Michał Kisielewicz SOME OPTIMAL CONTROL PROBLEMS FOR PARTIAL DIFFERENTIAL INCLUSIONS

Opucula Mathematica Vol. 28 No. 4 28 Dedicated to the memory of Profeor Andrzej Laota Michał Kiielewicz SOME OPTIMAL CONTROL PROBLEMS FOR PARTIAL DIFFERENTIAL INCLUSIONS Abtract. Partial differential incluion are conidered. In particular, baing on diffuion propertie of weak olution to tochatic differential incluion, ome exitence theorem and ome propertie of olution to partial differential incluion are given. Keyword: partial differential incluion, diffuion procee, exitence theorem. Mathematic Subject Claification: 93E3, 93C3. 1. INTRODUCTION The preent paper deal with the exitence of optimal olution to ome optimal control problem for partial differential incluion. Partial differential incluion conidered in the paper are invetigated by tochatic method connected with diffuion propertie of weak olution to tochatic differential incluion conidered in the author paper [9. In the recent year, ome propertie of partial differential incluion have been invetigated by G. Bartuzel and A. Fryzkowki ee [1 3 a application of ome general method of abtract differential incluion. Partial differential incluion conidered by G. Bartuzel and A. Fryzkowki have the form Du F u, where D i a partial differential operator and F i a given lower emicontinuou l..c. multifunction. In the preent paper we conider partial differential incluion of the form u tt, x L F G ut, x+ct, xut, x and ψt, x L F G ut, x+ct, xut, x, where c and ψ are given continuou function, u denote an unknown function and L F G i the et-valued partial differential operator generated by the given l..c. et-valued mapping F and G. Partial differential incluion conidered in the paper are invetigated together with ome initial and boundary condition and olution to uch initial and boundary valued problem are characteried by weak olution to tochatic differential incluion. Such approach lead to natural method of olving ome optimal control problem for ytem decribed by partial differential equation depending 57

58 Michał Kiielewicz on control parameter. Thee method ue the weak compactne with repect to ditribution of the et of all weak olution to tochatic differential incluion ee [7 and [8. In what follow we hall denote by C 1,2 Rn+1 and CR 2 n the pace of all function h C 1,2 R n+1, R and h C 2 R n, R, repectively with compact upport. By D and U we denote the boundarie of given domain D R n and U R n+1, repectively. By P F we hall denote a complete filtered probability pace Ω, F, F, P with a filtration F = F t t T atifying the uual condition, i.e., uch that F contain all A F uch that P A = and F t = ε> F t+ε for t [, T. We hall deal with et-valued mapping F : [, R n ClR n and G : [, R n ClR n m, where ClR n and ClR n m denote pace of all nonempty cloed ubet of R n and R n m, repectively. Given the et-valued mapping F and G, we hall denote by CF and CG et of all continuou elector f : [, R n R n and g : [, R n R n m of F and G, repectively. For a given n-dimenional tochatic proce X,x = X,x t t< on P F atifying X,x = x for, x [, R n, we hall denote by Y,x, an n + 1-dimenional tochatic proce defined on P F by Y,x = Y,x t t<, where Y,x t = + t, X,x + t for t <. For the tochatic procee given above, T > and a domain D R n we can define the firt exit time τd and τ U of X,x and Y,x from D and U =, T D, repectively, namely, τd = inf{r > : X,xr D} and τu = inf{t > : Y,xt U}. It can be verified ee [12, p. 226 that τu = τ D. In what follow we hall need ome continuou election theorem. We recall it ee [9, Th.3 in the general form. Let X, ρ, Y, and Z, be a Polih and Banach pace, repectively. Similarly a above by ClY we denote the pace of all nonempty cloed ubet of Y. Theorem 1 [9, Th. 3. Let λ : X Y Z and u : X Z be continuou and H : X ClY be l..c. uch that ux λx, Hx for x X. Aume λx, i affine and Hx i a convex ubet of Y for every x X. Then for every ε > there i a continuou function f ε : X Y uch that f ε x Hx and λx, f ε x ux ε for x X. 2. STOCHASTIC DIFFERENTIAL INCLUSIONS AND SET-VALUED PARTIAL DIFFERENTIAL OPERATOR Given et-valued meaurable and bounded mapping F : [, R n ClR n and G : [, R n ClR n m, by a tochatic differential incluion SDIF, G, we mean a relation x t x cl L 2 F τ, x τ dτ + Gτ, x τ db τ 1 which ha to be atified for every t < by a ytem P F, X, B coniting of a complete filtered probability pace P F, an L 2 -continuou F-nonanticipative n-dimenional tochatic proce X = Xt t< and m-dimenional F-Brownian

Some optimal control problem for partial differential incluion 59 motion B = B t t<. In a particular cae, when F and G take convex value, SDIF, G take the form x t x F τ, x τ dτ + Gτ, x τ db τ 2 and for it every olution P F, X, B the proce X i continuou. It can be treated a a random variable X : Ω, F C, βc, where C = C[,, R n and βc denote the Borel σ-algebra on C. We call the above ytem P F, X, B a weak olution to SDIF, G. We call a weak olution P F, X, B to 2 unique in law if for any other weak olution P F, X, B to SDIF, G there i P X 1 = P X 1, where P X 1 and P X 1 denote the ditribution of X and X, repectively, defined by P X 1 A = P X 1 A and P X 1 A = P X 1 A for A βc. In what follow, we hall conider tochatic differential incluion 2 together with an initial condition X = x a.., for fixed, x [, R n. Every weak olution P F, X, B to 2 will be identified with a pair X, B or imply with a tochatic proce X defined on P F. In what follow we hall denote by X,x F, G the et of all weak olution to tochatic differential incluion 2 atifying an initial condition x = x a.. We hall conider SDIF, G of the form 2 with F and G atifying the following condition A. Condition A: i F : [, R n ClR n and G : [, R n ClR n m are meaurable, bounded and take convex value, ii F t, and Gt, are continuou for every fixed t [,, iii G i diagonally convex-valued, i.e., for every t, x [, R n, the et DGt, x = {g g T : g Gt, x} i convex, where g T denote the tranpoition of g, iv G i uch that for every continuou elector σ of a multifunction DG defined above, there i a continuou elector g of G uch that g g T i uniformly poitive defined and σ = g g T, v F and G are continuou. Similarly a in [8 and [9 we can prove the following theorem. Theorem 2. Aume condition i iii of A are atified. Then for every, x [, R n the et X,x F, G i nonempty and weakly compact with repect to the convergence in ditribution. Theorem 3. Aume condition i iv of A are atified. Then for every f, g CF CG and, x [, R n there i X,x fg X,x F, G uch that a proce Y,x fg = Y,xt fg t< with Y,xt fg = + t, X,x fg + t for t < i an Itô diffuion uch that Y,x fg =, x, a.. Proof. Let f, g CF CG and, x [, R n be fixed. By virtue of [5, Th.IV.6.1 and Strook and Varadhan uniquene theorem ee [13 there i a unique in law weak olution P F, X,x, fg B to a tochatic differential equation x t =

51 Michał Kiielewicz x + fτ, x τ dτ + gτ, x τ db τ. Let a f = 1, f T T and b g =, g 1,..., g n T with, g i R 1 m, where =,..., and g i denote the i -th row of g for i = 1,..., n. Similarly a in [9, p.144 we can verify that the proce Y,x fg defined above i a unique in law weak olution to tochatic differential equation y t =, x+ a f y τ dτ + b gy τ db τ. Therefore ee [9, Y,x fg i an Itô diffuion on P F atifying Y,x fg =, x. Corollary 1. If condition i iv of A are atified, then for every f, g CF CG there i a nonempty et D fg R n+1 of function h : [, R n R uch that fg E,x hy,xt h, x lim t t 3 exit for every, x [, R n, where E,x denote the mean value operator with repect to a probability law Q,x of Y,x fg o that Q,x [Y,xt fg 1 E 1,..., Y,xt fg k E k = P [Y,xt fg 1 E 1,..., Y,xt fg k E k for t i < and E i βr n+1 with 1 i k. In what follow the above limit will be denoted by L C fg h, x and called the infiniteimal generator of Y fg,x. Similarly a in [12, Th.7.3.3, we can verify that for every h C 1,2 Rn+1 there i L C fg h, x = h t, x + n f i, x h x i, x + 1 2 i=1 n g g T ij, x h x ix j, x 4 i,j=1 for every, x [, R n. Hence in particular, for h C 2 R n we obtain L C fgh, x = n f i, xh x i x + 1 2 i=1 n g g T ij, xh x ix j x 5 i,j=1 for every, x [, R n. Similarly a in [12, Th.7.4.1, we obtain Dynkin formula [ hy [ τ E,x fg,xτ = h, x + E,x L C fg hy,xtdt 6 for h C 1,2 Rn+1,, x [, R n and a topping time τ uch that E,x [τ <. In particular, for h C 2 R n, one obtain E,x [ hx fg,x τ = hx + E,x [ τ L C fght, X fg,x tdt for x R n. Given the above et-valued mapping F and G we can define a et-valued partial differential operator on C 1,2 Rn+1 by etting L F G h, x = {Luv h, x : u F, x, v G, x}, 8 7

Some optimal control problem for partial differential incluion 511 where for, x [, R n. n L uv h, x = u i h xi, x + 1 2 i=1 n v v T ij h x ix j, x i,j=1 From Theorem 1, the next reult follow immediately. Theorem 4. Aume F and G atify condition i iv of A, let R + = [, and u, v : R + R + R n R be continuou and uch that ut, C 1,2 Rn+1 and vt,, x L F G ut,, x for, x [, R n and t [,. Then for every ε > there i f ε, g ε CF CG uch that g ε gε T i uniformly poitive defined and vt,, x L fεg ε ut,, x ε for, x [, R n and t [,. Proof. Let X = R + R + R n, Y = R n R n m, λt,, x, z, σ = n i=1 z iu x i t,, x + 1 n 2 i,j=1 σ iju x ix j t,, x, F t,, x = F πt,, x and Gt,, x = Gπt,, x for t,, x X, where π denote the orthogonal projection of R R R n onto R R n. From the propertie of v there follow vt,, x λt,, x, Ht,, x for t,, x X, where Ht,, x = F t,, x D Gt,, x. By virtue of Theorem 1, for every ε > there i a continuou elector h ε of H uch that vt,, x λt,, x, h ε t,, x ε for t,, x X. By the definition of H, there are continuou elector f ε and σ ε of F and D G, repectively, uch that hε = f ε, σ ε. From thi and iv it follow that for every, x [, R n there are f ε, x F, x and g ε, x G, x uch that f ε t,, x = f ε πt,, x = f ε, x and g ε t,, x = g ε πt,, x = g ε, x and vt,, x L fεg ε ut,, x ε for, x [, R n and t [,. In thi way we have defined, on [, R n, continuou function f ε and g ε, elector of F and G, uch that f ε t,, x = f ε, x and g ε t,, x = g ε, x. 3. INITIAL AND BOUNDARY VALUED PROBLEMS FOR PARTIAL DIFFERENTIAL INCLUSIONS Let F and G atify condition i iv of A and L F G be the et-valued partial differential operator on C 1,2 Rn+1 defined above. By L C F G h we hall denote a family L C F G = {LC fg : f, g CF CG}. For every f, g CF CG} and h D F G := {D fg : f, g CF CG} there i L C fg ht, x L C F G ht, x for t, x [, T R n. In [1, the following reult were proved. Theorem 5. Aume condition i iv of A are atified, T >, h C 1,2 Rn+1 and let c C[, T R n, R be bounded. For every, x [, T R n and every weak olution X,x to SDIF, G with an initial condition x = x a.., defined on a probability pace Ω, F, P, a function vt,, x = E,x [exp +t cτ, X,x τdτ h + t, X,x + t

512 Michał Kiielewicz atifie v tt,, x L C F G vt,, x c, xvt,, x for, x [, T R n, v,, x = h, x and t [, T, for, x [, T R n. 9 Theorem 6. Aume condition i, iii v of A are atified, T > and let h C 1,2 Rn+1. Suppoe c C[, T R n, R and v C 1,1,2 [, T [, T R n, R are bounded and uch that v tt,, x v t,, x L F G vt,, x c, xvt,, x, for, x [, T R n and t [, T, 1 v,, x = h, x for, x [, T R n. Then for every, x [, T R n there exit X,x X,x F, G defined on a probability pace Ω, F, P uch that [ vt,, x = Ẽ exp +t for, x [, T R n and t [, T. cτ, X,x τdτ h + t, X,x + t Theorem 7. Aume condition i, iii v of A are atified, T >, D i a bounded domain in R n and let Φ C, T D, R and u C, T D, R be bounded. If v C 1,2 Rn+1 i bounded and uch that { ut, x v tt, x L F G vt, x for t, x, T D, lim D x y vt, x = Φt, y for t, y, T D then for every x D there exit X,x X,x F, G defined on a probability pace Ω, F, P uch that vt, x = Ẽ[Φ τ T D, X,x τ T D Ẽ [ eτ T D ut, X,x tdt for t, x, T D, where τ T D = inf{r, T : X,x r D} T. Theorem 8. Aume condition i, iii v of A are atified, T >, D i a bounded domain in R n and let Φ C, T D, R, c C[, T D, R and u C, T D, R be bounded. If v C 1,2 Rn+1 i bounded and uch that { ut, x v t t, x L F G vt, x ct, xvt, x for t, x, T D, 11 lim x y vt, x = Φt, y for, y, T D

Some optimal control problem for partial differential incluion 513 then for every x D there exit X,x X,x F, G defined on a probability pace Ω, F, P uch that [ vt, x = Ẽ Ẽ Φ τ T D, X,x τ T Dexp { eτ T D [ ut, X,x texp eτ T D ct, X,x tdt } cz, X,x zdz dt for t, x, T D, where τ T D = inf{r, T : X,x r D} T. Remark 1. The above reult are alo true for T =, becaue by the boundedne of D there i τ D < a.., which implie that lim T τ T D = τ D a.. The cae T < i more applicable in practice. 4. EXISTENCE OF SOLUTION TO OPTIMAL CONTROL PROBLEMS FOR PARTIAL DIFFERENTIAL INCLUSIONS Given et-valued mapping F, G and function h, Φ, c and u, by ΛF, G, h, c we denote the et of all olution to initial valued problem: v tt,, x v t,, x L C F G vt,, x c, xvt,, x, for, x [, T R n and t [, T 12 v,, x = h, x for, x [, T R n, and by and ΓF, G, Φ, u, c the et of all olution to the boundary value problem { ut, x L C F Gvt, x ct, xvt, x for t, x, T D, 13 lim x y vt, x = Φt, y for, y, T D. Let u recall that by a olution to initial value problem 12 we mean a function v C 1 [, T [, T R n, R uch that vt, D F G for every t [, T and condition 12 are atified. Similarly by a olution to boundary valued problem 13 we mean a function v C 1 [, T R n uch that v D F G and conditin 13 are atified. In what follow, we hall denote the et ΛF, G, h, c C 1,1,2 [, T [, T R n, R and ΓF, G, Φ, u, c C 1,2 R n+1 by Λ C F, G, h, c and Γ C F, G, Φ, u, c, repectively. It i eay to ee that Λ C F, G, h, c and Γ C F, G, Φ, u, c are olution et to 1 and 11, repectively. Indeed, by the definition of L C F G and L F G there i L C F G ht, x = h t t, x + L F G ht, x for h C 1,2 R n+1 and t, x [, T R n. Let H : [, T R R be a given meaurable and uniformly integrably bounded function and let H and Z be mapping defined on ΛF, G, h, c and ΓF, G, Φ, u, c for fixed, x [, T R n by etting: Hv, x = T Ht, vt,, xdt for v ΛF, G, h, c

514 Michał Kiielewicz and Zvx = T Ht, vt, xdt for v ΓF, G, Φ, u, c. Theorem 9. Aume condition i, iii v of A are atified and let c C[, T R n, R be bounded. Let h C 1,2 R n+1 and aume H : [, T R R i meaurable, uniformly integrably bounded and uch that Ht, i continuou. If F and G are uch that for the h and c given above the et Λ C F, G, h, c i nonempty, then there i ṽ ΛF, G, h, c uch that Hṽ, x = inf{hv, x : v Λ C F, G, h, c} for every, x [, T R n. Proof. Let, x [, T R n be fixed. The et {Hv, x : v Λ C F, G, h, c} i nonempty and bounded, becaue there i a k L[, T, R + uch that Hv, x T ktdt for every v ΛC F, G, h, c. Therefore, there i a equence v n n=1 of Λ C F, G, h, c uch that α := inf{hv, x : v Λ C F, G, h, c = lim n Hv n, x. By virtue of Theorem 6 for every n = 1, 2,... and fixed, x [, T R n there i X,x n X,x F, G uch that v n t,, x = E [exp,x +t cτ, X,xτdτ n h + t, X n,x + t for t [, T. By the weak compactne of X,x F, G and [5,Th.I.2.1 there are a ubequence n k k=1 of n n=1, a probability pace Ω, F, P, tochatic procee X n k and X on Ω, F, P uch that P X n k = P X n k 1 for k = 1, 2,... and,x 1 up t T X n k t Xt, P - a.. Hence, in particular there follow v n k t,, x = E [exp,x [ = Ẽ exp +t +t cτ, X n k,xτdτ h + t, X n k,x + t = cτ, X n k τdτ h + t, Xn k + t. By the propertie of procee X n k, X and function c and h, thence there follow that [ lim k vn k t,, x = Ẽ exp +t cτ, Xτdτ h + t, X + t. Let [ ṽt,, x = Ẽ exp +t cτ, Xτdτ h + t, X + t. By virtue of Theorem 5, ṽ ΛF, G, h, c. Hence, by the propertie of the function H we get α = lim k Hv n k, x = Hṽ, x for, x [, T R n.

Some optimal control problem for partial differential incluion 515 In a imilar way, we can alo prove the following theorem. Theorem 1. Aume condition i, iii v of A are atified and let D be a bounded domain in R n. Let c C[, T R n, R, u C, T D and Φ C, T D, R be bounded. Aume H : [, T R R i meaurable, uniformly integrably bounded and uch that Ht, i continuou. If F and G are uch that for the Φ, u and c given above the et Γ C F, G, Φ, u, c i nonempty, then for every x R n there i X,x X,x F, G uch that Zṽx = inf{zvx : v Γ C F, G, Φ, u, c} for x R n, where ṽt, x = E,x [Φτ D, X,x τ D exp E,x { τd τd [ ut, X,x texp ct, X,x tdt for t, x, T D, where τ D = inf{r, T : X,x r D}. } cz, X,x zdz dt REFERENCES [1 G. Bartuzel, A. Fryzkowki, On the exitence of olution for incluion u F x, u, [in: R. Marz ed, Proc. of the Fourth Conf. on Numerical Treatment of Ordinary Differential Equation, Sek. Math. der Humboldt Univ. Berlin, 1984, 1 7. [2 G. Bartuzel, A. Fryzkowki, Abtract differential incluion Gwith ome application to partial differential one, Ann. Polon. Math. LIII 1991, 67 78. [3 G. Bartuzel, A. Fryzkowki, Stability of Principal Eigenvalue of Schrodinger type problem for differential incluion, Topol. Meth. Nonlinear Anal. 16 2, 181 194. [4 G. Bartuzel, A. Fryzkowki, A cla of retract in L p with ome application to differential incluion, Dicu. Math. Diff. Incl. Opt. 22 22, 213 224. [5 N. Ikeda, S. Watanabe, Stochatic Diferential Equation and Diffuion Procee, North Holand Publ., Amterdam, 1981. [6 M. Kiielewicz, Differential Incluion and Optimal Control, Kluwer Acad. Publ., New York, 1991. [7 M. Kiielewicz, Set-Valued tochatic integral and tochatic incluion, Stoch. Anal. Appl. 15 1997 5, 783 8. [8 M. Kiielewicz, Weak compactne of olution et to tochatic differential incluion with convex right hand ide, Topol. Math Nonlinear Anal. 18 21, 149 169. [9 M. Kiielewicz, Stochatic differential incluion and diffuion procee, Journal Math. Anal. Appl. 334 27, 139 154. [1 M. Kiielewicz, Stochatic repreentation of partial differential incluion, Journal Math. Anal. Appl. preented to publih. [11 E. Michael, Continuou election I, Ann. Math. 63 1956, 361 382.

516 Michał Kiielewicz [12 B. Økendal, Stochatic Differential Equation, Springer Verlag, Berlin-Heildelberg, 1998. [13 D.W. Strook, S.R.S Varadhan, Diffuion proce with continuou coeficient, I, II, Comm. Pure Appl. Math. 22 1969, 345 4, 479 53. Michał Kiielewicz M.Kiielewicz@wmie.uz.zgora.pl Univerity of Zielona Góra Faculty of Mathematic, Computer Science and Econometric ul. Podgórna 5, 65 246 Zielona Góra, Poland Received: February 9, 28. Accepted: July 4, 28.