Optimal control for a two-sidedly degenerate aggregation equation

. In this paper, we are concerned with the study of the mathematical analysis for an optimal control of a nonlocal degenerate aggregation model. This model describes the aggregation of organisms such as pedestrian movements, chemotaxis, animal swarming. We establish the well-posedness (existence and uniqueness) for the weak solution of the direct problem by means of an auxiliary nondegenerate aggregation equation, the Faedo–Galerkin method (for the existence result) and duality method (for the uniqueness). Moreover, for the adjoint problem, we prove the existence result of minimizers and ﬁrst-order necessary conditions. The main novelty of this work concerns the presence of a control to our nonlocal degenerate aggregation model. Our results are complemented with some numerical simulations.


Introduction
Nonlocal aggregation model has recently received great attention in biological applications.There exists a large variety of biological aggregation models such as the flocking of birds, aggregation of fish and the swarming insect.We refer the reader to a large number of works have focused on biological aggregation [6, 7, 10, 12, 16, 18-20, 22, 23].
In this paper, we consider the following nonlocal degenerate equation: Herein, Ω T := Ω ×(0, T ), Σ T := ∂Ω ×(0, T ), T > 0 is a fixed time, and Ω is a bounded domain in R 3 with Lipschitz boundary ∂Ω and outer unit normal η.In the model above, the density of the population is represented by u = u(x, t), a(u) is a density-dependent diffusion coefficient.Furthermore, K is the sensing (interaction) kernel that models the long-range attraction.In the convolution term, u is extended by zero, outside of Ω.More precisely, ∇K * u(x) = Ω ∇K(x − y)u(y) dy.
Note that system (1) arises in many models of biology and, in particular, in social organizations, which is one of the fundamental aspects of animal behaviors.
In this paper, we assume that the density-dependent diffusion coefficient a(u) degenerates for u = 0 and u = u.This means that the diffusion vanishes when u approaches values close to the threshold u and also in the absence of the population.This interpretation was proposed in [2] and in the references therein for the chemotaxis model.
To summarize, the following main assumptions are made: (A1) a ∈ C 1 ([0, 1]), a(0) = a(u) = 0 and a(s) > 0 for 0 < s < u; ) is a nonnegative radially nonincreasing function with the norm K C 2 (R 3 ) < ∞ and R 3 K(x) dx = 1.In addition, the reaction function F has the following form: where, α > 0 is the Malthusian growth coefficient, and w(•) (the control) is a nonnegative function of the intraspecific competition.
Regarding the degeneracy of the diffusion coefficient, a typical example of a is a(u) = u(u − u).Note that the degeneracy of the diffusion coefficient and the nonlocal term are major concerns for the mathematical and numerical treatment of equation (1).
To put this paper in the proper perspective, we mention that the nonlocal aggregation equation investigated analytically and numerically by many authors: [6,22] for the study of the pure aggregation equation, i.e., a(u) = 0 and F = 0, [8] for the existence result, [3,4] for the blowup of the solution and [13,14] for the analysis of the numerical simulation.
Many studies have focused on the competition between the degenerate diffusion as a repellent force and nonlocal aggregation terms as attractive force (see, e.g., [15]).This competition is observed in many biological phenomenon from social pattern formation to microbiological dynamics under chemotaxis force [5].From a mathematical perspective, Nonlinear Anal.Model.Control, 28(4):780-803, 2023 we mention, for example, the work [17] where the author proposed and proved the existence results of local and global solutions to a class of aggregation equations depending on attraction kernel regularity.In passing, we want to mention that the authors in [5] considered and studied the model (1) with F := 0 and a(u) = 0 for u = 0.
In our study, we are concerned with the mathematical analysis and numerical simulations of an optimal control problem arising in the study of population dynamics.Our model is governed by a degenerate aggregation-diffusion equation.To this model, we introduce a notion of a weak solution for the direct problem and prove its well-posedness.
Comparing to [5] (equation (1) with F := 0 and a(u) = 0 for u = 0), in this paper, we prove the existence of solutions by applying the Faedo-Galerkin method, deriving a priori estimates and then passing to the limit in the approximate solutions using monotonicity and compactness arguments.The uniqueness of these weak solutions is guaranteed by using the duality method.For the analysis of our optimal control problem, we use the Lagrangian framework in which the control problem is set as a constrained minimization problem.Note that, if there exists of a minimum to a suitable Lagrangian functional, it is a stationary point.
The numerical solution of our optimal control problem constrained by degenerate nonlocal aggregation model requires the proper discretization of the direct and the adjoint problems and the treatment of an optimization problem.From the standpoint of our specific application the main goal is to determine the control response to reduce the pattern formation generated by a nonlocal attraction term.More specifically, we are interested in determining the optimal intra species competition to insure a minimal pattern formation due the attraction force.
The structure of the paper is organized as follows.In Section 2, we present the main results, and we prove the well-posedness (existence and uniqueness) result to our degenerate aggregation model.Section 3 will be devoted to the optimal control problem.We present our optimal control approach, introduce a functional useful for minimize, prove the existence of the control, and we derive the adjoint-state problem.Finally, in Section 4, we introduce the numerical scheme for both direct and adjoint problem, present the optimal control procedure, and we demonstrate various realizations showing the effect of the optimal control solution on the overcrowding of the population.

Existence and uniqueness of weak solution 2.1 Weak solutions for the nonlocal degenerate equation
Before stating our main results, we give the definition of a weak solution for system (1).Definition 1.A weak solution of (1) is a nonnegative function u satisfying the following conditions: Here C w (0, T ; L 2 (Ω)) denotes the space of continuous functions with values in (a closed ball of) L 2 (Ω) endowed with the weak topology, and •, • denotes the duality pairing between H 1 (Ω) and (H 1 (Ω)) .
Our first result is the following well-posedness (existence and uniqueness) theorem for weak solutions.
Theorem 1. Assume that conditions (A1), (A2) hold and u 0 ∈ L ∞ (Ω) with 0 u 0 u, where u is a positive constant in R. Then there exists a unique weak solution to the nonlocal degenerate equation (1) in the sense of Definition 1.

Existence of weak solution
Note that a major difficulty for the analysis of equation ( 1) is the strong degeneracy of the diffusion term and the presence of the nonlocal term.To handle this difficulty, we replace the original diffusion term a(u) by a ε (u) = a(u) + ε (ε > 0) and consider the following nonlocal nondegenerate equation: To prove Theorem 1, we first prove existence of solutions to the nondegenerate problem (3) by using the Faedo-Galerkin method (in an appropriate functional setting).Convergence is achieved by means of a priori estimates and compactness arguments.
In what follows, we use the abbreviation "a.e." to denote "almost everywhere", which means that a property or condition holds for all points in a set, except for a set of points that has measure zero.

The Faedo-Galerkin solution
To construct our Faedo-Galerkin approximation, we employ a classical Hilbert basis, which is orthonormal in L 2 and orthogonal in H 1 .We look for solutions to the problem obtained from the projection of (3) onto the finite-dimensional subspace S n := span{e 1 , . . ., e n }.The approximate solution takes the following form: Herein, {e l } ∞ l=1 is an orthogonal basis in H 1 (Ω) and an orthonormal basis in L 2 (Ω).Our first goal is to determine the coefficients c n = {c n,l } n l=1 such that (l = 1, . . ., n) and with reference to the initial condition Herein, ε n := 1/n, n > 0.More explicitly, we can write (4) as an equation of ordinary differential equation (l = 1, . . ., n) where we have used the orthonormality of the basis.Observe that F l (l = 1, . . ., n) is a Carathéoodory function.Therefore, using the standard ODE theory, there exists an absolutely continuous functions {c n,l } n l=1 satisfying (5) for a.e.t ∈ [0, t ) for some t > 0. The next is to show that the local solution constructed above can be extended to the whole time interval [0, T ) (independent of n), but this can be done as in [1], so we omit the details.

Maximum principle
In this section, we prove that the solution of the nonlocal degenerate equation ( 2) satisfies the following version of maximum principle.
https://www.journals.vu.lt/nonlinear-analysisLemma 1. Assume that 0 u 0 u, then the solution u n to problem (3) satisfies 0 u n (x, t) e λt u for a.e.(x, t) ∈ Ω T , Proof.For technical reasons, we need to extend the function f (u) := αu − wu 2 so that it becomes measurable on Ω T , continuous with respect to u.We do this by setting (recall that α and w are nonnegative) Next, we define the following new variable ũn by setting u n = e λt ũn , where λ > 0 is defined in (7).It follows from (3) that ũn satisfies where ũ+ n = max{ũ n , 0}.Multiplying this equation by u − n = max{−ũ n , 0} and integrating over Ω, the result is Observe that Using this, we get easily from ( 9) Since the data u 0 is nonnegative, we deduce that ũ− n (x, t) = 0 for a.e.(t, x) ∈ Ω T .
Nonlinear Anal.Model.Control, 28(4):780-803, 2023 In the following step, we show that ũn (x, t) u, for a.e.(x, t) ∈ Ω T .To do this, it suffices to prove that (ũ n − u) + = 0. We multiply equation ( 8) by (ũ n − u) + , and we integrate over Ω to obtain where K n := ∇K * u n .Regarding the degenerate diffusion term, we have For the nonlocal term, we use an integration by part to deduce where we have used (recall that in (A2), K is radially nonincreasing) https://www.journals.vu.lt/nonlinear-analysis Collecting the previous estimates ( 11) and ( 12), we readily conclude from (10) Now, by the choice of λ in (7) we deduce from ( 13) Using that u 0 u, we conclude from this u n (t, •) e λt u in Ω for t ∈ (0, T ).This concludes the proof of the lemma.

A priori estimates
First, we denote To pass to the limit in (6) and prove the existence of the solution u, we need the following a priori estimates lemma.
Lemma 2. The solution u n to problem (3) satisfies for some constant C > 0 not depending of n.
Observe that A(u ε ) is uniformly bounded in S, where From the compact imbedding S ⊂ L 2 (Ω T ) we deduce that there exists a subsequence of Since A is monotone, we get A(u) = A. Therefore, Moreover, as A −1 is well defined and continuous, we apply the dominated convergence theorem to u n = A −1 (A(u n )) to obtain u n → u strongly in L 2 (Ω T ) and a.e. in Ω T .
Using this and the weak- * convergence of u n to u in L ∞ (Ω T ), we obtain With the above convergences, we are ready to identify the limit u as a weak solution of (1).Let ϕ ∈ L 2 (0, T ; H 1 (Ω)) be a test function in (6).By (20) it is clear that, as ε → 0, and , we also have that, as n → ∞, Nonlinear Anal.Model.Control, 28(4):780-803, 2023 Similarly, we have, as n → ∞, We have finally identified u as a solution of (1).

Uniqueness of the weak solution
In this section, we prove uniqueness of weak solutions to our nonlocal degenerate aggregation model, thereby completing the well-posedness analysis.The uniqueness proof of weak solutions is proved by using duality technique.First, we consider u 1 and u 2 two solutions of system (1).We set U = u 1 − u 2 , then U satisfies (for i = 1, 2) for a.e.t ∈ (0, T ).Since u 1 and u 2 are bounded in L ∞ , then we get from the theory of linear elliptic equations the existence, uniqueness and regularity of solution ϕ satisfying Note that from the boundary condition of ϕ in (22) and U (0, •) = 0 we deduce that Multiplying the second equation in ( 21) by ψ ∈ L 2 (0, T ; H 1 (Ω)) and integrating over Ω t := (0, t) × Ω, we get https://www.journals.vu.lt/nonlinear-analysisSince ϕ ∈ L 2 (0, T ; H 1 (Ω)), we can take ψ = ϕ in (24), and we obtain from ( 22), ( 23) and Since u 1 and u 2 are bounded in L ∞ , then there exist constants C 5 , C 6 > 0 such that Using (22)

The optimal control problem
In this subsection, we provide the existence of the solution for the optimal control problem of the nonlocal degenerate equation (1).We considered the following cost functional for the optimization of the population density and aggregation term: where ε 1 and ε 2 denote regularization parameters.Herein, Ω c ⊂ Ω is the control subdomain.We define the set of admissible controls U by where w ∈ R * + and w ∈ R * + are the minimal and the maximal intraspecific competition rates, respectively.We consider the following minimization problem: min w J(w, u) subject to (1). (31)

Existence of the control
In this subsection, we show the existence of the optimal solution w * ∈ U for the problem (31).
Proof.Let w n be a minimizing sequence of J such that Thanks to the definition of J, the sequence (w n ) n is bounded in L 2 (Ω T ).This implies that w n converge weakly to an w * .Let u n be a solution to problem (1) with respect to the control w n .Working exactly as in Section 2, we deduce the following convergence (upon a subsequence): u n → u * strongly in L q (Ω T ) and a.e. in Ω T for 1 q < ∞.
Note that, since the cost functional J(•, •) is continuous and convex on L 2 (Ω T ) × U, it follows that J(•, •) is weakly lower semicontinuous.Hence, by exploiting the strong convergence of u n combined with the weak lower semicontinuity of J we arrive to This implies the existence result of our optimal control solution (31). https://www.journals.vu.lt/nonlinear-analysis

Optimal conditions and dual problem
In this subsection, we derive the optimality conditions based on the Lagrangian formulation.We introduce the Lagrange functional L defined by (recall that A(u) = u 0 a(s) ds) The first-order optimality system characterizing the adjoint variables is given by the Lagrange multipliers, which result from equating the partial derivative of L with respect to u equal to zero: Observe that Next, we exploit (32) and (33) to deduce the adjoint equation of the nonlocal degenerate aggregation model ( 1) where To find the optimal conditions, we calculate the gradient of the functional J(w, u): https://www.journals.vu.lt/nonlinear-analysisTherefore, the optimality condition can be written as follows: Remark 1.Note that in aggregation equations, it is common to use an even kernel (K(−x) = K(x)).We mention that the gradient of an even function became odd (i.e., ∇K(−x) = −∇K(x).)

Numerical discretization
In this section, we present numerical methods to solve the nonlocal aggregation problem ( 1).We propose a numerical scheme to approximate the solution of the associated adjoint problem (34), and we implement the optimal control solver of the minimization problem (30).To approximate the solution of the direct problem (1) and adjoint problem (34), we will use the numerical scheme introduced in [9].First, let us consider a Cartesian mesh with the step h i in the direction i ∈ {1, . . ., d} and h = max i h i .Denote by (C J ) J∈Z d the space cells, where each cell C J has a center x J := (x 1 . . ., x d ) with x i = J i h i for i ∈ {1, . . ., d}.Next, we let e i the canonical basis of Z d , and we denote (u n J ) J∈Z d the approximation of cell average of u(t, •) at a given time t n = nτ .We propose the following numerical approximation for the direct problem (1) (recall that (s) + = max{0, s} and (s) − = max{0, −s} for a real number s): The numerical discrete aggregation and diffusion velocities are defined respectively by where for a pointy potential K. Now, for the solution of adjoint problem (34), we consider (p n J ) J∈Z the finite-volume approximation of cell average of p(t, •) at a given time t = nτ .We use the following Nonlinear Anal.Model.Control, 28(4):780-803, 2023 numerical approximation of the adjoint problem (34): where The term F n i,J can be computed as where the convection velocity is given by Following the establishment of the essential discretization of the direct and adjoint problems, we must develop a numerical approach to minimize the specified cost function (30).
It is common knowledge that the basic gradient descent approach does not ensure global convergence relative to the initial guest.Therefore, we implemented the nonlinear conjugate gradient technique [11] to achieve global convergence performance (see Algorithm 1).However, this class of methods has many limitations in terms of convergence to a global minimum.

Numerical simulations
In this section, we present an efficient implementation of the proposed numerical schemes (35), (36) with the optimal control Algorithm 1 to simulate the population dynamics under attractive forces.We focus our simulations on the effect of the optimal control (30) on the pattern formation induced by attractive forces under several initial conditions.
In the next subsections, we will present various tests and simulations.The first test is devoted to examine the effect of the aggregation and degenerate diffusion terms in the absence of the reaction term.Afterward, we will investigate the effect of the reaction term under various parameters.The final subsection focuses on evaluating the efficiency of the optimal control algorithm by considering two different initial conditions, and examining the impact of the resulting optimal control on the aggregation dynamics in each case.

Aggregation diffusion equation
In order to show the effect of the attractive force under a degenerate diffusion coefficient, we run different numerical simulations under different initial conditions.
Figure 1 presents the dynamic of a three group under attractive force.In Fig. 2, we depict the evolution of a randomly distributed initial population density.
It is well known that a diffusion process drives individuals of a given population towards lower densities according to gradient direction.In the other hand, the attraction force forms a velocity field that drives individuals from lower density groups to higher density groups.These two effects drive the solution to various forms of steady states.For example, in Fig. 1, we notice that a single group is formed from three different population densities.Starting from a randomly distributed initial condition with a sharper attraction  kernel (i.e., σ = 0.2), Fig. 2 shows the formation of several groups due to attraction forces.

Aggregation equation with nonlinear interaction term
To lessen the effect of over-crowding phenomenon, a degenerate diffusion plays a counter effect role.In more realistic phenomenon inspired from nature, the overcrowding effect comes with costs on the population.The mortality rate of the population rises due to some limited resources.This can be modeled by using logistic reaction term F (u, w) := αu − wu 2 .
Note that the logistic reaction term F (with α = 0.25 and w = 0.25) eliminates the aggregation phenomenon in Fig. 3.Moreover, the solution achieves in short time range a constant steady-state solution u ≡ 1.When α = 0.1 and w = 0.1, we observe that the evolution of the population is closer to the aggregation dynamics (see Fig. 4).In the last experiment of the direct problem (consult Fig. 5), we notice that the reaction term drives a random distributed initial condition to a more regular steady-state solution comparing to Fig. 2. We conclude that, according to the value of α, we can control the dominance of the logistic reaction term.
In the next subsection, we study and control the aggregation effect acting on the intraspecific competition rate w. https://www.journals.vu.lt/nonlinear-analysis

Optimal control simulation of the degenerate aggregation model
In this subsection, we implement several tests showing the efficiency of the proposed optimal control procedure to eliminate the hoarding effect.In each test, we plot the comparison between the controlled and the uncontrolled dynamics.In Figs. 6 and 8, we present a comparison between the controlled and uncontrolled dynamics of a given initial population density and a given attraction kernel.In the first and second rows of each figure, we illustrate the effects of an uncontrolled and controlled dynamic, respectively, and the third raw illustrates the development of the optimal solution w.The gradient of the functional, as depicted in Figs.7 and 9, is an important metric for evaluating the performance of the optimization process.A decrease in the gradient to a value less than 10 −5 indicates a significant improvement in the minimization process.This decrease in gradient is accompanied by a significant reduction in the cost functional, which approaches values close to 0. This leads to a reduced crowding effect compared https://www.journals.vu.lt/nonlinear-analysis  to the outcome of the direct problem, where the population dynamics is not guided by a control mechanism.Observations of the controlled solution in comparison to the uncontrolled solution in Fig. 6 reveal that the optimal control, represented by the control variable w, is effective in minimizing the attraction force between groups of individuals.By targeting the centers of the groups the optimal control minimizes the attraction force and prevents the formation of a single crowd at t = 3.0.This same effect is observed in Fig. 8 with a random initial distribution, where the optimal control decreases the gradient and the resulting attractive force preventing the formation of multiple crowds.These observations demonstrate the effectiveness of the optimal control approach in reducing crowding effects.

Conclusion
In this paper, we dealt with an optimal control to a two-sidedly degenerate aggregation equation with logistic source term.We provided a rigorous analysis of the mathematical model.We have proposed an optimal control procedure to reduce the over-crowding and pattern formation.We derived the adjoint state equation with the corresponding explicit formulation of the gradient of the cost functional.We showed a numerical simulation of the natural dynamics of different initial population under different attraction forces.Moreover, we have computed the optimal carrying capacity that reduces the pattern formation and the over-crowding effect.
Finally, we want to mention that the well-posedness of the adjoint problem (34) will be the subject of a forthcoming paper.

( 21 )
Now, we define the function ϕ solution of the problem

Figure 2 .
Figure 2. Evolution of the population dynamics using Gaussian attraction (with σ = 0.2) and a random initial population density u 0 .

Figure 5 .
Figure 5. Evolution of the population dynamics using the logistic source term F (with α = 0.01 and w = 0.01), the nonlocal pointy Gaussian potential (with σ = 0.2) and a random initial population density.

Figure 7 .
Figure 7.The outputs of the algorithm 1 with respect to Fig. 6 with ε 1 = 1 and ε 2 = 10 −8 .The L 2 -norm of the gradient of the cost functional at the left, the minimization values of the cost functional J(w, u) at the middle and the average of the optimal control w(t) = Ω w(x, t) dx at the right.

Figure 8 .
Figure 8.A comparison between controlled and uncontrolled dynamics acting on competition coefficient w where σ = 0.25, T = 4, α = 0.005.The initial condition u 0 (x, y) is a uniform random distribution.

Figure 9 .
Figure 9.The outputs of the algorithm 1 with respect to Fig. 8 with ε 1 = 1 and ε 2 = 10 −10 .The L 2 -norm of the gradient of the cost functional at the left, the minimization values of the cost functional J(w, u) at the middle and the average of the optimal control w(t) = Ω w(x, t) dx at the right.