Fractional Damping Through Restricted Calculus of Variations

We deliver a novel approach towards the variational description of Lagrangian mechanical systems subject to fractional damping by establishing a restricted Hamilton’s principle. Fractional damping is a particular instance of non-local (in time) damping, which is ubiquitous in mechanical engineering applications. The restricted Hamilton’s principle relies on including fractional derivatives to the state space, the doubling of curves (which implies an extra mirror system) and the restriction of the class of varied curves. We will obtain the correct dynamics and will show rigorously that the extra mirror dynamics is nothing but the principal one in reversed time; thus, the restricted Hamilton’s principle is not adding extra physics to the original system. The price to pay, on the other hand, is that the fractional damped dynamics is only a sufficient condition for the extremals of the action. In addition, we proceed to discretise the new principle. This discretisation provides a set of numerical integrators for the continuous dynamics that we denote Fractional Variational Integrators (FVIs). The discrete dynamics is obtained upon the same ingredients, say doubling of discrete curves and restriction of the discrete variations. We display the performance of the FVIs, which have local truncation order 1, in two examples. As other integrators with variational origin, for instance those generated by the discrete Lagrange–d’Alembert principle, they show a superior performance tracking the dissipative energy, in opposition to direct (order 1) discretisations of the dissipative equations, such as explicit and implicit Euler schemes.

1. We could be more precise on this: defining the usual Lebesgue functions as \(L_p[a,b]\), with \(1\le p\le \infty \), and \(I_{\pm }^{\alpha }(L_p[a,b])=\left\{ f\,\,|\,\,f=I_{\pm }^{\alpha }\phi \,\,,\phi \in L_p[a,b]\right\} \), with \(I_{\pm }^{\alpha }=D_{\pm }^{-\alpha }\), then the minimal condition for (4a) to hold is \(f\in I^{\alpha }_-(L_p[a,b])\) and \(g\in I^{\alpha }_+(L_q[a,b])\), with \(p^{-1}+q^{-1}\le 1+\alpha \) (Ferreira 2019; Samko 1993). A simple condition implying this is taking \(f,g\in AC^2([a,b])\), which simplifies considerably our further notation. On the other hand, (4b) holds if \(f\in I^{\alpha +\beta }_-(L_1[a,b])\), which is less restrictive than the previous one. Thus, it is enough to choose functions in \(AC^2([a,b])\).

2. Note that the usual requirements regarding regularity in the usual Hamilton’s principle are \(C^2\) curves and a \(C^2\) Lagrangian L (Abraham and Marsden 1978), since the usual Euler–Lagrange equations are written in terms of the second partial derivatives of L and the second total derivative of the curves with respect to time.

3. Observe that, according to (3), the only available value of \(D^{\alpha }_{-}x(t_0)\) is 0.

4. Particularly, it is a so-called inhomogeneous sequential fractional differential equation, i.e.

   $$\begin{aligned} \left( D^{n/q}_{-}+a_1D^{(n-1)/q}_{-}+\cdots +a_nD^0_{-}\right) \,x(t)=F(t), \end{aligned}$$

   with \((n,q)=(4,2)\), \(a_1=a_4=1\), \(a_2=a_3=0\) (Miller and Ross 1993).

This work has been funded by the EPSRC project: ‘’Fractional Variational Integration and Optimal Control”; ref: EP/P020402/1. FJ thanks Farhang Haddad Farshi for his assistance in the design of Figure 1.We are indebted to the referee, whose comments have improved notably this manuscript.

Authors and Affiliations

1. Departamento de Matemáticas Aplicadas a la Ingeniería Industrial, Universidad Politécnica de Madrid, José Gutiérrez Abascal 2, 28006, Madrid, Spain

   Fernando Jiménez

2. Department of Mathematics, University of Paderborn, Warburger Straße 100, 33098, Paderborn, Germany

   Sina Ober-Blöbaum

Corresponding author

Correspondence to Fernando Jiménez.

Communicated by Melvin Leok.

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Appendix: Existence and Uniqueness of Solutions of Fractional Differential Equations (31)

We study the existence and uniqueness of solutions of (31). It has been proven, Proposition 3.2, that x and y systems are equivalent in reversed directions of time. Thus, we focus on the x-system and set \(d=1\) for simplicity. We are considering \(L:T\mathbb {R}\rightarrow \mathbb {R}\) a \(C^2\) function; furthermore we shall consider \(\left( \partial ^2L/\partial \dot{x}\partial \dot{x}\right) ^{-1}\) smooth. All in all, (31a), can be expressed as

with initial condition \(x(t_0)=x_0,\,v(t_0)=v_0 \), \(\beta =2\alpha \) and \(f,{\bar{\rho }}:\mathbb {R}^2\rightarrow \mathbb {R}\) given by

with \(\rho \in \mathbb {R}_+\), adding up for the vector field \(F:\mathbb {R}^2\rightarrow \mathbb {R}^2\)

In proving the existence and uniqueness of solutions of (68), we shall take a local approach; in particular we consider the set \(\mathcal {B}=I_t\times I_x\times I_v\subset \mathbb {R}^3\), with \(I_t=[t_0-\delta _t, t_0+\delta _t],\,I_x=[x_0-\delta _x, x_0+\delta _x],\, I_v=[v_0-\delta _v,v_0+\delta _v]\), \(I_t\subset [a,b]\) and \(\delta _t,\delta _x,\delta _v\in \mathbb {R}_+\). We consider \(\mathbb {R}^d\) as a Banach space equipped with the norm

Given that, we define the cylinder \(\mathcal {C}=I_t\times \mathcal {D}\), with \(\mathcal {D}=\left\{ (x,v)\in \mathbb {R}^2\,|\,||(x,v)-\right. \left. (x_0,v_0)||\le \mathfrak {b}\right\} \), \(\mathfrak {b}=||(\delta _x,\delta _v)||.\) We establish the following hypotheses:

1. H1.

   \(||f(x,v)-f({\tilde{x}},{\tilde{v}})||\le K||(x,v)-({\tilde{x}},{\tilde{v}})||\) in \(\mathcal {D}\) for \(K\in \mathbb {R}_+\).

2. H2.

   \({\bar{\rho }}\) is continuous in \(\mathcal {D}\).

Note that these two hypotheses follow directly from the assumptions over L and \(\left( \partial ^2L/\partial \dot{x}\partial \dot{x}\right) ^{-1}\), say they are \(C^2\) and smooth, respectively. Given this, our strategy is to prove that the vector field (69) satisfies the required conditions to apply both Peano and Picard–Lindelöf theorems (Kartsatos 2005), ensuring the existence and uniqueness of solutions of (68) in \(\tilde{\mathcal {C}}={\tilde{I}}_t\times \mathcal {D}\), with \({\tilde{I}}_t=[t_0-{\tilde{\delta }}_t,t_0+{\tilde{\delta }}_t]\), \({\tilde{\delta }}_t=\,\)min\(\left\{ \delta _t\,,\,\mathfrak {b}/M\right\} \), where \(M\in \mathbb {R}_+\) is constructed in the proof below.

Proposition 6.1

Given the hypotheses H1, H2, the following is true: \(F:\mathcal {D}\rightarrow \mathbb {R}^2\)

1. (1)

   is bounded.

2. (2)

   is Lipschitz continuous.

Proof

1. (1)

   Let \((x,v)\in \mathcal {D}\), then \(||F(x,v)||=\)max \(\left\{ |v|, |f(x,v)+{\bar{\rho }}(x,v)D^{\beta }_{-}x|\right\} \). On the one hand, \(|v|\le |v_0+\mathfrak {b}|<\infty \). On the other, by H1 f is continuous in \(\mathcal {D}\) and therefore \(|f(x,v)|\le M_f<\infty \). By H2, \({\bar{\rho }}\) is also continuous; thus, \(|{\bar{\rho }}(x,v)|\le M_{{\bar{\rho }}}<\infty \). Now, we shall use the Caputo definition (??) of the fractional derivative in (69), just by using the relationship (??) and setting \(x_0=0.\) With that, in \(I_t\times \mathcal {D}\) we have

   $$\begin{aligned}&|f(x,v)+{\bar{\rho }}(x,v)D^{\beta }_{-}x|\le |f(x,v)|+|{\bar{\rho }}(x,v)||D^{\beta }_{-}x|\le M_{f}+M_{{\bar{\rho }}}|D^{\beta }_{-}x|\\&\quad =M_{f}+\frac{M_{{\bar{\rho }}}}{\Gamma (1-\beta )}\Big |\int _{t_0}^t(t-\tau )^{-\beta }\dot{x}(\tau )d\tau \Big |\\&\quad =M_{f}+\frac{M_{{\bar{\rho }}}}{\Gamma (1-\beta )}\Big |\int _{t_0}^t(t-\tau )^{-\beta }v(\tau )d\tau \Big |\\&\quad \le M_{f}+\frac{M_{{\bar{\rho }}}}{\Gamma (1-\beta )}\int _{t_0}^t|(t-\tau )^{-\beta }| |v(\tau )|d\tau \le M_{f}\\&\quad +\frac{M_{{\bar{\rho }}}|v_0+\mathfrak {b}|}{\Gamma (1-\beta )}\int _{t_0}^t|(t-\tau )^{-\beta }|d\tau \\&\quad \le M_{f}+\frac{M_{{\bar{\rho }}}|v_0+\mathfrak {b}|}{\Gamma (1-\beta )}\delta _t^{1-\beta }<\infty , \end{aligned}$$

   where, according to \(\beta =2\alpha \), we consider \(\alpha \in [0,1/2]\) in concordance with the restricted fractional Euler–Lagrange equations (31a). From this, it follows that \(||F(x,v)||\le M<\infty ,\) where \(M=\left\{ |v_0+\mathfrak {b}|\,\,\text{ or }\,\,M_{f}+\frac{M_{{\bar{\rho }}}|v_0+\mathfrak {b}|}{\Gamma (1-\beta )}\delta _t^{1-\beta }\right\} \), depending on whether the max \(\left\{ |v|, |f(x,v)+{\bar{\rho }}(x,v)D^{\beta }_{-}x|\right\} \) is achieved in the first or second entry.

2. (2)

   We have that

   $$\begin{aligned}&||F(x,v)-F({\tilde{x}},{\tilde{v}})||=||\big ( v-{\tilde{v}},\,f(x,v)-f({\tilde{x}},{\tilde{v}})\nonumber \\&\quad +{\bar{\rho }}(x,v)D^{\beta }_{-}x-{\bar{\rho }}({\tilde{x}},{\tilde{v}})D^{\beta }_{-}{\tilde{x}}\big )||, \end{aligned}$$

   (70)

   which is equal to the maximum of the absolute value of both entries. On the one hand,

   $$\begin{aligned} |v-{\tilde{v}}|\le ||(x,y)-({\tilde{x}},{\tilde{v}})||, \end{aligned}$$

   (71)

   by construction. On the other

   $$\begin{aligned}&|f(x,v)-f({\tilde{x}},{\tilde{v}})+{\bar{\rho }}(x,v)D^{\beta }_{-}x-{\bar{\rho }}({\tilde{x}},{\tilde{v}})D^{\beta }_{-}{\tilde{x}}|\le |f(x,v)-f({\tilde{x}},{\tilde{v}})|\\&\quad +|{\bar{\rho }}(x,v)D^{\beta }_{-}x-{\bar{\rho }}({\tilde{x}},{\tilde{v}})D^{\beta }_{-}{\tilde{x}}|\\&\quad \le K||(x,y)-({\tilde{x}},{\tilde{v}})||+M_{{\bar{\rho }}}|D^{\beta }_{-}x-D^{\beta }_{-}{\tilde{x}}|\\&\quad =K||(x,y)-({\tilde{x}},{\tilde{v}})||+\frac{M_{{\bar{\rho }}}}{\Gamma (1-\beta )}\big |\int _{t_0}^{t}(t-\tau )^{-\beta }(v(\tau )\\&\quad -{\tilde{v}}(\tau ))d\tau \big |\\&\quad \le K||(x,y)-({\tilde{x}},{\tilde{v}})||+\frac{M_{{\bar{\rho }}}}{\Gamma (1-\beta )}||(x,y)\\&\quad -({\tilde{x}},{\tilde{v}})||\int _{t_0}^t|(t-\tau )^{-\beta }|d\tau \\&\quad \le K||(x,y)-({\tilde{x}},{\tilde{v}})||+\frac{M_{{\bar{\rho }}}\delta _t^{1-\beta }}{\Gamma (1-\beta )}||(x,y)\\&\quad -({\tilde{x}},{\tilde{v}})||= \left( K+\frac{M_{{\bar{\rho }}}\delta _t^{1-\beta }}{\Gamma (1-\beta )}\right) \\&\quad ||(x,y)-({\tilde{x}},{\tilde{v}})||, \end{aligned}$$

   where we have used H1, H2 and (71). Thus, it follows that

   $$\begin{aligned} ||F(x,v)-F({\tilde{x}},{\tilde{v}})||\le {\tilde{K}}\,||(x,y)-({\tilde{x}},{\tilde{v}})||, \end{aligned}$$

   where \({\tilde{K}}=\left\{ 1\,\text{ or }\,K+\frac{M_{{\bar{\rho }}}\delta _t^{1-\beta }}{\Gamma (1-\beta )}\right\} \), depending on whether the maximum on the right-hand side of (70) is achieved on the first or second entry. In both cases, \({\tilde{K}}\in \mathbb {R}_+.\)

\(\square \)

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