Proceedings of Institute of Mthemtics of NAS of Ukrine 2004, Vol. 50, Prt 3, 1488 1495 The Role of Symmetry in the Regulrity Properties of Optiml Controls Delfim F.M. TORRES Deprtment of Mthemtics, University of Aveiro, 3810-193 Aveiro, Portugl E-mil: delfim@mt.u.pt The role of symmetry is well studied in physics nd economics, where mny gret contributions hve been mde. With the help of Emmy Noether s celebrted theorems, unified description of the subject cn be given within the mthemticl frmework of the clculus of vritions. It turns out tht Noether s principle cn be understood s specil ppliction of the Euler Lgrnge differentil equtions. We clim tht this modifiction of Noether s pproch hs the dvntge to put the role of symmetry on the bsis of the clculus of vritions, nd in key position to give nswers to some fundmentl questions. We will illustrte our point with the interply between the concept of invrince, the theory of optimlity, Tonelli existence conditions, nd the Lipschitzin regulrity of minimizers for the utonomous bsic problem of the clculus of vritions. We then proceed to the generl nonliner sitution, by introducing concept of symmetry for the problems of optiml control, nd extending the results of Emmy Noether to the more generl frmework of Pontrygin s mximum principle. With such tools, new results regrding Lipschitzin regulrity of the minimizing trjectories for optiml control problems with nonliner dynmics re obtined. 1 Introduction A nturl first step when deling with n optiml control problem is to pply the existence theorems. For mny interesting nd importnt optiml control problems, such s those of the clculus of vritions, the existence does not ssure the vlidity of first-order optimlity conditions. It my be the cse, for exmple, tht the minimizers whose existence is gurnteed by the existence theorem fil to stisfy Pontrygin s mximum principle. Obviously, the study of regulrity conditions which close the discrepncy between the hypotheses of existence nd necessry optimlity theories, is of crucil importnce. This pper contins survey of the recent results obtined by the uthor. We give new perspective on the results concerning the regulrity of the minimizing trjectories. Our clim is tht those results re fruit of the existence of certin symmetry properties, nd follow from pproprite conservtion lws. Such conservtion lws re explined by using the recent extensions to optiml control of the clssicl symmetry theorem of E. Noether. Emmy Amlie Noether ws the one to prove, in 1918, tht conservtion lws in the clculus of vritions re the mnifesttion of powerful nd universl principle: When system shows up symmetry with respect to prmeter-trnsformtion, then there exists conservtion lw for tht system ssocited with the invrince property. This ssertion comprises ll theorems on first integrls known to clssicl mechnics. Typicl ppliction of those results is to lower the order of the differentil equtions. Here we clim tht the conservtion lws re lso useful for different purposes. We clim tht they cn be used with success to prove Lipschitzin regulrity of the minimizing trjectories. Noether s principle is so deep nd rich, tht it cn be formlized s theorem in mny different contexts, nd in ech of such contexts under mny different ssumptions (see e.g. [12]).
The Role of Symmetry in the Regulrity Properties of Optiml Controls 1489 Formultions of Noether s symmetry principle to the more generl frmework of optiml control hve been recently obtined (see e.g. [7, 8, 10]). In this context conservtion lws re interpreted s quntities which re preserved long ll the Pontrygin extremls. They ply here fundmentl role in order to bring together the regulrity results in [6, 11]. 2 Optiml control nd the clculus of vritions The optiml control problem consists in finding r-vector function u( ), clled the control, nd the corresponding n-stte trjectory x( ), solution of dynmicl system described by n ordinry differentil equtions under specified boundry conditions, in such wy the pir (x( ),u( )) minimizes given integrl functionl: I[x( ),u( )] = b L (t, x(t),u(t)) dt min, ẋ (t) =ϕ (t, x (t),u(t)), (x(),x(b)) F, x ( ) X([, b]; R n ), u( ) U([, b]; Ω R r ). (1) The Lgrngin L nd the velocity vector ϕ re ssumed to be smooth functions: C 1 L :[, b] R n R r R, C 1 ϕ :[, b] R n R r R n. For the prticulr cse when there re no restrictions on the control vlue, Ω = R r, nd the control system is just ϕ(t, x, u) = u, one obtins the fundmentl problem of the clculus of vritions (r = n), I[x( ),u( )] = b L (t, x(t),u(t)) dt min, ẋ(t) =u(t), (x(),x(b)) F, x( ) X([, b]; R n ), which includes ll clssicl mechnics. Clerly, for the optiml control problem to be well defined, it is crucil to specify the clss of functions X nd U. This is ddressed in the following section. 3 Logicl-deductive pproch, discrepncy, nd bd behvior The logicl-deductive pproch for the resolution of n optimiztion problem is the following: 1. A solution exists for the problem. 2. The necessry optimlity conditions re pplicble nd they identify certin cndidtes (in the context of the clculus of vritions nd optiml control clled extremls). 3. Elimintion, if necessry, identifies the solution or solutions. The historicl fcts show tht mthemtics does not lwys dvnce using the logicl steps: both the clculus of vritions nd the mthemticl theory of optiml control hve born from the study of necessry optimlity conditions, the existence question being delyed. In the clculus of vritions the first obtined necessry conditions were the Euler Lgrnge differentil equtions, first proved by L. Euler in 1744, while the existence ws only ddressed two centuries lter, in 1911, by L. Tonelli. The mthemticl theory of optiml control, the modern fce of the clculus of vritions, ws born with the Pontrygin mximum principle, proved by V.G. Boltynski, R.V. Gmkrelidze, nd L.S. Pontrygin in 1956 [2], while existence ws solved three yers lter, in 1959, by A.F. Filippov [5]. It is very interesting the explntion why the study of necessry conditions hve been first in time. It turns out tht existence nd necessry optimlity theories rely on different clsses X nd U:
1490 D.F.M. Torres In the clculus of vritions the existence is ssured in the clss of bsolutely continuous functions (X = W 1,1 ), while the rguments which led to the necessry conditions demnd the minimizing trjectory to be t lest Lipschitzin (X W 1, ). In optiml control the existence is ssured in the clss of integrble controls (U = L 1 ), while the clssicl formultion of the Pontrygin mximum principle ssumes tht the optiml controls re essentilly bounded (U L ). The discrepncy between the hypotheses of existence nd optimlity theories is not the result of technicl detils, but hs n intrinsic nture. The gp is relity, even for problems very innocent in spect. For exmple, the minimizers predicted by the existence theory cn violte the necessry conditions even when the Lgrngin L is polynomil nd ϕ is liner. The first problem showing this possibility of bd behvior ws proposed by J. Bll nd V. Mizel in 1985. Exmple 1 (cf. [1]). For the Bll Mizel problem, 1 ( x 3 t 2 2 u 14 + ε u 2) dt min, 0 ẋ (t) =u (t), x(0) = 0, x(1) = k, it hppens: All hypotheses of Tonelli s existence theorem re stisfied. For some choices of the constnts ε nd k one hs the unique optiml control u (t) =kt 1/3. Pontrygin mximum principle (Euler Lgrnge eqution in integrl form) is not stisfied since ψ(t) =L x (t, x (t), ẋ (t)) = ct 4/3 is not integrble. The study of conditions which ssure the regulrity of the minimizers is importnt, nd excludes occurrence of bd behvior. 4 Regulrity of minimizers A regulrity result is ny ssertion ssuring tht the solution belongs to clss of functions more restrict thn the one considered in the formultion of the problem (e.g., x( ) W 1, insted of W 1,1 ;ũ( ) L insted of L 1 ). The objective is to find conditions beyond those of the existence theory, ssuring tht ll the minimizers stisfy the stndrd necessry optimlity conditions (e.g. the Pontrygin mximum principle or the Euler Lgrnge differentil equtions). A profound serch in the literture shows tht: The clculus of vritions is very rich in regulrity results. Regulrity results for optiml control problems re rrity. In this pper we comment on new pproch to estblish regulrity properties for the optiml controls. Our pproch hs its genesis in very beutiful regulrity theorem of 1985, due to F.H. Clrke nd R.B. Vinter. Theorem 1 (cf. [4]). Under the coercivity nd convexity conditions of Tonelli s existence theorem, ll the minimizing trjectories x( ) of the utonomous fundmentl problem of the clculus of vritions, I[x( ),u( )] = b L (x(t),u(t)) dt min, ẋ(t) =u(t), (x(),x(b)) F, x( ) W 1,1 ([, b]; R n ), re Lipschitzin: x( ) W 1, W 1,1.
The Role of Symmetry in the Regulrity Properties of Optiml Controls 1491 Theorem 1 is not vlid for utonomous optiml control problems with dynmics different from ẋ(t) = u(t). We lso remrk tht Theorem 1 is only vlid in the utonomous sitution, tht is, in the cse the Lgrngin L nd the velocity vector ϕ do not depend explicitly on the time vrible t. The proof of Clrke nd Vinter is very technicl, nd hs the focus on the use of nonsmooth nlysis. This, in our opinion, hides the crucil point nd the nturl question to be sked. 5 Autonomous problems...why specil? For utonomous problems, the Hmiltonin H(x, u, ψ 0,ψ)=ψ 0 L(x, u)+ψ ϕ(x, u) is preserved long the Pontrygin extremls (x( ),u( ),ψ 0,ψ( )) of the optiml control problem: H(x(t),u(t),ψ 0,ψ(t)) const, t [, b]. (2) This is consequence of the Pontrygin mximum principle, nd holds generlly when dmissible controls re bounded: u( ) L. The property is importnt in mny different contexts: In the clculus of vritions it corresponds to the 2 nd Erdmnn necessry condition. In clssicl mechnics hs the mening of conservtion of energy. In economy is interpreted hs the income/welth lw. The strting point for our pproch is the observtion tht Theorem 1 is strightforwrd consequence of the coercivity hypotheses of the existence theory, if one uses the fct tht, for the fundmentl utonomous problem of the clculus of vritions, (2) is vlid in the clss of bsolutely continuous dmissible trjectories (u( ) L 1 ): see [3, pp. 61 63]. 6 Wht bout the non utonomous problems? Property (2) dmits generliztion for the problems which depend explicitly on the time vrible t. If (x( ),u( ),ψ 0,ψ( )) is Pontrygin extreml, then H (t, x(t),u(t),ψ 0,ψ(t)) is n bsolutely continuous function in t, nd stisfies the equlity dh dt (t, x(t),u(t),ψ 0,ψ(t)) = H t (t, x(t),u(t),ψ 0,ψ(t)), (3) where on the left-hnd side we hve the totl derivtive with respect to t, nd on the right-hnd side the prtil derivtive of the Hmiltonin with respect to t. In the clculus of vritions (3) corresponds to the clssicl necessry condition of DuBois Reymond. We now give more generl formultion. Theorem 2 (cf. [9]). If F (t, x, u, ψ 0,ψ) is rel vlue function; continuously differentible with respect to t, x nd ψ, forfixedu; G( ) L 1 ([, b]; R) such tht (t,x,ψ) F (t, x(t),u(s),ψ 0,ψ(t)) G(t) (s, t [, b]), F (t, x(t),u(t),ψ 0,ψ(t)) = sup F (t, x(t),v,ψ 0,ψ(t)), v Ω then t F (t, x(t),u(t),ψ 0,ψ(t)) is n bsolute continuous function nd stisfies the equlity df dt = F t + F x H ψ F ψ H (4) x long ll the extremls. Choosing F = H in Theorem 2 we obtin equlity (3). In the clssicl context, where no dependence exists on the controls, (4) is the well-known reltion df dt = F t + {F, H}, where {F, H} denotes the Poisson brcket of the functions F nd H.
1492 D.F.M. Torres 7 Conservtion lws in optiml control Theorem 2 is very useful. From it one obtins necessry nd sufficient condition for function to be conservtion lw. Definition 1. AquntityF (t, x, u, ψ 0,ψ) which is preserved in t long ll the Pontrygin extremls (x( ),u( ),ψ 0,ψ( )) of the optiml control problem (1), F (t, x(t),u(t),ψ 0,ψ(t)) = const, (5) is clled first integrl. Eqution (5) is the corresponding conservtion lw. Corollry 1. Under the conditions of Theorem 2, F is first integrl if, nd only if, F t + F x H ψ F ψ H x =0. Equlity (2) is trivil consequence of Corollry 1: Exmple 2. The Hmiltonin H is first integrl if, nd only if, H t = 0. This mens tht the problem must be utonomous in order for the Hmiltonin to be first integrl. We hve reched the point where it is impossible not to think of the celebrted symmetry theorem of Emmy Amlie Noether (1882 1935). In 1918 E. Noether proved tht the conservtion lws of the clculus of vritions re fruit of the symmetries of the problems. For exmple, the utonomous fundmentl problem of the clculus of vritions is invrint under timetrnsltions t t + s. From this invrince it results, from Noether s symmetry theorem, the conservtion lw H = const. 8 Symmetry in optiml control To obtin formultion of the theorem of Emmy Noether in the more generl setting of optiml control, we begin by extending the concept of symmetry tht one finds in the clculus of vritions. We del with trnsformtions of the optiml control problem depending on time, stte, nd control vribles; nd we consider n invrince notion up to ddition of guge terms which, in generl, re non-liner with respect to the prmeters, time, stte, nd control vribles. Definition 2. The optiml control problem (1) is sid to be invrint under h s (t, x, u) = (h s t(t, x, u),h s x(t, x, u)) C 1 up to Φ s (t, x, u) C 1 ([, b], R n, Ω; R), if h 0 (t, x, u) =(t, x) nd for ll s =(s 1,...,s ρ ), s <ε, nd for ll β [, b], there exists u s ( ) L ([, b]; Ω) such tht h s t (β,x(β),u(β)) h s t (,x(),u()) β = L (t s,h s x(t s,x(t s ),u(t s )),u s (t s )) dt s ( L(t, x(t),u(t)) + d dt Φs (t, x(t),u(t)) ) dt, d dt s hs x(t s,x(t s ),u(t s )) = ϕ(t s,h s x(t s,x(t s ),u(t s )),u s (t s )). Equivlently, one sys tht h s is symmetry for problem (1). Theorem 3 (cf. [7]). If the optiml control problem (1) is invrint under the trnsformtions (h s t(t, x, u),h s x(t, x, u)) up to Φ s (t, x, u), in the sense of Definition 2, then the following ρ conservtion lws hold (k =1,...,ρ): ψ 0 Φ s (t, x(t),u(t)) s s=0 + ψ(t) h s k s x(t, x(t),u(t)) s=0 k H (t, x(t),u(t),ψ 0,ψ(t)) h s s t(t, x(t),u(t)) s=0 =const. (6) k
The Role of Symmetry in the Regulrity Properties of Optiml Controls 1493 Exmple 3. The utonomous optiml control problem is invrint under h s t = t+s nd h s x = x. Theorem 3 implies the conservtion lw (2). It is importnt to note tht the conservtion lws (6) re vlid long ll the Pontrygin extremls (x( ),u( ),ψ 0,ψ( )) of the problem: both for norml (ψ 0 0) nd bnorml ones (ψ 0 = 0). While bnorml extremls re not possibility for the fundmentl problem of the clculus of vritions (one cn lwys put the cost multiplier ψ 0 to be 1), they hppen to occur frequently in the generl sitution. The Mrtinet flt problem of sub-riemnnin geometry, considered in Exmple 4, is one such cse: it dmits bnorml minimizers. For complex situtions, the conditions found in [13] help to obtin the invrince trnsformtions. For the present purposes, it is enough to consider the homogeneous cse. Corollry 2. If there exist constnts α, β 1,...,β n, γ 1,...,γ r R, such tht for ll λ>0 ) L (λ α t, λ β 1 x 1,...,λ βn x n,λ γ 1 u 1,...,λ γr u r = λ α L (t, x 1,...,x n,u 1,...,u r ), ( ) ϕ i λ α t, λ β 1 x 1,...,λ βn x n,λ γ 1 u 1,...,λ γr u r = λ βi α ϕ i (t, x 1,...,x n,u 1,...,u r ), the following conservtion lw holds: n β i ψ i (t)x i (t) αh (t, x(t),u(t),ψ 0,ψ(t)) t const. i=1 Proof. Conclusion follows from Theorem 3 by choosing h s t =(s +1) α t, h s x i =(s +1) β i x i, i =1,...,n,ndu s k =(s +1)γ ku k, k =1,...,r. Exmple 4. Let us consider the Mrtinet flt problem of sub-riemnnin geometry: b (u 1 (t)) 2 +(u 2 (t)) 2 dt min, x 1 (t) =u 1 (t), x 2 (t) =u 2 (t), x 3 (t) =(x 2 (t)) 2 u 1 (t). With α =2,β 1 = β 2 =1,β 3 =3,γ 1 = γ 2 = 1, we get from Corollry 2 the conservtion lw ψ 1 x 1 (t)+ψ 2 (t)x 2 (t)+3ψ 3 x 3 (t) 2Ht const. The regulrity conditions we re looking for, re obtined with the help of Corollry 2. 9 Symmetry nd regulrity The regulrity results in [4, 6, 11] re obtined using very different techniques. In this pper we rgue tht from the point of view of symmetries, ll the proofs, which re completely different in detil, cn be summrized, in the lrge, by the sme steps. In fct the results follow from the sme ssumption: tht under the typicl coercivity conditions of the existence theory, the Lipschitzin regulrity conditions for the minimizing trjectories x( ) cn be obtined from the pplicbility conditions of the Pontrygin mximum principle to n equivlent uxiliry problem which possess rich symmetry properties. Our generl scheme to prove regulrity theorems is summrized in the following steps nd ingredients: 1. Reduce the problem to n equivlent form (equivlence in the sense of Elie Crtn, 1908) with richer set of extremls, nd possessing n pproprite symmetry. 2. Estblish the reltionship between the extremls of the problems (equivlence in the sense of Constntin Crthéodory, 1906).
1494 D.F.M. Torres 3. Impose the pplicbility conditions of the mximum principle to the uxiliry problem, nd use the conservtion lw relted to it. From the coercivity condition nd the Crthéodory equivlence, conclude tht ll (norml) minimizing controls of the optiml control problem (1) re essentilly bounded, nd the respective trjectories re Lipschitzin. Applying such lgorithm with the ides of R.V. Gmkrelidze nd Weierstrss, gives n explntion for the results in [6, 11]. 9.1 Regulrity from time-homogeneity Following R.V. Gmkrelidze, we mke reduction of problem (1) to n equivlent utonomous time-optiml problem with the controls tking vlues in sphere. Theorem 4 (cf. [6]). For the cse of control-ffine dynmics, ϕ (t, x, u) =f (t, x)+g (t, x) u, the coercivity hypotheses; complete rnk r of g(, ); nd the growth condition: there exist constnts γ, β, η nd µ, withγ>0, β<2 nd µ mx {β 2, 2}, such tht for ll t [, b], x R n nd u R r, ( L t + L x i + Lϕ t L t ϕ + Lϕ x i L x i ϕ ) u µ γl β + η, imply tht ll the minimizers ũ ( ) of the optiml control problem, which re not bnorml extreml controls, re essentilly bounded on [, b]. 9.2 Regulrity from control-homogeneity Writing the problem in prmetric form, n ide s old s Weierstrss, one gets control homogeneity. Applying the generl scheme, the following result comes out. Theorem 5 (cf. [11]). Under the coercivity hypothesis of the existence theorem, the growth conditions: there exist constnts c>0 nd k such tht L L t c L + k, ϕ x c L + k, ϕ i t c ϕ + k, x c ϕ i + k; imply tht ll the minimizing controls ũ( ) of the optiml control problem (1), which re not bnorml extreml controls, re essentilly bounded on [, b]. 10 Conclusions Regulrity theorems re very importnt becuse they imply tht ll the minimizers re Pontrygin extremls. New conditions of Lipschitzin regulrity of the minimizing trjectories, to brod clss of problems of optiml control, re estblished using pproprite conservtion lws. The results re vlid for nonliner control systems. Even for liner dynmics, for exmple for the problems of the clculus of vritions, one cn del with situtions for which the previously known regulrity conditions fil (cf. [6]). Our pproch does not require the Lgrngin L nd the velocity vector ϕ to be convex functions with respect to the control vribles. Acknowledgements This work hs been prtilly supported by the R&D unit Centre for Reserch in Optimiztion nd Control (CEOC) of the University of Aveiro, nd by the Project POCTI/MAT/41683/2001 Advnces in Nonliner Control nd Clculus of Vritions of the Portuguese Foundtion for Science nd Technology (FCT), Spiens 01, cofinnced by the Europen Community fund FEDER.
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