L t ϕ + L ϕ x i L x i ϕ L ( ϕ t + ϕ x i + ϕ ( L t + L x i ; Remrk 3. The result of Theorem 1 dmits generliztion for Lgrnge problems with dynmics which is nonliner in control. It will be ddressed in forthcoming pper Proof of the Theorem. We begin with n elementry observtion. Remrk 4. It suffices to prove Theorem 1 in the specil cse where in (H2 we put ζ = 0. Indeed, minimiztion of b L(t, x(t, u (t dt under conditions in (P is equivlent to b (L(t, x(t, u (t ζ dt min since the difference of the integrnds is constnt. Remind tht everywhere below the nottion ϕ (t, x, u stnds for f (t, x+ g (t, x u Reduction to time optiml problem. The following ide of trnsforming the vritionl problem into time-optiml control problem nd subsequent compctifiction of the control set hs been used erlier by R.V. Gmkrelidze ([13, Chp. 8] to prove some existence results. We introduce new time vrible τ (t = t L(θ, x(θ, u(θ dθ, t [, b],

6 6 ANDREI V. SARYCHEV AND DELFIM F. MARADO TORRES which is strictly monotonous bsolutely continuous function of t, for ny pir (x(t, u (t stisfying ẋ(t = ϕ (t, x(t, u (t. Obviously τ (b = T coincides with the vlue of the functionl of the originl problem. As fr s dτ(t = L(t, x(t, u(t > 0, dt then τ ( dmits monotonous inverse function t( defined on [0, T], such tht dt dτ (τ = 1 L(t(τ, x(t(τ, u(t(τ. Notice tht the inverse function t (, is lso bsolutely continuous. Obviously (3.1 dx (t(τ dτ = dx (t(τ dt dt(τ dτ = ϕ (t(τ, x(t(τ, u (t(τ L(t(τ, x(t(τ, u(t(τ. Tking τ s new time vrible, considering t(τ nd z(τ = x(t(τ s components of stte trjectory nd v(τ = u(t(τ s the control, we cn trnsform the problem (P into the following form (3.2 (3.3 ṫ(τ = ż(τ = T min 1 L(t(τ, z(τ, v(τ ϕ (t(τ, z(τ, v(τ L(t(τ, z(τ, v(τ, (3.4 v : IR IR m t(0 =, t(t = b z(0 = x, z(t = x b Compctifiction of the spce of dmissible controls. So fr the new control vrible tkes its vlues in IR m nd priori control v(τ cn be unbounded. This unboundedness is kind of fictitious s fr s the set {( 1 ϕ (t, z, v, L(t, z, v L(t, z, v : v IR m }, is bounded under the hypotheses (H1 nd (H2. This set (for fixed (t, z IRxIR n is not closed, but becomes compct if we dd to it the point (0, 0 IRxIR n which corresponds to the infinite vlue of the control v IR m. The set {( E(t, z = {(0, 0} 1 ϕ (t, z, v, L(t, z, v L(t, z, v : v IR m }, cn be represented s homeomorphic imge of the m-dimensionl sphere S m IR m. This homeomorphism is defined in stndrd wy: we fix point ŵ S m, clled north pole, nd consider the stereogrphic projection (3.5 π : S m \ {ŵ} IR m.

7 LIPSCHITZIAN REGULARITY FOR OPTIMAL CONTROL PROBLEMS 7 Obviously π S m \{ŵ} (3.6 w is continuous nd π (w = +. Functions w ŵ 1 L(t, z, π(w nd w ϕ (t, z, π(w L(t, z, π(w re continuous on S m \ {ŵ}, since L(t, z, π(w > 0. Due to the hypotheses (H1 nd (H2 nd therefore (3.7 ϕ (t, z, v = 0 v + θ ( v ϕ (t, z, v = 0. v + L(t, z, v Hence, one cn extend the functions defined by (3.6 up to the functions φ t, z ( nd h t, z ( which re continuous on the entire sphere S m (on the compctified spce IR m : 1 φ t, z if w ŵ (w = φ(t, z, w = L(t, z, π(w, 0 if w = ŵ ϕ (t, z, π(w h t, z if w ŵ (w = h(t, z, w = L(t, z, π(w. 0 if w = ŵ Given (H1, the mp w ( φ t, z (w, h t, z (w, of S m onto E(t, z, is continuous nd one-to-one nd therefore homeomorphism since S m is compct. Thus, we hve come to the utonomous optiml control problem: (3.8 T min (3.9 { ṫ(τ = φ(t(τ, z(τ, w(τ ż(τ = h(t(τ, z(τ, w(τ (3.10 w : IR S m t(0 =, t(t = b z(0 = x, z(t = x b, with compct set S m of vlues of control prmeters. We c tht every pir (x(, u ( stisfying ẋ = f (t, x + g (t, x u corresponds to trjectory (t (τ, z (τ, w (τ of the system (3.9 with w(τ ŵ for lmost ll τ [ 0, T] nd the trnsfer time T for this ltter solution equl to the vlue J [x(, u ( ]: T = J [x(, u ( ] = b L(t, x(t, u(t dt.

8 8 ANDREI V. SARYCHEV AND DELFIM F. MARADO TORRES Indeed we my define (t (τ, z (τ, w (τ setting t (τ n inverse function to τ (t = t z(τ = x(t(τ, w(τ = π 1 (u(t(τ, 0 τ T = J [x(, u ( ], L(θ, x(θ, u(θ dθ, where π 1 ( is the mpping inverse to (3.5. Function z( is bsolutely continuous since it is composition of n bsolutely continuous function with nother strictly monotonous bsolutely continuous function t (τ. Function w( is mesurble becuse π 1 ( is continuous function, u( is mesurble nd t( is strictly monotonous bsolutely continuous function. We lredy know tht dt (τ dτ dz (τ dτ = = 1 L(t(τ, z(τ, π (w(τ, ϕ (t (τ, z(τ, π (w(τ L(t(τ, z(τ, π (w(τ, nd w(τ ŵ for lmost ll τ [ 0, T], since u ( hs finite vlues lmost everywhere nd t ( is strictly monotonous. We shll show now tht every solution of (3.9, with w(τ ŵ.e., results from this correspondence. Tking the bsolutely continuous function τ(t, t b, which is the inverse of the strictly monotonous, bsolutely continuous function t(τ, 0 τ T, we set { x(t = z (τ(t u (t = π (w (τ (t, t b. The curve x( defined in this wy is bsolutely continuous (becuse z( nd τ( re bsolutely continuous nd τ( is strictly monotonous nd stisfies the boundry conditions x( = x nd x(b = x b. The function u ( is mesurble becuse π ( is continuous, w ( is mesurble nd τ ( is continuous nd monotonous. Also b u (t dt = = b T 0 π (w (τ (t dt = π (w (τ L(t(τ, z(τ, π (w(τ dτ. As fr s the ltter integrnd is bounded due to the coercivity condition, we conclude tht u ( is integrble on [, b]. Differentiting x(t with respect to t, we conclude tht dx (t dt = dz (τ(t dt = dz (τ(t dτ dτ(t dt = ϕ (t, x(t, u (t

9 LIPSCHITZIAN REGULARITY FOR OPTIMAL CONTROL PROBLEMS 9 for lmost ll t [, b]. Integrting one obtins dτ(t dt J [x(, u ( ] = = L(t, z (τ (t, π (w (τ(t = L(t, x(t, u (t b b dτ (t L(t, x(t, u(t dt = dt = dt = τ (b τ ( = T Continuous differentibility of the right-hnd side of (3.9. The functions φ(,, nd h(,, re continuous in {(t, z, w : t [, b], z IR n, w S m }. To pply the Pontrygin Mximum Principle (Theorem to the problem (3.8-(3.10, we need to ssure tht the right-hnd side of (3.9 is continuously differentible with respect to t nd z. Since L(,,, f (,, g (, re C 1 ; L(t, x, u > 0 for ll (t, x, u, π( is continuous, then we conclude t once tht φ z i (t, z, w, φ t (t, z, w, h z i (t, z, w nd h t (t, z, w re continuous for w ŵ. The only problem is the continuous differentibility t ŵ. Since φ(,, ŵ 0 nd h(,, ŵ 0, then we hve (from now on, when not indicted, L nd ϕ re evluted t (t, z, π(w L xi (t, z, π(w φ z i (t, z, w = L 2 if w ŵ (t, z, π(w, i = 1,..., n, 0 if w = ŵ { L ϕx i L x i ϕ h z i (t, z, w = L 2 if w ŵ, i = 1,..., n, 0 if w = ŵ L t (t, z, π(w φ t (t, z, w = L 2 if w ŵ (t, z, π(w, 0 if w = ŵ { L ϕt L t ϕ h t (t, z, w = L 2 if w ŵ. 0 if w = ŵ To verify the continuity of the derivtives t ŵ, we hve to prove tht for ll (t 0, z 0 [, b] x IR n (3.11 L x i (t, z, π(w ρ((t, z, w, (t 0, z 0, ŵ 0 L 2 (t, z, π(w = 0, (3.12 (3.13 (3.14 L t (t, z, π(w ρ((t, z, w,(t 0, z 0, ŵ 0 L 2 (t, z, π(w = 0, L ϕ x i L x i ϕ ρ((t, z, w, (t 0, z 0, ŵ 0 L 2 = 0, L ϕ t L t ϕ ρ((t, z, w,(t 0, z 0, ŵ 0 L 2 = 0,

10 10 ANDREI V. SARYCHEV AND DELFIM F. MARADO TORRES where ρ (, is distnce defined on [, b] xir n x S m. We shll see tht (3.11, (3.12, (3.13 nd (3.14 re true under our hypotheses. Let N (t, z, π(w denote ny of numertors in (3.11 (3.14. From the growth condition (H3, we obtin tht, for ll t [, b], x IR n nd u IR m, N (t, x, u γ L β (t, x, u u µ + η u µ. As long s 2 β > 0, one concludes Since by (H2 If we recll tht N (t, z, π(w L 2 (t, z, π(w γ 1 L(t, z, π(w 1 θ ( π(w, then N (t, z, π(w L 2 (t, z, π(w π(w µ L 2 β (t, z, π(w + η π(w µ L 2 (t, z, π(w. γ π(w + π(w µ θ 2 β ( π(w + η π(w µ θ 2 ( π(w. π(w θ ( π(w = 0, µ 2 β µ 2 2 β > 0, then we obtin N (t, z, π(w ρ((t, z, w,(t 0, z 0, ŵ 0 L 2 = 0, (t, z, π(w which proves the equlities (3.11, (3.12, (3.13 nd ( Pontrygin Mximum Principle nd Lipschitzin Regulrity. Let ( x(, ũ ( be minimizer of the originl problem (P nd ( t (, z (, w ( the correspondent minimizer for the time-optiml problem (3.8-(3.10 with w (τ ŵ lmost everywhere. Applying Pontrygin s Mximum Principle to the time-miniml problem (3.8-(3.10, we conclude tht there exists bsolutely continuous functions on [0, T] λ : IR IR, p : IR IR n, where T denotes the miniml time for the problem (3.8-(3.10 nd p (τ is row covector, not vnishing simultneously, stisfying: i: the Hmiltonin system { λ (τ = Ht (t (τ, z (τ, λ (τ, p (τ, w (τ ṗ (τ = H z (t (τ, z (τ, λ (τ, p (τ, w (τ with the Hmiltonin H(t, z, λ, p, w = λ φ (t, z, w + p, h(t, z, w ; ii: the mximlity condition ( c = sup w S m { λ (τ φ ( t(τ, z (τ, w + p (τ,h ( t(τ, z (τ, w }

11 (3.16 LIPSCHITZIAN REGULARITY FOR OPTIMAL CONTROL PROBLEMS 11.e. = λ (τ φ ( t(τ, z (τ, w (τ + p (τ, h ( t(τ, z (τ, w (τ, where c is constnt, z(τ = x ( t(τ nd w(τ = π 1 ( ũ( t(τ. We shll prove now tht, under the hypotheses of the theorem, c must be positive. Recll tht φ(t, z, w = 1 L(t, z, π(w, ϕ (t, z, π(w h(t, z, w = L(t, z, π(w lmost everywhere. In fct, if c = 0, then (3.15 implies (fter substitution v = π (w tht sup v IR m { λ (τ + p (τ, f ( t (τ, z (τ + p (τ g ( t (τ, z (τ, v } vnishes for lmost ll τ L ( t (τ, z (τ,v [ 0, T ]. This obviously implies (3.17 p (τ g ( t(τ, z (τ 0. At the sme time, since L is positive, λ (τ + p (τ, f ( t(τ, z (τ must be non-positive. Then, since finite vlue for v implies finite vlue for L ( t (τ, z (τ, v, one concludes (3.18 λ (τ + p (τ, f ( t (τ, z (τ = 0 The following proposition tell us tht (3.17 nd (3.18 imply tht ũ ( is n bnorml extreml control for (P. Proposition ( 1. If the equlities (3.17 nd (3.18 hold then the qudruple x(, ψ 0, ψ (, ũ ( defined s ( z ( τ (, 0, p ( τ (, ṽ ( τ (, where τ ( is the inverse function of t (, is n bnorml extreml for the problem (P..e. Proof. The respective Hmiltonin for the problem (3.8-(3.10 equls H (t, z, λ, p, v = λ + p, f (t, z + p g (t, z, v. L(t, z, v We hve to( verify the conditions (i nd (ii of the Theorem for the qudruple x(, 0, ψ (, ũ ( defined in the formultion of the Proposition. Let us introduce the bnorml Hmiltonin Direct computtion shows H = ψ, f (t, x + ψ, g (t, x u. x(t = d dt { z ( τ (t} = τ (t z ( τ (t ;

12 12 ANDREI V. SARYCHEV AND DELFIM F. MARADO TORRES z ( τ (t = f (t, x(t + g (t, x(t ũ (t ; L(t, x(t, ũ (t τ (t = L(t, x(t, ũ (t nd hence x(t = f (t, x(t + g (t, x(t ũ (t. Also ψ i (t = d dt p i ( τ (t = τ (t pi ( τ (t. Obviously ψ( is bsolutely continuous s composition of the bsolutely continuous function p( with the bsolutely continuous monotonous function τ(. For ll i = 1,..., n p i (τ = H = p(τ, f z i x i( t(τ, z(τ + p(τ, g x i( t(τ, z(τ ṽ(τ + L( t(τ, z(τ, ṽ(τ + (λ(τ+ p(τ, f( t(τ, z(τ + p(τ, g( t(τ, z(τ ṽ(τ L x i( t(τ, z(τ, ṽ(τ. L 2 ( t(τ, z(τ, ṽ(τ The second ddend vnishes by virtue of (3.17-(3.18 nd therefore ψ i (t = ψ (t, fx i (t, x(t ψ (t, gx i (t, x(t ũ (t = = H x i, so tht (i is fulfilled. On the other side, ψ (t, g (t, x(t u (t = p ( τ (t, g (t, z ( τ (t u (t 0, nd therefore ψ (t, f (t, x(t + g (t, x(t u (t = ψ (t, f (t, x(t does not depend in u, so the mximlity condition (ii is fulfilled trivilly (or bnormlly. This proves tht vnishing c corresponds to n bnorml extreml control of the problem (P. Thus, for minimizers which re not bnorml extreml controls, there holds c > 0. From (3.15 we obtin 0 < c.e. = λ(τ + p(τ, ϕ ( t(τ, z(τ, ṽ (τ L ( t(τ, z(τ, ṽ (τ L ( t(τ, z(τ, ṽ(τ = c 1 ( λ(τ + p(τ, ϕ ( t(τ, z(τ, ṽ (τ. Let λ (τ M nd p (τ M on [ Then for ny fixed τ 0, T ] [ 0, T ]. L ( t(τ, z(τ, ṽ(τ c 1 M ( 1 + ϕ ( t(τ, z(τ, ṽ(τ

13 LIPSCHITZIAN REGULARITY FOR OPTIMAL CONTROL PROBLEMS 13 nd hence (3.19 θ ( ṽ(τ ϕ ( t(τ, z(τ, ṽ(τ for lmost ll τ L ( t(τ, z(τ, ṽ(τ ϕ ( t(τ, z(τ, ṽ(τ c 1 M 1 + ϕ ( t(τ, z(τ, ṽ(τ ϕ ( t(τ, z(τ, ṽ(τ [ 0, T ]. The lst term of this inequlity cn be mjorized by 2c 1 M if ϕ ( t(τ, z(τ, ṽ(τ 1. From the growth condition (H2, nd from the fct tht g (t, x hs full column rnk, it follows tht ϕ (t, x, u = + nd from the linerity of ϕ (t, x, u with respect u + to u ṽ(τ + θ ( ṽ(τ ϕ ( t(τ, z(τ, ṽ(τ = +. Hence one cn find r 0 such tht r r 0 : ϕ ( t(τ, z(τ, r θ (r 1 nd ϕ ( t(τ, z(τ, r 2 c 1 M. Therefore for (3.19 to be stisfied there must be ṽ(τ r 0. The proof [ is now complete: ṽ ( = π ( w ( must be essentilly bounded (by r 0 on 0, T ], tht is, ũ ( = ṽ ( τ ( is essentilly bounded on [, b]. 4. Applictions to the Clculus of Vritions 4.1. Bsic problem of the Clculus of Vritions. The following result is n immedite Corollry of the Theorem 1. Theorem 2. Let L(,, be continuously differentible on IR x IR n x IR n,, b IR ( < b nd x, x b IR n. Consider the Bsic problem of the Clculus of Vritions: (4.1 J [x( ] = Under the hypotheses: b L(t, x(t, ẋ(t dt min x( = x, x(b = x b. (H1 coercivity: there is function θ : IR IR such tht L(t, x, u θ ( u > ζ, for ll (t, x, u IR 1+n+n, nd r + r θ (r = 0; ζ IR,

14 14 ANDREI V. SARYCHEV AND DELFIM F. MARADO TORRES (H2 growth condition: there re constnts γ, β, η nd µ, with γ > 0, β < 2 nd µ mx {β 1, 1}, such tht for ll t [, b], nd x, u IR n ( L t (t, x, u + L x i (t, x, u u µ γ L β (t, x, u + η, i {1,..., n}; ny minimizer of the problem in the clss of bsolutely continuous functions is Lipschitzin on [, b]. Remrk 5. There re no bnorml extremls in the Bsic Problem. Below we provide n exmple, which shows tht this result of Lipschitzin regulrity, is not covered by the previously obtined conditions, we re wre of. First we formulte Tonelli s Existence Theorem. If the Lgrngin L(,, is C 2 nd the following conditions hold: (T1: L(,, is coercive, i.e., there exist constnts, b > 0 nd c IR such tht L(t, x, v v 1+b + c, for ll (t, x, v; (T2: L vv (t, x, v 0 for ll (t, x, v; then solution to (4.1 exists in the clss of bsolutely continuous functions. The following regulrity results re due to F.H. Clrke nd R.B. Vinter (see [9], [7] nd re proven under weker hypotheses thn those we re considering here. Nmely, they re vlid when L is nonsmooth. Since nonsmoothness is not phenomenon we study, we restrict ourselves to the differentible cse. Regulrity Results. Let L(,,, in ddition to the hypotheses of Tonelli s existence Theorem, stisfy ny of the conditions (C1, (C2, (C3, (C4 or (C5: (C1: Lgrngin is utonomous (i.e., does not depend on t; (C2: there re k 0 IR nd k 1 ( integrble such tht, (t, x, v [, b] IR n IR n L t (t, x, v k 0 L(t, x, v + k 1 (t; (C3: there re k 0 IR nd k 1 (, k 2 ( integrble such tht, (t, x, v [, b] IR n IR n L x (t, x, v k 0 L(t, x, v + k 1 (t L v (t, x, v + k 2 (t; (C4: for ech fixed t, the function (x, v L(t, x, v is convex;

15 LIPSCHITZIAN REGULARITY FOR OPTIMAL CONTROL PROBLEMS 15 (C5: L vv (t, x, v > 0 nd there exist constnt k 0 such tht (t, x, v [, b] IR n IR n L 1 vv (t, x, v (L x (t, x, v L vt (t, x, v L vx (t, x, v v k 0 ( v 2+b + 1, where b is the positive constnt tht ppers in the coercivity condition (T1 of Tonelli s existence theorem; then every minimizer of the bsic problem of the Clculus of Vritions (4.1 is Lipschitzin. Notice tht (C1 is prticulr cse of condition (C2. The growth conditions (C3 nd (C5 re generliztions of clssicl conditions obtined respectively by Tonelli Morrey nd Bernstein (Loc. cit.. Now we provide n exmple of functionl which possesses minimizer, for which no one of the conditions (C1 (C5 is pplicble, while Theorem 2 is. Exmple. Let us consider the following problem (n = 1: (4.2 J [x( ] = 1 0 [ (ẋ e (ẋ 4 +1(t+ π 2 rctn x] dt min x(0 = x 0, x(1 = x 1. Denoting L(t, x, v = ( v e (v 4 +1(t+ π 2 rctn x, we conclude [ 132v v 2 ( 96v 6 ( L vv (t, x, v = (v v v2 t + π 2 rctnx + +16v 6 ( t + π 2 rctnx 2 ] L(t, x, v. Tonelli s theorem gurntees existence of minimizer ˆx( AC for the problem (4.2 s long s: L(,, C 2 ; For ll (t, x, v [0, 1] IR IR we hve: L vv (t, x, v 0. L(t, x, v > ( v v > 0; The ssumptions of the Theorem 2 re lso verifible with θ (r = r 4 + 1, β = 3 2, µ = 1, γ = 2 nd η = 0. Indeed: 2 L t (t, x, v = ( v L(t, x, v, L x (t, x, v = v4 + 1 L(t, x, v, 1 + x2

16 16 ANDREI V. SARYCHEV AND DELFIM F. MARADO TORRES ( L x (t, x, v + L t (t, x, v v 1/2 2 ( v 4 1/8 ( v L(t, x, v < < 2 ( v /8 L(t, x, v = 2 ( v /8 e (v 4 +1(t+ π 2 rctn x < < 2 ( v /8 e 3/2(v 4 +1(t+ π 2 rctn x = 2L 3/2 (t, x, v. Theorem 2 gurntees then, tht ll the minimizers of this functionl re Lipschitzin. As we shll see now, none of the conditions (C1 (C5 is pplicble to this exmple. Indeed: 1. The Lgrngin depends explicitly on t nd so the condition (C1 fils. 2. If (C2 were true, one might conclude L t (t, x, v L(t, x, v k 0 + k 1 (t L(t, x, v, which in our cse implies v k 0 + k 1 (t L(t, x, v, n inequlity which fils for v sufficiently lrge. 3. For (C3 to hold, we should hve for v > 0, x = 0 nd some t L x (t, 0, v v 7/2 L(t, 0, v k 0 v 7/2 + k 1 (t L v (t, 0, v v 7/2 L(t, 0, v + k 2 (t v 7/2 L(t, 0, v ; s fr s [ 12 v 3 ( L v (t, x, v = v v3 t + π 2 ] rctn x L(t, x, v, one obtins the inequlity v v 7/2 k 0 v 7/2 + k 1 (t 12 v 3 v ( 4v3 t + π 2 v 7/2 + k 2 (t v 7/2 L(t, 0, v which fils for v sufficiently lrge. 4. (C4 isn t true either: if we fix t nd v we come to the function x C 3 e C (B rctn x which is not convex: its second derivtive equls 2 x + C (1 + x 2 2 C4 C (B rctn x e which is not sign-definite. 5. Finlly (C5 fils, since we hve L vv (t, x, v = 0 for exmple for v = 0.

17 LIPSCHITZIAN REGULARITY FOR OPTIMAL CONTROL PROBLEMS Vritionl problems with higher order derivtives. Let us consider now the problem of the Clculus of Vritions with higher-order derivtives: b ( (4.3 L t, x(t, ẋ(t,..., x (m (t dt min (4.4 x( = x 0. x (m 1 ( = x m 1 x(b = x 0 b. x (m 1 (b = x m 1 b. We use the nottion W k,p (k = 1,...; 1 p for the clss of functions which re bsolutely continuous with their derivtives up to order k 1 nd hve k-th derivtive belonging to L p. Existence of minimizers for problem (4.3-(4.4 in the clss W m,1 ([, b], IR n, will follow from clssicl existence results, if we impose tht L(t, x 0,..., x m is convex with respect to x m (see [3]. One cn put question of whether every minimizer x( W m,1 hs essentilly bounded m th derivtive, i.e. belongs to W m,. A study of higher-order regulrity hs been done in 1990 by F.H. Clrke nd R.B. Vinter in [11], where they deduced condition of the Tonelli-Morrey-type: L xi (t, x 0,..., x m γ ( L(t, x 0,..., x m + x m + + η (t r (x 0,..., x m, with i = 0,..., m 1, γ being constnt, η n integrble function, r loclly bounded function. Once gin, we re ble to derive from our min result condition of new type lso for this cse. Theorem 3. Provided tht: for ll t [, b] nd x 0,..., x m IR n the function (t, x 0,..., x m L(t, x 0,..., x m is continuously differentible nd the following conditions hold (4.5 (H1 coercivity: there is function θ : IR IR nd constnt ζ such tht nd r + L(t, x 0,..., x m θ ( x m > ζ, r θ (r = 0; (H2 growth condition: for some constnts γ, β, η nd µ with γ > 0, β < 2 nd µ mx {β 1, 1} ( L t + L xi x m µ γ L β + η, i {0,..., m 1} ; then ll minimizers x( W m,1 ([, b], IR n of the vritionl problem (4.3-(4.4 belong to the clss W m, ([, b], IR n. Remrk 6. There re no bnorml extremls for problem (4.3-(4.4. Some corollries cn be esily derived.

18 18 ANDREI V. SARYCHEV AND DELFIM F. MARADO TORRES Corollry 2. Any minimizer x( of the functionl b ( J [x( ] = L x (m (t dt min, x( W m,1 ([, b], IR n, under the conditions (4.4, is contined in the spce W m, ([, b], IR n, provided L( is continuously differentible; nd for ll x m IR n nd some constnts ξ IR nd α ]1, + [, L(x m x m α + ξ. For m = 1 there exists much stronger result, thn this ltter corollry: if m = 1 nd L is utonomous, L = L(x, ẋ, coercive nd convex in ẋ, then ll the minimizers of the problem belong to W 1, (condition (C1 of Regulrity Results. The question of whether it cn be generlized onto the cse of higher-order functionls L ( x, ẋ,..., x (m remined open till recent time. It hs been shown in [15] tht utonomous higher-order functionls not only my possess minimizers belonging to W m,1 \W 1, but lso exhibit the Lvrentiev phenomenon: their infimum in W m,1 cn be strictly less thn the one in W m,. References 1. Ambrosio L, Ascenzi O, Buttzzo G (1989 Lipschitz regulrity for minimizers of integrl functionls with highly discontinuous integrnds. J Mth Anl Appl 142: Bernstein S (1912 Sur les qutions du clcul des vritions. Ann Sci Ecole Norm Sup 3: Cesri L (1983 Optimiztion Theory nd Applictions. Springer- Verlg, New York 4. Cheng C, Mizel VJ (1996 On the Lvrentiev phenomenon for optiml control problems with second-order dynmics. SIAM J Control Optimiztion 34: Clrke FH (1976 The Mximum Principle under miniml hypotheses. SIAM J Control nd Optimiztion 14: Clrke FH (1985 Existence nd Regulrity in the Smll in the Clculus of Vritions. J Differentil Equtions 59: Clrke FH (1989 Methods of Dynmic nd Nonsmooth Optimiztion. SIAM, Phildelphi 8. Clrke FH, Loewen PD (1989 Vritionl problems with Lipschitzin minimizers. Ann Inst Henri Poincr, Anl Nonlinire 6: Clrke FH, Vinter RB (1985 Regulrity properties of solutions to the bsic problem in the clculus of vritions. Trns Amer Mth Soc 289: Clrke FH, Vinter RB (1986 Regulrity of Solutions to Vritionl Problems with Polynomil Lgrngins. Bull Polish Acd Sci 34: Clrke FH, Vinter RB (1990 A regulrity theory for vritionl problems with higher order derivtives. Trns Amer Mth Soc 320:

19 LIPSCHITZIAN REGULARITY FOR OPTIMAL CONTROL PROBLEMS Clrke FH, Vinter RB (1990 Regulrity properties of optiml controls. SIAM J Control nd Optimiztion 28: Gmkrelidze RV (1978 Principles of Optiml Control Theory. Plenum Press, New York 14. Morrey CB (1966 Multiple integrls in the Clculus of Vritions. Springer, Berlin 15. Srychev AV (1997 First nd Second-Order Integrl Functionls of the Clculus of Vritions Which Exhibit the Lvrentiev Phenomenon. Journl of Dynmicl nd Control Systems 3: Sussmnn HJ (1994 A strong version of the Mximum Principle under wek hypotheses. Proc 33rd IEEE Conference on Decision nd Control, Orlndo Tonelli L (1915 Sur une mthode directe du clcul des vritions. Rend Circ Mt Plermo 39: Torres DM (1997 Lipschitzin Regulrity of Minimizers in the Clculus of Vritions nd Optiml Control (in Portuguese. MSc thesis, Univ Aveiro, Portugl Dep. of Mthemtics, Univ. of Aveiro, 3810 Aveiro, Portugl E-mil ddress: nsr@mt.u.pt E-mil ddress: delfim@mt.u.pt URL: