Hidden Shortcut Shrinks Huge Optimal Control Problems Without Extra Error, Study Shows

Mathematicians have found a way to make a large class of optimal control problems dramatically cheaper to solve without paying the usual price of approximation. In a study published in Computational Science and Engineering, Michael Kartmann and Stefan Volkwein of the University of Konstanz show that when a high-dimensional optimal control problem is compressed on the state side, a matching compression of the control side comes for free, inherited directly from the optimality conditions that define the solution. The result means engineers and scientists can optimize over a far smaller set of unknowns while introducing no additional approximation error from the control reduction itself.

The problems in question are linear-quadratic optimal control problems, in which one seeks a control signal that steers a system, governed by a partial differential equation, toward a desired state while keeping the control effort small. The cost functional balances the squared deviation of the observed state from a target trajectory against a penalty, scaled by a parameter beta, on the size of the control. Such problems arise across engineering and the sciences, and their difficulty stems from the fact that both the state space and the control space can be high-dimensional or even infinite-dimensional, making direct numerical solution expensive.

The key observation is algebraic rather than heuristic. The optimal control satisfies a first-order optimality condition that ties it to the adjoint variable, a Lagrange multiplier-like quantity solved backward in time: the control equals minus one over beta times the Riesz map applied to the dual operator acting on the adjoint. When the governing equation is projected onto a reduced state space built from a handful of basis vectors, the reduced adjoint lives in that same small space. Through the optimality condition, the reduced optimal control is then automatically a linear combination of a small number of basis functions in the control space. In other words, shrinking the state space implicitly shrinks the control space, and the two reduced problems are provably equivalent, with identical optimal states, adjoints, and controls.

The authors connect this idea to Hinze’s variational discretization, a well-known technique in finite element methods for control-constrained problems where the control is never discretized explicitly but recovered from the discrete adjoint. Their setting differs in two ways: they work with model order reduction rather than finite elements, and they treat unconstrained problems. That combination is what makes an explicit reduced linear control space possible, because without control constraints the optimality condition involves no projection onto an admissible set, so the reduced adjoint induces a genuine subspace on which optimization can be carried out exclusively.

Beyond the structural equivalence, the paper delivers certified error control. The researchers derive both lower and upper a posteriori bounds on the distance between the true full-order optimal control and any candidate control, based solely on the norm of the cost functional’s gradient at that candidate. A notable feature is that the bounds do not require the candidate to satisfy the reduced optimality condition, so they can be applied to every iterate of an iterative solver, which typically satisfies optimality only up to a tolerance. The evaluation does require full-order state and adjoint solves, so the bounds are not online-efficient, but in the adaptive setting those same full-order quantities are recycled as new data for updating the reduced model.

These estimates feed an online-adaptive algorithm built on proper orthogonal decomposition, a standard model reduction technique that extracts dominant basis functions from a set of solution snapshots. Starting from an initial guess, the algorithm checks the error bounds at the current iterate; if accuracy is insufficient, it enriches the snapshot set with the full-order state and adjoint just computed, rebuilds the reduced state and control spaces by POD, and solves the reduced problem, which is independent of the full-order dimension and therefore fast. The authors prove that this procedure converges to the full-order optimal control, and that for any positive tolerance it terminates after finitely many steps with a certified error guarantee.

The numerical experiments use a parabolic test problem on the unit square with a time-varying reaction coefficient, discretized with 10201 finite element degrees of freedom in space and 100 implicit Euler time steps. When the state-reduced and the combined control- and state-reduced problems were solved to a tight gradient tolerance, the control errors between the two models ranged from roughly ten to the minus sixteen to ten to the minus nine as the regularization parameter beta decreased, confirming the theoretical equivalence, and the combined reduction ran about nine times faster, a speed-up independent of beta.

Against the full-order model, the gains grew larger. Solving with a gradient tolerance of ten to the minus eight, the state-space-only reduced method achieved speed-ups up to 20.6, while the combined control- and state-reduced method reached up to 34.2 relative to the full-order gradient descent. The true control errors stayed between ten to the minus eight and ten to the minus five as beta ranged from ten to the minus one down to ten to the minus four, and the certified upper error bound held in every case. The speed-up increased as beta decreased, which the authors attribute to the problem becoming more ill-conditioned, so the reduced methods saved more outer iterations. The reduced basis sizes stabilized around twenty to forty after growing monotonically through the adaptive enrichment.

The study also verified the error estimator numerically across all iterations of the adaptive method, finding the upper bound sharp while the lower bound was less accurate because of a coarse approximation of the constant appearing in it. The authors note that the framework assumes the control enters the equation linearly and the cost is quadratic in the control, and they consider an extension to nonlinear partial differential equations feasible under those assumptions. Because the adaptive algorithm can also accelerate the offline training phase for parametrized problems, where many full-order solves would otherwise be needed, the approach could broaden the practical reach of certified reduced-order optimization for systems described by differential equations.

Subject of Research: Certified model order reduction of linear-quadratic optimal control problems via optimality-induced control space reduction

Article Title: Optimality-based control space reduction for high-dimensional control spaces

Article References: Kartmann, M., & Volkwein, S. (2026). Optimality-based control space reduction for high-dimensional control spaces. Computational Science and Engineering, 3(1), Article 5. https://doi.org/10.1007/s44207-026-00014-x

Image Credits: AI Generated

DOI: 10.1007/s44207-026-00014-x

Keywords: optimal control, model order reduction, reduced-order modeling, proper orthogonal decomposition, a posteriori error estimation, Galerkin methods, adjoint equations, linear-quadratic problems, adaptive algorithms, partial differential equations, numerical optimization, computational mathematics