Sparse Robust Optimal Control in Continuous-Time: A Computationally Viable Approach

Abstract: This article presents a novel, numerically viable algorithm for solving sparse robust optimal control problems in continuous time. We consider a constrained linear noisy system governed by an ordinary differential equation (ODE), with an $L<sup>1$-type objective function in line with the sparse optimal control literature. The resulting optimal control problem is shown to admit a semi-infinite programming (SIP) formulation. Building upon this insight, we develop a new framework that enables the computation of exact solutions -- to our knowledge, the first such achievement in the context of sparse optimal control. We demonstrate that a finite and computationally viable convex optimization problem can be solved to recover, in a lossless manner, both the optimal value and the corresponding optimizers of the original SIP, while also guaranteeing satisfaction of uncountably many constraints. We also show that the parameter-dependent noisy systems and the minimum attention problem fall into our framework and can be solved efficiently via our algorithm. The efficacy of our algorithm is illustrated through a benchmark numerical example.

• The paper formulates sparse robust control as a convex semi-infinite program and proves that targeted finite constraint sampling recovers the exact optimal value without conservatism, while regularized optimizers converge to an original solution.
• The method uses piecewise-constant control and disturbance dictionaries, supports process noise and parametric uncertainty, and achieves 99.9% empirical constraint satisfaction across 10,000 simulated disturbances in a spring-mass-damper benchmark.
• The proposed SparseRob architecture combines global optimization of the relaxed value function with regularization annealing, but its practicality depends on strict feasibility, unproven Lipschitz regularity, and improved scalability beyond the tested 250 decision variables.

Problem setting and motivation

The paper addresses the synthesis of sparse, robust control laws for continuous-time linear systems subject to process noise and parametric uncertainty. The underlying system is the constrained linear ODE $\dot{x}(t) = Ax(t) + Bu(t) + w(t)$ on a fixed horizon $[0,T]$, with box-constrained controls $u(t) \in [-\bar{u}^i, \bar{u}^i]$, box-bounded disturbances $w(t) \in \mathcal{W}$, compact convex state constraints $x(t) \in X$, and a terminal set $X_f$. Sparsity is promoted directly through an integral $\ell_1$ cost on the control trajectory, $$\mathsf{J}(u(\cdot)) = \int_0^T \|u(t)\|_{\ell_1}\,dt$$, rather than as a surrogate for an $\ell_0$ objective. The distinguishing feature of the problem is that all state and input constraints must hold for all $(t, w(\cdot))$ in an uncountable index set — a robust constraint satisfaction requirement that standard discretization-based direct methods cannot certify.

The authors' central claim is that this class of problems admits a convex semi-infinite program (CSIP) whose optimal value and optimizers can be recovered exactly — without conservatism — by solving a finite convex program. To their knowledge, this is the first such exactness guarantee in sparse optimal control under uncertainty; prior work either handles deterministic settings (e.g., continuous-time successive convexification [(2607.11827) references to elango2024successive], interior-point schemes for dynamic optimization), or provides only probabilistic guarantees via scenario programs (Zhang and Fujisaki's discrete-time, unconstrained treatment).

Parametrization and CSIP formulation

Both control and disturbance trajectories are parametrized by finite dictionaries of linearly independent piecewise constant functions: $[0,T]$0 with $[0,T]$1, and analogously $[0,T]$2 with $[0,T]$3. Piecewise constant dictionaries are justified on two grounds: they match digital actuator implementations, and piecewise constant functions are dense in $[0,T]$4 over the horizon, so any locally integrable admissible trajectory can be approximated arbitrarily well for large enough $[0,T]$5 and $[0,T]$6. The paper is careful to note that $[0,T]$7 is not a free numerical parameter but is tied to the temporal resolution of the hardware.

Two structural facts make the subsequent analysis tractable. First, the admissible parameter sets

$[0,T]$8

are compact and convex (Proposition 1). For the hyperrectangular control set, the semi-infinite condition $[0,T]$9 collapses via Hölder's inequality to finitely many $u(t) \in [-\bar{u}^i, \bar{u}^i]$0-ball constraints $u(t) \in [-\bar{u}^i, \bar{u}^i]$1. Second, because the cost carries no state-dependent term, the disturbance does not affect the objective value, so $u(t) \in [-\bar{u}^i, \bar{u}^i]$2; the min–max structure enters only through the constraints. The resulting program is finite-dimensional in the decision variable $u(t) \in [-\bar{u}^i, \bar{u}^i]$3 but retains uncountably many constraints indexed by $u(t) \in [-\bar{u}^i, \bar{u}^i]$4 — precisely the CSIP structure.

Exactness guarantees

Existence follows from Weierstrass arguments: the feasible set is a closed subset of the compact set $u(t) \in [-\bar{u}^i, \bar{u}^i]$5, and the objective is continuous and convex. A strictly convex regularizer $u(t) \in [-\bar{u}^i, \bar{u}^i]$6 with weight $u(t) \in [-\bar{u}^i, \bar{u}^i]$7 is added to obtain uniqueness of optimizers; notably, the authors concede that regularization is unnecessary if only the optimal value is sought ($u(t) \in [-\bar{u}^i, \bar{u}^i]$8 suffices for lossless value recovery), but it is essential for recovering optimizers, since the unregularized $u(t) \in [-\bar{u}^i, \bar{u}^i]$9 objective lacks strict convexity.

The main technical machinery rests on two results:

• Regularity: Under a Slater-like strict feasibility assumption (existence of a strictly feasible trajectory with margin into the interiors of $w(t) \in \mathcal{W}$0, $w(t) \in \mathcal{W}$1, and $w(t) \in \mathcal{W}$2 for all disturbances), the relaxed value function $w(t) \in \mathcal{W}$3 — obtained by enforcing constraints at only $w(t) \in \mathcal{W}$4 sampled index points — is jointly continuous on the compact index set, hence attains a global maximizer.
• Exactness (Theorem 1): At the maximizing index tuple $w(t) \in \mathcal{W}$5, the finite program reproduces the CSIP optimal value exactly, $w(t) \in \mathcal{W}$6, and the sequence of optimizers converges as $w(t) \in \mathcal{W}$7 to an optimizer of the original problem. The proof verifies the hypotheses of Das et al.'s targeted-sampling theorem for CSIPs, exploiting joint continuity of the constraint maps (a consequence of the piecewise constant parametrization, which renders the state solution affine in $w(t) \in \mathcal{W}$8 and $w(t) \in \mathcal{W}$9) and affinity of the constraint maps in the decision variable.

This is the strongest claim in the paper: a finite, computationally viable convex program yields the exact optimal value and convergent optimizers while certifying satisfaction of the full uncountable constraint family. No conservatism is introduced anywhere in the reduction.

Algorithmic architecture

The proposed $x(t) \in X$0 architecture alternates between an outer loop that anneals the regularization weight $x(t) \in X$1 and an inner loop that solves the global maximization of $x(t) \in X$2 over the index set using a black-box global optimizer. The framework is deliberately modular: simulated annealing is guaranteed convergent given continuity of $x(t) \in X$3 and a suitable cooling schedule; if Lipschitz continuity of $x(t) \in X$4 can be established (the authors note this requires smoothness assumptions on the constraint sets that are stated but not proved here), faster oracles such as SequOOL or LIPO become available with exponential or PAC-style regret bounds. Stopping criteria, selection rules, and regularization schedules are user-configurable metadata rather than fixed components.

Extensions

Two extensions inherit the exactness guarantees with minor modifications:

• Parameter-dependent systems: For $x(t) \in X$5 with $x(t) \in X$6 ranging over a compact convex polytope $x(t) \in X$7 and continuous $x(t) \in X$8, the index set grows to $x(t) \in X$9, and Theorem 2 establishes identical value-equality and optimizer-convergence results. The appendix verifies that joint continuity survives the matrix exponential dependence on $X_f$0.
• Robust minimum attention control: Minimizing $X_f$1 subject to a rate bound $X_f$2 reduces to the standard framework via the augmentation $X_f$3 with control $X_f$4, yielding Corollary 1 as a direct consequence.

A pointed discussion section argues that existing robust signal-processing tools (Ben-Tal–El Ghaoui–Nemirovski-style robust convex optimization) do not apply here for two reasons: the induced constraint matrices contain terms like $X_f$5 and convolution integrals, so the parameter dependence is far from affine and admits no tractable dual reformulation; and even constructing conservative affine-in-$X_f$6 approximations is nontrivial and risks infeasibility. By contrast, the present approach is exact by construction.

Numerical evidence

Experiments were run in Julia on commodity hardware (AMD Ryzen 5 5600H, 8 GB RAM) using simulated annealing as the inner oracle, on a spring-mass-damper benchmark with $X_f$7 basis functions:

• Process noise: With $X_f$8, state bound $X_f$9, terminal box $\ell_1$0, and $\ell_1$1, 9990 out of 10,000 (99.9%) simulated trajectories satisfied both state and terminal constraints across random disturbance realizations; the residual violations are attributed to numerical simulation error. Control activity concentrates near path-constraint boundaries and the terminal set, exhibiting the intended hands-off behavior. As $\ell_1$2 decreases, trajectories converge visibly, illustrating the regularization mechanism.
• Comparison against scenario optimization: Controls synthesized from 1,000 and 5,000 i.i.d. scenarios produced visible state and terminal constraint violations, consistent with the probabilistic (rather than almost-sure) nature of scenario guarantees. This contrast substantiates the paper's positioning against sampling-based alternatives.
• Parametric uncertainty: With $\ell_1$3 entering $\ell_1$4 and $\ell_1$5, the synthesized sparse control kept all 10,000 parameter realizations within state and terminal constraints.
• Minimum attention: The rate signal $\ell_1$6 exhibits clear sparsity, and the resulting trajectories remain constraint-satisfying under 10,000 disturbance realizations.

Limitations and open questions

Several caveats bear directly on the strength of the results. The strict feasibility (Slater-like) assumption is load-bearing for both continuity of $\ell_1$7 and the exactness theorem; how to verify it a priori for a given problem instance is not addressed. The claimed Lipschitz regularity of $\ell_1$8, which would unlock faster global optimization oracles, is deferred to smoothness assumptions on the constraint sets and left unproved. Scalability beyond the demonstrated dimensionality ($\ell_1$9 decision variables, low-dimensional dynamics) is untested, and the inner global optimization of $$\mathsf{J}(u(\cdot)) = \int_0^T \|u(t)\|_{\ell_1}\,dt$$0 remains the computational bottleneck — the authors themselves list gradient-based acceleration as future work. Finally, the extension to other sparsity-promoting structures, such as quasi-convex norms, is identified but not developed.

Conclusion

The paper formulates constrained sparse robust optimal control for noisy, possibly parameter-dependent continuous-time linear systems as a convex semi-infinite program and shows that a finite convex surrogate, solved at globally maximizing constraint indices, recovers the exact optimal value and convergent optimizers with no conservatism. The framework covers minimum attention control as a corollary, and benchmark experiments demonstrate near-perfect empirical constraint satisfaction (99.9% over 10,000 disturbance realizations) where scenario-based methods fail. The principal open issues are verifiability of the strict feasibility hypothesis, provable Lipschitz regularity of the relaxed value function, and computational scaling of the inner global optimization routine.