Fractional Tracking Quadratic Optimal Control

• Fractional tracking quadratic optimal control is defined as an optimal strategy employing Caputo and Riesz derivatives to manage tracking error and control cost in systems with memory effects.
• It utilizes nonlocal fractional operators and specialized discretization methods, such as finite element schemes on graded meshes and fractional finite differences, to ensure accurate numerical solutions.
• This framework has significant applications in anomalous diffusion, viscoelastic materials, and sub-diffusive transport, highlighting its practical importance in engineering systems with hereditary properties.

Fractional tracking quadratic optimal control refers to linear-quadratic optimal control and tracking problems for systems governed by fractional-order dynamics, typically in the sense of Caputo or Riesz derivatives, with a quadratic cost functional penalizing tracking error and control effort. Such control problems naturally arise in fractional diffusion, viscoelasticity, subdiffusive transport, and engineering systems with memory and hereditary effects.

1. Problem Formulation: Fractional Tracking LQ Control

Fractional tracking quadratic optimal control problems are characterized by the minimization of a cost functional of the form

$$J(u, z) = \frac{1}{2} \int_0^T \left( \|u(\cdot, t) - u_d(\cdot, t)\|_{L^2(\Omega)}^2 + \mu \|z(\cdot, t)\|_{L^2(\Omega)}^2 \right) dt$$

subject to fractional partial differential equation (PDE) constraints such as

$$\partial_t^\gamma u + L^s u = f + z,\quad \text{in}\;\Omega \times (0, T),$$

where $L^s$ denotes a fractional power of an elliptic operator (e.g., spectral or integral fractional Laplacian), $\gamma \in (0,1]$ is the Caputo time-fractional derivative order, and $z$ is the control. Box constraints $a \leq z \leq b$ are typically imposed almost everywhere in $\Omega \times (0,T)$, yielding an admissible set $Z_{\mathrm{ad}}$ of controls (Antil et al., 2015, Glusa et al., 2019, D'Elia et al., 2018).

This setting generalizes classical LQ tracking to fractional-in-time, space-fractional, and even combined space-time-fractional diffusions. The desired state $u_d$ can be prescribed on the full domain or restricted to a spatial subregion (“regional tracking”) (Ge et al., 2021).

2. State Equations and Fractional Operators

Fractional tracking control systems employ nonlocal operators in time (Caputo derivative) and/or space (fractional Laplacians):

• Time-fractional: The Caputo derivative of order $\gamma \in (0,1]$ is given by

$$\partial_t^\gamma u(t) = \frac{1}{\Gamma(1-\gamma)} \int_0^t (t - r)^{-\gamma} \frac{d}{dr} u(r) dr,$$

capturing memory-dependent effects as in anomalous subdiffusions (Antil et al., 2015, Ge et al., 2021).

• Space-fractional: The fractional Laplacian $(-\Delta)^s$ ($s \in (0,1)$) can be defined via spectral theory (as in bounded domains with Dirichlet conditions) or the singular integral (Riesz) form:

$$(-\Delta)^s w(x) = C(n,s)\,\mathrm{p.v.}\!\int_{\mathbb{R}^n} \frac{w(x)-w(y)}{|x-y|^{n+2s}} dy$$

(Glusa et al., 2019, D'Elia et al., 2018, Dohr et al., 2018).

For analytical and numerical tractability, space-fractional operators are often localized using extension techniques (e.g., Caffarelli–Silvestre extension), replacing the nonlocal PDE by a PDE posed on a semi-infinite cylinder with suitable boundary conditions (Antil et al., 2015, Antil et al., 2014).

3. First-Order Optimality System and Feedback Laws

The optimality conditions derive from Pontryagin’s minimum principle or direct variational calculus. The stationary point $(u^*, z^*)$ satisfies:

• State equation: Fractional PDE (direct problem).
• Adjoint equation: A backward-in-time fractional PDE (with the adjoint Caputo derivative and nonlocal spatial components).
• Gradient condition: For unconstrained control,

$z^* = -\frac{1}{\mu} p$

or, under constraints,

$$z^*(x, t) = \mathrm{proj}_{[a(x, t), b(x, t)]}\left( -\frac{1}{\mu} p(x, t) \right)$$

where $p(x, t)$ is the solution of the adjoint equation (Antil et al., 2015, Glusa et al., 2019, D'Elia et al., 2018, Antil et al., 2014). Analogous results hold in operator-theoretic and stochastic settings, with feedback control laws computed in closed form via adjoint processes or operator-valued Riccati equations (Malmir, 1 Feb 2026, Jha et al., 13 Apr 2025, Asadzade et al., 4 Jan 2026).

4. Discretization Schemes and Numerical Implementation

State-of-the-art discretizations are tailored to handle the regularity and anisotropy of fractional operators:

• Time discretization: Caputo time-fractional derivatives are discretized by fractional finite differences (e.g., Grünwald–Letnikov or L1 schemes) (Antil et al., 2015, Glusa et al., 2019).
• Spatial discretization: Finite element methods (FEM) on graded meshes or tensor-product meshes efficiently capture the singularity structure of the solution near boundaries and extended spatial variables (Antil et al., 2015, Antil et al., 2014, Dohr et al., 2018, D'Elia et al., 2018).
• Control discretization: Both variational (control remains in $L^2$) and fully discrete (piecewise constants or cellwise-projected controls) approaches are deployed. Both schemes admit explicit error estimates.
• Truncation: When using the extension approach, the infinite cylinder is truncated with a careful choice of length $Y \sim |\log N|$ (with $N$ the number of degrees of freedom) (Antil et al., 2015, Antil et al., 2014).
• Solver technology: Sinc quadratures, recycling Krylov subspace methods, AMG preconditioning, and panel clustering are used to solve the resulting large-scale linear systems efficiently (Dohr et al., 2018).

5. Error Estimates and Regularity Results

Rigorous a priori error estimates are derived for both the state and control variables. Representative results include:

Setting	State error ($L^2$ norm)	Control error ($L^2$ norm)	Key Conditions
Fully discrete, $s\in(0,1)$ (Antil et al., 2015)	$O(\tau + |\log N|^{2s} N^{-(1+s)/(n+1)})$	$O(\tau + |\log N|^{2s} N^{-1/(n+1)})$	$u_0\in H^{1+s}$, $f,u_d\in H^1$
Fully discrete (Glusa et al., 2019)	$O(\tau + h^{\min\{s+1/2-\epsilon,1\}})$	$O(\tau + h^{\min\{s+1/2-\epsilon,1\}})$	$y_0\in H^{s+1/2-\epsilon}$
Variational (D'Elia et al., 2018)	$O(h^{1/2+\beta-\epsilon})$	$O(h^{1/2+\beta-\epsilon})$	$\beta = \min\{s,1/2-\epsilon\}$

These rates reflect the nonlocality and reduced regularity of fractional equations: mesh grading and suitable selection of regularization are necessary for optimal convergence (Antil et al., 2014, D'Elia et al., 2018, Dohr et al., 2018).

6. Extensions: Stochastic, Nonlinear, and Regional Tracking

• Stochastic fractional control: Fractional stochastic evolution equations in Hilbert spaces are controlled using a stochastic maximum principle, with the optimal control law computed from the adjoint process via a coupled forward-backward stochastic differential equation (FBSDE) framework (Asadzade et al., 4 Jan 2026).
• Nonlinear and nonlocal initial conditions: Abstract operator-theoretic frameworks using Hammerstein-type equations enable the analysis of nonlinear systems with Caputo derivatives and nonlocal initial data, with existence, uniqueness, and characterization of optimal controls established via Gâteaux or Fréchet derivative calculus (Jha et al., 13 Apr 2025).
• Regional tracking: Optimal control targeting only subregions of the domain leverages eigenfunction expansions and the Hilbert uniqueness method (HUM) to derive explicit formulas for the control law, including series representations in the eigenbasis of the system operator (Ge et al., 2021).

7. Applications and Computational Studies

Fractional tracking quadratic optimal control is relevant in distributed parameter systems with memory and spatial heterogeneity, including anomalous diffusion, viscoelastic materials, and systems with long-range interactions. Computational studies validate error estimates and document the advantages of anisotropic mesh grading, nonuniform time stepping, and hierarchical solvers for high efficiency and accuracy (Antil et al., 2015, Glusa et al., 2019, Dohr et al., 2018, Antil et al., 2014).

Grid refinement studies confirm predicted convergence rates in both one and two spatial dimensions. Experiments with varying fractional orders ($s$, $\gamma$) demonstrate the adaptability of the framework to a wide range of dynamical regimes and problem data (Glusa et al., 2019, D'Elia et al., 2018, Ge et al., 2021).

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This field continues to advance toward unifying stochastic, nonlinear, discrete, and data-driven (e.g., FOLOC neural-operator) methodologies for fractional tracking optimal control, expanding the class of tractable systems and practical control strategies (Malmir, 1 Feb 2026, Zhang et al., 7 Feb 2025).