OPTIMAL CONTROL FOR A PARABOLIC SYSTEM MODELLING CHEMOTAXIS

Transcription

1 Trends in Mathematics Information Center for Mathematical Sciences Volume 6, Number 1, June, 23, Pages OPTIMAL CONTROL FOR A PARABOLIC SYSTEM MODELLING CHEMOTAXIS SANG UK RYU Abstract. We stud the optimal control and robust control problems for a parabolic sstem modelling chemotaxis. That is, we obtain uniqueness of the optimal control under some conditions. Moreover, we consider the uniqueness of the saddle point. 1. Introduction In this note we stud the optimal control problem for the Keller-Segel equations: (P Minimize J(u with the cost functional J(u of the form J(u = T T (u d 2 H 1 (Ω dt + γ u 2 H ε (Ω dt, where = (u is governed b the Keller-Segel equations (1.1 = a b { } t in Ω (, T ], = d + f g + u t in Ω (, T ], n = = n on Ω (, T ], (x, = (x, (x, = (x in Ω. u L2 (, T ; H ε (Ω, Here, Ω is a bounded region in R 2 of C 3 class. a, b, d, f, g are given positive numbers and γ is a given nonnegative number. u is a control function in some bounded subsets. ε is some fixed exponent such that < ε < 1/2. n = n(x is the outer normal vector at a boundar point x Ω and n denotes the differentiation along the vector n. (x and (x are nonnegative initial functions in L 2 (Ω and in H 1+ε (Ω, respectivel., are unknown functions of the Cauch problem (1.1. The Keller-Segel equations (1.1 was introduced b Keller and Segel [4] to describe the aggregation process of the cellular slime molds b chemical attraction. Unknown functions = (x, t and = (x, t denote the concentration of amoebae in Ω at time t and the concentration of chemical substance in Ω at time t, respectivel. The chemotactic term b { } indicates that the cells are sensitive to 2 Mathematics Subject Classification. 49J2, 49J35, 49K2. Ke words and phrases. Keller-Segel equations, Optimal control, Robust control, Saddle point. 45 c 23 Information Center for Mathematical Sciences

2 46 SANG UK RYU chemicals and are attracted b them, and the production term f indicates that the chemical substance is itself emitted b cells. Optimal control problem associated to nonlinear equations have alread studied b man authors ([1], [2], [3], [5]. Recentl, Ru and Yagi [5] studied the distributed optimal control problem for Keller-Segel equations of non-monotone tpe. Under the mild assumptions, this paper obtains the uniqueness of the optimal control. We also consider the robust control problem as a differential game finding the best control which takes into account the worst disturbance. 2. Mathematical setting Let us briefl recall the wa how to formulate (1.1 as a semilinear abstract differential equation in a Hilbert space. Let A 1 = a + a and A 2 = d + g be the Laplace operators equipped with the Neumann boundar conditions. The part of A i in L 2 (Ω is a positive definite self-adjoint operator in L 2 (Ω with the domain D(A i = Hn(Ω 2 = { H 2 (Ω; n = on Ω}. D(Aθ i = H2θ (Ω for θ < 3 4, and D(A θ i = H2θ n (Ω for 3 4 < θ 3 2 (see Triebel [8]. We introduce two product Hilbert spaces V H as V = H 1 (Ω D(A 1+ε/2 2 and H = L 2 (Ω D(A (1+ε/2 2, respectivel, where ε is some fixed exponent ε (, 1 2. B the identification of H and its dual H, we have: V H = H V. It is then seen that V = (H 1 (Ω D(A ε/2 2. The norms of V, H, and V are denoted b,, and, respectivel. The dualit product between V and V is denoted b,. We set a smmetric sesquilinear form on V V: a(y, Ỹ = ( A 1/2 1, A 1/2 1 ỹ L 2 + ( A 1+ε/2 2, A 1+ε/2 2 L 2, Y = (, Ỹ = (ỹ V. Obviousl, the form satisfies (2.1 a(y, Ỹ M Y Ỹ, Y, Ỹ V, (2.2 a(y, Y δ Y 2, Y V with ( some δ and M >. This form then defines a linear isomorphism A = A1 from V to V A, and the part of A in H is a positive definite self-adjoint 2 operator in H with the domain D(A = D(A 1 D(A (3+ε/2 2. (1.1 is, then, formulated as an abstract equation (2.3 dy + AY = F (Y + U(t, < t T, dt Y ( = Y in the space V. Here, F ( : V V is the mapping ( ( b { } + a F (Y =, Y = V. f U(t = ( u(t and Y = (. As verified in [5, Sec. 2], F ( satisfies the following conditions:

3 OPTIMAL CONTROL FOR A PARABOLIC SYSTEM MODELLING CHEMOTAXIS 47 (f.i For each η >, there exists an increasing continuous function φ η : [, [, such that F (Y η Y + φ η ( Y, Y V; (f.ii For each η >, there exists an increasing continuous function ψ η : [, [, such that F (Ỹ F (Y η Ỹ Y + ( Ỹ + Y + 1ψ η( Ỹ + Y Ỹ Y, Ỹ, Y V. Furthermore, F (Y is the first-order Fréchet differentiable with the derivative ( b { w} b {z } + az F (Y Z =, fz F ( satisfies the following estimates: (f.iii For each η >, there exists an increasing continuous functions µ η, ν : [, [, such that η Z P + ( Y + 1µ η ( Y Z P, Y, Z, P V, F (Y Z, P η Z P + ( Y + 1µ η ( Y Z P, Y, Z, P V, ν( Y Z P, Y, Z, P V. (f.iv F ( is continuous from H into L(V, V. We then obtain the following result. Theorem 2.1 ([5, Theorem 2.1]. Let (2.1, (2.2, (f.i, and (f.ii be satisfied. Then, for an U L 2 (, T ; V and Y H, there exists a unique weak solution Y H 1 (, T (Y, U; V C([, T (Y, U]; H L 2 (, T (Y, U; V to (2.3, the number T (Y, U > is determined b the norms U L 2 (,T ;V and Y. 3. Distributed control problem Let U = L 2 (, T ; V and U ad be closed, bounded and convex subset of U. The problem (P is obviousl formulated as follows: (P minimize J(U, where the cost functional J(U is of the form J(U = DY (U Y d 2 dt + γ U 2 dt, U U ad. Here, Y (U, U U ad, is the weak solution of (2.3 and is assumed to exist on a fixed interval [, S]. D ( ( = is a bounded operator from V into V and Yd = ( d is a fixed element of L 2 (, S; V with d L 2 (, T ; H 1 (Ω. γ is a nonnegative constant. Theorem 3.1 ([5, Theorem 4.1]. There exists an optimal control U U ad for (P such that J(U = min U U ad J(U.

4 48 SANG UK RYU To derive the uniqueness of optimal control for (P, the second order Fréchet derivative of the mapping F ( : V V is necessar. It is indeed observed b a direct calculation that ( 2b {z w} F (Y (Z, Z = and the following estimate: (f.v There exists N > such that, Y = (, Z = F (Y (Z, Z N Z Z, Y, Z V. ( z V. w Proposition 3.2. The mapping Y : U ad H 1 (, S; V C([, S]; H L 2 (, S; V is Gâteaux differentiable with respect to U. For V U ad, Y (UV = Z is the unique solution in H 1 (, S; V C([, S]; H L 2 (, S; V of the problem dz dt + AZ F (Y Z = V (t, < t S, Z( =. Moreover, there exists γ such that, for γ > γ, the mapping U J(U is strictl convex. Proof. The strict convexit of J as well as the existence of the Gâteaux derivatives are obtained in [6]. The main result is given b: Theorem 3.3. For all γ > γ, there exists a unique optimal control U U ad for (P. Remark 3.4. We can also obtain the strict convexit of the cost functional and the uniqueness of the optimal control if one can assume that S is sufficientl small. 4. Robust control problem The cost functional contains an additional term due to disturbance and it is given b: J(U, Λ = DY (U, Λ Y d 2 dt + Here, Y (U, Λ is the weak solution of [γ U 2 l Λ 2 ]dt. dy + AY = F (Y + U(t + Λ(t, < t T, dt Y ( = Y. U(t = ( ( u(t and Λ(t = λ(t. γ and l are nonnegative constants. As in the case of the optimal control problem, we assume that U ad and G ad are closed, bounded, and convex subsets of L 2 (, S; V. The main result is as follows (For the detailed proof, we refer to [7]. Theorem 4.1. There exist γ and l such that for γ > γ and l > l, U J(U, Λ is strictl convex lower semicontinuous and Λ J(U, Λ is strictl concave upper

5 OPTIMAL CONTROL FOR A PARABOLIC SYSTEM MODELLING CHEMOTAXIS 49 semicontinuous. Moreover, there exists a unique saddle point (U, Λ U ad G ad such that J(U, Λ J(U, Λ J(U, Λ (U, Λ U ad G ad. References [1] N. U. Ahmed and K. L. Teo, Optimal Control of Distributed Parameter Sstems, North- Holland, New York, [2] V. Barbu, Analsis and Control of Nonlinear Infinite Dimensional Sstems, Academic Press, Boston, [3] E. Casas and L. A. Fern andez, and J. Yong, Optimal control of quasilinear parabolic equations, Proc. Ro. Soc. Edinburgh Sect., 125 ( [4] K. L. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instabilit, Journal of Theoritical Biolog, 26 ( [5] S.-U. Ru and A. Yagi, Optimal control of Keller-Segel equations, J. Math. Anal. Appl., 256 ( [6] S.-U. Ru, Optimal control of some parabolic equations of non-monotone tpe, Preprint. [7] S.-U. Ru and J.-Y. Park, Robust control of Keller-Segel equations, Preprint. [8] H. Triebel, Interpolation theor, function spaces, differential operators, North-Holland, Amsterdam, Department of Mathematics and Information, Cheju National Universit, Cheju , Korea address: rusu81@ cheju.ac.kr