Majewski, Matematyka, Artykuły
[ Pobierz całość w formacie PDF ]
//-->Multidim Syst Sign Process (2013) 24:657–665DOI 10.1007/s11045-012-0207-2On the existence of an optimal solution of the Mayerproblem governed by 2D continuous counterpartof the Fornasini-Marchesini modelDorota Bors·Marek MajewskiReceived: 30 May 2012 / Revised: 4 October 2012 / Accepted: 10 October 2012 /Published online: 23 October 2012© The Author(s) 2012. This article is published with open access at Springerlink.comAbstractIn the paper the optimization problem described by some nonlinear hyperbolicequation being continuous counterpart of the Fornasini-Marchesini model is considered. Atheorem on the existence of at least one solution to this hyperbolic PDE is proved and someproperties of the set of all solutions are established. The existence of a solution to an optimiza-tion problem under appropriate assumptions is the main result of this paper. Some applicationof the obtained results to the process of gas filtration is also presented.KeywordsMayer problem·Continuous counterpart of the Fornasini-Marchesini model·Existence of optimal solutions1 IntroductionIn this paper we consider an optimal control problem governed by system of hyperbolicequations of the form∂2z(x,y)=f∂x∂ yx, y,∂z∂z(x,y),(x,y),z(x,y),u(x,y)∂x∂y(1)for almost every(x,y)∈P:=[0, 1]×[0, 1] with the cost indicator1J(z)=F t,ϕ (t) , ϕ (t) , ψ (t) , ψ (t)dt+gϕ (0) , ϕ (0) , ψ (0) ,whereϕ (t)=z(t,0) andψ (t)=z(0,t)for everyt∈[0, 1].D. Bors·M. Majewski (B)Faculty of Mathematics and Computer Science, University of Lodz, Lodz, Polande-mail: marmaj@math.uni.lodz.plD. Borse-mail: bors@math.uni.lodz.pl123658Multidim Syst Sign Process (2013) 24:657–665System (1) can be viewed as a continuous nonlinear version of the Fornasini-Marchesinimodel (cf.Fornasini and Marchesini 1978/1979; Kaczorek 1985; Klamka 1991),which iswell known in the theory of discrete multidimensional systems. It should be underlined thatsuch discrete systems play an important role in the theory of automatic control (cf.Fornasiniand Marchesini 1976).Moreover, continuous systems of the form specified by (1) can beused for modelling of the process of gas absorption (cf.Idczak et al. 1994; Tikhonov andSamarski 1990)for which some numerical results can be found inRehbock et al.(1998). Forrelated results on Fornasini-Marchesini models one can see Cheng et al. (2011), Yang et al.(2007), Idczak (2008).Furthermore, it should be noted that system (1) was investigated in many papers apartfrom the aforementioned ones. Specifically, the problem of the existence and uniquenessof solutions to (1) with boundary conditionsϕ (t)=z(t,0) andψ (t)=z(0,t)has beenproved for the linear case inIdczak and Walczak(2000) and for the nonlinear case inIdczakand Walczak(1994). Moreover, some results establishing the existence of optimal solutionsfor the problem governed by (1) can be found inIdczak and Walczak(1994) for the case ofthe Lagrange problem with controls with bounded variation, inIdczak et al.(1994) for thecase of the problem with the cost of rapid variation of control, and inMajewski(2006) forthe case of the Lagrange problem with integrable controls. It should be underlined that bothinIdczak and Walczak(2000) andIdczak and Walczak(1994) zero initial conditions wereconsidered. While in this paper the problem with general initial conditions are treated. Ourconsiderations involve the minimization of the cost functional which depends on the bound-ary values of the solutions to the PDE. The situation in which the boundary data appear inthe cost functional is referred to as the classical Mayer problem for ODEs. Our extensioncan be seen as a new contribution towards the Mayer problem governed by PDEs which canbe useful in many practical applications.The paper is organized as follows. In Sect.2,the optimization problem is formulatedand the space of solutions is defined. Section3is devoted to formulation of the requiredassumptions. Next, in Sect.4,the theorem on the existence of a solution to the system (1)is proved and some properties of the set of all solutions are stated. Subsequently, the mainresult of the paper can be proved, namely the theorem stating that under some assumptionsoptimal control problem possesses at least one solution. Finally, in Sect.5,an application ofthe obtained results to the process of gas filtration is presented.2 Formulation of the problemThe problem under consideration is as follows:Find a minimum of the functional1J(z)=F t,ϕ (t) , ϕ (t) , ψ (t) , ψ (t)dt+gϕ (0) , ϕ (0) , ψ (0) ,(2)subject to∂2z(x,y)=f∂x∂ yx, y,∂z∂z(x,y),(x,y),z(x,y),u(x,y)∂x∂yfor a.e.(x,y)∈P:=[0, 1]×[0, 1](3)whereϕ (t)=z(t,0) andψ (t)=z(0,t)fort∈[0, 1],123Multidim Syst Sign Process (2013) 24:657–665659z∈Z:=z∈AC P,RN:z(·,0),z(0,·) ∈H2[0, 1],RN,(4)(5)u∈U:=u:P→RM:uis measurable andu(x,y)∈Ufor a.e.(x,y)∈PwhereU⊂RMis a given compact set.In the definition ofZgiven in (4),AC P,RNdenotes the set of absolutely continuousfunctions of two variables defined onP.A functionz:P→Ris said to be absolutelycontinuous onPif1.the associated functionFzof an interval defined by the formulaFz([x1,x2]×[y1,y2])=z(x2,y2)−z(x1,y2)+z(x1,y1)−z(x2,y1)2.for all intervals [x1,x2]×[y1,y2]⊂Pis an absolutely continuous function of aninterval (seeŁojasiewicz(1988) for details),the functionsz(·,0) andz(0,·)are absolutely continuous on [0, 1].A functionz=(z1, . . . ,zN):P→RNis said to be absolutely continuous onPifall coordinates functionsziare absolutely continuous onPfori=1,. . .N.In the paperWalczak(1987), the author proved that a functionz:P→RNis absolutely continuous if1 2and only if there exist functionslz∈L1P,RN,lz,lz∈L1[0, 1],RN, and a constantc∈RNsuch thatxyxy1lz(s)ds+2lz(t)dt+cz(x,y)=lz(s,t) dsdt+(6)for all(x,y)∈P.Moreover, an absolutely continuous functionzhaving the representation(6) possesses, in the classical sense, the partial derivatives∂z(x,y)=∂x∂z(x,y)=∂yy1lz(x,t) dt+lz(x) ,x2lz(s,y) ds+lz(y) ,∂2z(x,y)=lz(x,y)∂x∂ yfor a.e.(x,y)∈P.It is obvious thatz∈Zif and only if it has the following representationxyz(x,y)=l(s,t) dsdt+ϕ (x)+ψ (y)−z(0,0) for(x,y)∈P,(7)∂zϕ (x) ,∂y(x,y)=∂∂x∂zy(s,y) ds+ψ (y)for a.e.(x,y)∈P.ByH2[0, 1],RNwe denote the space of absolutely continuous functions defined on[0, 1] such thatxis absolutely continuous andx∈L2[0, 1],RN.x2wherel∈L1P,RN, ϕ, ψ∈H2[0, 1],RNandϕ (0)=ψ (0) .Furthermore,we have thatϕ (x)=z(x,0), ψ (y)=z(0,y)forx, y∈[0, 1] andzpossesses22y2∂z ∂z∂zderivatives∂∂x∂zy,∂x,∂yand∂∂x∂zy(x,y)=l(x,y),∂x(x,y)=∂∂x∂zy(x,t) dt+123660Multidim Syst Sign Process (2013) 24:657–6653 Basic assumptionsIn the paper we shall use the following assumptions.(A1) The functionP(x,y)→f(x,y, z1,z2,z, u)∈RNis measurable for(z1,z2,z, u)∈RN×RN×RN×RMand the functionRMu→f(x,y, z1,z2,z, u)∈RNis continuous for(z1,z2,z)∈RN×RN×RNand a.e.(x,y)∈P.(A2) There exists a constantL>0 such that|f(x,y, z1,z2,z, u)−f(x,y,w1, w2, w,u)|≤L(|z−w|+ |z1−w1| + |z2−w2|)for(z1,z2,z), (w1, w2, w)∈RN×RN×RN,u∈Uand a.e.(x,y)∈P.(A3) There existsb>0 such that|f(x,y,0, 0, 0,u)|≤bfor a.e.(x,y)∈Pandu∈U.(A4) The function[0, 1]t→F(t, v)∈RNis measurable for everyv∈R4Nand the functionR4Nv→F(t, v)∈RNis continuous for a.e.t∈[0, 1].(A5) For every bounded setB⊂R4Nthere is a functionυB∈L1[0,1],R+such thatF(t, v)≤υB(t)for a.e.t∈[0, 1] and everyv∈B.(A6) There are positive constantsαiand functionsβi∈L2([0,1],R) , γi∈L1([0,1],R) ,i=1, 2, 3, 4 such that4F(t, v1, v2, v3, v4)≥i=1αi|vi|2+βi(t)|vi| +γi(t)for a.e.t∈[0, 1] and everyvi∈RN,i=1, 2, 3, 4.(A7) The functiong:R3N→Ris lower semicontinuous and coercive, i.e.g(v)→ ∞if|v| → ∞.4 Existence of solution and the main resultTo begin with we shall prove the theorem on the existence of solution to the system (3). Wealso formulate some properties of the set of all solutions.Theorem 1Let assumptions (A1)–(A4) be satisfied. Then, for each control u∈U, and eachϕ, ψ∈H2[0, 1],RNsuch thatϕ (0)=ψ (0)there exists a unique solution zu,ϕ,ψ∈Zto(3) satisfying condition zu,ϕ,ψ(x,0)=ϕ (x) ,and zu,ϕ,ψ(0,y)=ψ (y)for x, y∈[0, 1].123Multidim Syst Sign Process (2013) 24:657–665661Moreover, for any c>there existsρ >such that ifϕ, ψ∈H2[0, 1],RN, ϕ (0)=ψ (0)and|ϕ(x)| ,|ψ(x)| , ϕ (x) , ψ (x)≤c for x∈[0, 1],then∂zu,ϕ,ψ∂zu,ϕ,ψ∂2zu,ϕ,ψ(x,y),(x,y),(x,y),zu,ϕ,ψ(x,y)≤ρ∂x∂ y∂x∂yfor a.e.(x,y)∈P and u∈U.ProofFor a fixedu∈Uandϕ, ψ∈H2[0, 1],RNsuch thatϕ (0)=ψ (0),consider theoperatorT:L1P,RN→L1P,RNdefined by⎛yxT(l) (x,y)=f⎝x, y,xyl(x,t) dt+ϕ (x) ,l(s,y) ds+ψ (y) ,⎞l(x1,y1)d x1dy1+ϕ (x)+ψ (y)−ϕ (0) ,u(x,y)⎠.It can be proved by applying the Banach Contraction Principle, in the same manner as inIdczak and Walczak(1994), that the operatorTpossesses a unique fixed pointl˜ ∈L1P,RNand consequently, if we definexyzu,ϕ,ψ(x,y):=l˜(s,t) dsdt+ϕ (x)+ψ (y)−ϕ (0) ,(x,y)∈Pwe have thatzu,ϕ,ψ∈Zis the unique solution to (3) satisfying conditionsϕ (x)=zu,ϕ,ψ(x,0) andψ (y)=zu,ϕ,ψ(0,y)forx, y∈[0, 1].Moreover, from the proof of Banach Contraction Principle it follows that forln:=Tn(0) ,we get thatln→l˜inL1P,RN. Next, fork≥2, by (A2)-(A3), it is possible to show that|lk(x,y)−lk−1(x,y)|is bounded by a sum of 3k−1terms each of them is a product ofLk−1and some multiple integral. In each of this multiple integral we have at leastk−2integra-2tions with respect to variable which appears as the upper limit of the integration. Therefore,using the Cauchy formula for multiple integral we obtain|lk(x,y)−lk−1(x,y)|≤(3L)k−1c1k−22!for a.e.(x,y)∈Pandk≥2, wherec1is independent of(x,y)andk.Passing then, ifnecessary, to a subsequence, we get the following estimatejl˜(x,y)≤lim∞j→∞|lk(x,y)−lk−1(x,y)|+ |l1(x,y)|≤k=2k=2c1(3L)k−1k−22!+c2for a.e.(x,y)∈P,wherec2is independent of(x,y)andk.Eventually,∂2zu,ϕ,ψ(x,y)≤ρ1<∞,∂x∂ yxyzu,ϕ,ψ(x,y)≤l˜(s,t) dsdt+ |ϕ(x)|+ |ψ(y)|+ |ϕ(0)|≤ρ1+3c:=ρ,123
[ Pobierz całość w formacie PDF ]