Page 459 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 459
LYSIS, CONTROL AND INVERSE PROBLEMS FOR PARTIAL DIFFERENTIAL
EQUATIONS (MS-22)
nomenon is called chemo-repulsion.
In this work, we want to study an optimal control problem for the (repulsive) Keller-Segel
model and a bilinear control acting on the chemical equation in a 2D and 3D domains. The
system can be written as:
∂tu − ∆u − ∇ · (u ∇v) = 0 in Ω × (0, T ),
∂tv − ∆v + v = u + f v 1Ωc in Ω × (0, T ),
(1)
∂nu = ∂nv = 0 on ∂Ω × (0, T ),
in Ω,
u(0, ·) = u0 ≥ 0, v(0, ·) = v0 ≥ 0
being f : Qc := (0, T ) × Ωc → R (the control) with Ωc ⊂ Ω ⊂ Rn (n = 2, 3) the con-
trol domain, and the state u, v : Q := (0, T ) × Ωc → R+ the celular density and chemical
concentration, respectively. Here, n is the outward unit normal vector to ∂Ω.
The existence and uniqueness of global in time weak solution (u, v) for the uncontrolled
system is known (see for instance [1, 4]).
In this work we study an optimal control problem subject to a chemo-repulsion system with
linear production term, and in which a bilinear control acts injecting or extracting chemical
substance on a subdomain of control Ωc ⊂ Ω. Existence of weak solutions are stablished (in
the 3D case by using a regularity criterion), and, as a consequence, a global optimal solution
together with first-order optimality conditions for local optimal solutions are deduced.
The results presented in this talk are based on [2, 3].
References
[1] Cieslak, T., Laurençot, P., Morales-Rodrigo, C.: Global existence and convergence to
steady states in a chemorepulsion system. Parabolic and Navier-Stokes equations. Part 1,
105-117, Banach Center Publ., 81, Part 1, Polish Acad. Sci. Inst. Math., Warsaw, 2008.
[2] Guillén-González, F.; Mallea-Zepeda, E.; Rodríguez-Bellido, M.A. Optimal bilinear con-
trol problem related to a chemo-repulsion system in 2D domains. ESAIM Control Optim.
Calc. Var. 26 (2020), Paper No. 29, 21 pp.
[3] Guillén-González, F.; Mallea-Zepeda, E.; Rodríguez-Bellido, M.A. A regularity criterion
for a 3D chemo-repulsion system and its application to a bilinear optimal control problem.
SIAM J. Control Optim. 58 (2020), no. 3, 1457–1490.
[4] Guillén-González, F.; Rodríguez-Bellido, M. A.; Rueda-Gómez, D. A. Unconditionally
energy stable fully discrete schemes for a chemo-repulsion model. Math. Comp. 88 (2019),
no. 319, 2069–2099.
[5] Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J.
Theo. Biol. Vol. 26, 399-415 (1970).
457
EQUATIONS (MS-22)
nomenon is called chemo-repulsion.
In this work, we want to study an optimal control problem for the (repulsive) Keller-Segel
model and a bilinear control acting on the chemical equation in a 2D and 3D domains. The
system can be written as:
∂tu − ∆u − ∇ · (u ∇v) = 0 in Ω × (0, T ),
∂tv − ∆v + v = u + f v 1Ωc in Ω × (0, T ),
(1)
∂nu = ∂nv = 0 on ∂Ω × (0, T ),
in Ω,
u(0, ·) = u0 ≥ 0, v(0, ·) = v0 ≥ 0
being f : Qc := (0, T ) × Ωc → R (the control) with Ωc ⊂ Ω ⊂ Rn (n = 2, 3) the con-
trol domain, and the state u, v : Q := (0, T ) × Ωc → R+ the celular density and chemical
concentration, respectively. Here, n is the outward unit normal vector to ∂Ω.
The existence and uniqueness of global in time weak solution (u, v) for the uncontrolled
system is known (see for instance [1, 4]).
In this work we study an optimal control problem subject to a chemo-repulsion system with
linear production term, and in which a bilinear control acts injecting or extracting chemical
substance on a subdomain of control Ωc ⊂ Ω. Existence of weak solutions are stablished (in
the 3D case by using a regularity criterion), and, as a consequence, a global optimal solution
together with first-order optimality conditions for local optimal solutions are deduced.
The results presented in this talk are based on [2, 3].
References
[1] Cieslak, T., Laurençot, P., Morales-Rodrigo, C.: Global existence and convergence to
steady states in a chemorepulsion system. Parabolic and Navier-Stokes equations. Part 1,
105-117, Banach Center Publ., 81, Part 1, Polish Acad. Sci. Inst. Math., Warsaw, 2008.
[2] Guillén-González, F.; Mallea-Zepeda, E.; Rodríguez-Bellido, M.A. Optimal bilinear con-
trol problem related to a chemo-repulsion system in 2D domains. ESAIM Control Optim.
Calc. Var. 26 (2020), Paper No. 29, 21 pp.
[3] Guillén-González, F.; Mallea-Zepeda, E.; Rodríguez-Bellido, M.A. A regularity criterion
for a 3D chemo-repulsion system and its application to a bilinear optimal control problem.
SIAM J. Control Optim. 58 (2020), no. 3, 1457–1490.
[4] Guillén-González, F.; Rodríguez-Bellido, M. A.; Rueda-Gómez, D. A. Unconditionally
energy stable fully discrete schemes for a chemo-repulsion model. Math. Comp. 88 (2019),
no. 319, 2069–2099.
[5] Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J.
Theo. Biol. Vol. 26, 399-415 (1970).
457
