Numerical method for a class of optimal control problems subject to nonsmooth functional constraints

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1 Journal of Computatonal and Appled Mathematcs 17 (008) Numercal method for a class of optmal control problems subject to nonsmooth functonal constrants C.Z. Wu a, K.L. Teo b,, Y Zhao c a School of Mathematcs and Computer Scence, Chongqng Normal Unversty, Chongqng , Chna b Department of Mathematcs and Statstcs, Curtn Unversty of Technology, Perth, Western Australa, Australa c Department of Mathematcs, Zhongshan Unversty, Chna Receved 13 September 005; receved n revsed form December 005 Abstract In ths paper, we consder a class of optmal control problems whch s governed by nonsmooth functonal nequalty constrants nvolvng convoluton. Frst, we transform t nto an equvalent optmal control problem wth smooth functonal nequalty constrants at the expense of doublng the dmenson of the control varables. Then, usng the Chebyshev polynomal approxmaton of the control varables, we obtan an sem-nfnte quadratc programmng problem. At last, we use the dual parametrzaton technque to solve the problem. 007 Elsever B.V. All rghts reserved. MSC: 49N99; 90C34; 90C0 Keywords: Optmal control; Functonal nequalty constrants; Sem-nfnte programmng; Chebyshev seres; Dual parametrzaton 1. Introducton Optmal control problems arse n a wde varety of dscplnes. Apart from tradtonal areas such as aerospace engneerng, robotcs and chemcal engneerng, optmal control theory has also been used wth great success n areas as dverse as economcs to bomedcne. In practce, many optmal control problems are subject to constrants on the state and/or control varables. It s well known that constraned optmal control problems are very dffcult to solve. In partcular, ther analytcal solutons are n many cases out of the queston. Thus, numercal methods are needed for solvng many of these real world problems. There are now many numercal methods avalable n the lterature for varous optmal control problems. See, for example [1,13,,5,7,4,15,11]. In partcular, an effcent optmal control software package, known as MISER3. [6], was developed based on the control parametrzaton technque [13] and control parametrzaton enhancng transform [14]. For constraned optmal control problems, there are varous types of constrants such as bound constrants on the control varables and contnuous state nequalty constrants (see [13]). In [13], a constrant transcrpton method s ntroduced to deal wth the optmal control problems subject to contnuous state nequalty constrants. In [17], an optmal desgn of an envelop-constraned flter wth nput uncertanty s consdered, where the constrants are expressed Correspondng author. Tel.: ; fax: E-mal address: k.l.teo@curtn.edu.au (K.L. Teo) /$ - see front matter 007 Elsever B.V. All rghts reserved. do: /j.cam

2 31 C.Z. Wu et al. / Journal of Computatonal and Appled Mathematcs 17 (008) by nonsmooth functonal nequalty constrants nvolvng convoluton. Ths class of optmal envelope-constraned flter desgn problems has been extensvely studed by many researchers n the past two decades. See, for example [17,3,16]. In ths paper, we consder a class of optmal control problems subject to the above mentoned nonsmooth functonal nequalty constrants. These constrants, whch are expressed n terms of convoluton, are to be satsfed at each tme pont. Ths optmal control problem, whch can be consdered as a generalzed optmal envelope-constraned flter desgn problem, cannot be solved as such usng any standard optmal control technques. Here, we shall make use of the dea of Teo et al. [1] to transform these nonsmooth functonal nequalty constrants nto equvalent smooth functonal nequalty constrants at the expense of doublng the dmenson of the control varables, whch leads to an equvalent optmal control problem subject to smooth functonal nequalty constrants. Consequently, by usng the Chebyshev polynomal approxmaton of the control varables, we obtan an approxmate optmal control problem, whch s further reduced nto an equvalent sem-nfnte quadratc programmng problem. Then, the dual parametrzaton technque obtaned n [8] s used to develop an effcent method for solvng the equvalent sem-nfnte programmng problem. The convergence analyss supportng ths computatonal method wll also be presented. The rest of the paper s organzed as follows. In Secton, the problem to be studed s formulated, whch s an optmal control problem subject to nonsmooth functonal nequalty constrants nvolvng convoluton. In Secton 3, the dea appeared n [1] s used to transform ths nonsmooth functonal nequalty constraned optmal control problem nto an equvalent optmal control problem subject to smooth functonal constrants. Then, by usng the Chebyshev polynomal approxmaton of the control varables, we convert the optmal control problem nto an equvalent semnfnte programmng problem. Subsequently, the dual parametrzaton technque reported n [8] (or ts mproved verson reported n [9]) s used to develop an effcent computatonal method for solvng the sem-nfnte programmng problem. In Secton 4, a numercal example s solved usng the proposed approach. Secton 5 concludes the paper.. Problem formulaton Consder a process descrbed by the followng system of lnear ordnary dfferental equatons defned on a fxed tme nterval (,t f ]: ẋ(t) = A(t)x(t) B(t)u(t), (.1a) where x =[x 1,x,...,x n ] R n, u=[u 1,u,...,u m ] R m are, respectvely, the state and control vectors; the superscrpt denotes the transpose. The ntal condton for the dfferental (.1a) s x( ) = x 0, (.1b) where x 0 =[x 0 1,x0,...,x0 n ] R n s a gven vector. Defne U ={v =[v 1,v,...,v m ] R m : a v b,= 1,...,m}, where a and b, = 1,...,m, are real numbers. A contnuous functon u =[u 1,u,...,u m ] from [0,T] nto R m s sad to be an admssble control f u(t) U for t (0,T]. Let U be the class of all such admssble controls. For each u U, let x( u), t [0,T], be the soluton of dfferental equaton (.1a) wth ntal condton (.1b) correspondng to the control u U. We consder the followng nonsmoothng functonal nequalty constrant on the control varables: φ tf m (t τ)u(τ) dτ d(t) θ (t τ) u (τ) dτ ε(t) 0, (.) where φ(t), ε(t), and θ (t), = 1,...,m, are gven real-valued functons. If u U s such that the constrant (.) s satsfed, then t s called a feasble control. Let F be the class of all such feasble controls.

3 C.Z. Wu et al. / Journal of Computatonal and Appled Mathematcs 17 (008) We may now state our optmal control problem as follows: Problem (P). Gven system (.1), fnd a control u F such that the cost functonal J (u) = x (t f )R 0 x(t f ) x (τ)r 1 (t τ)x(t) dτ dt u (τ)r (t τ)u(t) dτ dt (.3) s mnmzed over F, where R 0, R 1 and R are gven real-valued functons. The followng condtons are mposed n the rest of the paper. (A1) There exsts a soluton for Problem (P). (A) The matrx valued functons A(t) and B(t) are contnuously dfferentable wth respect to t. (A3) The functons φ(t), ε(t), d(t) and θ (t), = 1,...,m, are all contnuously dfferentable wth respect to t. (A4) The ntegral kernel operators K 1 x(t) = R 1 (t τ)x(τ) dτ, K u(t) = R (t τ)u(τ) dτ are all postve defnte and the matrx R 0 s also postve defnte,.e., x(t)r 1 (t τ)x(τ) dτ dt 0, u(t)r (t τ)u(τ) dτ dt 0. (A5) The slater condton s satsfed,.e., there exsts a control u(t), such that φ tf m (t τ)u(τ) dτ dτ d(t) θ (t τ) u (τ) dτ ε(t) < 0 for all t [,t f ], where and u(t) = 0 for all t [,t f ], a <u (t)<b for t [,t f ], = 1,...,m. Remark.1. Note that Problem (P) wth A(t) = 0, n = 1, b(t)= 1, R 0 = 0 and R 1 = 0 reduces to the optmal desgn of envelope-constraned flters wth nput uncertanty consdered n [17]. Ths class of envelope-constraned flter desgn problems wth nput uncertanty has been extensvely studed n the past two decades, see, for example [9 11]. The functonal nequalty constrants (.), whch are nondfferentable wth respect to u, are n the form of convoluton of the control varable u. The cost functonal s n the form of double ntegral. Thus, Problem (P ) cannot be solved drectly by standard optmal control methods such as those developed n [13]. 3. Problem transformaton In ths secton, we shall show that the nondfferentable nequalty constrant on the control varable can be converted nto equvalent dfferentable nequalty constrants. For ths, we need the followng lemma, whch s quoted from [1]. Lemma 3.1. For each x R n, let p : R n R n, and q: R n R n be defned by p(x) =[max{x 1, 0}, max{x, 0},...,max{x n, 0}], q(x) =[max{ x 1, 0}, max{ x, 0},...,max{ x n, 0}],

4 314 C.Z. Wu et al. / Journal of Computatonal and Appled Mathematcs 17 (008) Then (1) p(x) 0, q(x) 0, () p(x) q(x) = 0, (3) p(x) q(x) = x, p(x) q(x) = x =[ x 1, x,..., x n ], (4) for each x, y R n wth x, y 0, there exsts a unque nonnegatve functon r : R n R n R n, such that x y =x y r(x,y), where x =[x 1,x,...,x n ] 0 means x 0,= 1,...,n. We may now defne formally the followng optmal control problem. Problem (EP). Gven the system ẋ(t) = A(t)x(t) B(t)(u 1 (t) u (t)), t (,t f ], (3.1a) wth the ntal condton x( ) = x 0 (3.1b) fnd a contnuous control ũ(t) =[(u 1 (t)),(u (t)) ] C(,t f ; R m ), where C(,t f ; R m ) s the set of contnuous functons from [,t f ] to R m, such that the cost functonal J(ũ) = x (t f )R 0 x(t f ) x (τ)r 1 (t τ)x(t) dτ dt (u 1 (τ) u (τ)) R (t τ)(u 1 (t) u (t)) dτ dt (3.) s mnmzed subject to the functonal nequalty constrants: φ (t τ)(u 1 (τ) u (τ)) dτ d(t) φ (t τ)(u 1 (τ) u (τ)) dτ d(t) a [I m, I m ]ũ(t) b, m θ (t τ)(u 1 (τ) u (τ)) dτ ε(t) 0, (3.3a) m θ (t τ)(u 1 (τ) u (τ)) dτ ε(t) 0, (3.3b) (3.3c) β ũ(t) =[(u 1 (t)),(u (t)) ] 0 for all t [,t f ] (3.3d) where I m denotes the m m dentty matrx, u =[u 1,u,...,u m ], = 1,, β =[β 1,...,β m ], β = a b, = 1,...,m, and β m = a b, = 1,...,m. Now we have the followng theorem: Theorem 3.1. Problem (P) s equvalent to Problem (EP) n the sense that f ũ =[u 1,,u, ] s the optmal soluton of Problem (EP), then u = u 1, u, s an optmal soluton of Problem (P). On the other hand, f u s an optmal soluton of Problem (P),then [p(u ), q(u )] s an optmal soluton of Problem (EP) and J(u ) = J(ũ ). Proof. Suppose that ũ = ((u 1, ),(u, ) ) s an optmal soluton of Problem (EP), and u s an optmal soluton of Problem (P ). Let u = u 1, u,, ũ =[p (u ), q (u )].

5 C.Z. Wu et al. / Journal of Computatonal and Appled Mathematcs 17 (008) Clearly, u U, and ũ C(,t f ; R m ) s such that (3.3c) and (3.3d) are satsfed. Note that φ tf m (t τ)u(τ) dτ d(t) θ (t τ) u (τ) dτ ε(t) = = 0. φ (t τ)(u 1, (τ) u, (τ)) dτ d(t) m θ (t τ) u 1, (τ) u, (τ) dτ ε(t) φ (t τ)(u 1, (τ) u, (τ)) dτ d(t) m θ (t τ)(u 1, (τ) u, (τ) r(u 1, (τ), u, (τ))) dτ ε(t) φ (t τ)(u 1, (τ) u, (τ)) dτ d(t) m θ (t τ)(u 1, (τ) u, (τ)) dτ ε(t) Ths mples that u = u 1, u, satsfes the nequalty constrant (.) of Problem (P), and hence u F. At the same tme, the soluton x( u 1,,u, ) of the dynamcal system (3.1a) and (3.1b) s the same as the soluton x( u) of the dynamcal system (.1a) and (.1b). Thus, J (u) = x (t f u)r 0 x(t f u) x (τ u)r 1 (t τ)x(t u) dτ dt u (τ)r (t τ)u(t) dτ dt = x (t f u 1,,u, )R 0 x(t f u 1,,u, ) x (τ u 1,,u, )R 1 (t τ) x(t u 1,,u, ) dτ dt (u 1, (τ) u, (τ)) R (t τ)(u 1, (t) u, (t)) dτ dt = J(u 1,,u, ) and hence J(u ) J (u) = J(u 1,,u, ). (3.4) By the same token, we can show that J(u 1,,u, ) J(ũ) = J(u ). (3.5) Therefore, the concluson of the theorem follows from (3.4) and (3.5).

6 316 C.Z. Wu et al. / Journal of Computatonal and Appled Mathematcs 17 (008) Denote [ ] Im B(t) = B(t)[I m, I m ], R (t τ) =, R (t τ)[i m, I m ], I m φ (t τ) = φ (t τ)[i m, I m ], θ (t τ)[i m,i m ]. Then, Problem (EP) can be wrtten more concsely as Gven the dynamcal system ẋ(t) = A(t)x(t) B(t)ũ(t), t (,t f ], (3.6a) wth the ntal condton x( ) = x 0 (3.6b) fnd a contnuous control ũ(t) =[(u 1 (t)),(u (t)) ] such that the cost functonal J(ũ) = x (t f )R 0 x(t f ) x (τ)r 1 (t τ)x(t) dτ dt s mnmzed over C(,t f ; R m ) and subject to the functonal nequalty constrants: ( φ(t τ)) ũ(τ) dτ d(t) ( φ(t τ)) ũ(τ) dτ d(t) a [I m, I m ]ũ(t) b, ũ (τ) R (t τ)ũ(τ) dτ dt (3.7) (θ(t τ)) ũ(τ) dτ ε(t) 0, (3.8a) (θ(t τ)) ũ(τ)dτ ε(t) 0 (3.8b) (3.8c) β ũ(t) =[(u 1 (t)),(u (t)) ] 0 for all t [,t f ]. (3.8d) Ths problem s agan denoted by Problem (EP). We have the followng theorem. Theorem 3.. The cost functonal J(ũ) s a strctly convex quadratc functonal of u on C(,t f ; R m ), and the Slater condton s stll satsfed for Problem (EP),.e., there exts a ũ C(,t f ; R m ), such that the followng condtons are satsfed: φ (t τ)ũ(τ) dτ d(t) φ (t τ)ũ(τ) dτ d(t) a<[i m, I m ]ũ(t) < b, θ (t τ)ũ(τ) dτ ε(t) < 0, β > ũ(t) =[u 1 (t), u (t)] > 0 for all t [,t f ]. θ (t τ)ũ(τ) dτ ε(t) < 0, Proof. By Assumpton (A5), the ntegral kernel operator K u(t) s postve defnte. Thus, the ntegral kernel operator K u(t) = R (t τ)ũ(τ) s also postve defnte. Note the form of the cost functonal J(ũ), t s easy to verfy the valdty of the concluson of the theorem. The Slater condton s clearly satsfed.

7 C.Z. Wu et al. / Journal of Computatonal and Appled Mathematcs 17 (008) The method of Chebyshev approxmaton In ths secton, we shall frst approxmate the optmal control problem (EP) whch s a sem-nfnte quadratc programmng problem by Chebyshev seres approxmaton, then we use the dual parametrzaton method developed n [7] to deal wth the functonal nequalty constrants. Let Φ(t, τ) denote the fundamental matrx of the dynamcal system (3.6). For a gven control ũ C(,t f ; R m ), the correspondng trajectory of dynamcal system (3.6) s gven by t x(t ũ) = Φ(t, )x 0 Φ(t, τ) B(τ)ũ(τ) dτ. (4.1) To convert the optmal control problem (EP) nto a quadratc programmng problem, we parameterze each element ũ (t), =1,...,m, of the control varable ũ(t) by a fnte length Chebyshev seres of the frst class, {T n (t)=cos(n cos t)}, wth unknown parameters. But before approxmatng the control varables, t s necessary to transform the tme horzon t [,t f ] of Problem (EP) nto the new nterval s [, 1], because the Chebyshev seres are defned on the nterval [, 1]. Ths can be acheved by usng the scalng t = t f (s 1). For the smplcty of the notaton, we assume that = and t f = 1. Approxmate each element ũ (t), = 1,...,m, of the control varable ũ(t) s approxmated by a fnte Chebyshev seres wth unknown parameters gves ũ (t α N, ) = N j=0 α N, j T j (t), = 1,...,m, (4.) where T j (t) s the jth order Chebyshev polynomal of the frst type and α N, j, = 1,...,m; j = 1,...,N, α N, = [α N, 1, α N,,...α N, N ] are the unknown parameters. Defne and α N =[α N,1 1, α N,1,...,α N,1 N, αn, 1, α N,,...,α N, N,...,αN,m 1, α N,m,...,α N,m N ũ(t α N ) =[ũ 1 (t α N,1 ), ũ (t α N, ),...,ũ m (t α N,m )], where ũ (t α N, ), = 1,...,m, are gven by (4.). Clearly, ũ( α N ) depends on the choce of α N. Thus, the soluton of (4.1) correspondng to ũ( α N ) can be wrtten as x( α N ). Now, by approxmaton the control n the form of (4.), Problem (EP) reduces to the followng sequence of sem-nfnte quadratc programmng problem. Problem (EP(N)). mn J N (α N ) s.t h N (α N,t) 0,t [, 1], (4.3) where α N R mn, J N (α N ) = x (1 ũ(t α N ))R 0 x(1 ũ(t α N )) x (τ ũ(t α N ))R 1 (t τ)x(t ũ(t α N )) dτ dt ũ (τ α N ) R (t τ)ũ(τ α N ) dτ dt, ]

8 318 C.Z. Wu et al. / Journal of Computatonal and Appled Mathematcs 17 (008) φ (t τ)ũ(τ α N ) dτ d(t) θ (t τ)ũ(τ α N ) dτ ε(t) h N (α N,t)= [I m, I m ]ũ(t α N ) b [ I m,i m ]ũ(t α N ) a ũ(t α N ) β ũ(t α N ) φ (t τ)ũ(τ α N ) dτ d(t) θ (t τ)ũ(τ α N ) dτ ε(t). Theorem 4.1. Consder Problem (EP(N)). Then, J N (α N ) = 1 (αn ) Q N α N (p N ) α N e N, (4.4) and h N (α N,t)= A N (t)α N q(t), (4.5) where [ ] 1 Q N = F(1, τ)(i m T N (τ)) dτ R 0 F(1, τ)(i m T N (τ)) dτ [ t τ F (t, s)(i m T N (s)) ds] R 1 (t τ) F(τ, s)(i m T N (s)) ds dτ dt (I m T N (t)) R (t τ)(i m T N (τ)) dτ dt, (p N ) = [Φ(1, )x 0 ] F(1, τ)(i m T N (τ)) dτ τ [Φ(t, )x 0 ] R 1 (t τ) F(τ, s)(i m T N (s)) ds dτ dt, e N =[Φ(1, )x 0 ] R 0 Φ(1, )x 0 [Φ(t, )x 0 ] R 1 (t τ)φ(τ, )x 0 dτ dt, [ φ (t τ) θ (t τ)](i m T N (τ)) dτ [ φ (t τ) θ (t τ)](i m T N (τ)) dτ A N (t) = [I m, I m ](I m T N (t)), [ I m,i m ](I m T N (t)) I m T N (t) I m T N (t) q(t) =[d(t) ε(t), d(t) ε(t), b, a,β, 0], T N (t) =[T 1 (t), T (t),...,t N (t)], F(t,τ) = Φ(t, τ) B(τ), and denotes the Kronecker product.

9 C.Z. Wu et al. / Journal of Computatonal and Appled Mathematcs 17 (008) Proof. Note that (4.) can be wrtten n terms of the Kronecker product as ũ(t α N ) = (I m T N (t))α N, where I m denotes the m m dentty matrx. By drect computaton and usng (4.1), t follows that J N (α N ) and h N (α N,t)are gven by (4.4) and (4.5), respectvely. Remark 4.1. Note that the cost functon (4.4) s a strctly postve quadratc functon, whle the constrants gven by (4.5) are lnear. Thus, Problem (EP(N)) s a strctly convex quadratc sem-nfnte programmng problem. The next theorem shows that the Slater condton s satsfed f (A5) holds. Theorem 4.. Suppose that (A5) s satsfed. Then, there exsts an nteger N 0 such that for N>N 0, the Slater condton s also satsfed for Problem (EP(N)),.e., there exsts an α N R mn, such that h N (α N,t)= A N (t)α N q(t)<0 for all t [, 1]. Proof. Snce (A5) s satsfed, t follows from Theorem 4. that there exsts a control ũ C(, 1; R m ), such that and φ (t τ)ũ(τ α N ) dτ d(t) θ (t τ)ũ(τ α N ) dτ ε(t) < 0, φ (t τ)ũ(τ α N ) dτ d(t) θ (t τ)ũ(τ α N ) dτ ε(t) < 0, a<[i m, I m ]ũ(t α N )<b, β > ũ(t) α N =[u 1 (t αn ), u (t αn )] > 0 for all t [,t f ]. Snce ũ C(, 1; R m ) and φ(t) and θ(t) are contnuous on the nterval [, 1], there exsts a δ, 1 > δ > 0, such that and φ (t τ)ũ(τ α N ) dτ d(t) θ (t τ)ũ(τ α N ) dτ ε(t) δ, φ (t τ)ũ(τ α N ) dτ d(t) θ (t τ)ũ(τ α N ) dτ ε(t) δ, ũ(t α N ) δ for all t [, 1], max φ (t τ) θ (t τ) dτ 1 t 1 4mδ, max t 1 φ (t τ) θ (t τ) dτ 1 4mδ, = 1,...,m, = 1,...,m. By Jackson Theorem ([18, p. 19, Chapter ]), there exsts an nteger N 0, such that for N N 0, max ũ (t) ũ (t α N, ) δ δ t 1, = 1,...,m,

10 30 C.Z. Wu et al. / Journal of Computatonal and Appled Mathematcs 17 (008) and the followng nequaltes are satsfed: a<[i m, I m ]ũ(t α N )<b, β > ũ(t α N ) =[(u 1 (t α N )),(u (t α N )) ] > 0 for all t [,t f ]. Therefore, we have = φ (t τ)ũ(τ α N ) dτ d(t) m θ (t τ)ũ(τ α N ) dτ ε(t) ( φ (t τ) θ (t τ))(ũ (τ α N, ) (ũ (t α N, ) ũ (t α N, ))) dτ d(t) ε(t) ( φ (t τ) θ (t τ))ũ(τ α N ) d(t) ε(t) m φ (t τ) θ (t τ) ũ (t α N, ) ũ (t α N, ) ( φ (t τ) θ (t τ))ũ(τ) d(t) ε(t) δ δ = δ. m 1 4mδ δ By the same token, we obtan φ (t τ)ũ(τ) dτ d(t) θ (t τ)ũ(τ) dτ ε(t) δ. Ths completes the proof. Snce e N s a constant n Problem (EP(N)), t can be dropped. Hence, we only need to consder the followng equvalent problem: Problem (MEP(N)). mn J N (α N ) = 1 (αn ) Q N α N p N αn (4.6) s.t. h N (α N,t)= A N (t)α N q(t) 0. (4.7) The Dorn s dual of Problem (MEP(N)) s Problem (DMEP(N)). mn α N,ν 1 (αn ) Q N α N s.t. Q N α N p N ν N M ([, 1]), α N R mn, q (t) dν N (t) (4.8) A N (t) dν N (t) = 0, (4.9) where M ([, 1]) s the set of all nonnegatve bounded regular Borel measures on [, 1].

11 C.Z. Wu et al. / Journal of Computatonal and Appled Mathematcs 17 (008) To proceed further, we need some crucal results obtaned n [9]. They are presented n the followng theorem. Theorem 4.3. Let the Slater constrant qualfcaton be satsfed. The mnmum of Problem (MEP(N)) s acheved at α N f and only f α N s feasble and there exsts a ν N M ([, 1]), such that Q N α N p N A N (t) dν N (t) = 0, ν N 0. (A N (t)αn q(t))dν N (t) = 0, Proof. See the proof gven for Lemma.1 of [9]. Theorem 4.4. Let Assumpton (A5) be satsfed, and assume that the mnmum of Problem (MEP(N)) s acheved at α N. Then the soluton of the dual problem (DMEP(N)) contans a soluton (α N, ν N ) of whch the measure ν N has a fnte support of no more than mn ponts. Proof. See the proof gven for Theorem.1 of [9]. On the bass of the above theorems, solvng the sem-nfnte programmng problem (MEP(N)) s equvalent to solvng the dual problem (DMEP(N)), whch s a fnte dmensonal optmzaton problem. If we restrct the dscrete measure wth fnte support of k mn ponts, then Problem (DMEP(N)) s reduced to the followng Problem ((DMEP(N)) k. Problem ((DMEP(N)) k. mn α N,ν,τ JN k (αn, ν N, τ k ) = 1 k (αn ) Q N α N q(t ) ν s.t. Q N α N p N ν 0, k A N (t )ν = 0, t 1, = 1,,...,k, where ν = ν(t ), τ k = (t 1,t,...,t k ) and ν N = (ν 1, ν,...,ν k ). Once a soluton (α N, ν N, τ k ) of Problem (DMEP(N)) k whch satsfes the condton n Step 7 of Algorthm, s acheved, then ũ(t α N ) = (I m T N (t))α N s an optmal soluton of Problem (EP(N)). Next theorem presents the relatonshp between the soluton of Problem (EP) and that of Problem (EP(N)). Theorem 4.5. Let ũ (t) be the optmal soluton of Problem (EP), and α N, the optmal soluton of Problem (EP(N)). Then, J(α N ) J(ũ (t)) as N. For suffcently large N, ũ(t α N ) = (I m T N (t))α N can be vewed as an optmal soluton of Problem (EP). Proof. Let α,n =[α 1,, α,,...,α N, ], ũ,n (t) = N n=1 α n, T n (t). It s clear that J(ũ (t)) J(ũ,N (t)) = J(α N ) J(α,N ). On the other hand, we know that J(ũ,N (t)) J(ũ (t)). Thus, the proof s complete.

12 3 C.Z. Wu et al. / Journal of Computatonal and Appled Mathematcs 17 (008) We now propose an algorthm to solve Problem (P) as follows: Algorthm 1. Transform Problem (P) wth nonsmooth nequalty constrants nto equvalent Problem (EP).. Rescale tme t [,t f ] nto the new tme scale s [, 1]. 3. Set N = Approxmate the control varable ũ(t) n the form of (4.). Then, express Problem (EP) n the form of (4.5) and (4.6). 5. Set k = Compute the optmal soluton (α N, ν k N, τ k ) and J N k (αn, ν k N, τ k ) of Problem ((DMEP(N)) k. 7. If k and JN k J N k s suffcently small, set αn = α N and goto Step 8, otherwse, set k = k 1, goto Step Compute J N (α N ). 9. If N and J N (α N ) J N (α N ) s suffcently small, goto Step 10. Otherwse, set N = N 1, goto Step Let ũ = (I m T N (t))α N. Then,u =[I m, I m ]ũ s an approxmate optmal soluton of Problem (P). 5. Numercal example To llustrate the effcency of the proposed method, we wll present a numercal example n ths secton. Example 5.1. Gven the dynamcal system ẋ 1 (t) = x (t), ẋ (t) = x 1 (t) u(t), (5.1a) (5.1b) wth ntal condton x 1 () = 0, x () =. (5.1c) The cost functonal s J (u) = x 1 (1) x (1) u(t)e (t τ) u(τ) dτ dt ( ) t τ (x 1 (t) x (t)) cos (x 1 (τ) x (τ)) dτ dt. (5.) Our problem s to fnd a control u C[,t f ], such that J (u) s mnmzed subject to the dynamcal system (5.1a) (5.1c), and the followng functonal nonsmooth nequalty constrant: (t τ) u(τ) dτ 1. (5.3) We transform t nto the followng equvalent optmal control problem wth smooth nequalty constrants. Problem (EEP). The dynamcal system s ] [ ][ ] [ẋ1 (t) 0 1 x1 (t) = ẋ (t) 0 x (t) wth ntal condton x(0) =[0, ]. The cost functonal s J(ũ) = x (1)R 0 x(1) [ ][ ] u1 (t), u (t) x(t)r 1 (t τ)x(τ) dτ dt ũ(t)r (t τ)ũ(τ) dτ dt

13 C.Z. Wu et al. / Journal of Computatonal and Appled Mathematcs 17 (008) and the constrants are where (t τ) [1, 1]ũ(τ) dτ 1, ũ(t) 0, t [, 1], x(t) =[x 1 (t), x (t)], ũ(t) =[u 1 (t), u (t)], ( ) ( ) t τ t τ [ ] cos cos 1 0 R 0 =, R (t τ) = ( ) ( ) t τ t τ, cos cos [ ] e (t τ) e R (t τ) = (t τ). e (t τ) e (t τ) We approxmate the control varable ũ(t) n Problem (EEP) by an Nth order Chebyshev seres, and the trajectory x(t) s obtaned from (4.1). Problem (EEP) s transformed nto the standard form expressed by (4.4) and (4.5) as stated n Theorem 4.1. Then, by usng the dual parametrzaton method developed n [7], Problem (EEP) can be solved. We solve the problem for N = 6, 8, 10. The results are gven as follows: Case N = 6: The value of the cost functonal s , α 6 =[0.0369, , , , , 0.083, , , , , 0.058, 0.085]; Case N = 8: The value of the cost functonal s , α 8 =[0.0337, 0.038, 0.058, , 0.064, 0.096, , , , 0.041, 0.058, , 0.064, 0.096, , ]. Case N = 10: The value of the cost functonal s , α 10 =[0.034, , 0.058, , 0.06, 0.098, 0.093, , 0.004, , 0.058, , 0.065, 0.087, 0.093, , , 0.001, , ]. The correspondng optmal controls obtaned are depcted n Fgs From the numercal experence, the method s developed n the paper s effectve N=6 0 u(t) t Fg. 1. The optmal control when N = 6.

14 34 C.Z. Wu et al. / Journal of Computatonal and Appled Mathematcs 17 (008) N= u(t) Fg.. The optmal control when N = 8. t N= t Fg. 3. The optmal control when N = Concluson In ths paper, we consdered an optmal control problem subjected to nonsmooth contnuous nequalty constrants. We frst showed that t s equvalent to an optmal control problem subject to smooth contnuous nequalty constrants. Then, we developed an effcent algorthm to solve the equvalent problem usng the parametrzaton of control varable by fnte Chebyshev seres expanson. A numercal example was then solved usng the method proposed. We see that the method s hghly effcent. Acknowledgements The authors would lke to thank the anonymous referees for ther detaled comments.

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