Control Tasks in Dynamical Systems

• Control tasks are formalized objectives in dynamical systems, defined by cost functions, constraints, and execution contexts such as real-time or multi-agent environments.
• They integrate optimization, scheduling, and multi-objective composition methods to guarantee performance, stability, safety, and adaptability under practical constraints.
• Algorithmic foundations and resource management techniques, including neural feedback scheduling and real-time MPC, enable robust control in complex, cyber-physical systems.

A control task is a formalized goal or objective in dynamical systems, robotics, or cyber-physical systems, defined by a cost function, constraints, and often an execution context (e.g., real-time, multi-agent, or resource-constrained environments). Control tasks encompass tracking, stabilization, constraint satisfaction, optimization, and composition of multiple behaviors, and are central to fields such as optimal control, reinforcement learning, robot autonomy, and embedded systems. The control task abstraction enables the development and analysis of algorithms that guarantee specified properties—performance, stability, safety, or adaptability—while coping with real-world complexities such as uncertainty, nonlinearity, partial observability, interaction, and computation-resource limits.

1. Mathematical Formulation of Control Tasks

The canonical control task is posed as a constrained optimization problem over the space of admissible control policies $u:[0,T]\rightarrow\mathcal{U}$ for a dynamical system $x_{t+1}=f(x_t,u_t)$:

$$\min_{u_{0:T-1}} \sum_{t=0}^{T-1} \ell(x_t, u_t) + \Psi(x_T) \quad \text{s.t.} \quad x_{t+1} = f(x_t,u_t), \quad x_0 \in \mathcal{X}_0, \ (x_t,u_t) \in \mathcal{Z}_t$$

Here, $\ell$ is the stage cost (e.g., tracking error, energy), $\Psi$ a terminal cost, and $\mathcal{Z}_t$ aggregates state and input constraints (actuation, safety, visibility). In reinforcement learning (RL), the objective is typically to find a policy $\pi(a|s)$ or $\mu(s)$ maximizing return or minimizing expected cumulative cost.

Certain classes of tasks introduce additional structure:

• In real-time or embedded settings, one further constrains sampling periods, execution budgets, or computational delay (e.g., $\Delta T_k$, utilization $U_R$) (0805.3062).
• In multi-task robotics, control tasks can be composed (parallel or prioritized), with each subtask $i$ expressed via cost $J_i$, CLF $V_i$, or temporally specified goals (e.g., Linear Temporal Logic) (Li et al., 2019, Kantaros et al., 2018, Tahmid et al., 1 Apr 2025).
• In partially observable or memory-based formulations, the task incorporates latent state inference (e.g., via variational models) (Han et al., 2019).

2. Classification and Composition Strategies

Control tasks are frequently categorized by their structure and operational requirements:

• Single-objective tasks: Focused on regulation or tracking with well-defined constraints.
• Multi-objective or concurrent tasks: Require simultaneous satisfaction of multiple objectives, potentially with priorities or stack-based execution (Li et al., 2019, Tahmid et al., 1 Apr 2025). This includes task stacking (Stack-of-Tasks), subtask decoupling (via CLF, RMPs, or interference-penalized value functions), and temporal sequencing.
• Closed-set vs. open-set tasks: Closed-set tasks allow only a fixed catalog at design/training time, while open-set tasks support compositional, on-demand specification of arbitrary constraint/task combinations at execution (e.g., via programmable atomic constraints over generative priors) (Liu et al., 29 May 2024).
• Optimization-based control tasks: Rely on real-time or iterative solution of mathematical programs—often MPC or convex QPs—with provisions for resource constraints and robustness to early termination (Hosseinzadeh et al., 2022, Li et al., 2021).

Control task composition introduces mechanisms for:

• Task priority via null-space projections, sequential QPs, or stacked constraint hierarchies (Domínguez et al., 2022, Li et al., 2019).
• Stability and safety guarantees by constructing shared Lyapunov (or CLF) certificates integrating all sub-tasks (Li et al., 2019).
• Value or policy independence through cost design (e.g., penalizing Lie-derivative alignment of value gradients, as in (Tahmid et al., 1 Apr 2025)).

3. Scheduling and Resource Management in Control Tasks

In embedded, cyber-physical, and networked control systems, resources such as CPU, bandwidth, and energy impose constraints on task execution:

• Feedback Scheduling: Dynamically allocates sampling periods $h_i$, ensuring CPU utilization $\sum_i C_i/h_i\leq U_R$, and updating periods in response to measured execution times and available budget (0805.3062).
• Neural Feedback Scheduling (NFS): Employs neural networks (trained offline via optimal solutions to the scheduling problem) for online, low-overhead adaptation of task periods, matching near-optimal performance at a small fraction of the computational cost (0805.3062).
• Optimization under real-time constraints: Onboard control tasks are often implemented with algorithms that guarantee feasibility even under abrupt preemption (e.g., via primal-dual flows that maintain invariance of constraints upon early termination) (Hosseinzadeh et al., 2022).
• Cloud-assisted control: Integrates high-fidelity, delayed cloud computation with simplified, fast local controllers, fusing their outputs to optimize performance while robustly satisfying constraints over finite horizons in the presence of communication delays and model mismatch (Li et al., 2021).

Resource management in these contexts is a control task in itself, often composed with lower-level regulation/tracking tasks.

4. Algorithmic and Theoretical Foundations

Designing algorithms for control tasks demands strong guarantees and careful tradeoffs:

• Stability and feasibility: Ensured via recursive feasibility (e.g., in LMPC, where safe sets and terminal costs are built from trajectory data (Rosolia et al., 2016)), CLF or Lyapunov-based composition (Li et al., 2019), or input-to-state stability (ISS) type arguments in switching/fused architectures (Li et al., 2021).
• Scheduling optimality and suboptimality bounds: While neural and approximative methods reduce computation, tight approximation-error bounds are often lacking, posing open questions regarding suboptimality gaps (0805.3062).
• Exploration and expressiveness in RL-based tasks: Specialized policy architectures (e.g., multi-style exploration as in CCEP (Li et al., 2023)), adaptive discretization (as in Growing Q-Networks (Seyde et al., 5 Apr 2024)), and value function independence enforcement (Tahmid et al., 1 Apr 2025) are employed to address the challenges of redundancy and multi-task exploration in high-dimensional control.

The following table summarizes key guarantees from select control task methodologies:

Method	Core guarantee	Limitation/Open issue
NFS (0805.3062)	Near-optimal performance, low overhead	Requires retraining on task changes
Early-term MPC (Hosseinzadeh et al., 2022)	Feasibility at any iteration	Optimality improves but not guaranteed
RMPflow-CLF (Li et al., 2019)	Stability under task composition	Design of CLF per subtask needed
LMPC (Rosolia et al., 2016)	Asymptotic performance improvement	Requires at least one successful trial
Open-set motion (Liu et al., 29 May 2024)	Arbitrary constraint composition	Scalability to real-time not yet shown

5. Experimental Validation Across Domains

Control task frameworks are validated in diverse domains, illustrating the breadth and expressivity of the control task abstraction:

• Embedded multitasking loops: Demonstrated on multitask LQG inverted pendulum systems; NFS shrinks scheduling overhead by $8\times$ while matching optimal cost/error (0805.3062).
• Resource-constrained cyber-physical systems: Robust-to-early-termination methods yield control performance within 4–9% of the ideal, even under non-implementable sampling rates (Hosseinzadeh et al., 2022).
• Robot manipulation and interaction: Adaptive MPC frameworks achieve two- to fourfold improvements in interaction error and generalize across diverse manipulation primitives without manual retuning (Minniti et al., 2021).
• Multi-task and multi-agent coordination: CLF-based subtask generators handle complex formations, cooperative avoidance, or morphological reconfiguration, and maintain global stability under concurrent execution (Li et al., 2019, Tahmid et al., 1 Apr 2025).
• Open-set motion generation: Unified programmable error functions over frozen generative priors yield coherent novel skills, compositional constraint satisfaction, and task generalization beyond pre-trained datasets (Liu et al., 29 May 2024).

6. Limitations, Open Challenges, and Future Perspectives

Despite the wide applicability and empirical performance of control task methodologies, several open issues remain:

• Scalability and adaptation: Generating dense training sets for neural approximators, ensuring generalization to unmodeled workloads, and online or continuous learning in nonstationary environments are active research areas (0805.3062, Liu et al., 29 May 2024).
• Theoretical performance bounds: Many practical methods lack tight, dimension-independent suboptimality or feasibility guarantees—particularly true for black-box or compositional approaches.
• Expressivity vs real-time constraints: Highly general control task formulations (e.g., open-set programmable motion) often remain computationally intensive, limiting their applicability to real-time or reactive control (Liu et al., 29 May 2024).
• Integration of learning and classical control: Combining RL-based skill discovery, symbolic task composition, and scheduling with classical control and stability analysis is an emerging direction, with challenges in formal specification and safety assurance (Srivastava et al., 2022, Tahmid et al., 1 Apr 2025).
• Human-in-the-loop and explainability: For assistive or teaching tasks, decomposing control tasks into interpretable skills mapped to human-understandable curricula offers significant gains, but adaptive and safety guarantees remain to be generalized (Srivastava et al., 2022).

A plausible implication is that future control task research will increasingly rely on hybrid architectures that unify symbolic, learning-based, and optimization-driven components, with interfaces for dynamic task specification, compositionality, and robust safety/performance guarantees under real-world constraints.