Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 58, pp. 1–16. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu CONTROLLABILITY AND STABILIZATION OF A NONLINEAR HIERARCHICAL AGE-STRUCTURED COMPETING SYSTEM ZE-RONG HE, NAN ZHOU Abstract. This article concerns the approximate controllability of a biolog- ical system, which is composed of two hierarchical age-structured competing species. Basing on a controllability result of linear system, we prove that the nonlinear system is approximately controllable by means of a fixed point theorem for multi-valued mappings. To fix a suitable control policy, we deal with an optimal control problem and established the existence of the unique optimal strategy. In addition, the stabilization problem of the system is also considered. 1. Introduction We consider the dynamics of a biological system consisting of two competing age-structured populations, in which the vital rates of individuals of age a mainly depend on the size of elders. Let p(a, t) be the age-specific density of a population at moment t and A the maximum age, then ∫ A a p(r, t)dr represents the number of the individuals with ages larger than a. According to the balance law of continuous age-structured population system, we derive the following dynamical model for the evolution of a competing community: ∂p1 ∂t + ∂p1 ∂a = −[µ1(a) +m1(E(p1)(a, t)) + f1(E(p2)(a, t))]p1(a, t) + u1(a, t), (a, t) ∈ Q1, ∂p2 ∂t + ∂p2 ∂a = −[µ2(a) +m2(E(p2)(a, t)) + f2(E(p1)(a, t))]p2(a, t) + u2(a, t), (a, t) ∈ Q2, pi(0, t) = ∫ Ai 0 βi(a,E(pi)(a, t))pi(a, t)da, t ∈ (0, T ); i = 1, 2, pi(a, 0) = p0i (a), a ∈ [0, Ai], i = 1, 2, E(pi)(a, t) = α ∫ a 0 pi(r, t)dr + ∫ Ai a pi(r, t)dr, 0 ≤ α < 1, where Qi = (0, Ai)×(0, T ), Ai is the maximum age of the individuals in population i and T the control horizon. pi(a, t) stands for the density of the population i. 2010 Mathematics Subject Classification. 92D05, 47D06, 35B35. Key words and phrases. Hierarchy of age; population system; competition; controllability; hfill Fan-Glicksberg fixed points. c©2020 Texas State University. Submitted November 19, 2019. Published June 11, 2020. 1 2 Z.-R. HE, N. ZHOU EJDE-2020/58 E(pi)(a, t) (with the weight α) describes the age-specific instant environment within the population i. µi(a) and mi(E(pi)(a, t)) denotes the natural death rate and the additional mortality caused by over-crowding, respectively. fi(E(pj)(a, t)), j 6= i, shows the competition effect of population j on population i, and the control variable ui(a, t) is the migration rate. βi(a,E(pi)(a, t)) is the fertility and p0i (a) the initial age distribution. As one can see, in the extremal case of α = 0, the vital rates of an individual of age a depends only on elders except its age, which reveals the internal hierarchy of ages among the individuals. Controllability has been an interesting and challenging topic for infinite-dimen- sional systems. This is also the case in the study of structured population models, and a number of excellent results have been achieved by researchers, see for exam- ple [2, 3, 5, 6, 8, 12, 13, 21, 29, 30, 31] and their references. Most of the existing works in the literature focus on linear systems, which are, of course, necessary and fundamental. However, because realistic models in almost all practical situations are nonlinear, the investigation of controllability for nonlinear systems is of signifi- cance, and which is the main concern in the present article. We are also interested in stabilizing the competing populations, which makes sense in the control of pest or invading species. One can refer to [4, 18, 19, 24, 26, 28, 30] and the references cited for some related works. During the previous four decades, the socialization in biological populations has attracted much attention of ecologists and mathematicians, see [10, 25] for eco- logical investigations and [1, 7, 9, 14, 15, 16, 17, 20, 22, 23, 27, 31] for analysis in mathematical modelling. To the best of our knowledge, almost all works on hierarchical population models are limited in single-species and directed to the dy- namical behaviors, results in control problems on multi-species systems are quite rare. Motivated by the observation, we in this article deal with controllability and stabilization problems for a hierarchical competing model. This paper is organized as follows: in the next section we present the normal- ization for the above model and assumptions posed on the parameters. The third section is devoted to the theoretic proof of the approximate controllability, which is followed by an appropriate choice in section 4. The stabilization of the zero state in a finite period is treated in section 5 by means of an optimal control problem. Some remarks are included in the final section. EJDE-2020/58 CONTROLLABILITY OF HIERARCHICAL SYSTEMS 3 2. Model normalization and assumptions Denote A = max(A1, A2) and Q = (0, A) × (0, T ). The natural zero extension to functions and parameters in the above model reaches the normal system ∂p1 ∂t + ∂p1 ∂a = −[µ1(a) +m1(E(p1)(a, t)) + f1(E(p2)(a, t))]p1(a, t) + u1(a, t), (a, t) ∈ Q, ∂p2 ∂t + ∂p2 ∂a = −[µ2(a) +m2(E(p2)(a, t)) + f2(E(p1)(a, t))]p2(a, t) + u2(a, t), (a, t) ∈ Q, pi(0, t) = ∫ A 0 βi(a,E(pi)(a, t))pi(a, t)da, t ∈ (0, T ); i = 1, 2, pi(a, 0) = p0i (a), a ∈ [0, A]; i = 1, 2, E(pi)(a, t) = α ∫ a 0 pi(r, t)dr + ∫ A a pi(r, t)dr, 0 ≤ α < 1. (2.1) To meet the need of forthcoming theoretic analysis, we propose the following assumptions for the model parameters (i = 1, 2), which are biologically meaningful: (A1) A and T are finite positive constants; (A2) µi(a) > 0 for for all a ∈ (0, A), µi ∈ L1 loc[0, A) and ∫ A 0 µi(a)da = +∞; (A3) 0 ≤ mi(x) ≤ m̄i for all x ≥ 0 with constant m̄i > 0; (A4) 0 ≤ fi(x) ≤ Fi for all x ≥ 0 with constant Fi > 0; (A5) u ∈ U := {(u1, u2) : |ui(a, t)| ≤ Ui, a.e. (a, t) ∈ Q, i = 1, 2}, with constant Ui > 0; (A6) 0 ≤ βi(a, x) ≤ Bi for all (a, x) ∈ [0, A]× [0,+∞) with constant Bi > 0; (A7) 0 ≤ p0i (a) ≤ P ∗i for all a ∈ [0, A] with constant P ∗i > 0. 3. Approximate Controllability Definition 3.1. System (2.1) is said to be approximately controllable on [0, T ] if, for any given initial distribution p0 = (p01, p 0 2) ∈ (L∞[0, A])2, target p̄ = (p̄1, p̄2) ∈ (L∞[0, A])2 and ε, 0 < ε << 1, there exists u = (u1, u2) ∈ U , such that the solution pu(a, t) to (2.1) meets ‖pu(·, T )− p̄‖{L2[0,A]}2 ≤ ε. Let p(0,0) be the solution of the system (2.1) corresponding to u = (0, 0). Without loss of generality, we suppose that ‖p(0,0)(·, T ) − p̄‖ > 1. We note that if E(p) = (E(p1), E(p2)) in the functions mi, fi, βi are fixed as P = (P1, P2), then system 4 Z.-R. HE, N. ZHOU EJDE-2020/58 (2.1) reduces to ∂p1 ∂t + ∂p1 ∂a = −[µ1(a) +m1(P1(a, t)) + f1(P2(a, t))]p1(a, t) + u1(a, t), (a, t) ∈ Q, ∂p2 ∂t + ∂p2 ∂a = −[µ2(a) +m2(P2(a, t)) + f2(P1(a, t))]p1(a, t) + u2(a, t), (a, t) ∈ Q, pi(0, t) = ∫ A 0 βi(a, Pi(a, t))pi(a, t)da, t ∈ (0, T ); i = 1, 2, pi(a, 0) = p0i (a), a ∈ [0, A], i = 1, 2. (3.1) It is readily seen that the system (3.1) can be divided into two independent subsys- tems, which are approximately controllable according to a result in the literature [8]; that is, there is ui such that ‖pui i (·, T ;P )− p̄i‖L2[0,A] ≤ ε 2 , i = 1, 2. (3.2) We hope to extend the controllability of system (3.1) to system (2.1) by means of fixed point approach. To do so, we need some analysis on the linear system (3.1). In what follows, we adopt the notation: Mi(a, t;P ) = µi(a) +mi(Pi(a, t)) + fi(Pj(a, t)), i, j = 1, 2, i 6= j. Lemma 3.2. For any given P = (P1, P2) ∈ (L∞(Q))2, and all u = (u1, u2) ∈ U , the solution pu(·, ·;P ) = (pu1 1 (·, ·;P ), pu2 2 (·, ·;P )) to system (3.1) satisfies ‖E(pui i (·, ·;P ))‖L∞(Q) + ‖E(pui i (·, ·;P ))t‖L∞(Q) + ‖E(pui i (·, ·;P ))a‖L∞(Q) ≤ C1, where E(p)t and E(p)a stand for the partial derivatives with respect to t and a respectively, the constant C1 is independent of P and u. Proof. It follows from (3.1) and the method of characteristics that pui i (a, t;P ) = { p0i (a− t)Πi(a, t, t;P ) + Πui (a, t, t;P ), a ≥ t, bui i (t− a;P )Πi(a, t, a;P ) + Πui(a, t, a;P ), a < t, (3.3) where bui i (t;P ) = pui i (0, t;P ), (3.4) Πi(a, t, s;P ) = exp { − ∫ s 0 Mi(a− τ, t− τ ;P )dτ } , (3.5) Πui (a, t, s;P ) = ∫ s 0 exp { − ∫ τ 0 Mi(a− r, t− r;P )dr } ui(a− τ, t− τ)dτ. (3.6) By (3.3)-(3.4) and the third equation in (3.1), we claim that the population fertility bui i solves the Volterra integral equation bui i (t;P ) = Fi(t;P ) + ∫ t 0 Ki(t, s;P )bui i (t− s;P )ds, t ∈ (0, T ), (3.7) EJDE-2020/58 CONTROLLABILITY OF HIERARCHICAL SYSTEMS 5 where Fi(t;P ) =  ∫ A−t 0 βi(a+ t, Pi(a+ t, t))p0i (a)Πi(a+ t, t, t;P )da + ∫ t 0 βi(a, Pi(a, t))Πui (a, t, a;P )da + ∫ A t βi(a, Pi(a, t))Πui (a, t, a;P )da, 0 < t < A,∫ A 0 βi(a, Pi(a, t))Πui(a, t, a;P )da, A ≤ t ≤ T ; (3.8) Ki(t, s;P ) = { βi(s, Pi(s, t))Πi(s, t, s;P ), s < t, 0, s ≥ t. (3.9) Here we consider the case T > A only. The other case has a similar process. From (3.5)–(3.6) and (3.8)–(3.9), we have Πi(a, t, s;P ) ≤ 1, |Πui (a, t, s;P )| ≤ AUi, |Fi(t;P )| ≤ max(ABiP ∗ i +ATBiUi, AUiBi), 0 ≤ Ki(t, s;P ) ≤ Bi. Combining (3.7) with the above estimations, we derive that bui i (t;P ) ≤ C2, with the constant C2 independent of P, u. Therefore, the expression (3.3) and the definition of E(pi) guarantee the uniform boundedness of E(pui i )(a, t;P ) and E(pui i )a(a, t;P ). To estimate E(pui i )t(a, t;P ), we use (3.3) to obtain that, if 0 < t ≤ a then (with the symbol P omitted) E(pui i )(a, t) = α ∫ t 0 [bui i (t− r)Πi(r, t, r) + Πui(r, t, r)]dr + α ∫ a t [p0i (r − t)Πi(r, t, t) + Πui (r, t, t)]dr + ∫ A a [p0i (r − t)Πi(r, t, t) + Πui (r, t, t)]dr; (3.10) if 0 < a < t < A, then E(pui i )(a, t) = α ∫ a 0 [bui i (t− r)Πi(r, t, r) + Πui(r, t, r)]dr + ∫ t a [bui i (t− r)Πi(r, t, r) + Πui (r, t, r)]dr + ∫ A t [p0i (r − t)Πi(r, t, t) + Πui (r, t, t)]dr; (3.11) and if t ≥ A, then E(pui i )(a, t) = α ∫ a 0 [bui i (t− r)Πi(r, t, r) + Πui(r, t, r)]dr + ∫ A a [bui i (t− r)Πi(r, t, r) + Πui (r, t, r)]dr. (3.12) 6 Z.-R. HE, N. ZHOU EJDE-2020/58 Making appropriate changes of integration variables, we are able to write that, if 0 < t ≤ a, then E(pui i )(a, t) = α ∫ t 0 bui i (τ) exp { − ∫ t τ Mi(v − τ, v)dv } dτ + α ∫ a−t 0 p0i (v) exp { − ∫ t 0 Mi(v + τ, τ)dτ } dv + ∫ A−t a−t p0i (v) exp { − ∫ t 0 Mi(v + τ, τ)dτ } dv + α ∫ t 0 ∫ s 0 exp { − ∫ t v Mi(θ − s, θ)dθ } ui(v − s, v)dvds + α ∫ a−t 0 ∫ t 0 exp { − ∫ t v Mi(θ + s, θ)dθ } ui(v + s, v)dvds + ∫ A−t a−t ∫ t 0 exp { − ∫ t v Mi(θ + s, θ)dθ } ui(v + s, v)dvds. (3.13) Consequently, for 0 < t ≤ a, we have ∂E(pui i ) ∂t = −α ∫ t 0 bui i (τ)Mi(t− τ, t) exp { − ∫ t τ Mi(v − τ, v)dv } dτ + αbui i (t)− αp0i (a− t) exp { − ∫ t 0 Mi(a− t+ τ, τ)dτ } − α ∫ a−t 0 p0i (v)Mi(v + t, t) exp { − ∫ t 0 Mi(v + τ, τ)dτ } dv − p0i (A− t) exp { − ∫ t 0 Mi(A− t+ τ, τ)dτ } + p0i (a− t) exp { − ∫ t 0 Mi(a− t+ τ, τ)dτ } − ∫ A−t a−t p0(v)Mi(v + t, t) exp { − ∫ t 0 Mi(v + τ, τ)dτ } dv + α ∫ t 0 exp { − ∫ t v Mi(θ − t, θ)dθ } ui(v − t, v)dv − α ∫ t 0 ∫ s 0 Mi(t− s, t) exp { − ∫ t v Mi(θ − s, θ)dθ } ui(v − s, v)dvds − α ∫ t 0 exp { − ∫ t v Mi(θ + a− t, θ)dθ } ui(v + a− t, v)dv + α ∫ a−t 0 [ui(t+ s, t) − ∫ t 0 ui(v + s, s)Mi(t+ s, t) exp { − ∫ t v Mi(θ + s, θ)dθ } ]ds − ∫ t 0 exp { − ∫ t v Mi(θ +A− t, θ)dθ } ui(v +A− t, v)dv + ∫ t 0 exp { − ∫ t v Mi(θ + a− t, θ)dθ } ui(v + a− t, v)dv EJDE-2020/58 CONTROLLABILITY OF HIERARCHICAL SYSTEMS 7 + ∫ A−t a−t [ui(t+ s, t) − ∫ t 0 ui(v + s, v)Mi(t+ s, t) exp { − ∫ t v Mi(θ + s, θ)dθ } dv]ds. One can handle the cases 0 < a < t < A and t ≥ A similarly. The assumptions tell us that E(pui i (·, ·;P ))t is uniformly bounded in P and u. � The following three results will also be used in the proof of the controllability. Lemma 3.3. The set K = { (P1, P2) ∈ (L∞(Q))2 : ‖Pi‖L∞(Q)+‖ ∂Pi ∂t ‖L∞(Q)+‖ ∂Pi ∂a ‖L∞(Q) ≤ C1, i = 1, 2 } is compact in (C(Q))2, with the constant C1 given in Lemma 3.2. Proof. Note that Q̄ = [0, A]× [0, T ] is compact in R2. The structure of K implies that every element in K must be a continuous function on Q̄, and every sequence in K must be uniformly bounded and equi-continuous. The conclusion follows from the Arzela-Ascoli theorem. � Lemma 3.4. The solution pu(·, ·;P ) for the linear system (3.1) depends continu- ously on u ∈ U . Proof. For any given ur = (ur1, u r 2) ∈ U , r = 1, 2, the expression (3.3) implies that, if a ≥ t then |pu 1 i i (a, t;P )− pu 2 i i (a, t;P )| = |Πu1 i (a, t, t;P )−Πu2 i (a, t, t;P )| ≤ ∫ t 0 |u1i (a− τ, t− τ)− u2i (a− τ, t− τ)|dτ ≤ T ‖ u1 − u2 ‖(L∞(Q))2 ; (3.14) If a < t, then |pu 1 i i (a, t;P )− pu 2 i i (a, t;P )| = |bu 1 i i (t− a;P )− bu 2 i i (t− a;P )| + ∫ a 0 |u1i (a− τ, t− τ)− u2i (a− τ, t− τ)|dτ. (3.15) On the other hand, one can see from (3.7) that |bu 1 i i (t;P )− bu 2 i i (t;P )| ≤ |Fu 1 i i (t;P )− Fu 2 i i (t;P )|+ ∫ t 0 Bi|b u1 i i (t− s;P )− bu 2 i i (t− s;P )|ds ≤ ABi‖u1i − u2i ‖(L∞(Q))2 +Bi ∫ t 0 |bu 1 i i (t− s;P )− bu 2 i i (t− s;P )|ds. (3.16) Applying the Bellmann inequality, we derive from (3.16) that |bu 1 i i (t;P )− bu 2 i i (t;P )| ≤ ABi exp{TBi} ‖ u1 − u2 ‖(L∞(Q))2 . (3.17) Substituting (3.17) into (3.15), one gets that, if a < t, then |pu 1 i i (a, t;P )− pu 2 i i (a, t;P )| ≤ A(1 +Bi exp{TBi}) ‖ u1 − u2 ‖(L∞(Q))2 . (3.18) 8 Z.-R. HE, N. ZHOU EJDE-2020/58 Finally, combining (3.14) with (3.18), we arrive at ‖pu 1 (·, ·;P )− pu 2 (·, ·;P )‖(L∞(Q))2 ≤ C1 ‖ u1 − u2 ‖(L∞(Q))2 , where the constant C1 is independent of u1, u2 and P . � For the sake of convenience, we list the following existence result [31, P. 452]. Lemma 3.5 (Ky Fan-Glicksberg). Suppose that the following conditions are satis- fied: (1) Subset K is nonempty, compact and convex in a locally convex space X; (2) Set-valued mapping G : K → 2K is upper-continuous; (3) G(x) is nonempty, closed and convex for each x ∈ K. Then G has at least one fixed point. Now we are ready to show the following result. Theorem 3.6. System (2.1) is approximately controllable in (L∞(Q))2. Proof. Let X = (L∞(Q))2, and K be given in Lemma 3.3. Define the set-valued mapping G : K → 2X , for P ∈ K, G(P ) = {E(pu(·, ·;P )) : u ∈ U such that ‖pu(·, T ;P )− p̄‖(L∞([0,A]))2 ≤ ε}, where pu(a, t;P )) = (pu1 1 (a, t;P )), pu2 2 (a, t;P ))) is the solution to the linear system (3.1). The conclusion in Lemma 3.2 implies that G(P ) ∈ 2K for each P ∈ K. It is clear that K is convex. Lemma 3.3 shows that the condition (1) in Lemma 3.5 is satisfied. According to the main result in [8], G(P ) 6= ∅. Since the control variable u = (u1, u2) serves as the nonhomogeneous term, G(P ) is convex. More- over, Lemma 3.4 assures the closedness of G(P ). Hence condition (3) in Lemma 3.5 is true. It remains to prove that G is upper-continuous. Let Pn = (Pn1 , P n 2 ) → P = (P1, P2) in (L∞(Q))2, and E(pu n ) = (E(p un 1 1 ), E(p un 2 2 ))→ h = (h1, h2) in (L∞(Q))2 for some sequence {un = (un1 , u n 2 )} such that E(pu n ) ∈ G(Pn). We need to show h ∈ G(P ). By (3.3), we derive that p un i i (a, t;Pn) = { p0i (a− t)Πi(a, t, t;P n) + Πun i (a, t, t;Pn), a ≥ t, b un i i (t− a;Pn)Πi(a, t, a;Pn) + Πun i (a, t, a;Pn), a < t, (3.19) Note that uni ∈ L∞(Q) ⊂ L2(Q) and |uni (a, t)| ≤ Ui, i = 1, 2. There exists a subsequence (denoted still by {uni }) such that uni → ui weakly. On the other hand, it follows from (3.17) that {bu n i i } is bounded uniformly. One can extract a subsequence (denoted still by {bu n i i }) such that {bu n i i } converges to some bi in the weak-star sense. Passing into the limit in the right side of (3.19), we define the function pi(a, t) := { p0i (a− t)Πi(a, t, t;P ) + Πui (a, t, t;P ), a ≥ t, bi(t− a)Πi(a, t, a;P ) + Πui (a, t, a;P ), a < t. (3.20) Consequently, pi(0, t) = bi(t) = ∫ A 0 βi(a, P (a, t))pi(a, t)da. We are sure that the function given by (3.20) is the solution for the system (3.1). Furthermore, E(pu n ) ∈ G(Pn) implies that the control functions uni must meet ‖pu n (·, T ;Pn)− p̄‖(L∞([0,A]))2 ≤ ε. EJDE-2020/58 CONTROLLABILITY OF HIERARCHICAL SYSTEMS 9 Passing to the limit in the above inequality, we have ‖p(·, T ;P )− p̄‖(L∞([0,A]))2 ≤ ε. In a word, h = E(p(·, ·;P )) and h ∈ G(P ). Therefore, the mapping given above satisfies all the conditions in Lemma 3.5 and has at least one fixed point, which establishes the approximate controllability of system (2.1). � 4. Choice of controls Theorem 3.6 implies that there is at least one control u = (u1, u2) ∈ U , such that the corresponding state of the system (2.1) at the moment T approaches a prescribed target p̄ arbitrarily. Generally speaking, there are many (even infinite) controls of such type. Which control should we choose to adjust the population states? To seek an appropriate control, we consider the following optimal control prob- lem: (P1) Find u∗ = (u∗1, u ∗ 2) ∈ U , such that J(u∗) ≤ J(u) for all u ∈ U , with U = {(u1, u2) ∈ (L2(Q))2 : |ui(a, t)| ≤ Ui, a.e. (a, t) ∈ Q; i = 1, 2}, and J(u) = 1 2 2∑ j=1 {∫ Q u2j (a, t) da dt+ k ∫ A 0 [pj(a, T )− p̄j(a)]2da } , (4.1) where k is the penalty parameter, (u, p) is subject to the system (2.1), and p̄ is the target. As in the proof of Lemma 3.4, we can show the following result. Lemma 4.1. The solutions pu to the system (2.1) are continuous in u. Without loss of generality, we in what follows assume that m1(x) = m2(x) ≡ 0. Theorem 4.2. Any optimal pair (u∗, p∗) of problem (P1) must be in the form u∗j = Fj(k|qj |), j = 1, 2, where the function Fj is given by Fj(x) =  0, x < 0, Uj , x > Uj , x, 0 ≤ x ≤ Uj , 10 Z.-R. HE, N. ZHOU EJDE-2020/58 and the adjoint variables qj solve the system ∂q1 ∂t + ∂q1 ∂a = [µ1 + f1(E(p∗2))]q1 − q1(0, t)[β1(a,E(p∗1)) + Ẽ1] + Ē2, ∂q2 ∂t + ∂q2 ∂a = [µ2 + f2(E(p∗1))]q2 − q2(0, t)[β2(a,E(p∗2)) + Ẽ2] + Ē1, qj(a, T ) = p̄j(a)− p∗j (a, T ), a ∈ (0, A), j = 1, 2, qj(A, t) = 0, t ∈ (0, T ), j = 1, 2, Ẽj(a, t) = ∫ a 0 p∗j (r, t) ∂βj ∂x (r, E(p∗j )(r, t))dr + α ∫ A a p∗j (r, t) ∂βj ∂x (r, E(p∗j )(r, t))dr, Ēi(a, t) = ∫ a 0 [f ′ i (E(p∗j ))p ∗ i qi](r, t)dr + α ∫ A a [f ′ i (E(p∗j ))p ∗ i qi](r, t)dr, i 6= j, i, j = 1, 2, (4.2) in which the variables (a, t) are omitted in the main equations. Proof. Let (u∗, p∗) be an optimal pair for (P1). Then for any given v ∈ TU (u∗) (the tangent cone to U at u∗), one infers that u∗ + εv ∈ U for ε small enough. Consequently J(u∗) ≤ J(u∗ + εv); that is, 2∑ j=1 ∫ Q u∗2j (a, t) da dt+ k 2∑ j=1 ∫ A 0 [p∗j (a, T )− p̄j(a)]2da ≤ 2∑ j=1 ∫ Q [u∗j (a, t) + εvj(a, t)] 2 da dt+ k 2∑ j=1 ∫ A 0 [pεj(a, T )− p̄j(a)]2da, (4.3) where pε = (pε1, p ε 2) denotes the solution of (2.1) corresponding to u = u∗ + εv. By Lemma 4.1, we derive that 2∑ j=1 {∫ Q (u∗jvj)(a, t) da dt+ k ∫ A 0 zj(a, T )[p∗j (a, T )− p̄j(a)]da } ≥ 0, (4.4) where z(a, t) = (z1(a, t), z2(a, t)) := limε→0+ ε −1[pε(a, t) − p∗(a, t)] solves the fol- lowing system (with (a, t) in the main equations omitted) ∂z1 ∂t + ∂z1 ∂a = −[µ1(a) + f1(E(p∗2))]z1 − f ′ 1(E(p∗2))p∗1E(z2) + v1, ∂z2 ∂t + ∂z2 ∂a = −[µ2(a) + f2(E(p∗1))]z2 − f ′ 2(E(p∗1))p∗2E(z1) + v2, zi(0, t) = ∫ A 0 [p∗iE(zi) ∂βi ∂x (a,E(p∗i )) + βi(a,E(p∗i ))zi](a, t)da, i = 1, 2, zi(a, 0) = 0, a ∈ [0, A], i = 1, 2. (4.5) We note that, for given p∗, the existence of the limit limε→0+ ε −1[pε(a, t)− p∗(a, t)] and the well-posedness of linear system (4.5) can be treated by a standard way [3]. EJDE-2020/58 CONTROLLABILITY OF HIERARCHICAL SYSTEMS 11 Multiplying the i-th equation in the system (4.5) with qi(a, t) and integrating over Q, we obtain that (with a use of (4.2)) 2∑ j=1 ∫ A 0 zj(a, T )[p̄j(a)− p∗j (a, T )]da = 2∑ j=1 ∫ Q qj(a, t)vj(a, t) da dt. (4.6) Combining (4.6) with (4.4), we see that 2∑ j=1 ∫ Q [(kqj − u∗j )vj ](a, t) da dt ≤ 0 (4.7) holds for every v ∈ TU (u∗). Therefore, (kq1 − u∗1, kq2 − u∗2) ∈ NU (u∗), the normal cone to U at u∗. A use of the characteristics of normal vectors derives the conclusion. � Theorem 4.3. The optimal control problem (P1) has one and only one solution. Proof. Let ui = (ui1, u i 2) ∈ U , i = 1, 2 be fixed and arbitrary, with u1 6= u2. For ε ∈ (0, 1), define the real-valued function H(ε) := J(εu1 + (1− ε)u2). Next we show that H ′(ε) is strictly monotone increasing. Denote by pε and pε+δ (with δ > 0 small enough) the solutions of (2.1) corre- sponding to εu1 + (1− ε)u2 and (ε+ δ)u1 + [1− (ε+ δ)]u2, respectively. Then H ′(ε) = lim δ→0 1 δ { J [(ε+ δ)u1 + (1− (ε+ δ))u2]− J(εu1 + (1− ε)u2) } = lim δ→0 1 δ {1 2 2∑ j=1 ∫ Q [(ε+ δ)u1j + (1− (ε+ δ))u2j ] 2(a, t) da dt + k 2 2∑ j=1 ∫ A 0 [pε+δj (a, T )− p̄j(a)]2 da − 1 2 2∑ j=1 ∫ Q [εu1j + (1− ε)u2j ]2(a, t) da dt − k 2 2∑ j=1 ∫ A 0 [pεj(a, T )− p̄j(a)]2 da } = 2∑ j=1 ∫ Q [(εu1j + (1− ε)u2j )(u1j − u2j )](a, t) da dt + k 2∑ j=1 ∫ A 0 zεj (a, T )[pεj(a, T )− p̄j(a)] da , (4.8) where (zε1, z ε 2) is the solution of (4.5) corresponding to u∗ = εu1+(1−ε)u2. Treating in a similar manner as in the proof of (4.6), we have 2∑ j=1 ∫ A 0 zεj (a, T )[pεj(a, T )− p̄j(a)] da = 2∑ j=1 ∫ Q [(u1j − u2j )pεjqεj ](a, t) da dt, (4.9) 12 Z.-R. HE, N. ZHOU EJDE-2020/58 where qε is solution of the system (4.2) corresponding to u∗ = εu1 + (1− ε)u2 and p∗ = pε. Combining (4.8) and (4.9) yields H ′(ε) = 2∑ j=1 ∫ Q {(u1j − u2j )[kpεjqεj + (εu1j + (1− ε)u2j )]}(a, t) da dt. (4.10) For any εi ∈ (0, 1), i = 1, 2, with ε1 6= ε2, we have (ε1 − ε2)[H ′(ε1)−H ′(ε2)] = (ε1 − ε2) 2∑ j=1 ∫ Q {(u1j − u2j )[kp ε1 j q ε1 j − kp ε2 j q ε2 j + (ε1 − ε2)(u1j − u2j )]}(a, t) da dt = k(ε1 − ε2) 2∑ j=1 ∫ Q [(u1j − u2j )(p ε1 j q ε1 j − p ε2 j q ε2 j )](a, t) da dt + (ε1 − ε2)2 2∑ j=1 ∫ Q (u1j − u2j )2(a, t) da dt =: ∆ + (ε1 − ε2)2 2∑ j=1 ∫ Q (u1j − u2j )2(a, t) da dt . (4.11) From the continuity and bounded-ness of p and q (let the upper bounds be C3 and C4), it follows that ∆ = k(ε1 − ε2) 2∑ j=1 ∫ Q [(u1j − u2j )(p ε1 j q ε1 j − p ε2 j q ε2 j )](a, t) da dt ≥ −k|ε1 − ε2| 2∑ j=1 ∫ Q [|u1j − u2j | |p ε1 j q ε1 j − p ε2 j q ε2 j |](a, t) da dt = −k|ε1 − ε2| 2∑ j=1 ∫ Q [|u1j − u2j | |p ε1 j (qε1j − q ε2 j ) + qε2j (pε1j − p ε2 j )|](a, t) da dt ≥ −k|ε1 − ε2| { 2∑ j=1 ∫ Q [|u1j − u2j |p ε1 j |q ε1 j − q ε2 j |](a, t) da dt + 2∑ j=1 ∫ Q [|u1j − u2j ||q ε2 j ||p ε1 j − p ε2 j |](a, t) da dt } ≥ −k|ε1 − ε2| { C3 2∑ j=1 [ ∫ Q (u1j − u2j )2(a, t) da dt ∫ Q (qε1j − q ε2 j )2(a, t) da dt ]1/2 + C4 2∑ j=1 [ ∫ Q (u1j − u2j )2(a, t) da dt ∫ Q (pε1j − p ε2 j )2(a, t) da dt ]1/2} ≥ −k(ε1 − ε2)2(C3 √ C2T + C4 √ C1T ) 2∑ j=1 ∫ Q (u1j − u2j )2(a, t) da dt. EJDE-2020/58 CONTROLLABILITY OF HIERARCHICAL SYSTEMS 13 Inserting the above expression into (4.11), we see that (ε1 − ε2) [ H ′(ε1)−H ′(ε2) ] ≥ (ε1 − ε2)2[1− k √ T (C3 √ C2 + C4 √ C1)] 2∑ j=1 ∫ Q (u1j − u2j )2(a, t) da dt. If T is small enough, then (ε1 − ε2) [ H ′(ε1) − H ′(ε2) ] > 0. Therefore H(ε) is strictly convex down, and there exists only one ε∗ ∈ (0, 1), which minimizes H(ε). Consequently, the functional J(u) will be minimized at ε∗u1+(1−ε∗)u2. The proof is complete. � 5. Stabilization of the zero state Firstly, we consider the stability of the zero solution to the system ∂p1 ∂t + ∂p1 ∂a = −[µ1(a) + f1(E(p2)(a, t))]p1(a, t), (a, t) ∈ Q, ∂p2 ∂t + ∂p2 ∂a = −[µ2(a) + f2(E(p1)(a, t))]p2(a, t), (a, t) ∈ Q, pi(0, t) = ∫ A 0 βi(a,E(pi)(a, t))pi(a, t)da, t ∈ (0, T ); i = 1, 2, pi(a, 0) = p0i (a), a ∈ [0, A]; i = 1, 2, E(pi)(a, t) = α ∫ a 0 pi(r, t)dr + ∫ A a pi(r, t)dr, (a, t) ∈ Q. (5.1) The linearization of this system about (0, 0) is ∂p1 ∂t + ∂p1 ∂a = −[µ1(a) + f1(0)]p1(a, t), (a, t) ∈ Q, ∂p2 ∂t + ∂p2 ∂a = −[µ2(a) + f2(0)]p2(a, t), (a, t) ∈ Q, pi(0, t) = ∫ A 0 βi(a, 0)pi(a, t)da, t ∈ (0, T ); i = 1, 2, pi(a, 0) = p0i (a), a ∈ [0, A], i = 1, 2. (5.2) Considering the solutions to (5.2) in the form pi(a, t) = wi(a) exp{λt}, i = 1, 2, we have λwi(a) + w ′ i(a) = −[µi(a) + fi(0)]wi(a), a ∈ (0, A), i = 1, 2. (5.3) Consequently, wi(a) = wi(0) exp { − ∫ a 0 [λ+ µi(r) + fi(0)]dr } . (5.4) Substituting this expression into the third equation of (5.2), we derive the charac- teristic equation∫ A 0 βi(a, 0) exp { − ∫ a 0 [λ+ µi(r) + fi(0)]dr } da = 1. (5.5) It can be readily seen that if there exists an i ∈ {1, 2}, such that∫ A 0 βi(a, 0) exp { − ∫ a 0 [µi(r) + fi(0)]dr } da > 1, (5.6) 14 Z.-R. HE, N. ZHOU EJDE-2020/58 then equation (5.5) has at least one positive real solution, which means the zero solution to (5.1) is unstable. Next, we investigate the problem of how to keep the zero state stable by means of appropriate migration strategies. In a finite period [0, T ], its specific form is (P2) Find u∗ ∈ U , such that J(u∗) ≤ J(u) for every u ∈ U , with J(u) = 1 2 2∑ j=1 ∫ Q {u2j (a, t) + kp2j (a, t)} da dt, and (u, p) is subject to (5.1), and k is the penalty parameter. As in Theorem 4.2, we are able to describe the optimal policies as follows. Theorem 5.1. Problem (P2) has a unique solution. If (u∗, p∗) is an optimal pair to the problem, then there are variables (q1, q2) solving the adjoint system (with arguments (a, t) omitted in the main equations) ∂q1 ∂t + ∂q1 ∂a = [µ1 + f1(E(p∗2))]q1 − q1(0, t)[β1(a,E(p∗1)) + Ẽ1] + Ē2 + kp∗1, ∂q2 ∂t + ∂q2 ∂a = [µ2 + f2(E(p∗1))]q2 − q2(0, t)[β2(a,E(p∗2)) + Ẽ2] + Ē1 + kp∗2, qj(a, T ) = qj(A, t) = 0, a ∈ (0, A), t ∈ (0, T ), j = 1, 2, (5.7) such that u∗i = F(qi), i = 1, 2. Here, the functions F , Ẽi, Ēi are given by (4.2). Concluding remarks. We have established the approximate controllability for the hierarchical competitive system (2.1) in Theorem 3.6, which means that there exist migrations (u1, u2) such that the populations distribution (p1(a, T ), p2(a, T )) approaches arbitrarily to the prescribed target. With a proper selection of k, The- orems 4.2 and 4.3 demonstrate that the migrations can be uniquely determined by equations (2.1), (4.2) and the feedback control law in Theorem 4.2. Further- more, the control policy may be approximated by means of the conjugate gradient algorithms [31, P. 29]. Therefore, we have sufficient theoretic and computational preparations for the population adjustment. A similar understanding applies to the stabilization of the trivial state. Acknowledgments. This work was supported by National Natural Science Foun- dation of China (11871185), and by Zhejiang Provincial Natural Science Foundation of China (LY18A010010). 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N.; Stabilizing a mathematical model of population system. Journal of the Franklin Institute, 2011, 348: 2744-2758. [31] Zeidler, E.; Nonlinear Functional Analysis and Its Applications. 1986, Springer-Verlag, Berlin. Ze-Rong He (corresponding author) Department of Mathematics, Institute of Operational Research and Cybernetics, Hangzhou Dianzi University, Hangzhou 310018, China Email address: zrhe@hdu.edu.cn 16 Z.-R. HE, N. ZHOU EJDE-2020/58 Nan Zhou Department of Mathematics, Institute of Operational Research and Cybernetics, Hangzhou Dianzi University, Hangzhou 310018, China Email address: 1749521930@qq.com 1. Introduction 2. Model normalization and assumptions 3. Approximate Controllability 4. Choice of controls 5. Stabilization of the zero state Concluding remarks Acknowledgments References