Resolution methods for mathematical models based on differential equations with Stieltjes derivates

Electronic Journal of Qualitative Theory of Differential Equations 2019, No. 72, 1–15; https://doi.org/10.14232/ejqtde.2019.1.72 www.math.u-szeged.hu/ejqtde/ Resolution methods for mathematical models based on differential equations with Stieltjes derivatives Rodrigo López Pouso and Ignacio Márquez AlbésB Universidade de Santiago de Compostela, R. Lope Gómez de Marzoa, Santiago de Compostela, Spain Received 15 March 2019, appeared 2 October 2019 Communicated by Gennaro Infante Abstract. Stieltjes differential equations, i.e. differential equations with usual derivatives replaced by derivatives with respect to given functions (derivators), are useful to model processes which exhibit dead times and/or sudden changes. These advantages of Stieltjes equations are exploited in this paper in the analysis of two real life models: first, the frictionless motion of a vehicle equipped with an electric engine and, second, the evolution of populations of cyanobacteria Spirullina plantensis in semicontinuous cultivation processes. Furthermore, this is not only a paper on applications of known results. For the adequate analysis of our mathematical models we first deduce the solution formula for Stieltjes equations with separate variables. Finally, we show that differential equations with Stieltjes derivatives reduce to ODEs when the derivator is continuous, thus obtaining another resolution method for more general cases. Keywords: Stieltjes differential equations, dynamic equations, separation of variables, biological models. 2010 Mathematics Subject Classification: 34A36, 34K05, 34K05. 1 Introduction and preliminary results In this paper we will obtain resolution methods for differential equations with Stieltjes derivatives to be applied in the exact computation of solutions of two mathematical models. This will be done in two different ways. We will first study Stieltjes differential equations with separate variables and later we show that, in general, Stieltjes differential equations are equivalent to ODEs under certain hypotheses. Consider a Stieltjes differential system x0 g(t) = f(t,x(t)),t∈I= [t0,t0+T],x(t0) = x0(1.1) where T>0 and x0∈Rare fixed, f:I×R→Ris a given function, and x0 g(t)stands for the g-derivative of the unknown with respect to a nondecreasing and left-continuous derivator g:R→R. The precise definition and background on g-derivatives are collected in Section 2. BCorresponding author. Email: [email protected] 2R. López Pouso and I. Márquez Albés It is shown in [3, Section 3], see also [1, Section 8], that this kind of equation contains as particular cases ∆-differential equations on time scales or differential equations with countably many impulses. In Section 3, we look at (1.1) in the particular case of separable variables, namely x0 g(t) = c(t)f(x(t)),t≥t0,x(t0) = x0, for which, under certain hypotheses, we obtain an explicit solution using the chain rule for g-derivatives. Then, in Section 4 we introduce and study our mathematical models in the form of these problems. We present a model for the motion of a vehicle and another one for a bacteria population. These examples are meant to show the interest of Stieltjes differential equations for modelling different processes which could hardly be studied by means of ordinary differential equations. It was precisely in the analysis of those models where we found the main idea for Section 5: Stieltjes differential equations of the form of (1.1) reduce to ODEs when the derivator is continuous, a most useful result in the exact computation of solutions to Stieltjes equations. 2 Preliminares Let g:R→Rbe a nondecreasing and left-continuous function. Let us recall the definition of the g-derivative introduced in [3]. To that end, we first introduce the following two sets: the set of points around which gis constant, Cg={s∈R:gis constant on (s−ε,s+ε)for some ε>0}, and the set of discontinuity points of gthat can be written as Dg={s∈R:g(s+)−g(s)>0}, where g(s+)denotes the limit of gat sfrom the right. Now the g-derivative of a function x:R→Rat a point t∈R\Cgis x0 g(t) =        lim s→t x(s)−x(t) g(s)−g(t), if t6∈ Dg, x(t+)−x(t) g(t+)−g(t), if t∈Dgand t<t0+T, provided that the corresponding limit exists. Notice that we do not define g-derivatives at points t∈Cg, nor it is necessary because Cg is a null-measure set for µg(the Lebesgue–Stieltjes measure induced by g), see [3, Proposition 2.5]. Therefore, the differential equation in (1.1) is not really defined for t∈I∩Cg. Roughly speaking, connected components of Cgcorrespond to negligible times, i.e. lapses when our system does not evolve at all. In turn, discontinuities of gcorrespond with times when sudden changes occur and which are usually introduced in models in the form of impulses. For the remaining set of times I\(Cg∪Dg)we note that different slopes of the derivator gcorrespond to different influences of the corresponding times, namely, the bigger the slope of gthe more important the corresponding times are for the process. Resolution methods for equations with Stieltjes derivatives 3 Finally, we recover an interesting set introduced in [3]. By definition, the set Cgis open in the usual topology, so it can be uniquely expressed as the countable union of open disjoint intervals, say Cg=[ n∈N (an,bn). (2.1) Without loss of generality, we can assume that an<an+1for all n∈N. The set Ngis defined then as the endpoints of such intervals that are continuity point of g, that is Ng={an,bn:n∈N} \ Dg. We also define N− g={an:n∈N} \ Dgand N+ g={bn:n∈N} \ Dg. Clearly, Ng=N− g∪N+ g. Remark 2.1. Note that if u6∈ Cg∪Ng∪Dg, then g(v)6=g(u)for v=u. Hence, if gis continuous, u6∈ Cg∪Ngimplies that g(v)6=g(u)for v=u. By a solution of (1.1), we mean a function x:[t0,t0+T]→Rsuch that x(t0) = x0and xis g-absolutely continuous function in the sense of the definition included in the following Fundamental Theorem of Calculus for the Lebesgue–Stieltjes integral [3, Theorem 5.4]. Theorem 2.2 (Fundamental Theorem of Calculus for the Lebesgue–Stieltjes integral).Let a,b∈ R, a <b, and F :[a,b]→R. The following conditions are equivalent. (1) The function F is absolutely continuous with respect to gon [a,b](or g-absolutely continuous) according to the following definition: to each ε>0there is some δ>0such that, for any family {(an,bn)}m n=1of pairwise disjoint open subintervals of [a,b], the inequality m ∑ n=1 (g(bn)−g(an)) <δ implies m ∑ n=1 |F(bn)−F(an)|<ε. (2) The function F fulfills the following properties: (a) There exists F0 g(t)for g-almost all t ∈[a,b)(i.e., for all t except on a set of µgmeasure zero); (b) F0 g∈ L1 g([a,b)), the set of Lebesgue–Stieltjes integrable functions with respect to µg; and (c) For each t ∈[a,b], we have F(t) = F(a) + Z[a,t)F0 g(s)dµg. (2.2) In this paper we consider integration in the Lebesgue–Stieltjes sense mainly, and we shall call “g-measurable” any function (or set) which is measurable with respect to the Lebesgue– Stieltjes σ-algebra generated by g. Moreover, integrals such as that in (2.2) shall be denoted also as Z[a,t)F0 g(s)dg(s). For the particular case of g(t) = t, we have that µg=m, the usual Lebesgue measure, for which we use the notation Zt aF0(s)ds. For properties of g-absolutely continuous functions we refer readers to [1,3]. For convenience of readers, we include the following results. 4R. López Pouso and I. Márquez Albés Proposition 2.3 ([3, Proposition 5.3]).If F is g-absolutely continuous on [a,b], then it has bounded variation and it is continuous from the left at every t ∈[a,b). Moreover, F is continuous in [a,b]\Dg, where Dgis the set of discontinuity points of g, and if g is constant on some interval (α,β)⊂[a,b], then F is constant on (α,β)as well. Proposition 2.4 ([1, Proposition 5.3]).Let F1:[a,b]→Rbe g-absolutely continuous. Assume that F1([a,b]) ⊂[c,d]for some c,d∈R, c <d, and let F2:[c,d]→Rsatisfy a Lipschitz condition on [c,d]. Then the composition F2◦F1is g-absolutely continuous on [a,b]. Finally, we recall the chain rule for g-derivatives and g-differentiation of indefinite integrals, which we shall use in order to obtain the formula of the solution of a problem with separate variables. Theorem 2.5 (Chain rule for g-derivatives [3, Theorem 2.3]).Let f be a real-valued real function defined on a neighborhood of t ∈R\Dg, and let h be another function defined in a neighborhood of f(t). The following results hold for the g-derivative of the composition h ◦f at t: 1. If there exist h0(f(t)) and f 0 g(t), then there exists (h◦f)0 g(t) = h0(f(t)) f0 g(t). 2. If there exist h0 g(f(t)), g0(f(t)), and f 0 g(t), then there exists (h◦f)0 g(t) = h0 g(f(t))g0(f(t)) f0 g(t). Theorem 2.6 ([3, Theorem 2.4, Proposition 5.2]).Assume that c :[a,b)→Ris integrable on [a,b) with respect to µgand consider its indefinite Lebesgue–Stieltjes integral C(t) = Z[a,t)c dµgfor all t ∈[a,b]. Then C is g-absolutely continuous on [a,b]and there is a g-measurable set N ⊂[a,b]such that µg(N) = 0and C0 g(t) = c(t)for all t ∈[a,b]\N. 3 Separation of variables This section is devoted to the explicit resolution of the separable initial value problem x0 g(t) = c(t)f(x(t)),t≥t0,x(t0) = x0. (3.1) Note that for the particular case of f(x) = xthe problem has been solved in [1]. As in the ODE case, problem (3.1) can be solved with an exponential map, which we recall here for the convenience of the reader. Definition 3.1. Let c∈ L1 g([a,b)) be such that c(t)g(t+)−g(t)>−1 for every t∈[a,b)∩Dg, (3.2) and ∑ t∈[a,b)∩Dglog 1+c(t)(g(t+)−g(t))<∞. (3.3) Resolution methods for equations with Stieltjes derivatives 5 We define ec(·,a):[a,b]→(0, ∞)by ec(t,a) = eR[a,t)e c(s)dµg, (3.4) where e c(t) =    c(t)if t∈[a,b]\Dg, log 1+c(t)(g(t+)−g(t)) g(t+)−g(t)if t∈[a,b)∩Dg. (3.5) Proposition 3.2 ([1, Lemma 6.3]).Let c ∈ L1 g([a,b)) satisfy (3.2)and (3.3). Then for every xa∈R the mapping t 7→ xaec(t,a)is g-absolutely continuous and solves the initial value problem x0 g(t) = c(t)x(t)for g-almost all t ∈[a,b), x(a) = xa.(3.6) It is important to note that chas to be redefined at discontinuity points of g. Bearing this idea in mind, we consider the particular case of (3.1) corresponding to a continuous derivator g. Theorem 3.3. Let c ∈ L1 g,loc([t0,+∞)) (i.e., c is g-integrable on compact subsets of [t0,∞)) and assume that there is some R >0such that f is continuous and positive on J = (x0−R,x0+R). Define F(x) = Zx x0 dr f(r)for every x ∈J. (3.7) If there exists r >0such that Z[t0,t)c(s)dg(s)∈F(J)for all t ∈[t0,t0+r), and g is continuous on [t0,t0+r),(3.8) then a solution of (3.1)is given by the following formula: x(t) = F−1Z[t0,t)c(s)dg(s)for all t ∈[t0,t0+r).(3.9) Proof. Clearly, x(t0) = x0. Since F−1is locally Lipschitzian, we can deduce from Proposition 2.4 and Theorem 2.6 that xis g-absolutely continuous on any interval [t0,t0+s],s∈(0, r). In particular, there exists x0 g(t)for g-almost all t∈[t0,t0+r). Using the chain rule (Theorem 2.5) and Theorem 2.6, we compute for g-almost all t∈[t0,t0+r) x0 g(t) = (F−1)0Z[t0,t)c(s)dg(s)Z[t0,·)c(s)dg(s)0 g (t) = 1 F0(x(t))c(t) = f(x(t))c(t). Remark 3.4. Formula (3.9) is equivalent to Zx(t) x0 dr f(r)=Z[t0,t)c(s)dg(s). Theorem 3.3 is false, in general, when gis discontinuous. Indeed, observe that (3.9) does not give (3.4) when fis the identity and ghas at least one discontinuity point in (t0,t0+r). At discontinuity points of gwe cannot use Theorem 2.5 to compute derivatives of compositions 6R. López Pouso and I. Márquez Albés by means of the chain rule, so we need an alternative approach and an alternative formula for the solutions. Here and henceforth, we assume that gis discontinuous exactly at the points of a sequence {τk}∞ k=1, where t0<τ1<τ2<· · · . Assuming that gis continuous at the initial time t0is not really a restriction, see [1, Section 5]. Remark 3.5. In general, Dgis just a countable set. Here we assume that Dgis discrete, i.e. all its elements are isolated points. We have no solution formula for the general case. Solving (3.1) on the interval [t0,τ1)can be done with the aid of Theorem 3.3, because gis continuous on [t0,τ1). Furthermore, since solutions are continuous from the left everywhere, we get the solution on [t0,τ1]with the same formula. Specifically, under suitable conditions, a solution of (3.1) on the interval [t0,τ1]is implicitly given by the expression Zx(t) x0 dr f(r)=Z[t0,t)c(s)dg(s)for all t∈[t0,τ1]. (3.10) Obtaining the solution formula on the right of τ1is a matter of induction. First, according to the definition of g-derivative at discontinuity points, the differential equation in (3.1) for t=τ1reads simply as follows: x(τ+ 1) = x(τ1) + c(τ1)f(x(τ1))(g(τ+ 1)−g(τ1)) ≡x1. (3.11) Therefore, we have to solve another initial value problem x0 g(t) = c(t)f(x(t)),t∈(τ1,τ2],x(τ+ 1) = x1, (3.12) by means of (3.9), with obvious modifications: under suitable conditions (see Remark 3.6), a solution of (3.12) is defined by Zx(t) x1 dr f(r)=Z(τ1,t)c(s)dg(s)for all t∈(τ1,τ2], (3.13) where x1is defined in (3.11). Remark 3.6. Formula (3.13) gives a solution of (3.12) provided that, for instance, fis continuous and positive on J= (x1−R,x1+R), for some R>0, and Z(τ1,t)c(s)dg(s)∈F(J)for all t∈(τ1,τ2], where F(x) = Rx x1dr/f(r),x∈J. Summing up, a solution of (3.1) can be recursively computed as follows: define x(t)on [t0,τ1]by means of (3.10); assume that we have defined x(t)on [t0,τk], for some k∈ {1, 2, . . . }, then compute the number xk=x(τk) + c(τk)f(x(τk))(g(τ+ k)−g(τk)), (3.14) and define x(t)implicitly on (τk,τk+1]by the expression Zx(t) xk dr f(r)=Z(τk,t)c(s)dg(s)for all t∈(τk,τk+1]. (3.15) Resolution methods for equations with Stieltjes derivatives 7 4 Real life applications 4.1 An irregularly forced frictionless motion We want to set up a simple model for the motion of a vehicle impulsed by an electric engine which we can turn on and off as often as we please. We disregard any other force. In particular, the speed increases when the engine is turned on, and the vehicle never slows down, it just keeps its speed when the engine is turned off. Let g(t)denote the number of seconds that the engine has been on until time t. This function g(t)is continuous, nondecreasing, and constant on the time intervals when the engine is turned off. Let s(t)denote the vehicle’s speed after tseconds. For simplicity, we assume that speed increases on every time interval [t,t+h],h>0, at a rate proportional to the time the engine has been on during that time interval. Moreover, we consider that accelerating the vehicle is harder at very slow or at very high speeds, so we assume a proportionality “constant” which depends on s(t), at least for small values of h>0. This leads to s(t+h)−s(t) = f(s(t))(g(t+h)−g(t)), which, considering the limit as h→0+, yields the g-differential model s0 g(t) = f(s(t)),t≥0, s(0) = s0. (4.1) We suggest using logistic-type functions like f(s) = αmax{0, (s+β)(smax −s)}, for s≥0, where α,βand smax are positive constants. Observe that f(s)>0 for s∈[0, smax)and f(s) = 0 for s≥smax, which means that the engine can accelerate the vehicle only when its speed belongs to the interval [0, smax). Observe also that, in case β<smax (which we assume from now on), fattains a maximum at (smax −β)/2, which means that the engine is more efficient when the vehicle is moving at that specific value of speed. As an instance, we consider f(s) = max{0, (s+1)(2−s)}(4.2) and we solve (4.1) for s0=0. We note that fis continuous and positive for s∈[0, 2), and a solution is implicitly given by Zs(t) 0 dr (r+1)(2−r)=Z[0,t)dg(s) = g(t). Elementary computations yield the solution s(t) = 2e3g(t)−2 e3g(t)+2,t≥0. (4.3) Keeping the engine on at every time corresponds to the derivator g(t) = tfor all t≥0. The corresponding solution is displayed in Figure 4.1. 8R. López Pouso and I. Márquez Albés Figure 4.1: Solution (4.3) for g(t) = t(engine constantly on). If we turn the device off for t∈[1/2, 1]∪[3/2, 2], then we should take g(t) =            t, for t∈[0, 1/2], 1/2, for t∈[1/2, 1], t−1/2, for t∈[1, 3/2], 3/2, for t∈[3/2, 2]. The graph of gand the solution given by (4.3) can be seen in Figures 4.2 and 4.3, respectively. Figure 4.2: Graph of g(t). Resolution methods for equations with Stieltjes derivatives 9 Figure 4.3: Solution (4.3) with dead times (engine off). 4.2 Semicontinuous culture systems Semicontinuous cultivation is a system to produce bacteria in which a portion of the culture medium is periodically removed and the remaining culture is used as the starting point for continuation of the culture. For the model in this section we take into account [5], where semicontinuous cultivation of the cyanobacteria Spirulina plantensis is studied. We highlight the following important feature of the system in [5]: illumination was controlled to have a 12 hours light/dark photoperiod, which resulted in two different reproduction phases every day. Bearing the above considerations in mind, we shall set up a mathematical model for the production of Spirulina plantensis in a semicontinuous culture system. First, we consider days as our time units. We assign light periods to be the time intervals [k,k+1/2),k=0, 1, 2, . . . , while dark periods are [k+1/2, k+1),k=0, 1,2, . . . . Second, we assume that half of the culture is removed every 10 days and immediately refilled with new nutrients so that the remaining bacteria start reproducing again. Finally, we take as derivator a nondecreasing function g:[0, ∞)→[0, ∞), continuous everywhere with the exception of the positive multiples of 10, and such that g0(t) = (1, if t∈(k,k+1/2),k=0, 1, 2, . . . , 1/2, if t∈(k+1/2, k+1),k=0, 1, 2, . . . , and g((10k)+)−g(10k) = 1 for k=1, 2, 3, . . . A concise explicit expression for g(t)can be obtained by defining first its values on the first day, namely h(t) = (t, if t∈[0, 1/2), t/2 +1/4, if t∈(1/2, 1], and then we can define the remaining values by “periodicity”, and introducing jump discon-