Closed-Form Control Law

• Closed-form control laws are explicit feedback functions that map system states to control inputs without relying on runtime optimization.
• They are derived using techniques like HJB, H∞ methods, and barrier functions to ensure stability, robustness, and constraint satisfaction.
• Their computational efficiency and analytical structure make them ideal for real-time, embedded, and large-scale distributed control applications.

A closed-form control law is an explicit, analytic feedback function or control policy for a dynamical system, typically derived without the need for numerical optimization or iterative schemes at run time. Such laws are of central importance in control theory, both for their interpretability and for their computational efficiency, and they appear in a diversity of modern control disciplines including optimal, robust, safe, and constraint-driven feedback design. Closed-form solutions are especially valued in real-time, embedded, or large-scale applications where on-the-fly optimization is computationally prohibitive.

1. Foundational Principles and Mathematical Structure

Closed-form control laws arise when the solution to a control synthesis problem—such as optimal trajectory tracking, constraint enforcement, or system stabilization—can be written as an explicit function of the current state (and possibly time or external parameters). Canonical examples include solutions to the Linear Quadratic Regulator (LQR), $H_\infty$ optimal control, and control-Lyapunov/Barrier Function methods.

A prototypical closed-form law for an input-affine system,

$\dot{x} = f(x) + g(x)\,u,$

has the generic structure

$u^*(x) = \mathcal{F}(x),$

where $\mathcal{F}(x)$ is an explicit expression derived from problem data (dynamics, cost, constraints). In advanced frameworks, $\mathcal{F}$ may incorporate state-dependent feedback matrices, resolvents, or parameter sensitivities, but it avoids any online optimization—even for nonlinear, hybrid, or constraint-laden systems.

2. Key Closed-Form Designs in Contemporary Research

Recent research has produced several fundamentally distinct classes of closed-form control laws:

• Optimal control via solution of the Hamilton–Jacobi–Bellman (HJB) equation: For specific nonlinear input-affine systems and quadratic cost, explicit analytical solutions to the HJB equation yield optimal feedback laws with precise stability certificates. For example, in "Closed Form HJB Solution for Continuous-Time Optimal Control of a Non-Linear Input-Affine System" (Vyas et al., 26 Nov 2025), the optimal control is given by

$$u^*(x) = - R^{-\frac{1}{2}}\,\frac{P(x)^\top x}{\|P(x)^\top x\|}\,\sqrt{x^\top[Q_0 + \gamma P(x)P(x)^\top]x},$$

with $P(x)$ defined by the system drift and input matrices.

• Closed-form $H_\infty$ optimal control: For certain classes of LTI systems, the $H_\infty$ optimal controller can be given as a static state feedback gain derived in closed algebraic form, without Riccati equations. For $A$, $B$ matrices:

$K^* = -N(j\omega_0)^\top M(j\omega_0)^{-T},$

where $M(s)$ and $N(s)$ are frequency-domain plant matrices, and $\omega_0$ is the critical frequency (Bergeling et al., 2019).

• Closed-form CBF-based safe control: For robotic platforms subject to multiple safety constraints, recent works have produced analytic barrier-function-based feedbacks that enforce forward invariance of constraint sets without per-step quadratic programming. E.g., for the Stewart platform:

$F_{\rm cf} = F_{\rm des} + F_{\rm safe},$

where $F_{\rm safe}$ involves explicit algebraic expressions in the Lie derivatives of soft-min barrier aggregations across all constraints (Cinun et al., 11 Dec 2025).

• Sequential action control (SAC) for nonlinear and nonsmooth systems: This approach produces closed-form optimal actions by minimizing a local quadratic surrogate for cost-sensitivity, updating the control via

$$u^*_2(t) = [\Gamma(t)^\top\Gamma(t) + R]^{-1}\Gamma(t)^\top\alpha_d,$$

at every candidate action insertion time, where $\Gamma(t)$ depends on the adjoint and system linearization (Ansari et al., 2017).

3. Constraint-Aware and Robust Closed-Form Laws

Constraint enforcement in closed-form feedback has advanced via:

• Consolidated barrier techniques: Multiple hard and soft state constraints (including time-varying and conflicting ones) can be aggregated through log-sum-exp functions and reciprocal barriers, then enforced by gradient-based feedback that is explicitly adjustable for prioritization (e.g., safety-critical over performance) and dynamic relaxation of soft constraints. The entire law is explicit in constraint functions and gradients (Mehdifar et al., 13 Oct 2025).
• Hybrid and multi-mode systems: For hybrid systems with both flows and jumps, pointwise minimum-norm closed-form laws are computable from control Lyapunov function decrease constraints in both the continuous and discrete regimes (Sanfelice, 2020).
• Robustness and input-to-state stability: Closed-form controllers with analytic stability certificates are now available for uncertain systems under both state and input perturbations, often with explicit formulas for input-to-state stability (ISS) gain bounds (Kumar et al., 28 Nov 2025, Rai et al., 11 Oct 2025).

4. Closed-Form Solutions in Large-Scale and Distributed Systems

In distributed or high-dimensional settings, closed-form controllers are critical to ensure scalability and locality:

• For state-coupled systems typical of infrastructure networks (power, water, HVAC), optimal and robust control laws are constructed to respect sparsity and distributed execution. For example, network LTI systems admit controllers of the form $K^* = B^\top A^{-1}$, where sparsity matches the physical interconnection, enabling each actuator to use only local state (Bergeling et al., 2019).
• Models coupling deterministic plants and external stochastic processes (e.g., linear MDPs) admit closed-form, parameterized feedbacks that blend LQR-type gains with terms dependent on exogenous feature states. These are explicit once a set of weights are learned via least squares (Makdah et al., 24 Aug 2025).

5. Sensitivity, Performance Index Shaping, and Parameterization

Closed-form feedbacks can be analytically differentiated and shaped with respect to problem data and objectives:

• Neighboring extremal optimal control (NEOC): The sensitivity of a closed-form optimal control law to small parameter changes in dynamics or cost is determined by solving a linear PDE in the value function's parameter gradient, yielding

$$u_{\mathrm{NE}}(x, \bar\alpha + \delta\alpha) = u^*(x, \bar\alpha) + \delta u(x, \bar\alpha, \delta\alpha),$$

where $\delta u$ has an explicit linear form in parameter increments (Rai et al., 2023).

• Performance index shaping: Closed-form analytical relationships between cost function modifications and the resulting optimal control laws have been established. For LTI systems, shaping $Q\mapsto Q+\Delta Q$ in the quadratic form yields an immediate update to the feedback:

$$\Delta K = R^{-1}B^\top \Delta P, \quad \Delta P = \int_0^\infty e^{(A-BK_0)^\top t} \Delta Q e^{(A-BK_0) t} dt,$$

and for nonlinear systems a corresponding PDE in the value function's "shaping term" provides a feedback update $$u_{p}^* = u_0^* - \frac12 R^{-1} g(x)^\top \nabla h$$ (Rai et al., 11 Oct 2025).

6. Theoretical Guarantees: Stability, Convergence, and Optimality

Closed-form control laws are typically accompanied by analytically derived guarantees:

• Lyapunov stability is established via candidate functions constructed from the value function or the barrier functions embedded in the law itself. Global asymptotic stability or uniform global attractivity can often be proven directly from the closed-feedback expressions (Vyas et al., 26 Nov 2025, Kumar et al., 28 Nov 2025, Mehdifar et al., 13 Oct 2025).
• Input-to-state stability and robustness: For systems under exogenous disturbance or model uncertainty, closed-form feedbacks provide explicit ISS gain bounds or worst-case transient bounds directly computable from feedback matrices and state/parameter norms (Makdah et al., 24 Aug 2025, Rai et al., 11 Oct 2025).
• Optimality: For knowledge classes (LQR, $H_\infty$), closed-form laws achieve performance bounds or minimize cost functionals directly without numerical solution of Riccati or SDP problems (Bergeling et al., 2019).

7. Practical Implementation and Scalability

Analytic closed-form controllers are particularly significant in embedded, resource-constrained, or latency-critical scenarios. Their key advantages, as repeatedly documented in recent research:

• Real-time feasibility: Closed-form designs avoid iterative optimization, allowing execution rates up to kilohertz even for high-dimensional, nonlinear, or hybrid systems (Ansari et al., 2017, Cinun et al., 11 Dec 2025).
• Computational complexity: The computation per control step is typically $O(1)$ or $O(m^2)$ (where $m$ is the number of constraints), in contrast to cubic or superlinear scaling for online optimization (Cinun et al., 11 Dec 2025).
• Structural transparency: The analytic dependence of the law on problem parameters allows for sensitivity analysis, designer-guided trade-off shaping, and explicit enforcement of safety or performance certifications at implementation time (Rai et al., 11 Oct 2025, Rai et al., 2023).

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In summary, closed-form control laws form a cornerstone of modern control theory for their tractability, interpretability, and real-time suitability across the spectrum from optimal, robust, and safe control to distributed and parameter-sensitive feedback design. Contemporary arXiv research illustrates the systematic extension of closed-form feedback synthesis to nonlinear, constrained, large-scale, and data-driven systems, underpinning both practical applications and foundational advances in control science (Makdah et al., 24 Aug 2025, Mehdifar et al., 13 Oct 2025, Vyas et al., 26 Nov 2025, Cinun et al., 11 Dec 2025, Bergeling et al., 2019, Rai et al., 2023, Rai et al., 11 Oct 2025, Kumar et al., 28 Nov 2025, Ansari et al., 2017, Sanfelice, 2020, Lin et al., 2021, Tafazoli, 2018).