Optimal strategies for a time-dependent harvesting problem

We focus on an optimal control problem, introduced by Bressan and Shen as a model for fish harvesting. We consider the time-dependent case and we establish existence and uniqueness of an optimal strategy, and sufficient conditions for optimality. We also consider a related differential game that models the situation where there are several competing fish companies and we prove existence of Nash equilibria. From the technical viewpoint, the most relevant point is establishing the uniqueness result. This amounts to prove precise a-priori estimates for solutions of suitable parabolic equations with measure-valued coefficients. All the analysis is developed in the case when the fishing domain is one-dimensional.


Introduction
This paper deals with a model for fish harvesting introduced by Bressan and Shen in [5]. The model involves an optimization problem for a payoff functional representing the profit of the fish company. We consider the time-dependent case and prove existence and (local) uniqueness of optimal strategies. We also exhibit sufficient conditions for optimality and establish the existence of Nash equilibria in the case where there are several competing players, i.e. fish companies. We always focus on the case when the fishing domain is one-dimensional.
Before discussing our results, we go over the main features of the model introduced in [5]. Consider a one-dimensional fishing domain, modeled by the interval ]0, R[, and a time interval ]0, T [. We denote by ϕ = ϕ(t, x) the density of fish at time t ∈]0, T [ at the point x ∈]0, R[. We assume that, when no fishing activity is conducted, the evolution of the fish population is modelled by the parabolic equation In the above expression, c is a nonnegative, lower semicontinuous function representing the cost of the fishing effort. One could for instance have a cost c which is monotone increasing with respect to the distance of the point x from the fish company hub. Also, the presence of a natural park where no fishing is allowed can be modeled by setting c(t, x) = +∞ in that region. We can now define our payoff functional by setting In the above expression, Ψ is a nondecreasing, convex function (the simplest possible choice is the identity). The function ϕ is the solution of the initial-boundary value problem obtained by coupling (1.4) with (1.2) and (1.3). Note, in particular, that ϕ depends on µ and hence the functional J is nonlinear.
In the present paper we focus on the problem of maximizing the payoff functional J, i.e. finding an optimal fishing strategy µ. We impose the constraints In the above expression, the nonnegative function b models the maximum amount of harvesting power within the capabilities of the fish company. In practice, it may for instance depend on the number of fishermen and on the size of the fishing boats. To solve the above optimization problem we actually search for optimal strategies that are not necessarily functions, but more generally nonnegative Radon measures. This is motivated by two main considerations: • from the analytic viewpoint, we remark that the functional J has only linear growth with respect to µ, and hence we expect that, in general, an optimal strategy µ does not belong to Note that a quadratic harvesting cost such likê T 0ˆR 0 c(t, x)µ 2 (t, x) dtdx is entirely natural from the mathematical viewpoint and it would give an optimal strategy µ opt ∈ L 2 (]0, T [×]0, R[). However, the linear cost (1.5) provides a more realistic model. • From the modeling viewpoint, it is reasonable to expect that, when for instance there is a natural park, the optimal strategy concentrates the fishing effort at the park border. An explicit analytic example where the optimal strategy contains atomic parts concentrated at discontinuity points of c is exhibited in [6].
As mentioned before, the above model for fish harvesting was introduced by Bressan and Shen in [5].
In [5] the analysis focuses on the one-dimensional, steady state when (1.4) reduces to a second order ordinary differential equation. The authors establish existence and local uniqueness of optimal strategies and discuss the related differential game showing existence of Nash equilibria. See also [6] for related results. In [4] Bressan, Coclite and Shen established existence of optimal strategies for the steady case by considering multidimensional fishing domains. Finally, in [10] Coclite and Garavello established existence of optimal strategies in multi-dimensional domains in the time-dependent case.
The main results of the present paper are the following: • we establish existence, uniqueness and stability of weak solutions (in the sense of Definition 3.1 in § 3) of the parabolic equation (1.4) in the case when µ is a Radon measure, see Theorem 3.2.
We basically follow the same strategy as in [10], but we can impose much weaker assumptions on the coefficient µ owing to the fact that the domain is one-dimensional. • We establish existence of an optimal strategy µ for the payoff functional J in (1.6) subject to the constraints (1.7). We also establish sufficient conditions for optimality (see Theorem 4.1) and we show that the optimal strategy is locally unique, i.e. it is unique in the class of measures with sufficiently small total variation. The uniqueness result is stated as Theorem 5.1. Note that, while the existence proof is the same as in [4,10], the uniqueness proof is, from the technical viewpoint, the most relevant result of the present paper. • By relying on the above local uniqueness result we establish existence of Nash equilibria for a differential game modeling the case where there are several competing fish companies that exploit the same environment, see Theorem 7.1 for the precise result. The local uniqueness of optimal strategies was established by Bressan and Shen [5] in the steady case. The main novelties of our analysis compared to the one in [5] are the following: • in both cases, the main point of the argument is showing that the functional J is locally concave. This amounts to establish suitable a-priori estimates on the solutions of parabolic equations with measured-valued coefficients similar to (1.4). However, as mentioned before, in the steady case the parabolic equation (1.4) reduces to a second order ordinary differential equation: this makes the analysis considerably simpler than the time-dependent case. In particular, in the time-dependent case we establish precise estimates on solutions of parabolic equations with measured-valued coefficients by making extensive use of the Duhamel representation formula. • The analysis in [5] is based on a technical assumption, i.e. condition [5, (5.15)]. In the timedependent case we replace [5, (5.15)] with (7.5), namely with the requirement that the initial fish density distribution is sufficiently close, in the H 1 norm, to the constant h, which represents the maximal fish density supported by the environment. The exposition is organized as follows. In §2 we establish existence, uniqueness and stability results for a nonlinear parabolic problem with smooth coefficients. These results are pivotal to the analysis in §3, where we establish existence, uniqueness and stability results for the initial-boundary value problem (1.2), (1.3), (1.4) in the case when µ is a given nonnegative Radon measure. In §4 we prove existence of an optimal strategy maximizing (1.6) subject to the constraint (1.7). We also establish sufficient conditions for optimality. In §5 we establish the local uniqueness of the optimal strategy, while §6 is devoted to the proof of a technical result which is pivotal to the existence proof. Finally, in §7 we introduce a differential game modeling the situation where there are several competing fish companies and we establish existence of Nash equilibria. Finally, in the Appendix we collect some results concerning the Duhamel representation formula and the fundamental solutions of the heat equation that we use in the paper.
Notation. For the reader's convenience, we collect here the main notation used in the present paper.
Throughout the paper, C(a 1 , . . . , a k ) denotes a constant which only depends on the quantities a 1 , . . . , a k : its precise value can vary from occurrence to occurrence. Also, K denotes a universal constant (i.e., a number) and again its precise value can vary from occurrence to occurrence.
General mathematical symbols.
Note that the Sobolev space H 1 (]0, R[) compactly embeds into C 0 ([0, R]) and we have the inequality • • a.e. x, a.e. (t, x): for L 1 almost every x, for L 2 almost every (t, x). Here L 1 , L 2 denote the Lebesgue measure on R 1 and R 2 , respectively.
To conclude, we recall that, owing to the Hölder inequality, Notation introduced in the present paper.
• T : the length of the time interval where we set our problem.
• R: the length of the space interval where we set our problem.

A nonlinear parabolic problem with smooth coefficients
In this section we focus on the nonlinear parabolic problem x ∈ ]0, R[.
In the previous expression a :]0, T [×]0, R[→ R is a nonnegative, smooth function. The nonlinear source term f satisfies the following hypothesis.
and there are a constant α 1 > 0 and a continuous, We first provide the definition of weak solution of (2.1).
Definition 2.1. We term weak solution of the initial-boundary value problem (2.1) a function Also, we require that the following equality holds for every test function v ∈ C ∞ c (] − ∞, T [×R): Note that (2.4) implies that, by possibly changing the value of ϕ(t, ·) in a negligible set of times, we can assume that ϕ ∈ C 0 [0, T ]; L 2 (]0, R[) , see for instance [18,Theorem 7.22]. In the following, we will always identify ϕ satisfying (2.4) with its L 2 -continuous representative. In this way we can define the value ϕ(t, ·) for every t ∈ [0, T ]. We can now state the main result of this section, which establishes existence, uniqueness and stability for the initial-boundary value problem (2.1).
Theorem 2.1. Let hypothesis (H.1) hold and assume furthermore that Then the initial-boundary value problem (2.1) admits a unique weak solution. Also, this solution enjoys the following properties: first, Second, we have stability with respect to the initial datum and with respect to the coefficient a. More precisely, let ϕ be the solution of the initial-boundary value problem x ∈ ]0, R[, whereâ satisfies the same hypotheses as a andφ 0 satisfies (2.6). Then (2.11) In the previous expression, the constant F is defined by setting The proof of Theorem 2.1 is organized as follows. In Subsection 2.1 we establish the stability estimate, which implies uniqueness of weak solutions. In Subsection 2.2 we establish existence of a weak solution of the initial-boundary problem (2.1) by relying on an iteration algorithm.
Next, we point out that (2.12) ≤ α 1 |ϕ| + F and the above inequality implies (2. 16) We conclude the proof by proceeding according to the following steps.
Step 1: we give a formal proof of uniqueness and stability. We proceed formally, i.e. we pretend that everything is sufficiently regular to have that all the following manipulations are justified. We refer to Step 2 below for the rigorous justification of our argument. We multiply the equation at the first line of (2.14) times ψ, we integrate with respect to space and we use the homogeneous Neumann boundary conditions. We get Next, we recall thatâ ≥ 0 by assumption and we infer that Hence, from (2.17) we get (2.18) Next, we use the Young Inequality: we fix a parameter γ > 0, to be determined in the following, and we point out that (2.19) We now choose γ in such a way that we recall that and by using (2.18) we arrive at (2.21) Owing to the Gronwall Lemma, the above inequality implies that The same inequalities hold also forφ. To establish uniqueness, we take ϕ 0 ≡φ 0 , which implies ψ 0 ≡ 0, and a ≡â. From (2.22) we get ψ ≡ 0, which owing to (2.13) implies ϕ ≡φ and hence establishes uniqueness.
To establish the stability estimate (2.11) we use the uniform bound (2.8) and from (2.22) we get (2.23) By plugging the above inequality into (2.21) and integrating in time we eventually arrive at (2.11).
Step 2: we provide the rigorous justification of our argument. First, we recall the definition (2.13) and we point out that, for every test (2.24) We fix t ∈]0, T ] and choose the test function v by setting where z j is determined in the following and w n is a smooth cut-off function with compact support such that We let n → +∞ and by using the fact that ψ ∈ C 0 ([0, T ], L 2 (]0, R[)) we infer from (2.24) the following equalityˆR (2.26) Next, we point out that and we choose a sequence of test functions z j in such a way that We let j → +∞ and by arguing as in Step 1 and using (2.27) we infer from (2.26) the inequalitŷ We can then argue as in Step 1 and establish the uniqueness of weak solutions and the stability estimate (2.11).

2.2.
Existence. In this paragraph we establish existence of a weak solution of the initial-boundary value problem (2.1). More precisely, we proceed as follows.
• § 2.2.1: we define an approximation algorithm and we establish a-priori bounds on the approximate solutions. To simplify the analysis, in § 2.2.1 we assume that the initial datum ϕ 0 is smooth. This hypothesis will be eventually removed in § 2.2.3. • § 2.2.2: we establish compactness of the approximate solutions, we pass to the limit and we establish existence of a weak solution of (2.1). • § 2.2.3: we conclude the proof of Theorem 2.1 by establishing the maximum principle (2.8) and by removing the assumption that ϕ 0 is smooth.
2.2.1. Construction of approximate solutions. In this paragraph we define the iteration algorithm that we will use to establish the existence of a solution ϕ as in the statement of Theorem 2.1. First, we term ϕ 1 the solution of the following linear parabolic initial-boundary value problem: x, ϕ 0 (x) . Next, we argue iteratively: we assume that the function ϕ n : [0, T ]×[0, R] → R is given and we define the function f n : We term ϕ n+1 the solution of the following linear parabolic initial-boundary value problem: x ∈]0, R[ , The main ingredient in the iteration argument is the following lemma.
Lemma 2.2. Let the same assumptions as in the statement of Theorem 2.1 hold. Assume furthermore that • the initial datum ϕ 0 ∈ C ∞ ([0, R]); • the function ϕ n satisfies Then the initial-boundary value problem (2.30) admits a unique classical solution ϕ n+1 ∈ C ∞ ([0, T ] × [0, R]), which furthermore satisfies the following estimates: first, Second, Proof. We proceed according to the following steps.
Step 1: we establish existence of a classical solution. First, we recall that by assumption the function ϕ n is smooth. The very definition (2.29) of f n and (2.31) implies that f n is also smooth, and henceforth bounded on [0, T ] × [0, R]. The same holds for the coefficient a, which is smooth by assumption (2.7). Existence and uniqueness of a classical solution of the initial-boundary value problem (2.1) follows then from classical results on linear parabolic equations, see for instance [15, §7.1].
Step 2: we establish (2.33). First, we multiply the equation at the first line of (2.30) times ϕ n+1 and we integrate in space and we use the homogeneous Neumann boundary conditions. We arrive at (2.34) Next, we recall the definition (2.29) of f n , the fact that by assumption a n ≥ 0 and hypothesis (2.3).
We conclude that Owing to condition (2.2), where F is the same as in (2.12). Hence, f n ≤ F and from (2.34) we arrive at and by plugging the above inequality into (2.35) and integrating with respect to time we arrive at (2.33).
Step 3: we establish (2.32). First, we define the C 2 function β : R → R by setting We multiply the equation at the first line of (2.1) times β ′ (ϕ n+1 ) and we integrate in space. By using the relation β ′ (w)w = 4β(w) and proceeding as at the previous step we infer that By assumption (2.6), ϕ 0 ≥ 0 and hence β(ϕ 0 ) = 0. This implies that by combining (2.37) with the Gronwall Lemma we get that β(ϕ n+1 ) ≡ 0, namely ϕ n+1 ≥ 0. This concludes the proof of the lemma.

Limit analysis.
This paragraph aims at establishing the following result.
Assume that the hypotheses of the Theorem 2.1 are satisfied and furthermore that . Let ϕ n be the sequence of functions recursively defined in § 2.2.1. Then for some limit function ϕ satisfying Also, ϕ is a weak solution of (2.1).
Proof. We proceed according to the following steps.
Step 1: we establish compactness of the sequence ϕ n . We first outline our argument: we want to apply the Aubin-Lions Lemma [19]. We recall the Rellich Theorem, which gives that the inclusion endowed with the standard dual norm. By putting together all the previous considerations we conclude that we have the following chain of inclusions, and the first inclusion is compact and the second is continuous. Owing to the Aubin-Lions Lemma, to establish the compactness of the sequence {ϕ n } in Condition (2.40) immediately follows from (2.33). To establish (2.41) we multiply the equation at the first line of (2.30) times a test function v ∈ H 1 (]0, R[) and we integrate with respect to x. By using the Integration by Parts Formula and recalling that ϕ n+1 satisfies homogeneous boundary conditions we obtainˆR Owing to (2.15), (2.43) By plugging the previous inequality into (2.42) and using (2.33) we conclude that This establishes (2.41) and hence shows that the sequence {ϕ n } is compact in Step 2: we show that every accumulation point ϕ is a weak solution of (2.1). Owing to Step 1 we have that, up to subsequences, for some accumulation point ϕ. Note that, owing to (2.33), the sequence {∂ x ϕ n } is uniformly bounded in L 2 (]0, T [×]0, R[) and hence, by the Urysohn Lemma, Owing to (2.33) and to the lower semicontinuity of the norm with respect to weak convergence, we have that ϕ satisfies (2.39) and hence, in particular, .
We now show that ϕ is a weak solution of (2.1). Owing to the uniqueness result established in § 2.1, this implies that the accumulation point ϕ is unique and hence that the convergence (2.45) holds for the whole sequence {ϕ n }.
First, we point out that, owing to (2.2), and hence, owing to (2.45), To conclude, we fix a test function v ∈ C ∞ c (] − ∞, T [×R), we use the equation at the first line of (2.30), we integrate in space and time and we arrive at Owing to (2.45), (2.46) and (2.47), we can pass to the limit in all the terms in the previous expression and obtain (2.5). This shows that ϕ is a weak solution of (2.1) and concludes the proof of Lemma 2.3.

2.2.3.
Conclusion of the proof of Theorem 2.1. First, we establish (2.8). The bound from below is a direct consequence of (2.32) and (2.45). To establish the bound from above, we recall that a(t, x) ≥ 0 for every (t, x). Next, we point out that, owing to (2.3) and (2.9), f (t, x, M ) ≤ 0. These considerations imply that the function identically equal to M is a supersolution of the initial-boundary value problem (2.1) and hence that ϕ ≤ M .
To conclude the proof of Theorem 2.1, we are left to remove the assumption that ϕ 0 ∈ C ∞ ([0, R]) from the existence part. We fix ϕ 0 ∈ L ∞ (]0, R[), ϕ 0 ≥ 0, and we construct a sequence of functions {ϕ 0k } such that We apply Lemma 2.3 and we term ϕ k the corresponding sequence of solutions of the initial-boundary value problem (2.1) with initial condition given by ϕ 0k . Note that ϕ k satisfieŝ T 0ˆR 0 Next, we chooseâ = a and we use the stability estimate (2.11). We conclude that the sequence {ϕ k } is a Cauchy sequence in L 2 (]0, T [; H 1 (]0, R[). We term ϕ its limit and by passing to the limit in (2.49) we get (2.5). This shows that ϕ is a weak solution of (2.1) and hence concludes the proof of Theorem 2.1.

A parabolic problem with measure-valued coefficients
In this section we establish the well-posedness of the following parabolic initial-boundary value problem with a time-dependent, measure-valued coefficient µ t : We introduce the following hypothesis on the coefficient µ t . (H. 2) The measured valued coefficient µ t satisfies the following conditions: In the following, we denote by L ∞ (]0, T [; M(]0, R[)) the space of time-dependent measures satisfying (H2). Also, we denote by µ the measure defined by setting where ½ E denotes the characteristic function of E. Also, Proof. First, we recall that, owing to the inclusion , the function ϕ(t, ·) is continuous for a.e. t ∈]0, T [. Next, we introduce a convolution kernel and we construct a sequence of continuous functions such that Owing to (3.7), we have Since the functions ϕ n are continuous and henceforth µ-measurable, the function ϕ is µ-measurable. We now show that ϕ is µ-summable. Owing to the analysis in [1, §2.5], we have . Owing to (3.8), the limit is ϕ, which is henceforth a µ-summable function. Finally, by passing to the limit in the equalitŷ we arrive at (3.4) and (3.5) follows by arguing as in (3.9).
We can now provide the definition of weak solution of (3.1). (3.10) Note that the second term in the above expression is well defined owing to Lemma 3.1. We now state the main result of the present section.
Theorem 3.2. Assume (H.1) and (H.2) and assume furthermore that the initial datum ϕ 0 satisfies (2.6). Then the initial-boundary value problem (3.1) admits a unique weak solution. Also, this solution enjoys the following properties: first, where M is the same constant as in (2.9). Second, we have stability with respect to the initial datum and with respect to the coefficient µ. Namely, if we term ϕ the solution of the initial-boundary value problem Proof. To establish the uniqueness of the solution of (3.1) and the stability estimate (3.13) we can follow the same argument as in § 2.1. We are left to establish existence and we proceed according to the following steps.
Step 1: we construct a sequence of approximate solutions. First, we take a sequence {a j } such that for every j we have We apply Theorem 2.1 and we term ϕ j the solution of the initial-boundary value problem (2.1) in the case when a = a j . By applying the stability estimate (2.11) withâ = a j ,φ 0 = 0 we obtain that Step 2: we establish compactness of the sequence {ϕ j }. We apply the Aubin-Lions Lemma. First, we point out that we have the following chain of inclusions and the first inclusion is compact and the second is continuous. Owing to the Aubin-Lions Lemma, to establish the compactness of the sequence To establish (3.16) we can use (3.15). To establish (3.17), we use the equation at the first line of (2.1), repeat the same computations as in Step 1 of the proof of Lemma 2.3, arrive at (2.44) and then use the forth estimate in (3.14).
By relying on the Aubin-Lions Lemma we conclude that, up to subsequences, we have ) and hence, in particular, By extracting, if needed, a further subsequence, we have and, owing to (2.8), this implies, in particular, that ϕ satisfies (3.11). Finally, we recall that ∂ x ϕ j is uniformly bounded in L 2 (]0, T [×]0, R[) owing to (3.15) and we infer that, by the Urysohn Lemma, By combining (3.15), (3.20) and the lower-semicontinuity of the norm with respect two weak convergence we infer that ϕ satisfies Step 3: we show that the accumulation point ϕ is a weak solution of (3.1). First, we point out that ϕ ∈ L 2 (]0, T [; H 1 (]0, R[)) owing to (3.22). Next, we fix a test function v ∈ C ∞ (] − ∞, T [×R) and we recall that ϕ j satisfies (2.5). We now pass to the limit in each of the terms in (2.5). Owing to (3.19) and (3.21), we have Next, we point out that ≤ α 1 |ϕ j − ϕ| and hence, owing to (3.19), Finally, we consider the last term. First, we point out that . (3.25) We first deal with the term T j 1 : we recall that, owing to (3.18), the function ϕ(t, ·) is continuous for a.e. t ∈]0, T [. We recall the third property in (3.14) and we conclude that as j → +∞ T j 1 (t) → 0 for a.e. t ∈]0, T [. Next, we point out that for a.e. t ∈]0, T [ Owing to the Lebesgue Dominated Convergence Theorem, this implies that Next, we deal with the term T j 2 : we first point out that and hence, owing to (3.20), for a.e. t ∈]0, T [, T j 2 (t) → 0 as j → +∞. By using again (3.27) we get and hence, by applying the Lebesgue Dominated Convergence Theorem, we get By combining (3.25), (3.26) and (3.28) we get that We recall (3.23) and (3.24) and we eventually conclude that ϕ satisfies (3.10). This concludes the proof of Theorem 3.2.

Necessary conditions for optimality
This section aims at discussing existence of optimal strategies µ for the payoff functional In the above expression, ϕ is the weak solution of the initial-boundary value problem (3.1). Note that ϕ depends on µ, and hence the functional J is nonlinear. The functions c and Ψ satisfy the following hypotheses. In the following we aim at discussing the existence and uniqueness of an optimal µ for J under the constraint for some suitable constant b 0 > 0.
Note that b is µ t -integrable because it is lower semicontinuous and positive, and hence the integral at the left hand side of (4.2) is well-defined. Note furthermore that by combining (4.3) with (4.2) we get We now focus on the problem The following result establishes the existence of an optimal strategy µ * for problem (4.5).
x ∈]0, R[ and satisfies (4.7) T 0ˆR 0 The function g : [0, T ] × [0, R] × R → R at the first line of (4.6), is defined by setting Proof. Fix ν as in the statement of the theorem and, for every real number ε > 0, consider the quantity In the previous expression, we term ϕ ε the solution of the initial-boundary problem (3.1) in the case µ = µ * + εν. The heuristic idea to establish (4.6) is to differentiate both J(µ * + εν) and ϕ ε with respect to the variable ε. The rigorous proof is organized into the following steps.
Step 1: we construct an approximate derivative of ϕ * . For every ε > 0, define Note that ψ ε is the weak solution, in the sense of Definition 3.1, of the initial-boundary value problem (4.10) In other words, for every test We now provide a formal argument, which can be made rigorous by following the same argument as in Step 2 of § 2.1. We multiply the equation at the first line of (4.10) times ψ ε and integrate by parts to get (4.14) Next, we fix a constant k > 0 (to be determined in the following), we use the Young Inequality and we infer that (4.15) Next, we choose the constant k in such a way that C(M, R)k = 1 and by combining (4.14) and (4.15) we arrive at (4.16) Owing to the Gronwall Lemma and to the fact that ψ ε (t = 0) ≡ 0, the above inequality implies that    Owing to the Aubin-Lions Lemma, we infer that the family {ψ ε } is compact in L 2 (]0, T [; C 0 ([0, R])) and hence that there is a sequence ψ ε k and a function ϕ 1 ∈ L 2 (]0, T [; C 0 (]0, R[)) such that Step 3: we show that the accumulation point ϕ 1 is a weak solution of (4.6). We argue by passing to the limit in each of the terms in (4.12). First, we point out that by using (4.21) and (4.23) we get Next, we point out that  We now want to show that (4.26) To this end, we recall that ϕ ε is the weak solution of the initial-boundary problem (3.1) in the case when µ = µ * + εν and we apply the stability estimate (3.13) with ϕ = ϕ * ,φ = ϕ ε , µ = µ * ,μ = µ * + εν. By using (1.8), we infer that and by arguing as in (4.25) we arrive at (4.26). We are left with the last term in (4.12). To handle it, we recall that 0 ≤ ϕ * , ϕ ε ≤ M owing to (3.11), we use the Taylor formula with Lagrange remainder and we infer that Owing to (4.8), (4.9) and (4.11), this implies that x, y)||ψ ε ||ϕ ε − ϕ * | and hence that and show that ϕ 1 is a weak solution of (4.6).
Step 4: we establish (4.7). We recall that ε k > 0, that µ * is an optimal strategy and µ * + ε k ν is a competitor provided that ν ≥ 0 is sufficiently small. We conclude that By using the explicit expression of J(µ * + ε k ν) we infer By combining the two above expressions we get that is (4.7). This concludes the proof of Theorem 4.2.

Uniqueness of optimal solutions
In this section we discuss the uniqueness of optimal solutions of the optimization problem (4.5). We refine hypothesis (H.1) by introducing the following condition. We recall that the function ∂ ϕ f is continuous and by recalling (2.2) we define the constant Also, in the following we term α 2 a Lipschitz constant for ∂ ϕ f , namely The main result of the present section states that, if the initial datum ϕ 0 is sufficiently small, then the solution of the optimization problem (4.5) is unique within a class of measures with sufficiently small total variation. Proof of Theorem 5.1. We fix two points of maximumμ andμ satisfying (5.5) and we argue by contradiction: we assume thatμ =μ. We set We define the (signed) measure ν t ∈ L ∞ (]0, T [; M(]0, R[)) by setting We define the map j : [0, ε] → R by setting and we point out that by construction j attains its maximum at both ζ = 0 and ζ = ε. Next, we use Lemma 5.2 below and we conclude that the map j is continuous and concave on [0, ε]. This contradicts the fact that j attains its maximum at ζ = ε and hence concludes the proof of Theorem 5.1. Proof. First, we point out that the map j is continuous: this can be seen by using the stability estimate (3.13). Also, by arguing as in the proof of Theorem 4.2 we infer that j is twice differentiable and that, for every ζ ∈]0, ε[, we have provided that the measure µ * is given by (5.11) µ * :=μ + ζν and the functions ϕ 1 and ϕ 2 are defined as follows. The function ϕ 1 is the solution of (5.12) x ∈]0, R[ and ϕ 2 is the weak solution of the initial-boundary value problem In the above equation, ϕ * is the weak solution of (5.14) x ∈ ]0, R[. By using the convexity of the function Ψ, we infer from (5.10) that The proof of Lemma 5.3 is rather long and technical and it is established in § 6.

Proof of Lemma 5.3
To establish the proof of Lemma 5.3 we proceed as follows: in §6.1, §6.2 and §6.3 we provide a formal argument which is completely justified only in the case when all the functions are sufficiently regular. In §6.4 we first conclude this formal argument by (formally) establishing (5.15) and then we explain how the formal argument can be made rigorous by relying on an approximation argument. To simplify notation, in the formal argument given in §6.1, §6. 6.1. Proof of Lemma 5.3: estimates on ϕ * . We first control the distance of the function ϕ * from the constant h. Lemma 6.1. Let ϕ * be the weak solution of the initial-boundary value problem (3.1), then In particular, there is a threshold δ, which only depends on α 1 , M, T, R and h such that, if (5.4) holds, then Proof. Owing to the Duhamel Representation Formula (see the Appendix) we have x, y)f s, y, ϕ * (s, y) ϕ * (s, y)dyds.
Under the same assumptions as in the statement of Lemma 6.1, we have Proof. By differentiating the representation formula (6.3) with respect to x and using (A.8) we get x, y)f s, y, ϕ * ϕ * (s, y)dyds.
6.2. Proof of Lemma 5.3: estimates on ϕ 1 . We can now control the first term in (5.15).  Proof. We argue according to the following steps.
Step 1: we find a more convenient expression for the left hand side of (6.11). Owing to (6.2), the function ϕ * is bounded away from 0. We recall that by definition g(·, ·, ϕ) = f (·, ·, ϕ)ϕ, which implies that We divide the equation at the first line of (4.6) times ϕ * and we use the above expression for ∂ ϕ g. We eventually get By using the above expression for ν t , we can formally rewrite the left hand side of (6.11) aŝ x, ϕ * )ϕ 2 1 dxdt.

(6.16)
Next, we use the Young Inequality and we get for some a > 0 to be determined in the following. We plug the above inequality into (6.16) and we get Owing to (5.2), we get Next, we choose a = 2 and by recalling (3.11) we infer By combining the above equation with (6.18) and plugging the result in (6.17) we arrive at To control A 1 we use (6.6) and argue as follows: (6.20) To control A 2 we use (6.1) and we get (6.21) By plugging (6.20) and (6.21) into (6.19) we eventually arrive at that is (6.11).
6.3. Proof of Lemma 5.3: estimates on ϕ 2 . We now establish a control on the L ∞ norm of ϕ 2 . Proof. First, we recall that, since g(t, x, ϕ) = f (t, x, ϕ)ϕ, then   From (6.25) we get Next, we point out that by definition (6.26) the function I is nondecreasing and we apply the Gronwall Lemma. We get Finally, we conclude the proof of Lemma 6.4 by time integrating the above inequality and using Lemma 6.5 below.
To conclude the proof of Lemma 6.4 we are left to establish the following result. Proof. First, we use the formula in the proof of Lemma 6.3 for ν t , we recall that I is defined as in (6.26) and we getˆT 0 I(t)dt We now separately control the terms J 1 , . . . , J 5 . First, we consider the term J 1 : we point out that . .
We now focus on the term J 11 defined in (6.32). First, we point out that We infer that    We now control J 111 : Next, we control J 112 : By combining (6.36) and (6.37) and recalling that δ ≤ 1 we get (6.38) . We now control J 12 : we recall that J 12 is defined as in (6.31) and that ϕ * satisfies (5.14). We get To control J 121 , we first point out that This implies J 121   We now focus on J 122 , which is defined in (6.39). We control it by arguing as follows: .

(6.46)
To control J 123 , we first point out that Next, we recall that J 123 is defined in (6.39) and we control it by arguing as follows: Finally, we recall (6.33) and we conclude that (6.50) .
We now focus on the term J 2 . We recall that J 2 is defined as in (6.29) and we preliminary point out that Owing to the Integration by Parts Formula, this implieŝ To control J 21 we argue as follows:  Next, we control J 22 by arguing as follows: .
Finally, we control J 23 :  . We now focus on the term J 3 , which is defined as in (6.29), and we control it by arguing as follows: Recalling that J 4 is defined in (6.29), we have: .
Finally, for the term J 5 , defined in (6.29), we have:  and this concludes the proof of Lemma 6.5.
6.4. Conclusion of the proof of Lemma 5.3. We use (6.22) and we get Next, we combine (6.11) with the above inequality and we conclude that In the previous expression, the quantity at the right hand side is negative provided that the constant δ is sufficiently small. This establishes (5.15) and concludes the formal proof of Lemma 5.3.
To complete the proof of Lemma 5.3 we are left to make rigorous the formal argument given so far. To this end, we rely on an approximation argument. First, we recall the equality and we point out that, by passing to the limit in the inequality (4.18), we get that ϕ 1 satisfieŝ By relying on analogous computations and by using (6.63) we infer that We term ϕ * k , ϕ 1k and ϕ 2k the corresponding solutions of the initial-boundary value problems (5.14), (5.12) and (5.13), respectively. Since the coefficient µ * k and ν k and the initial datum ϕ 0k are all smooth, then one can show that the solutions ϕ * k , ϕ 1k and ϕ 2k are also smooth. This implies that the formal argument given at the previous paragraphs is completely justified and one gets provided that the constant δ is sufficiently small. Next, we point out that ϕ * k , ϕ 1k and ϕ 2k satisfies the inequality (2.39), (6.63) and (6.64), respectively. By arguing as in the proof of Theorem 3.2, one can show that In particular, the above convergence results imply that and, by arguing as in the estimate of (3.25), that By passing to the limit in (6.68) we get

Solutions of the differential game and Nash equilibria
This section aims at the discussing the differential game modeling the case when there are several competing fish companies and at establishing the existence of Nash equilibria. More precisely, we define our differential game as follows: we assume that there are m > 1 players (i.e., fish companies) and we denote by µ i the fishing intensity of the i-th company. We term ϕ the fish population density and we consider the initial-boundary value problem The goal of the i-th player (i.e., fish company) is to maximize his payoff J i , which is defined by setting The admissible controls satisfy µ i ∈ L ∞ (]0, T [; M + (]0, R[)) and the constraint Proof. We follow the same argument as in [5, §6] and, to simplify the exposition, we only discuss the case when m = 2 and we assume c 1 = c 2 , Ψ 1 = Ψ 2 . The proof straightforwardly extends to the general case. We proceed according to the following steps.
Next, we fix a small constant δ > 0. The precise value of δ will be determined in the following. We define the set C δ by setting By using the same argument as in the proof of Theorem 4.1 (existence) and Theorem 5.1 (uniqueness) we arrive at the following result.
By relying on Lemma 7.2 we can define the map T by setting T : C δ × C δ → C δ × C δ T (µ 1 , µ 2 ) := µ opt (µ 2 ), µ opt (µ 1 ) . (7.11) We now show that C δ is compact with respect to the weak- * convergence. First, we fix a sequence {µ n } in C δ and we recall that the Borel measure µ n on ]0, T [×]0, R[ is defined by setting (7.12) µ n (E) :=ˆT Note that, if µ n ∈ C δ , then the total variation |µ n | ≤ δT . Hence, there is a Borel measure µ such that, up to subsequences, µ n * . We now have to show that the limit measure µ ∈ C δ , namely it admits a representation like (7.12). To this end, we term π the projection In the above expression, π ♯ µ n denotes the push-forward of the measure µ n and L 1 ]0,T [ the restriction of the Lebesgue measure. Also, the density f n is given by for some accumulation point f satisfying f L ∞ (]0,T [) ≤ δ. By passing to the weak * limit on both sizes of the equality (7.13) we obtain π ♯ µ = f L 1 ]0,T [ . By using the Disintegration Theorem [1, Theorem 2.28] and recalling the inequality f L ∞ (]0,T [) ≤ δ we eventually conclude that µ ∈ C δ . This implies that the set C δ is compact. Also, it is obviously convex. Owing to the Schauder-Tychonoff Fixed Point Theorem, if the map T defined as in (7.11) is continuous, then it admits a fixed point, which is by construction a Nash equilibrium in the sense of Definition 7.1. Hence, the proof of Theorem 7.1 boils down to the proof of the continuity of T .
Step 2: we prove that the map T defined as in (7.11) is continuous with respect to the weak- * convergence. To prove the continuity of T it suffices to show that the map η → µ opt (η) defined as in the statement of Lemma 7.2 is continuous. Hence, we fix (7.14) σ n * ⇀ σ.
The above chain of inequalities implies (7.17) and hence concludes the proof of Theorem 7.1.