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Discrete-Time Fractional Variational Problems
Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres
TL;DR
Discrete fractional variational problems lack the optimality theory available for continuous time, motivating a formulation on the time scale (hZ)a. The paper introduces left and right fractional operators and derives necessary variational conditions, finding convergence to classical discrete and fractional continuous-time solutions while using a Legendre condition to filter Euler–Lagrange candidates.
Problem
Discrete-time fractional systems have received less study, while existing necessary optimality results address only continuous-time fractional variational problems.
Method
The paper introduces left and right fractional sums and differences on (hZ)a and proves fractional summation by parts together with first- and second-order necessary optimality conditions.
Results
The solutions converge to classical discrete-time solutions at integer fractional orders and to fractional Riemann–Liouville continuous solutions as h→0.
Takeaways & Limitations
The Legendre condition can help determine whether candidates from the fractional Euler–Lagrange equation are solutions of the variational problem.
Takeaways & Limitations
Generalization to an arbitrary time scale remains open because the proofs rely on constant graininess in (hZ)a.
Abstract
from arXiv · showhide
We introduce a discrete-time fractional calculus of variations on the time scale $h\mathbb{Z}$, $h > 0$. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. They show that solutions to the considered fractional problems become the classical discrete-time solutions when the fractional order of the discrete-derivatives are integer values, and that they converge to the fractional continuous-time solutions when $h$ tends to zero. Our Legendre type condition is useful to eliminate false candidates identified via the Euler-Lagrange fractional equation.
1. Introduction
The paper develops a discrete-time fractional calculus of variations on (hZ)a to address limited optimality theory for discrete fractional systems. It introduces fractional operators and necessary conditions, with results connecting discrete fractional problems to classical discrete-time and continuous-time cases.
- Fractional calculus extends derivatives to arbitrary orders and has applications across physics, chemistry, biology, economics, and control theory.
- Discrete-time fractional systems remain less studied than their continuous-time counterparts.
- As h tends to zero, computer simulations recover previous fractional continuous-time results.
- The paper develops a fractional variational theory on (hZ)a, introducing left and right fractional sums and differences.
- The theory establishes a fractional h-summation-by-parts formula and first- and second-order necessary optimality conditions.
- Generalizing the results to an arbitrary time scale remains open because the proofs rely on constant graininess in (hZ)a.
2. Preliminaries
The preliminaries establish time-scale notation and specialize the framework to the discrete time scale (hZ)a with constant graininess h. They define the associated delta calculus, linear dynamic-equation tools, and fractional h-sums that support the later variational theory.
- A time scale is a nonempty closed subset of R equipped with forward and backward jump operators and a graininess function.
- The preliminaries review delta differentiability, higher-order delta derivatives, product rules, rd-continuity, antiderivatives, and delta integration.
- The paper extends discrete fractional operators from Z_a to the more general scaled time scale (hZ)a.
- Linear dynamic-equation preliminaries include solution definitions, regressivity, Cauchy functions, fundamental systems, and generalized polynomials.
- On (hZ)a, the jump operators satisfy σ(t)=t+h and ρ(t)=t−h, while the graininess is constantly h.
- For this discrete time scale, the delta derivative becomes the usual discrete forward difference and the delta integral becomes an h-sum.
- Left and right fractional h-sums of positive order are introduced as operators on functions over the discrete time scale.
3. Main Results
The paper develops fractional variational problems on the time scale hZ and derives first- and second-order necessary optimality conditions. A fractional h-summation-by-parts formula supports the Euler-Lagrange equation, natural boundary conditions, and Legendre condition, with classical discrete-time specializations.
- Fractional h-summation by parts: A fractional h-summation-by-parts formula is established to derive necessary optimality conditions in the time-scale setting.The formula is presented for fractional sums and differences with 0 < α ≤ 1.
- Fractional variational problem: The considered problem minimizes or maximizes a functional involving fractional differences subject to prescribed boundary conditions.Local minima are defined using admissible variations that vanish at both endpoints.
- First-order optimality condition: A local minimum satisfies the h-fractional Euler-Lagrange equation, which provides a first-order necessary optimality condition.The result is obtained by applying the fractional summation-by-parts formula to the first variation.
- Classical limits: The classical discrete Euler-Lagrange equation is recovered when α = 1 and the Lagrangian does not depend on the additional fractional-difference argument.Taking h = 1 yields the usual discrete Euler-Lagrange equation.
- First-order optimality condition: When an endpoint is free, additional natural boundary conditions accompany the h-fractional Euler-Lagrange equation.Separate supplementary conditions are given for a free initial or terminal value.
- Second-order optimality condition: A local minimum also satisfies an h-fractional Legendre inequality as a second-order necessary condition.For h tending to zero, the inequality coincides with the classical condition Lvv[ŷ](t) ≥ 0.
4. Examples
The examples show convergence from discrete-time fractional extremals to continuous-time fractional solutions as h decreases, and to classical discrete-time solutions as fractional orders approach integers. They also demonstrate that the fractional Legendre condition filters Euler–Lagrange candidates.
- Example 4.1: Each h in Example 4.1 yields a unique h-fractional Euler–Lagrange extremal satisfying the h-fractional Legendre condition.Figure 1 compares extremals for h = 0.50, 0.125, 0.0625, and 1/30 with the continuous solution.
- Example 4.3: The Legendre condition is useful for eliminating false candidates produced by the fractional Euler–Lagrange equation.Example 4.3 explicitly uses the condition to distinguish candidates before selecting the extremal.
- Example 4.3: For problem (36) with α = 0.8, β = 0.5, h = 0.25, and θ = 1, eight Euler–Lagrange extremals reduce to two candidates satisfying the Legendre condition.Comparing the functional values of the two admissible candidates identifies candidate number five as the desired extremal.
- Example 4.3: For problem (36) with α = 0.3, h = 0.1, and θ = 0, sixteen Euler–Lagrange extremals reduce to one satisfying the fractional Legendre condition.The selected extremal is candidate number six on Table 2.
- Examples 4.1–4.2: As h tends to zero, the discrete-time fractional extremals converge to corresponding fractional Riemann–Liouville continuous-time solutions.This behavior is illustrated for the examples corresponding to problems (33) and (35).
- Examples 4.1–4.2: For integer fractional order, the solutions converge to classical discrete-time solutions.The examples report this convergence as the fractional order of the discrete derivatives approaches integer values.
5. Conclusion
The paper develops a discrete fractional variational calculus on (hZ)a with necessary optimality conditions, whose solutions connect discrete, continuous, and integer-order cases. Its Legendre condition helps assess Euler–Lagrange candidates, while existence theory and broader problem classes remain open directions.
- 5. Conclusion: The work introduces left and right discrete-time fractional derivatives and a fractional summation-by-parts formula for constructing variational optimality conditions.The resulting framework includes an Euler–Lagrange equation, fractional natural boundary conditions, and a second-order Legendre necessary condition.
- 5. Conclusion: Solutions converge to classical discrete-time solutions as the fractional derivative orders approach integer values and to fractional Riemann–Liouville continuous solutions as h → 0.
- 5. Conclusion: The Legendre condition can eliminate Euler–Lagrange candidates that fail it and distinguish candidates for minimizers from candidates for maximizers.When several Euler–Lagrange candidates exist, only some may satisfy the Legendre condition.
- 5. Conclusion: The theory remains incomplete, including the unresolved question of existence for solutions to the discrete fractional Euler–Lagrange equations.Further work also includes variable-endpoint, isoperimetric, higher-order, and sufficient optimality conditions.