Well posedness and control in renewal equations with nonlocal boundary conditions

A large class of biological models leads to initial boundary value problems for nonhomogeneous balance laws, possibly with nonlocal boundary conditions. Here, for these equations, a general well posedness result is proved, a full set of stability estimates is provided, and sample control problems are tackled.


INTRODUCTION
Several biological models lead to systems of renewal equations that fit into the following general initial boundary value problem (IBVP) x ∈ R + a subinterval J ′ ⊆ J, refer to Section 4 for further information on BV functions. As it is usual in the present context, we regard BV as a subset of L ∞ , so that BV functions need not be in L 1 . When a BV function is considered, we refer to its right continuous representative, and, where appropriate, we adopt the usual notation u(x±) = lim →x ± u( ). We now deal with an analytic framework where the models described in Sections 3.1 and 3.2 can be settled, their well posedness proved, and the corresponding optimization problems tackled. Indeed, the models in Section 3 all fit in the general IBVP (1.1)-(1.2) while the quantities to be optimized can be written as in (1.3).
We provide here the basic statements ensuring the well posedness and the stability of (1.1)-(1.2), by which we mean the existence of a solution, its uniqueness, and continuous dependence both with respect to the initial datum and to the terms appearing in the equation. Below, solution to (1.1) is understood in the following sense, see also Colombo and Garavello 5 or the more general Bardos et al 15 x)] dt dx = 0; 2. u(0, x) = u o (x) for a.e. x ∈ R + ; 3. for every i ∈ {1, … , n} and for a.e. t ∈ I, lim The following result ensures the well posedness of the IBVP for (1.1)-(1.2). variations of g and m, such that for any initial data u ′ o , u ′′ o ∈ (L 1 ∩ BV)(R + ; R n ), the corresponding solutions u ′ and u ′′ satisfy ) . The above result admits an immediate extension to the case u o with possibly unbounded total variation. Indeed, in the case ≡ 0, a straightforward approximation argument based on (2.1) ensures the existence of a solution to (1.1)-(1.2). This argument works also in the case ≠ 0; however, a merely L 1 function can hardly be defined a solution to (1.1)-(1.2), due to the lack of meaning of punctual values such as u(t,x i ).
We now state the stability of solutions to (1.1) with respect to initial and boundary data, extending Theorem 2.5 by Colombo and Garavello. 5

Theorem 2.3. Let both systems
satisfy the assumptions of Theorem 2.2. Then, the corresponding solutions u ′ and u ′′ satisfy
The proofs, which extend those in Colombo and Garavello 5, Theorem 2.4, proposition 2.6 , are deferred to Section 4. Observe that in view of the application of the above results to control problems, we need to relax the regularity assumption on f and m to (M) m ∈ L 1 (I × R + ; R) and for all t ∈ I, ‖m(t, ·)‖ L ∞ (R + ;R) + TV (m(t, ·)) ≤ M.
Note also that it is possible to trivially extend Definition 2.1 with assumptions (f) and (m) replaced respectively by (F) and (M).

Corollary 2.4. Theorems 2.2 and 2.3 remain true if assumption (f) is replaced by (F) and assumption (m) is replaced by (M).
The proof is deferred to Section 4. Once the Lipschitz estimate (2.4) is available, various techniques can be followed to actually search for the control parameters that optimize a Lipschitz cost of the type (1.3), see the next section for related examples. We refer, for instance, to Jones et al 17

APPLICATIONS
The present construction, based on Theorems 2.2 and 2.3, comprehends and extends previous results found in the literature. Indeed, a variety of models fit in (1.1), such as the one devoted to the management of renewable resources in. 5,7,8 We refer also to other studies 1,6,19,20 for other situations fitting into (1.1)- (1.2). Further examples also comprised in (1.1)-(1.2) are found, for instance, in Perthame. 2, Chapter 3 We consider below in detail applications taken from the more applied literature that specifically take advantage of the extensions introduced in the present paper.

Cell growth and cancer control
Consider the following model for the evolution of cancer cells developed in Billy et al 20 : where i is the transition rate in the type i cells without circadian control; h i , c i are the effects of the natural 24-h-periodic circadian cycle and G is the control at the cell level of the drug infusion, which we also assume to be 24-h-periodic.
The transition functions for healthy and cancer cells h i and c i are provided by experimental measurements, see for instance the green lines in Billy et al, 6, figs 9 and 10 respectively.
with all other w i being set to 0, system ( Once (3.2) and (3.3) are given, verifying that (b), (f), (g), and (m) hold is immediate and, hence, omitted. For completeness, we recall that Clairambault et al, 9,Theorem 3.1 at least in the autonomous case, covers the well posedness of (3.1). The proof therein is obtained by means of an entirely different technique, relying on the adjoint equation, see Perthame. 2, section 3.2 Note that the assumption that 1 and 2 vanish for a > is obviously satisfied in any real situation, since it is clearly consistent with the finite life span of cells.
Our task is to choose the function G so that the target functional defined by is maximal for a given T. For instance, in Billy et al, 6 the choice T = 12 days was adopted.
An application of Corollary 2.4, in particular of the estimate (2.4), ensures that the variation in the cost (3.4) due to different drug infusion schedules G 1 and G 2 can be estimated by for a suitable constant (1) depending on (T) in (2.4), on T and on norms of 1 , 2 , h 1 , h 2 , c 1 , c 2 . Therefore, any strongly convergent minimizing sequence of drug infusion schedules actually yields a minimal cost.

Age-structured population economics
The model introduced in Feichtinger et al 21 fits into the form , (3.5) where N = N(t, a) is the number of individuals of age a at time t, K = K(t, a) is the physical capital stock, = (t, a) is the fertility rate at time t of individuals of age a, so that the integral ∫ R + (t, a) N(t, a)da measures the amount of newborn individuals. The term  is the investment in capital goods, the dependence of  on K and N being nonlocal, with assigned weights p and q. The functions = (t, a) is the mortality rate and = (t, a) is the depreciation rate of the physical capital. The term (t) o (a) is the (positive) net migration, with o (a) assigned while (t) acts as a control parameter, see Feichtinger et al. 21,Section 2 We now show that the model (3.5) fits in (1.1)-(1.2).
Following Feichtinger et al, 21 the function is chosen to maximize the social welfare, measured through a functional of the general type Here,  is related to the instantaneous welfare and depends on the functions a → N(t, a) and a → K(t, a), defined for a ∈ R + . Refer to Feichtinger et al 21 for a more detailed expression of the cost functional. Obvious regularity assumptions on  ensure the strong L 1 continuity of  , thanks to Corollary 2.4. Once Lipschitz regularity is available, various procedures to actually find optimal controls are outlined in the literature, see for instance Jones et al 17 and Malherbe and Vayatis. 18 In specific applications, where depends on a finite number of parameters, this procedure also ensures the existence of an optimal control.

A juvenile-adult model with metamorphosis
Consider a species where juveniles (J) develop into adults (A) through a metamorphosis at age a = a max . Calling as usual g, the growth function and J , respectively A , the juvenile, respectively adult, mortality, both time and age dependent, we are lead to the following system, which was introduced in Ackleh and Deng 1, Formula (2.1) : (3.8) The general form of (1.1)-(1.2) also comprises (3.8). Indeed, problem (3.8) fits into (1.1)-(1.2) setting n = 2, x = a, and where  = [0, a max − a min ]. Both Theorems 2.2 and 2.3 then apply and ensure the well posedness and the stability of (3.8) under assumptions slightly different from those in Ackleh and Deng, 1 simplifying the result in to Colombo and Garavello, 5, section 3.1 to which we refer for the details.

Optimal control in biological resources' management
An extension of the above system (3.8) allows to model an economic/industrial exploitation of a biological resource. Assume that at age a =ā, juveniles (J) are selected into those that are going to be used for reproduction (R) and those that are bred to be sold (S). Along the lines of Colombo and Garavello, 5, section 3.3 one is thus lead to the following model: where we used essentially the same notation as in the preceding Section 3.1. The percentage of juveniles selected for the market is quantified by the, here time dependent, parameter System (3.10) fits into (1.1)-(1.2) setting n = 3, x = a, U ≡ (U 1 , U 2 , U 3 ) and The same remarks in Colombo and Garavello, 5 but based on the present extension provided by Theorems 2.2 and 2.3, ensure the well posedness of (3.10) and its stability with respect to boundary data and parameters. Note in particular that this opens the way to discussing optimal control problems where has to be chosen to maximize a suitable functional modeling the long-term income due to selling the selected S individuals.

TECHNICAL DETAILS
This section contains the technical details, both proofs of Theorems 2.2 and 2.3, and the proof of Corollary 2.4.

Elementary estimates on BV functions
We now recall elementary estimates on BV functions, see also Colombo and Garavello. 5

Characteristic curves
If g satisfies (g), we introduce the globally defined maps, see Bressan and Piccoli, 25, Chapter 3 We collect here results about the differentiability of the maps (4.7), deduced by classical results about ordinary differential equations. Lemma 4.1. Let g satisfy (g). With the notation (4.7), for all t, t o ∈ I and x, x o ∈ R + , the following relations hold: If moreover g is of class C 1 , (4.10) Proof. The relations (4.8) and (4.10) are classical, see for instance Hartman. 26, Chapter 5, Section 3 The equality (4.9) follows from the monotonicity of x → T(0; t, x).

The scalar renewal equation
We consider the following IBVP for a linear nonhomogeneous scalar balance law, or renewal equation, see also Perthame 2, Chapter 3 : under the assumptions (f), (g), and (m) from Section 2, together with b ∈ BV loc (I; R).
There is a wide literature on (4.11), we refer in particular to Colombo and Garavello 22, Definition 2 ; see also other studies. 2,15,16,24,27,28 Recall the expression of the solution, based on the notation (4.7): (4.13) Lemma 4.2 ( 22, Lemma 2 ). Let (g) and (m) hold. Then,  defined in (4.13) satisfies the following estimates, for x ∈ R + and , t ∈ I with ≤ t: 14) The following Lemma summarizes various properties of the solution to (4.11), see also Perthame. 2 (a) for all ∈ C 1 c (I ×R (SP.2) For every t ∈ I, the following a-priori estimates hold:

(SP.3)
For every t ∈ I, the following total variation estimate holds: where (t) is a nondecreasing continuous function of t, depending also onǧ, G 1 , G ∞ and M, satisfying   The next result deals with the stability properties of (4.11).
The following monotonicity property holds:

Different notions of solutions for (4.11)
We now deal with the different definitions of solutions available in the BV loc select (4.12) as solution to (4.11). Proof. Let u 1 and u 2 be two weak solutions to (4.11) and call w = u 2 − u 1 . Then, by (a) in (SP.1), for all ∈ C 1 c (I × R Step 1: Fixt ∈I and x 1 , x 2 ∈ R + with x 2 > x 1 ≥ (t). We prove that for all ∈ C 1 c (R + ; R) with spt ⊆ [x 1 , Choose a test function (t, x) = (t) (t, x), where ∈ C 1 c (I ×R while approximates the characteristic function of the time interval [0,t], that is, Choosing as test function, we have that for all positive and sufficiently small , where we used (b) in (SP.1).
Step 2: Fixx ∈R + and t 1 , t 2 ∈ I with t 2 > t 1 ≥ Γ(x). We now prove that for all ∈ C 1 Proceed as in the step above. Define (t, x) = (x) (t, x), where ∈ C 1 c (I ×R while approximates the characteristic function of the space interval [0,x], that is, Choosing as a test function, we have that for all positive and sufficiently small , where we used (c) in (SP.1). Step 3: Extension to a general m. Fix a sequence m k of continuous maps converging to m in L 1 and apply the above Steps 1 and 2 to each m k . Call w k the corresponding difference of solutions as above, and in both cases The next lemma bridges (4.11) to IBVPs set on bounded domains, so that we can then exploit results proved in this latter setting.

The map u
3. The map u ∶ I × R + → R is such that for all > 0, the restriction u = u |I×[0, ] is a Kružkov solution to (4.28) in the sense that for a.e. t ∈ I, the traces u(t, 0 +) and u(t, −) are well defined, and moreover, for all k ∈ R and for any test The proof is standard and hence omitted. The above allows to exploit known results on the solutions to IBVP for balance laws on bounded domains. In particular, the total variation estimates (SP. 3
This completes the proof of (4.30). Proof of (4.31). Fix t ∈ [0, T 1 ] and i ∈ {1, … , n}. Denoting i (t) = X i (t; 0, 0), we have We estimate separately both terms in the right-hand side of (4.33), starting from the second one: Pass to the first term in the right-hand side of (4.33). By (b), we can define, for all s ∈ [0, t], Moreover, Using (4.33), we deduce that Applying Grönwall lemma, we obtain that proving (4.31).

Proofs of the main results
This part deals with the proof of Theorems 2.2 and 2.3.

Proof of Theorem 2.2 and proof of
Note that this choice implies that i (T 1 ) ≤ i (T 1 ) <x i for every i = 1, … , n, and that i (T 1 ) < , where we use the notation (4.7). Below, we inductively construct the solution u to (1.1) on the time intervals [(k − 1)T 1 , kT 1 ], with k ∈ N ∖ {0}. Consider k = 0, and for every i ∈ {1, … , n}, define the sets and define, for every i ∈ {1, … , n} and (t, x) ∈ B i 1 , using the notation (4.13),   Assume now k > 0, with (k + 1)T 1 < t * , and suppose that u is defined in the time interval [0, kT 1 ]. Define, for every i ∈ {1, … , n}, the sets and for every i ∈ {1, … , n} and (t, x) ∈ B i k+1 , using the notation (4.13), . Then, by (4.40) and Colombo and Garavello,22,section 4.3 . Iterate now on [T 1 , 2T 1 ], [2T 1 , 3T 1 ], … to obtain global uniqueness on I × R + . L 1 well posedness and stability estimates w.r.t. , , w. Fix t ∈ I, t ≤ T 1 as defined in (4.37). Consider two different initial conditions, namely, u ′ o and u ′′ o , and two different sets of data, namely, ′ , ′ , w ′ and ′′ , ′′ , w ′′ with Consider the second term in the right-hand side of (4.42). If x > i (t), then the construction in the first part of the proof implies that where, for ∈ I, . Therefore, where , defined in Lemma 4.7, does not depend on . Using (4.42), we deduce that By Grönwall lemma, we obtain the existence of a positive constant , depending only on T 1 and on the constants defined in (b), (f), (g), and (m), such that Consider the first term in the right-hand side of (4.45). If x < i (t), then the construction in the first part of the proof implies that where, for ∈ I, . (4.48) Preliminary, we claim that, for t ∈ I, t ≤ T 1 , Finally, by (4.45), we have that By Grönwall lemma, we obtain the existence of a positive constant , depending only on T 1 and on the constants defined in (b), (f), (g), and (m), such that ] .
Iterating on [0, and bound both terms in the right-hand side above. Consider first the ith component of the second one. Use (4.38) and (4.14): Consider now the first term in the right-hand side of (4.51), using (4.14) and (4.40), Therefore, Iterating on the time intervals [0, T 1 ], [T 1 , 2T 1 ], and so on permits to conclude the proof of (2.2).

Positivity
This point directly follows from the explicit expressions (4.38), (4.39), and (4.40). Assumption (4.36) does not hold Fix an initial condition u o ∈ ( L 1 ∩ BV ) (R + ; R n ) and data , , and w satisfying (b). Consider, for every k ∈ N ∖{0}, the sequence w k where . We prove that u k is a Cauchy sequence in C 0 ( I; L 1 (R + ; R n ) ) .
Proof of Corollary 2.4. Let k be a sequence of positive mollifiers in C ∞ c converging to the Dirac delta centered at the origin. Assume that f and m satisfy (F) and (M). Define k = * k and m k = m * k . Then, the computations related to f being entirely similar, m k ∈ C 0 (I × R + ; R) by the standard properties of the convolution. Moreover, for t ∈ I, ‖m k (t, ·)‖ L ∞ (R + ;R) ≤ ‖m k ‖ L ∞ (I×R + ;R) ≤ ‖m‖ L ∞ (I×R + ;R) ‖ k ‖ L 1 (I×R + ;R) = ‖m‖ L ∞ (I×R + ;R) and similar, elementary but lengthier, computations ensure that an analogous inequality holds between the total variations. For every k ∈ N, Theorem 2.2 can be applied with the source term f k and the mortality function m k ,