A Local Variational Problem of Second Order for a Class of Optimal Control Problems with Nonsmooth Objective Function

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1 A Local Varatonal Problem of Second Order for a Class of Optmal Control Problems wth Nonsmooth Objectve Functon Alexander P. Afanasev Insttute for Informaton Transmsson Problems, Russan Academy of Scences, Nakhmovsky Prospekt 36-1, Moscow , Russa Natonal Research Unversty Hgher School of Economcs, Myasntskaya str., 2, Moscow 11, Russa Natonal Unversty of Scence and Technology MISS, Lennsky Prospekt 4, Moscow 11949, Russa Lomonosov Moscow State Unversty, GSP-1, Vorobevy Gory, Moscow , Russa apa@tp.ru Elena V. Putlna Insttute for Informaton Transmsson Problems, Russan Academy of Scences, Nakhmovsky Prospekt 36-1, Moscow , Russa elena@tp.ru 1 Introducton Abstract An optmal control problem wth a nonsmooth objectve functon defned on a polytope s studed. A scheme reducng the orgnal problem to the problem of maxmzng the sum of matrx elements lyng above the man dagonal s constructed. Necessary condtons of optmalty for ths fnte-dmensonal optmzaton problem are establshed. A method of searchng for the global maxmum s suggested. Optmal control problems wth nonsmooth objectve functons of the form J[u] = mn{g x(τ), = 1, N} max, (1) u under consderaton n ths work arse to study of the stuaton of swtchng durng the contact of the optmum trajectory wth phase and functonal constrants. These stuatons are typcal for optmal control problems wth mxed constrants (see [Mlyutn et al., 24]). Moreover, the need to descrbe the stuaton of swtchng arses n the contnuaton of solutons on the parameter (see [Dkusar et al., 21]) and when formng local varatonal problems (see [Afanasev et al., 199]). Frst-order degeneraton n the objectve functon (1) s studed n (see [Afanasev, 1998]). In ths paper, we consder the second-degeneraton for a local varatonal problem wth the objectve functon of the form (1). Copyrght c by the paper s authors. Copyng permtted for prvate and academc purposes. In: Yu. G. Evtushenko, M. Yu. Khachay, O. V. Khamsov, Yu. A. Kochetov, V.U. Malkova, M.A. Posypkn (eds.): Proceedngs of the OPTIMA-217 Conference, Petrovac, Montenegro, 2-Oct-217, publshed at 1

2 2 Formulaton and Dscusson of the Problem We wll study the problem J[u] = mn 1 P x(t), u(t) dt max, = 1, R, (2) u ẋ(t) = u(t), (3) u(t) M, (4) x( ): [, 1] R n, u( ): [, 1] R n, (5) M - s a polyhedron x() =, x(1) X, X M. Ths problem arses n the study of the followng mportant class of problems J[u] = mn {G (x(t )), = 1, p} max u, (6) ẋ(t) = u(t), (7) u(t) U(x(t)), U(x(t)) = {u: K(x(t)) = u(t) = L(x(t)), M(x(t)) N(x(t))}, (8) where M( ) s an m n matrx, L( ) s an m 1 matrx, K( ) s an k n matrx, N( ) s an k 1 matrx. Wth the mplementaton of the procedure of contnuaton of the optmal trajectory (see[afanasev et al., 199]), where T s a parameter, there arses a need n lftng the degeneracy G (x(τ)) = G j (x(τ)), j, j p, p p. Wthout loss of generalty, we may assume that p = p. Ths case s studed n detal n (see [Afanasev et al., 199]). The functons G (x(τ)) were presented n the form T grad G (x(t)), ẋ(t) dt, and the consderaton was lmted to the study of the lnear programmng problem of the form u n+1 max u n+1 a, u, p, where a = grad G (x ), u M, M polyhedron, u R n. The second-order degeneraton occurs when grad G (x ) =, P. To remove degeneracy, let us put nto consderaton x grad G (ẋ) = 2 G (x) x 2 u, x=x x=x and, under the assumpton 2 G (x) x 2 x=x {} n n, P, we ntroduce the notaton 2 G (x) x 2 x=x = P. And the objectve functon of problem (2) (5) became J[v] = mn 1 P x(t), u(t) dt max u, P. Further we assume that P s a skew-symmetrc matrx, because a symmetrc part of the matrx we may to take out from under the ntegral sgn. 2

3 3 Reducton of the Problem Denote by {R = (r1, r2,..., rn), = 1, N} the set of vertces of a polyhedron M. Then, by the Caratheodory theorem, P N U(t) = R v (t), v (t), v (t) = 1. Let v (t) = ẏ (t). =1 The vectors R can be consdered as columns of the matrx R = {R } N n, and t s easy to see that x(t) = R t v(τ)dτ, y() =, Integrants of the objectve functon take the form Fnally, the problem (2) - (5) takes the form: =1 C y(t), v(t), where C = R T AR. J[v] = mn 1 C y(t), v(t) dt max v, (9) ẏ(t) = v(t), (1) v(t) S, where S smplex, y() =, Ry(1) = x(1) X, (11) 4 Fnte-dmensonal Analogue of Problem (9) - (11) Let v(t) on the nterval [, 1] takes fnte number of values of L. Then where t =, t L = 1. It s clear that, when t [t 1, t ], y(t) = v when t [t, t 1 ] v 1 when t [t 1, t 2 ]... v(t) = v when t [t 1, t ]... v L when t [t, t L ] t 1 v(τ)dτ = v α (t α t α 1 ) + v (t t ). α= Let us ntroduce the notaton v β (t β+1 t β ) = y β. Substtute the obtaned y(t) and y β, β =, L nto the objectve functon (9) nto constrants (1) and (11)), and get α 1 J[v] = mn C y α, y β max (12) {y } β= y α = y(1), Ry(1) X (13) α= y(1), 1 N = 1, y(1), 1 N = (1, 1,..., 1 }{{} )T, (14) where C = R T P R. Ths problem s easly rewrtten n the form of a problem of mathematcal programmng wth a quadratc (blnear) constrants. Usng the skew-symmetrc property of matrces C, we can derve the followng equalty, α 1 C y α, y β = β= L C y α, y β 3

4 and fnally have Z Z max, (15) L C y α, y β, (16) L y α = y(1), (17) α= y(1), 1 N = 1, y(1), R y(1) Z. (18) To study propertes of the soluton of problem (15) - (18) we may to relax t by droppng constrants (18). Then, the followng theorem s vald. T h e o r e m 1. Among the solutons of problem (15) - (16) (y(1) s f parameter) there s always such that y α, y β = for any α β Proof. Let us replace problem (15) - (18) by the set of blnear programmng problems wth lnear constrants of the form L C y α, y β max (19) L α= y α = y(1), = 1, p (2) and let us solve each task separately, and among the solutons of problem (15) - (17) we wll choose the one whch has the maxmum value of the objectve functon. (Obvously that ths technque makes sense to use only n the proof). Reasonable computatonal procedures wll be dscussed later. Thus, the task s came down to the problem of the form (19) - (2). Ths problem s nvestgated n detal n (see[afanasyev, 1993]) where s set to among solutons to the problem of the form (19) - (2) there s always such that, y α, y β = V α β Ths theorem has mportant corollary. C o r o l l a r y 1. Let the vector y(1) R N has a N N non-zero components. Then, whatever L (the number of splts when constructng dfference schemes), the soluton of the problem (15) - (17) (and hence (15) - (18) can be realzed no more than N of mutually orthogonal vectors J α. 5 The Theorem on the Bypass of the Vertces of the Polyhedron Let us return to problem (2) - (5). T h e o r e m 2. The soluton of the problem (2) - (5) exsts and can be always realzed at the vertces of the polyhedron M. and the number of swtchngs does not exceed N, where N s the number of vertces of polyhedron M. Proof. Consder the sequence v L (t) of smple (step) functons convergng to a measurable functon v(t), whch gves the optmal soluton to the problem (9) - (11). lm L yl (t) = lm L lm L vl (t) = v (t), (21) t v L (τ)dτ y (t) = t v (τ)dτ. (22) Let to each L there s a task (12) - (14), the soluton of whch, begnnng wth number N due to a corollary of Theorem 1) does not depend on L. Therefore y (t) = {y α }, α = 1, N. That s, the soluton of problem (9) - (11) s completely determned by the soluton of the problem (12) - (14) (fnte-dmensonal analogue). The transton from problem (9) - (11) to problem (2) - (5) s obvous. Theorem s proven. 4

5 6 Propertes of the Soluton of the Dfference Problem (12) - (14) By theorem 1, a vector y α has only one nonzero component all nonzero components of vectors are the components y α of the vector y(1). The essence problem (12) - (14) s to fnd the sutable traversal of vertces of the polyhedron M. Let us we go back to the maxmzaton of the expresson L C y α, y β where N y α = y(1) n vew of the above. t s clear how the orgnal problem s reduced to the followng: consder the matrx {y α C αβ y β } N N and put the task of fndng such a permutaton rows and columns wth the same names to maxmze the sum of above-dagonal elements n the resultng matrx. Thus we can formulate the followng optmzaton problem. Suppose that there s a set of skew-symmetrc matrces A = {a αβ } n n, = 1, k. Let S(A ) denote the sum of the above-dagonal elements. It s necessary to fnd a set of permutatons of rows and columns wth the same names such that mn S(A ) max. (23) In (see[afanasev et al., 199]) t s shown that, when = 1, then nonnegatvty of all partal sums of abovedagonal elements along each strngs s the necessary and suffcent condton for the maxmum,.e. α=γ a αβ, α = 1, n, β = 1, n, γ = 1, n. (24) Generally speakng, they are local maxmum condtons. The fact that the problem (22) s a global optmzaton problem shows the followng example Matrx wth the sum of the above-dagonal elements = 12 satsfyng the condton (23), by usng the permutaton (3, 4, 1, 2) can be reduced to the matrx wth the sum of the above-dagonal elements = 2 whch also satsfy condton (23). Thus, even n the case of = 1, to solve problem (22) should be nvolved methods of search for global extremum. However, the use of condtons(23) n problem (22) when > 1 allows one to apply the method of branches and boundares. Acknowledgements Ths work was supported by Russan Scentfc Foundaton, Project References [Mlyutn et al., 24] Mlyutn A. A., Dmtruk A. V., Osmolovsk N. P. (24). Maxmum prncple n optmal control Moscow State Unversty, Faculty of Mechancs and Mathematcs. Moscow, 168 p. [Dkusar et al., 21] Dkusar V. V., Koshka M., Fgura A. (21). A Method of contnuaton on a parameter n solvng boundary value problems n optmal control. Dfferents. equatons, 37:4, , 5

6 [Afanasev et al., 199] Afanasev A. P., Dkusar V. V., Mlyutn A. A., Chukanov S. V. (199) A necessary condton n optmal control, Moscow: Nauka. [Afanasev, 1998] Afanasev A. P. (1998) On local varatonal problems n the presence of functonal constrants. Dfferents. equatons, volume 34:11, [Afanasyev, 1993] Afanasyev A. P. (1993) Generalzed sopermetrc problem on a polyhedron Dfferents. Equatons, Vol. 29:11,