FIRST AND SECOND ORDER NECESSARY OPTIMALITY CONDITIONS FOR DISCRETE OPTIMAL CONTROL PROBLEMS

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1 Yugoslav Journal of Operatons Research 6 (6), umber, 53-6 FIRST D SECOD ORDER ECESSRY OPTIMLITY CODITIOS FOR DISCRETE OPTIML COTROL PROBLEMS Boban MRIKOVIĆ Faculty of Mnng and Geology, Unversty of Belgrade Belgrade, Serba mboban@veratnet Receved: October 4 / ccepted: June 5 bstract: Dscrete optmal control problems wth varyng endponts are consdered Frst and second order necessary optmalty condtons are obtaned wthout normalty assumptons Keywords: Dscrete optmal control, mathematcal programmng, abnormal extremal ITRODUCTIO Consder dscrete optmal control problem wth varyng endponts mnmze f( x, u) ; ( = x = + ϕ( x, u), =,, () K( x, x ) =, (3) n r where f ( xu, ): R R R s twce contnuously dfferentable functon, ϕ ( x, u): n r n n n k R R R and K( x, x ): R R R are twce contnuously dfferentable n r mappngs Here x R s state varable, u R s a control parameter, s gven number of steps Vector ε = ( x, x,, x n ) s called a trajectory, ω = ( u, u,, u s called a control, x s a startng pont and x s an end pont of correspondng trajectory

2 54 B Marnkovć / Frst and Second Order ecessary Optmalty Condtons Let x be a startng pont and let be a control Then the par ( x, ω) defnes the correspondng drectory ε = ( x, x,, x n ) If the condton (3) s satsfed then we say that the par ( x, ω ) s feasble The dscrete optmzaton problem s to mnmze the functon J ( x, ω) = f( x, u) = on the set of feasble pars The am of ths paper s to obtan frst and second order necessary optmalty condtons for the problem (-(3) wthout normalty assumptons FIRST ORDER ECESSRY OPTIMLITY CODITIOS Let ( x, ω) be a feasble par and let ε = ( x, x,, x ) be a correspondng trajectory Suppose that the par ( x, ω ) s optmal Put ϕ ϕ = Ck, = Dk u and k k k ϕ ϕ ϕ = C,, k = D k = Mk u xu n( + r Let ( hv, ), h= ( h, h,, h) R, v= ( v, v,, v R, be the vector for ' ' n( + ) r whch there exsts a vector ( h, v ) R R such the followng condtons are satsfed: k k k s s j j k k s= j= j< s k, (4) h = C h + C D v + D v, k =, K ( x, x)( h, h) =, (5) ( x, x ) C [ h ] D [ v ] M [ h, v ] = h C h D v, k =,, (6) ' ' ' k k k k k k k k+ k k k k Here [-,-] stands for the arguments of a blnear form (or, more generally, blnear mappng) Denote by K set of all vectors (h,v) for whch the precedng condtons are satsfed Defne the functons ϕ ϕ H ( xu,, pq,, λ, hv, ) = p, ϕ( xu, ) + q, ( xuh, ) + ( xuv, ) u λ f ( x, u), =,,

3 B Marnkovć / Frst and Second Order ecessary Optmalty Condtons 55 where K l ( x, x, λ, λ, h, h ) = λ, K( x, x ) + λ, ( x, x )( h, h ), ( x, x ),, k n λ R λ λ R, p, q R Theorem : Let ( x, ω) be the optmal soluton for the problem (-(3) Then there exsts Lagrange multpler λ = λ λ λ pq λ R λ λ R p R q R λ + ( q, λ ) such that for every ( hv, ) Kthe followng condtons are satsfed: () x k n( + n (,,,, ),,,,,, l p = ( x, x, λ, λ, h, h ), (7) H p (, = x u, p+, q+, λ, h, v), =,, (8) l p = ( x, x, λ, λ, h, h ), (9) x H ( x, u, p, q,, h, v ),, + + λ = ( u () There exsts vector n( + r ( h, v ) R R such that p,, k = h k Ck h k Dk v k k =, ( λ = K ( x, x ) h, ( x () K Cq+ ( x, x ) λ =, (3) * * q C q =, k =,, (4) * k k k+ K * q ( x, x) λ =, (5) * k D q =, k =,, (6) k Proof: We shall formulate the problem (-(3) as a mathematcal programmng problem, and we shall apply results from []

4 56 B Marnkovć / Frst and Second Order ecessary Optmalty Condtons Defne the functons and by r n f( εω, ): R R R, F( εω, ): R R R, =,, n( + r n( + F R R R n( + r n ( εω, ):, f ( εω, ) = f ( x, u ), = F ( εω, ) = x ϕ ( x, u ), =,, + + F( εω, ) = ( F ( εω, ),, F ( εω, )) Consder the followng mathematcal programmng problem: Mnmze f ( εω, ); (7) F( εω, ) =, (8) K( x, x ) = (9) The pont ( ε, w ) s the local mnmum for precedng problem Consder the operator F( εω, ): R R R R n( + r n k gven by F( εω, ) = ( F( εω, ), K( x, x )) Defne the Lagrange vakov functon by n( + r n+ k L( εωλ,,, h, v): R R R R Κ R F L( εωλ,,, h, v) = λ f( εω, ) + p, F( εω, ) + q, ( εω, )( h, v) ( εω, ), where n + k λ = ( λ, pq, ), λ R, pq, R and K n( + s the set of all ( hv, ) R R such that the followng condtons are satsfed: F ( εω, )( hv, ) =, ( εω, ) ) F F ( εω, )[( hv, ),( hv, )] m ( εω, ) ( εω, ) ( εω, ) ote that the vector (h,v) s the parameter n the Lagrange vakov functon r

5 B Marnkovć / Frst and Second Order ecessary Optmalty Condtons 57 F Put = ( εω, ) ( εω, ) From [], theorem, we have that λ, λ, λ + q such that for every ( hv, ) K, From the fact that L ( εωλ,,, h, v) =, p m, q ( m) () ( εω, ) εωλ λ ϕ λ + + = = L (,,, h, v) = f ( x, u ) + p, x ( x, u ) +, K( x, x ) + K +, +, (, )(, ), q+ h+ Ch Dv λ x x h h = ( x, x ) where p = ( p,, p, λ and q = ( q,, q, ) λ, we have L l = ( x, x,,, h, h ) H ( x, u, p, q,, h, v) =, ( L λ λ λ L H = p ( x, u, p+, q+, λ, h, v) = =,, () l = p + ( x, x, λ, λ, h, h ) =, (3) Put L H = ( x, u, p +, q +, λ, h, v ) = =, (4) u u H λ p = ( x, u, p, q,, h, v) (5) x From (5) and ( we have l p = ( x, x, λ, λ, h, h ) x Obvously that from (), (3) and (4) we obtan that (8), (9) and ( hold Put p = ( p, p,, p ), q = ( q, q,, q ) and λ = ( λ, λ, λ, pq, ) We proved that for consdered λ hold (7), (8), (9) and (

6 58 B Marnkovć / Frst and Second Order ecessary Optmalty Condtons From the fact that p m we have that there exsts vector n( + r ( h, v ) R R such that ( and ( hold Snce q ( m) we have that q = or, n other words, we obtan that (3)-(6) hold From hv (, ) = we obtan the system of the equatons hk Ck hk Dk vk =, k =, (6) wth the boundary condtons K ( x, x)( h, h) = ( x, x ) Solvng the equatons (6) we obtan that k = k k s s j j k k s= j= j< s k, h = C h + C D v + D v, k =, holds It s easy to see that from F ( εω, ) ( εω, )[( hv, ),( hv, )] m we obtan the equaton (6) We conclude that K = K 3 SECOD ORDER ECESSRY OPTIMLITY CODITIOS Suppose that the functon f ( xu, ) and mappngs ϕ( x, u) and K( x, x ) are the three tmes contnuously dfferentable Put l l λ H H q (,,,,, ), (,,,,,, ) + = x x λ h h = x u p+ λ h v x x 3 x x 3 For a gven Lagrange multpler λ = ( λ, λ, λ, pq, ) defne the blnear form Where l l l Ω [( h, v),( h, v)] = [ h, h ] + [ h, h ] + [ h, h ] x H H H [, ] [, ] [, ] h h v v h v = = u = xu, H H H H,, and u u are ntroduced analogously as above

7 B Marnkovć / Frst and Second Order ecessary Optmalty Condtons 59 Theorem Let ( x, ω) be the optmal soluton for the problem (-(3) Then there exsts Lagrange multpler λ = ( λ, λ, λ, pq, ) such that the assertons of the theorem hold, and for every ( hv, ) Kholds: Ω [( hv, ),( hv, )] (7) λ Proof: nalogously as n the proof of theorem we consder the mathematcal programmng problem (7)-(9) It s easy to see that for Lagrange vakov functon hold: L L, (, j) : j,(, j) (, );, j u j j From [], theorem, and from precedng facts we obtan that the asserton of theorem hold 4 COCLUDIG REMRKS Frst we shall compare the number of varables and number of equatons from theorem We have that the number of varables ε, ω, ( h, v ) andλ from the theorem s equal to: ( n ( + + r) + + ( n+ k) From (), (3), and (8)-(7) we obtan: + k + n( + + r + n + k + n( + + r n equatons Snce λ then, wthout loss of generalty, we may take that λ = It follows that the number of varables s equal to the number of the equatons and we have complete system of the equatons assocated wth ε, ω, ( h, v ) andλ The par ( x, ω) s sad to be extreme for the problem (-(3) f feasble and f we have satsfed condtons from theorem as n [3] we shall defne the normal extreme for problem (-(3) Defnton : The extremal s to be normal f n k m = R + F and abnormal otherwse ote that = ( εω, ) ( εω, ) Frst we suppose that the extreme ( x, ω) s normal Then we have that ( q, λ ) = and we obtan classcal frst order optmalty condtons whch are known for

8 6 B Marnkovć / Frst and Second Order ecessary Optmalty Condtons a long tme (see [4, 7, 8]) lso theorem becomes a known second order optmalty condtons (see [9]) Suppose that the extreme ( x, ω) s abnormal We shall defne two regular constrant mappng for the problem (-(3) Denote by K the set ( hv, ) F K = ( hv, ): hv (, ), ( εω, )[( hv, ),( hv, )] m ( hv, ) = ( εω, ) Defnton : The constrant mappng F( εω, ) s sad to be - regular at the pont ( εω, ) wth respect to a drecton ( hv, ) Kf co dm K( hv = n + k, ) Suppose the extreme ( x, ω ) s abnormal and that for every ( hv, ) Kthe mappng F( εω, ) s not -regular at the pont ( x, ω) wth respect to a drecton ( hv, ) Then we have that assertons of the theorem and theorem are satsfed for every mnmzng functon f It follows that n that case we have trval theorem The most nterestng case s when the mappng F( εω, ) s -regular at the pont ( εω, ) wth respect to a drecton ( hv, ) K Then we obtan nontrval frst and second order optmalty condtons for abnormal extremes For detals of the precedng facts we refer to [] REFERECES [] vakov, ER, "Extremum condtons n smooth problems wth equalty type of constrants", Comput Math and Math Phys, 5 (5) (985) (n Russan) [] rutyunov, V, Optmalty condtons: bnormal and Degenerate Problems, Moscow, 997 (n Russan) [3] rutyunov, V, "Second order condtons n optmal control problems", Doklady Math, 37 ( () -3 (n Russan) [4] Boltyansk, VG, Optmal Control of Dscrete Systems, Moscow, 973 (n Russan) [5] Vaslev, FP, umercal Methods of Optmal Control Problems, Moscow, 988 (n Russan) [6] Vaslev, FP, Optmzaton Methods, Moscow, (n Russan) [7] Ioffe, D, and Thomrov, VM, Theory of Extremal Problems, Moscow 974 (n Russan) [8] Propo, I, Elements of the Theory of Optmal Dscrete Processes, Moscow, 973 (n Russan) [9] Hlscher, R, and Zedan, V, "Dscrete optmal control: second order optmalty condtons", J Dffer Equatons ppl, Vol8 ( ()