AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation97439410.1155/2009/974394974394Research ArticleNecessary Conditions for a Class of Optimal Control Problems on Time ScalesZhanZaidongWeiW.KottaÜlleDepartment of MathematicsGuizhou UniversityGuiyang Guizhou 550025Chinagzu.edu.cn20090803200920091510200806122008250220092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Based on the Gateaux differential on time scales, we investigate and establish necessary conditions for Lagrange optimal control problems on time scales. Moreover, we present an economic model to demonstrate the effectiveness of our results.

1. Introduction

In this paper, we consider optimal control problem (P). Find u0Uad such that J(u0(·))J(u(·)),uUad, where J is the cost functional given by minimizeJ(u(·)):=[0,σ(T))𝕋l(x(t;u),u(t))Δt, and x(·;u)AC([0,T]𝕋,) is a solution corresponding to the control u𝒰ad of the following equation: xΔ(t)=p(t)x(t)+f(t)+u(t),forΔa.e.,t[0,ρ(T)]𝕋,x(0)=x0, where 𝕋 is a bounded time scale, [0,T]𝕋=[0,T]𝕋 and σ(T):=max𝕋. The admissible control set is 𝒰ad:={u(t),t[0,T]𝕋,uisΔ‐measurableandu(t)𝒰}. Here, the control set 𝒰 is a bounded, closed, and convex subset of .

Time scale calculus was initiated by Hilger in his Ph.D. thesis in 1988  in order to unite two existing approaches of dynamic models-difference and differential equations into a general framework, which can be used to model dynamic processes whose time domains are more complex than the set of integers (difference equations) or real numbers (differential equation). There are many potential applications for this relatively new theory. The optimal control problems on time scales are also an interesting topic, and many researchers are working in this area. Existing results on the literature of time scales are restricted to problems of the calculus of variations, which were introduced by Bohner  and by Hilscher and Zeidan . There are many opportunities for applications in economics [4, 5]. More general optimal control problems on time scales were studied in [6, 7].

To the best of our knowledge, it seems that there is not too much work about the necessary conditions of optimal control problems on time scales by adapting the method of calculus of variations. That motivates us to investigate new necessary conditions of optimal control problem on time scales. In this paper, based on the Gateaux differential on time scales, we establish necessary conditions for Lagrange optimal control problems on time scales. Moreover, we present an economic model to demonstrate our results.

The paper is organized as follows. We present some necessary preliminary definitions and results about the time scales 𝕋 in Section 2. In Section 3, based on the existence and uniqueness of solutions of a linear dynamic equation on time scales, we derive existence and uniqueness of system solutions for the controlled system. Then, we prove the minimum principle on time scales for the optimal control problem (P) in Section 4. Finally, in Section 5, an example is given to demonstrate our results.

2. Preliminaries

A time scale 𝕋 is a closed nonempty subset of . The two most popular examples are 𝕋= and 𝕋=. The forward and backward jump operators σ,ρ:𝕋𝕋 are defined by σ(t)=inf{s𝕋:s>t},ρ(t)=sup{s𝕋:s<t}. We put inf =sup𝕋 and sup=inf 𝕋, where denotes the empty set. If there is the finite max𝕋, then σ(max𝕋)=max𝕋, and if there exists the finite min 𝕋, then ρ(min 𝕋)=min 𝕋. The graininess function μ:𝕋[0,+) is μ(t):=σ(t)-t. A point t𝕋 is called left-dense (left-scattered, right-dense, and right-scattered) if ρ(t)=t (ρ(t)<t, σ(t)=t, and σ(t)>t) holds. If 𝕋 has a left-scattered maximum value M, then we denote 𝕋k:=𝕋-{M}. Otherwise, 𝕋k:=𝕋.

Definitions and propositions of Lebesgue Δ-measure μΔ and Lebesgue integral can be seen in .

Definition 2.1.

Let P denote a proposition with respect to t𝕋 and A a subset of 𝕋. If there exists E1A with μΔ(E1)=0 such that P holds on AE1, then P is said to hold Δa.e., on A.

Remark 2.2.

For each t0𝕋{max𝕋}, the single-point set {t0} is Δ-measurable, and its Δ-measure is given by μΔ({t0})=σ(t0)-t0=μ(t0).

Obviously, E1A does not have any right-scattered points. For a set E𝕋, define the Lebesgue Δ-integral of f over E by Ef(t)Δt and let fL𝕋1(E,) (see ).

Lemma 2.3 (see [<xref ref-type="bibr" rid="B8">8</xref>]).

Let f:[a,b)𝕋. f̃:[a,b) is the extension of f to real interval [a,b], defined by f̃(t):={f(t)ift[a,b)𝕋,f(ti)ift(ti,σ(ti)),forsomeiI, where {ti}iI,I is the index of the set of all right-scattered points of [a,b]𝕋. Then, fL𝕋1([a,b)𝕋,) if and only if f̃L1([a,b],). In this case, [a,b)𝕋f(t)Δt=[a,b]f̃(t)dt.

Definition 2.4.

Suppose that f:[a,b)𝕋. fL𝕋([a,b)𝕋,), if there exists a constant C such that |f(t)|CΔa.e.t[a,b)𝕋.

Definition 2.5 (see [<xref ref-type="bibr" rid="B10">10</xref>]).

A function f:𝕋 is said to be absolutely continuous on 𝕋 if for every given constant ε>0, there is a constant δ>0 such that if {[ak,bk)𝕋}k=1n, with ak,bk𝕋, is a finite pairwise disjoint family of subintervals of 𝕋 satisfying k=1n(bk-ak)<δ, then k=1n|f(bk)-f(ak)|<ε. If 𝕋=[a,b]𝕋, then we denote all absolutely continuous functions on [a,b]𝕋 as AC([a,b]𝕋,).

Lemma 2.6.

If f is Lebesgue Δ-integrable on [a,b)𝕋, then the integral F(t)=[a,t)𝕋f(l)Δl,t[a,b)𝕋 is absolutely continuous on [a,b]𝕋. Moreover, FΔ(t)=f(t),for  Δ‐a.e.  t[a,b)𝕋.

Proof.

F(t)=[a,t)𝕋f(l)Δl=[a,t]f̃(l)dl where f̃(l) is introduced in (2.3) and f̃L1([a,t],) from Lemma 2.3. Now, by the standard Lebesgue integration theory, F(·) is an absolutely continuous function on the real interval [a,b] and F'(t)=f̃(t),a.e.t[a,b]. Using Definition 2.5, F(t)=[a,t)𝕋f(l)Δl is also absolutely continuous on [a,b]𝕋.

Let F be differentiable at t for t[a,b)𝕋. If t is right-scattered, that is, t=ti for some {ti}iI, it follows from the continuity of F at t that FΔ(t)=F(σ(ti))-F(ti)μ(ti)=[ti,σ(ti))f̃(s)dsσ(ti)-ti=[ti,σ(ti))f(s)dsσ(ti)-ti=f(ti)=F'(ti). If t is right dense, limst,s𝕋F(t)-F(s)t-s=limstF(t)-F(s)t-s=F'(t). Hence, F is Δ-differentiable at t and FΔ(t)=F'(t). That is, E1:={t[a,b)𝕋:FΔ(t)}{t[a,b):F'(t)}=:E2. The continuity of F guarantees that F is Δ-differentiable at every right-scattered point ti. Moreover, (2.9) implies λ(E2)=0. We deduce that E1 does not contain any right-scattered points and μΔ(E1)=λ(E1)=0. Hence, F is Δ-differentiable Δ-a.e., on [a,b)𝕋 and FΔ(t)=F'(t)=f̃(t)=f(t)forΔa.e.t[a,b)𝕋. The proof is complete.

It follows from Definition 2.5 and Lemma 2.6 that one can easy to prove the following integration by parts formula on time scales.

Lemma 2.7.

If f,g:[a,b]𝕋 are absolutely continuous functions on [a,b]𝕋, then f·g is absolutely continuous on [a,b]𝕋 and the following equality is valid: [a,b)𝕋(fΔg+fσgΔ)(s)Δs=f(b)g(b)-f(a)g(a)+[a,b)𝕋(fgΔ+fΔgσ)(s)Δs.

Let C([a,b]𝕋,) denote the linear space of all continuous functions f:[a,b]𝕋 on time scale 𝕋 with the maximum norm fC=maxt[a,b]𝕋|f(t)|. The following statement can be understood as a time scale version of the Arzela-Ascoli theorem.

Lemma 2.8 (see [<xref ref-type="bibr" rid="B11">11</xref>] (Arzela-Ascoli theorem)).

Let X be a subset of C([a,b]𝕋,) satisfying the following conditions:

X is bounded;

for any given ε>0, there exists δ>0 such that t1,t2[a,b]𝕋,|t1-t2|<δ implies |f(t1)-f(t2)|<ε for all fX (i.e., the functions in X are equicontinuous).

Then, X is relatively compact.

3. Existence and Uniqueness of Solutions for a Controlled System Equation

In order to derive necessary conditions, we prove the existence and uniqueness of solutions for controlled system equation (1.3).

Definition 3.1.

A function xAC([0,T]𝕋,) is said to be a solution of problem (1.3) if

x is Δ-differentiable Δ-a.e.on[0,T)𝕋 and xΔL𝕋1([0,T)𝕋,);

x(0)=x0 and xΔ(t)=p(t)x(t)+f(t)+u(t),Δa.e.on[0,T)𝕋.

We assume the following.

p is regressive rd-continuous function and fL𝕋1([0,T)𝕋,).

The scalar functions l(x,u) along with their partial derivation {lx,lu} are continuous and uniformly bounded on ×𝒰 for almost all t[0,T]𝕋.

Theorem 3.2 (existence and uniqueness of solutions for the controlled system equation).

If assumption [HF] holds, for any u𝒰ad, problem (1.3) has a unique solution in [0,T]𝕋 which given by x(t)=ep(t,0)x0+[0,t)𝕋ep(t,σ(τ))(f(τ)+u(τ))Δτ,t[0,T]𝕋.

Proof.

For conciseness, we just give a brief proof. Define a function F as F(t)=f(t)+u(t),t[0,T)𝕋. Then, problem (1.3) is equivalent to xΔ(t)=p(t)x(t)+F(t),forΔa.e.,t[0,ρ(T)]𝕋,x(0)=x0. Since FL𝕋1([0,T)𝕋,), there exists a sequence {Fn} in C([0,T]𝕋,) such that F-FnL𝕋10. Therefore, the Cauchy problem xnΔ=p(t)xn+Fn(t),xn(0)=x0,t[0,ρ(T)]𝕋, has an unique classical solution given by xn(t)=ep(t,0)x0+[0,t)𝕋ep(t,σ(τ))Fn(τ)Δτ,t[0,T]𝕋. Now, we define x(t)=ep(t,0)x0+[0,t)𝕋ep(t,σ(τ))F(τ)Δτ,t[0,T]𝕋. Then, xn-xC=maxt[0,T]𝕋|xn(t)-x(t)|[0,T)𝕋|ep(t,σ(τ))|·|Fn(τ)-F(τ)|Δτsupt,τ[0,T]𝕋|ep(t,τ)|·F-FnL𝕋10, and Lemma 2.6 can be applied to testify that x tailors to Definition 3.1.

Let M1=supt[0,T]𝕋|ep(t,0)|,M2=supt,τ[0,T]𝕋|ep(t,τ)|. Define the Hamiltonian H(x,ψσ,u) as H(x,ψσ,u)=l(x,u)+ψσ(px+f+u).

4. Necessary Conditions for Optimal Control Problem (P)

In this section, we will present the minimum principle on time scales for the optimal control problem (P). Theorem 4.1 (minimum principle on time scales).

Suppose that [HF] and [HL] hold. If u0 is an optimal solution for problem (P) and x0(·;u0) is an optimal trajectory corresponding to u0, then it is necessary that there exists a function ψAC([σ(0),σ(T)]𝕋,) satisfying the following conditions: [0,σ(T))𝕋Hu(x0(t),ψσ(t),u0(t)),u(t)-u0(t)Δt0,u𝒰 ad ,ψΔ(t)=-Hx(x0(t),ψσ(t),u0(t))=-p(t)ψσ(t)-lx(x0(t),u0(t)),t[σ(0),T]𝕋,ψ(σ(T))=0.

Proof.

This theorem can be proved in the following several steps.

(i) For all ε[0,1] and for all u𝒰ad, define uε=u0+ε(u-u0). Since 𝒰 is a bounded closed convex set, then 𝒰<ad is also a closed convex subset of L𝕋([0,T]𝕋,) and uε𝒰ad. Because u0𝒰ad is optimal, J(u0(·))J(uε(·)),ε[0,1],u𝒰ad.limε0uε(t)=u0(t),on[0,T]𝕋. (ii) Now, we verify that {xε(·;uε)} converges to x0(·;u0) in C([0,T]𝕋,) as ε0 by using Arzela-Ascoli theorem (Lemma 2.8). By boundedness of 𝒰ad, we have

|xε(t;uε)|=|ep(t,0)x0+[0,t)𝕋ep(t,σ(τ))[f(τ)+uε(τ)]Δτ||ep(t,0)x0|+[0,t)𝕋|ep(t,σ(τ))[f(τ)+uε(τ)]|ΔτM1|x0|+M2[0,T)𝕋|[f(τ)+uε(τ)]|ΔτM1|x0|+M2[[0,T)𝕋|f(τ)|Δτ+[0,T)𝕋|uε(τ)|Δτ]M.{xε(·;uε)} is uniformly bounded on [0,T]𝕋.

Taking arbitrary points t1 and t2 of the segment [0,T]𝕋 and using the absolutely continuity of integral and the boundedness of 𝒰ad, we obtain |(xε(t1;uε)-xε(t2;uε))||ep(t1,t2)-1|·|[ep(t2,0)x0+[0,t2)𝕋ep(t2,σ(τ))[f(τ)+uε(τ)]Δτ]|+|ep(t1,t2)|·|[t2,t1)𝕋ep(t2,σ(τ))[f(τ)+uε(τ)]Δτ|. Since ep(t1,t2)1,|(xε(t1;uε)-xε(t2;uε))|0,as|t1-t2|0. Hence, {xε(·;uε)} is equicontinuous in [0,T]𝕋.

It follows from (4.5) that |xε(t;uε)-x0(t;u0)|=|[0,t)𝕋ep(t,σ(τ))(uε(τ)-u0(τ))Δτ|[0,t)𝕋|ep(t,σ(τ))|·|(uε(τ)-u0(τ))|ΔτM2[0,T)𝕋|(uε(τ)-u0(τ))|Δτ0,asε0. By Arzela-Ascoli theorem (Lemma 2.8), we obtain xεx0inC([0,T]𝕋,). (iii) Denote y(t):=limε0xε(t)-x0(t)ε. Then, y satisfies the following initial value problem: yΔ(t)=p(t)y(t)+(u(t)-u0(t)),forΔa.e.,t[0,ρ(T)]𝕋 with y(0)=0. We call (4.12) and (4.13) the variational equations.

(iv) We calculate the Gateaux differential of J at u0𝒰ad in the direction u-u0. It follows from hypotheses [HL], Lemma 2.3, and (4.4) that 0limε0J(uε(·))-J(u0(·))ε=limε0[0,σ(T))𝕋l(xε(t;uε),uε(t))-l(x0(t;u0),u0(t))εΔt=limε0[0,σ(T)]l(xε(t;uε)̃,uε(t)̃)-l(x0(t;u0)̃,u0(t)̃)εdt=limε0[0,σ(T)]{01[lx(x0¯+θ(xε̃-x0̃),u0̃+θε(ũ-u0̃)),xε̃-x0̃ε+lu(x0̃+θ(xε̃-x0̃),u0̃+θε(ũ-u0̃)),ũ-u0̃xε̃-x0̃ε]dθ}dt=[0,σ(T)][lx(x0(t)̃,u0̃),y(t)̃+lu(x0(t)̃,u0̃),u(t)̃-u0(t)̃]dt=[0,σ(T))𝕋[lx(x0(t),u0),y(t)+lu(x0(t),u0),u(t)-u0(t)]Δt. Here, the “title’’ is the corresponding extension function in Lemma 2.3. That is, uε(t)̃:={uε(t)=u0(t)+ε(u(t)-u0(t))  ift[0,T]𝕋,uε(ti)=u0(ti)+ε(u(ti)-u0(ti))ift(ti,σ(ti)),forsomeiI, where {ti}iI,I, is the set of all right-scattered points of [0,T]𝕋. Obviously uε(t)̃=u0(t)̃+ε(u(t)̃-u0(t)̃),t[0,T]. By the variational equations (4.12) and (4.13), we define an operator T1:LT1([0,T)𝕋,)C([0,T]𝕋,) as y(t):=T1(u-u0)(t)=[0,t)𝕋ep(t,σ(τ))[u(τ)-u0(τ)]Δτ,t[0,T]𝕋. Then, T1 is a continuous linear operator. Furthermore, due to the uniform bound of lx, T2:C([0,T]𝕋,), given by T2y:=[0,σ(T))𝕋lx(x0(t),u0(t)),y(t)Δt, is also a linear continuous functional. Hence, T2T1:LT1([0,T)𝕋,) defined by T2T1(u-u0)=[0,σ(T))𝕋lx(x0(t),u0(t)),y(t)Δt is a bounded linear functional.

By the Riesz representation theorem (see [12, Theorem  2.34]), there is a ψσLT([0,T]𝕋,) such that [0,σ(T))𝕋lx(x0(t),u0(t)),y(t)Δt=[0,σ(T))𝕋u(t)-u0(t),ψσ(t)Δt. Using (4.20), (4.14), and (3.9), we obtain 0[0,σ(T))𝕋[lx(x0(t),u0(t)),y(t)+lu(x0(t),u0(t)),u(t)-u0(t)]Δt=[0,σ(T))𝕋[lu(x0(t),u0(t))+ψσ(t),u(t)-u0(t)]Δt=[0,σ(T))𝕋Hu(x0(t),ψσ(t),u0(t)),u(t)-u0(t)Δt,u𝒰ad. Hence we have derived the necessary condition (4.1).

(v) Now, we can claim that ψAC([0,σ(T)]𝕋,) and the last part of necessary conditions are true. Using Lemma 2.7, (4.12), and (4.13) as well as (4.20), we obtain T2(y)=[0,σ(T))𝕋lx(x0(t),u0(t)),y(t)Δt=[0,σ(T))𝕋yΔ(t)-p(t)y(t),ψσ(t)Δt=[0,σ(T))𝕋yΔ(t),ψσ(t)Δt-[0,σ(T))𝕋p(t)y(t),ψσ(t)Δt=y(σ(T))ψ(σ(T))-[0,σ(T))𝕋y(t),ψΔ(t)Δt-[0,σ(T))𝕋y(t),p(t)ψσ(t)Δt=y(σ(T))ψ(σ(T))-[0,σ(T))𝕋y(t),ψΔ(t)+p(t)ψσ(t)Δt. From the first and the last equalities, we have y(σ(T))ψ(σ(T))-[0,σ(T))𝕋y(t),ψΔ(t)+p(t)ψσ(t)+lx(x0(t),u0(t))Δt=0. Hence, similar to Theorem 3.2, one may choose ψσ as the solution of the following backward problem: ψΔ(t)=-p(t)ψσ(t)-lx(x0(t),u0(t)),forΔa.e.t[σ(0),T]𝕋,ψ(σ(T))=0. This completes the proof.

Remark 4.2.

If the control set 𝒰=, then (4.1) reduces to Hu(x0(t),ψσ(t),u0(t))=0,        Δa.e.on[0,T]𝕋.

5. Example (A Model in Economics)

In this section, for illustration, we will apply Theorem 4.1 to the following economics model. This model had been discussed by the method of Nabla version calculus of variation on time scales (see [4, 13]). We briefly present it here. A consumer is seeking to maximize his lifetime utility subject to certain constraints. During each period in his life, a consumer has to make a decision regarding how much to consume and how much to spend. Utility is the value function of the consumer that one wants to maximize. It can depend on numerous variables, in this simple example, it depends only on the consumption of some generic production C. Utility function u(C) abides by the Law of Diminishing Marginal Utility, that is to say, u'(C)>0 and u''(C)<0.

5.1. Discrete Time Model

A representative consumer has to make decisions not just about one period but about the sequence of C's: C0,C1,,CT. The problem is to find a consumption path that would maximize lifetime utility U as follows: maxU(C)=t=0T(11+δ)tu(Ct), where Ct is the consumption during period t, u is one-period utility, and 0<δ<1 is the (constant) discount rate. We assume that the future consumption is less than the current consumption, so we discount the future at the rate δ. The consumer is limited by the budget constraints: At+1=(1+r)At+Yt-Ct,AT(11+r)T0, where At+1 is the amount of assets held at the beginning of period t+1, Yt is the income received in period t, and r is the constant interest rate. AT(1/(1+r))T0 that can be interpreted as “we are not allowed to borrow without limit.”

5.2. Continuous Time Model

The same problem can be solved in a continuous time case, where lifetime utility is the sum of instantaneous utilities: maxU(C)=0Tu(C(t))e-δtdt with respect to the path {C(t),t[0,T]} subject to the constraint A'(t)=A(t)r+Y(t)-C(t).

5.3. Time Scale Calculus Model

A consumer receives income at one time point, asset holdings are adjusted at a different time point, and consumption takes place at another time point. Consumption and saving decisions can be modeled to occur with arbitrary, time-varying frequency. Hence, the time scale version of this model can de described by maxU(C)=[0,σ(T))𝕋u(C(t))ê-δ(t,0)Δt, subject to the budget constraint AΔ(t)=rA(t)+Y(t)-C(t),t[0,ρ(T)], where ê-δ(t,0) is the Nabla exponential function of -δ, ê-δ(t,0)exp([0,t)𝕋ξ̂ν(τ)(-δ)τ). Note that (see  for more details) ê-δ(t,0){e-δtif𝕋=,(11+δ)tif𝕋=. Now, we use Theorem 4.1 to solve this model. The Hamiltonian H(A,ψσ,C)=-u(C(t))ê-δ(t,0)+ψσ(t)(rA(t)+Y(t)-C(t)). Optimal consumption satisfies the following necessary conditions: -u'(C(t))ê-δ(t,0)-ψσ(t)=0,ψΔ(t)=-rψσ(t). If 𝕋=, then (5.8), (5.10) imply C'(t)=(δ-r)u'(C(t))u''(C(t)). Due to u'(C)>0 and u''(C)<0, it shows that Ct'>0 if r>δ. Hence, the consumer will wait to consume.

If 𝕋=, then (5.8), (5.10) imply u'(Ct)=1+r1+δu'(Ct+1). It follows from u'(C)>0 and u''(C)<0 that if u'(Ct+1)<u'(Ct), then Ct+1>Ct. Therefore if the interest rate r is higher than the future's discount rate δ, the consumer will wait to consume until next periods. Therefore, we obtain the same results as .

Acknowledgments

This work is supported by the National Science Foundation of China under Grant 10661004, the Key Project of Chinese Ministry of Education (no. 2071041) and the Talents Foundation of Guizhou University under Grant 2007043, and the technological innovation fund project of Guizhou University under Grant 2007011.

HilgerS.Analysis on measure chains—a unified approach to continuous and discrete calculusResults in Mathematics1990181-21856MR1066641ZBL0722.39001BohnerM.Calculus of variations on time scalesDynamic Systems and Applications2004133-4339349MR2106410ZBL1069.39019HilscherR.ZeidanV.Calculus of variations on time scales: weak local piecewise Crd1 solutions with variable endpointsJournal of Mathematical Analysis and Applications20042891143166MR202053310.1016/j.jmaa.2003.09.031ZBL1043.49004AticiF. M.BilesD. C.LebedinskyA.An application of time scales to economicsMathematical and Computer Modelling2006437-8718726MR221831510.1016/j.mcm.2005.08.014AticiF. M.UysalF.A production-inventory model of HMMS on time scalesApplied Mathematics Letters2008213236243MR2433734ZBL1153.9000110.1016/j.aml.2007.03.013FerreiraR. A. C.TorresD. F. M.Higher-order calculus of variations on time scalesMathematical Control Theory and Finance2008New York, NY, USASpringer149159HilscherR.ZeidanV.Weak maximum principle and accessory problem for control problems on time scalesNonlinear Analysis: Theory, Methods & Applications20097093209322610.1016/j.na.2008.04.025CabadaA.ViveroD. R.Expression of the Lebesgue Δ-integral on time scales as a usual Lebesgue integral: application to the calculus of Δ-antiderivativesMathematical and Computer Modelling2006431-2194207ZBL1092.39017MR220670210.1016/j.mcm.2005.09.028GuseinovG. Sh.Integration on time scalesJournal of Mathematical Analysis and Applications20032851107127MR200014310.1016/S0022-247X(03)00361-5ZBL1039.26007CabadaA.ViveroD. R.Criterions for absolute continuity on time scalesJournal of Difference Equations and Applications2005111110131028MR217411210.1080/10236190500272830ZBL1081.39011JiangL.ZhouZ.Existence of weak solutions of two-point boundary value problems for second-order dynamic equations on time scalesNonlinear Analysis: Theory, Methods & Applications20086941376138810.1016/j.na.2007.06.034MR2426698AdamsR. A.Sobolev Spaces19756New York, NY, USAAcademic Pressxviii+268Pure and Applied MathematicsMR0450957ZBL0314.46030MartinsN.TorresD. F. M.Calculus of variations on time scales with nabla derivativesNonlinear Analysis: Theory, Methods & Applications. In press10.1016/j.na.2008.11.035BohnerM.PetersonA.Advances in Dynamic Equations on Time Scales2003Boston, Mass, USABirkhäuserxii+348MR1962542ZBL1025.34001