Fractional Optimal Control Problems

• Fractional optimal control problems are governed by nonlocal fractional differential operators, modeling anomalous diffusion, memory effects, and long-range interactions.
• The Caffarelli–Silvestre extension converts the nonlocal problem into a local degenerate elliptic PDE, enabling the application of standard finite element methods.
• Advanced numerical schemes with anisotropic FE discretization yield quasi-optimal error estimates while enforcing sparsity through L1 controls and box constraints.

A fractional optimal control problem is a class of control problem in which the system dynamics are governed by fractional differential operators—typically, nonlocal or non-integer order elliptic or parabolic operators such as the fractional Laplacian—subject to optimization of a prescribed cost functional and various constraints. These problems arise in the modeling, control, and inversion of processes characterized by anomalous diffusion, heavy-tail behaviors, and long-range interactions, and they generalize classical @@@@1@@@@ to the nonlocal and memory-dependent regime.

1. Problem Formulation and Core Structure

For a bounded polygonal domain $\Omega \subset \mathbb{R}^n, n\ge 1$, and a fixed order $s \in (0,1)$, the prototypical fractional optimal control problem is to minimize a functional

$$J(u,z) = \frac12 \|u - u_d\|^2_{L^2(\Omega)} + \frac{\sigma}{2}\|z\|_{L^2(\Omega)}^2 + \nu\|z\|_{L^1(\Omega)}$$

subject to the fractional state equation

$$(-\Delta)^s u = z \quad \text{in }\Omega, \qquad u = 0 \quad \text{on }\partial\Omega,$$

and box-constraints on the control,

$$a \leq z(x) \leq b \quad \text{a.e. in }\Omega, \quad a<0<b.$$

Here, $u:\Omega\to\mathbb{R}$ is the state variable, $z$ is the distributed control, $u_d\in L^2(\Omega)$ is a given desired target, and $\sigma, \nu > 0$. The term $\nu\|z\|_{L^1}$ enforces sparsity in $z$. The admissible set for the control is $Z = \{z\in L^2(\Omega): a \le z \le b\}$ (Otárola et al., 2017).

The fractional Laplacian $(-\Delta)^s$ is understood in the spectral or integral sense, and the problem may be interpreted as a nonlocal elliptic-PDE-constrained optimization with built-in sparsity control.

2. Local Realization via the Caffarelli–Silvestre Extension

A key technical step is reduction of the nonlocal optimal control problem to a local, but degenerate elliptic PDE in "one-more-dimension," using the Caffarelli–Silvestre extension: $$\begin{align*} \alpha &= 1 - 2s \in (-1,1), \ \text{Find}\quad &U: \Omega\times[0,\infty) \to \mathbb{R} \:\text{such that} \ -{\rm div}_{(x,y)}(y^\alpha \nabla U) &= 0 \quad \text{in } \Omega \times (0, \infty) \ U &= 0 \quad \text{on } \partial\Omega \times [0, \infty) \ -\lim_{y\to 0^+} y^\alpha \partial_y U(x,0) &= d_s z(x), \quad d_s=2^\alpha\Gamma(1-s)/\Gamma(s). \end{align*}$$ The trace $u(x)=U(x,0)$ recovers the original state, and the Dirichlet-to-Neumann map provides the fractional operator (Otárola et al., 2017).

This extension results in a uniformly (or degenerate) elliptic PDE with a weighted measure in the extended cylinder $\mathcal{C} := \Omega \times (0,\infty)$. Exponential decay in the extended variable justifies truncation of the domain to $\mathcal{C}_Y = \Omega\times (0,Y)$ with negligible error.

3. Existence, Regularity, and Optimality System

The reduced functional $f(z) = J(Sz, z)$, with $S: L^2(\Omega)\to H = (L^2,H_0^1)_s$ the fractional control-to-state map, is strictly convex, coercive on the convex set $Z$, and admits a unique minimizer $\bar z$ and state $\bar u = S\bar z$. The first-order optimality conditions exploit convex subdifferential calculus to yield

$$\bar p + \sigma \bar z + \nu \bar \lambda \in N_Z(\bar z),$$

where $\bar p = S^*(\bar u-u_d)$ is the adjoint (fractional) state, $\bar\lambda \in \partial|\cdot|(\bar z)$ is an $L^1$-subgradient, and $N_Z$ is the normal cone. In pointwise projection form,

$$\bar z(x) = \operatorname{Proj}_{[a,b]}\left( -\frac{1}{\sigma}(\bar p(x) + \nu \bar\lambda(x)) \right).$$

Sparsity is characterized by $\bar z(x) = 0 \Leftrightarrow |\bar p(x)| \leq \nu$ (Otárola et al., 2017).

Regularity theory shows that for $u_d \in H^{1-s}(\Omega)$,

$$\bar z, \bar \lambda \in H_0^1(\Omega),\qquad \bar u \in H^{\min\{1+2s,2\}}(\Omega),\qquad \bar p\in H^{\min\{1+s,2\}}(\Omega),$$

which is crucial for subsequent error analysis in discretization.

4. Numerical Approximation: Truncation and Finite Element Discretization

The rapid exponential decay of $U$ in the $y$-direction suggests truncating $\mathcal{C}$ to finite height $Y \sim |\log N|$, with $N$ the number of degrees of freedom. The finite element discretization is constructed on an anisotropic tensor product mesh:

• State variable $U$: approximated in $V_h$ of continuous piecewise linear (in both $x$ and $y$), vanishing on the lateral boundary and the truncation face $\{y=Y\}$
• Control variable $z$: discretized as piecewise constant on $\Omega$, obeying $a\le z_h\le b$

The fully discrete problem is: $$\text{Minimize } J(\operatorname{tr} V, z_h) \ \text{subject to } a_Y(V,W) = (z_h, \operatorname{tr} W)\;\forall W \in V_h,\;\;a\le z_h\le b,$$ where $a_Y$ is the weighted bilinear form on the truncated cylinder, and $\operatorname{tr} V = V(\cdot,0)$.

A quasi-optimal a priori estimate is established: $$\|\bar z - \bar Z\|_{L^2(\Omega)} + \|\bar u - \operatorname{tr} \bar V\|_{H^s(\Omega)} \lesssim |\log N|^{2s} N^{-1/(n+1)},$$ demonstrating that the error is only logarithmically suboptimal relative to the best possible algebraic rate dictated by the (anisotropic) finite element method on the extended domain.

5. Variational Analysis, Extensions, and Impact

The fractional optimal control framework accommodates several critical generalizations:

• Sparsity-promoting $L^1$-terms for control selection;
• Box or more general constraints on controls;
• Nonlocal-to-local reduction via the extension method, enabling use of standard FE tools;
• Generalization to elliptic operators with variable coefficients via spectral or extension approaches (Otarola, 2015).

The structure of the optimality system (state equation, adjoint equation, projection formula, and sparsity condition) mirrors classical PDE-constrained optimization but crucially incorporates nonlocal or degenerate features both analytically and numerically.

Fractional optimal control is central in fields modeling anomalous transport, inverse problems with nonlocal priors, and sparse actuator selection in extended dynamical systems. The combination of Dirichlet-to-Neumann realization, rigorous functional analysis, and sharp FE error analysis underpins robust numerical practice, enabling adaptive mesh refinement and scalable computing in high-dimensional and complex domains.

6. References and Comparative Theoretical Landscape

The foundational theory and numerical scheme for sparse fractional optimal control is thoroughly developed in (Otárola et al., 2017), with extensions to nonuniformly elliptic and curved domains, improved regularity analysis, and higher-order finite element discretization found in (Otarola, 2015). These works build upon the Caffarelli–Silvestre extension and modern convex variational techniques to achieve both theoretical and practical advances in nonlocal PDE-constrained optimization.

The underlying methodologies connect with and extend the classical optimal control theory to nonlocal settings, provide sparse solutions via nondifferentiable functionals, and support simulation-driven applications in systems with long-range interactions or fractional diffusion phenomena.