A Semi-smooth Newton Method for the Constrained Optimal Control of Continuous-Time Linear Systems

Dominic Liao-McPherson; Marco M. Nicotra; Simon J. Jones

arxiv: 2605.06991 · v1 · submitted 2026-05-07 · 🧮 math.OC · cs.SY· eess.SY

A Semi-smooth Newton Method for the Constrained Optimal Control of Continuous-Time Linear Systems

Simon J. Jones , Dominic Liao-McPherson , Marco M. Nicotra This is my paper

Pith reviewed 2026-05-11 02:00 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY

keywords constrained optimal controlsemi-smooth Newton methoddifferential Riccati equationKKT conditionscontinuous-time systemscomplementarity functionsuperlinear convergence

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The pith

A semi-smooth Newton method reduces constrained optimal control of linear systems to solving modified differential Riccati equations at each iteration.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper presents a method to solve constrained optimal control problems for continuous-time linear systems without discretizing the dynamics upfront. It reformulates the problem by embedding the KKT optimality conditions into a non-smooth complementarity function, turning it into a root-finding problem in Banach space. This is solved using a semi-smooth Newton method, where each update is found by solving a differential Riccati equation whose cost terms are adjusted using the current constraint multipliers. The method is shown to converge superlinearly, limited only by the accuracy of the ODE solver used. Readers interested in numerical optimal control would care because it offers an indirect, function-space approach that handles constraints naturally.

Core claim

The paper establishes that the Newton step for the non-smooth root-finding problem can be obtained by solving a modified differential Riccati equation, in which the quadratic cost is reweighted at every iteration based on the values of the constraint multipliers. This allows the constrained problem to be handled directly in continuous time.

What carries the argument

The non-smooth complementarity function embedding the KKT conditions, which allows the semi-smooth Newton iteration to be computed via a reweighted differential Riccati equation.

If this is right

The method achieves superlinear convergence up to the tolerance of the ODE solver.
It applies directly to continuous-time linear systems without requiring time discretization.
Constraint multipliers are incorporated by reweighting the cost at each Newton iteration.
Numerical examples demonstrate effectiveness for problems with state and input constraints.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This framework could potentially be extended to systems with nonlinear dynamics by incorporating appropriate linearizations.
It provides an alternative to discretization-based methods that might preserve more structure of the continuous problem.
The approach might integrate with existing Riccati solvers for efficient implementation.

Load-bearing premise

The non-smooth complementarity function can embed the KKT conditions in Banach space without needing additional regularity assumptions on the constraint sets or cost functionals.

What would settle it

Observing whether the computed control trajectory satisfies the original constraints to within the expected tolerance or whether the convergence rate remains superlinear when the ODE solver is made more accurate.

Figures

Figures reproduced from arXiv: 2605.06991 by Dominic Liao-McPherson, Marco M. Nicotra, Simon J. Jones.

**Figure 2.** Figure 2: ||F(z)||2 versus number of solver iterations for both min and Fischer-Burmeister NCP functions. Regularization parameter δ was chosen experimentally. For min function: δ = 1 × 10−1 . For Fischer-Burmeister function: δ = 1 × 10−2 . optimization in constrained robotic systems. IEEE Transactions on Robotics, 39(1), 183–202. Bryson, A.E. (2018). Applied optimal control: optimization, estimation and control. … view at source ↗

read the original abstract

This paper details a novel indirect method for solving constrained optimal control problems (OCPs) directly in continuous-time function space. The KKT conditions are embedded in a non-smooth complementarity function, which enables their reformulation as a rootfinding problem in Banach space. This problem is then solved using a non-smooth Newton method. Finally, the paper shows that the Newton update can be obtained by solving a modified differential Riccati equation, where the cost terms are reweighted at every iteration based on the constraint multipliers. Numerical simulations show the effectiveness of the method, which converges superlinearly up to the tolerance of the ODE solver.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a function-space semi-smooth Newton method for constrained continuous-time linear-quadratic control where each step reduces to a multiplier-reweighted differential Riccati equation, but the numerics are light on detail and the Banach-space regularity assumptions need explicit checking.

read the letter

The core contribution is a direct function-space solver for constrained linear-quadratic optimal control. They embed the KKT conditions into a non-smooth complementarity equation on the trajectory space, apply a semi-smooth Newton iteration, and show that the resulting linear system reduces to a differential Riccati equation whose quadratic cost matrices are updated by the current multiplier values. This keeps the method in continuous time rather than discretizing the dynamics first, which is the main practical hook for control applications that care about accuracy between sample times. The derivation itself looks clean on the abstract level and follows standard steps for non-smooth Newton on complementarity problems. The claim of superlinear convergence up to ODE tolerance is plausible if the operator is indeed semi-smooth and the linearization stays well-posed. That is the part that works. The simulations are described only at the level of “they converge superlinearly,” with no problem dimensions, no baseline solvers, no timing data, and no mention of how the method behaves when constraints become active over intervals. Without those, it is difficult to judge whether the approach is competitive with existing indirect or direct methods on realistic instances. On the theory side, the reduction to the Riccati form requires that the chosen complementarity function exactly reproduces the KKT conditions in the chosen Banach space and that its semi-smoothness holds without extra constraint qualifications. The stress-test note flags this correctly; if the paper does not state and verify the needed regularity conditions for general convex constraint sets, the convergence guarantee rests on an assumption that may not be automatic. The work is aimed at researchers in optimal control who already use Riccati-based methods and want to handle state or control inequalities without first discretizing. A reader who cares about function-space formulations or semi-smooth Newton in infinite dimensions will get something concrete from it. It is coherent enough and technically grounded enough to deserve a serious referee rather than a desk reject, even though the numerical section will probably need expansion and the regularity hypotheses will need to be spelled out clearly.

Referee Report

2 major / 2 minor

Summary. The paper proposes an indirect method for constrained optimal control problems of continuous-time linear systems. The KKT conditions are embedded into a non-smooth complementarity function on a Banach space of trajectories, yielding a root-finding problem that is solved by a semi-smooth Newton method. The central technical claim is that each Newton step reduces exactly to the solution of a modified differential Riccati equation whose quadratic cost matrices are reweighted by the current constraint multipliers. Numerical simulations are reported to exhibit superlinear convergence up to the tolerance of the underlying ODE solver.

Significance. If the reduction to the reweighted Riccati equation is rigorously justified and the complementarity embedding is exact, the approach would provide a discretization-free indirect solver for constrained linear-quadratic OCPs that reuses existing Riccati solvers at each iteration. This could be valuable for high-accuracy trajectory optimization where mesh refinement is undesirable. The absence of free parameters in the derivation and the explicit link to a standard Riccati structure are positive features.

major comments (2)

[Abstract and the complementarity reformulation section] The claim that the zero set of the non-smooth complementarity operator coincides exactly with the KKT points in Banach space (and that the operator is semi-smooth) is load-bearing for both the correctness of the Newton linearization and the superlinear convergence result. No constraint qualifications, regularity assumptions on the cost functionals, or hypotheses on the closed convex constraint sets are stated; these properties do not hold automatically for arbitrary data and must be supplied before the reduction to the modified Riccati equation can be asserted.
[Numerical simulations] Numerical simulations section: the abstract states that the method converges superlinearly up to ODE tolerance, yet no concrete problem instances, state/control dimensions, constraint types, baseline comparisons (e.g., against direct collocation or standard Riccati-based solvers), error bars, or quantitative tables of iteration counts versus residual are supplied. Without these details the empirical support for the central claim remains unverifiable.

minor comments (2)

[Preliminaries] The precise definition of the Banach space (e.g., whether it is L^2 or a Sobolev space) and the precise form of the complementarity function should be written explicitly with all function-space norms.
[Newton step derivation] The reweighting rule for the cost matrices in the modified Riccati equation should be displayed as a numbered equation with an explicit dependence on the current multiplier trajectory.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the recommendation for major revision. We address each major comment below and will incorporate the necessary changes in the revised manuscript.

read point-by-point responses

Referee: [Abstract and the complementarity reformulation section] The claim that the zero set of the non-smooth complementarity operator coincides exactly with the KKT points in Banach space (and that the operator is semi-smooth) is load-bearing for both the correctness of the Newton linearization and the superlinear convergence result. No constraint qualifications, regularity assumptions on the cost functionals, or hypotheses on the closed convex constraint sets are stated; these properties do not hold automatically for arbitrary data and must be supplied before the reduction to the modified Riccati equation can be asserted.

Authors: We agree that explicit statements of the required assumptions are essential for the validity of the claims. In the revised version, we will add a dedicated subsection in the complementarity reformulation section detailing the constraint qualifications (such as a Slater-type condition for the convex constraint sets) and regularity assumptions on the cost functional (strong convexity of the quadratic cost) that guarantee the semi-smoothness of the operator and the exact coincidence of its zero set with the KKT points. This will rigorously support the reduction to the reweighted differential Riccati equation. revision: yes
Referee: [Numerical simulations] Numerical simulations section: the abstract states that the method converges superlinearly up to ODE tolerance, yet no concrete problem instances, state/control dimensions, constraint types, baseline comparisons (e.g., against direct collocation or standard Riccati-based solvers), error bars, or quantitative tables of iteration counts versus residual are supplied. Without these details the empirical support for the central claim remains unverifiable.

Authors: We acknowledge the need for more detailed empirical evidence. The revised manuscript will include an expanded numerical simulations section with specific problem instances, including state and control dimensions, types of constraints, quantitative tables showing iteration counts and residuals, comparisons with baseline methods such as direct collocation and standard Riccati solvers, and error bars from repeated simulations. This will substantiate the superlinear convergence claim up to ODE solver tolerance. revision: yes

Circularity Check

0 steps flagged

No circularity: Newton update derived directly from complementarity reformulation

full rationale

The paper's central derivation applies a non-smooth Newton step to a root-finding problem obtained by embedding the KKT conditions via a complementarity operator in Banach space. This produces a linearization that reduces to a modified differential Riccati equation with multiplier-dependent reweighting of the cost matrices. No step reduces to a fitted parameter, self-definition, or load-bearing self-citation; the reduction follows from standard semi-smooth Newton theory applied to the reformulated system without presupposing the Riccati form as input. The result is self-contained and independent of the target claim.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is limited to elements explicitly invoked there; the method rests on standard optimal-control KKT theory and non-smooth analysis without introducing new fitted constants or postulated entities.

axioms (1)

domain assumption KKT conditions for the constrained OCP can be embedded in a non-smooth complementarity function suitable for Banach-space root-finding.
This embedding is presented as the enabling step that converts the OCP into a root-finding problem.

pith-pipeline@v0.9.0 · 5413 in / 1380 out tokens · 55405 ms · 2026-05-11T02:00:35.801696+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

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Liao-McPherson, D. and Kolmanovsky, I. (2020). Fbstab: A proximally stabilized semismooth algorithm for con- vex quadratic programming.Automatica, 113, 108801. Løwenstein, K.F., Bernardini, D., and Patrinos, P. (2024). Qpalm-ocp: A newton-type proximal augmented la- grangian solver tailored for quadratic programs arising in model predictive control.IEEE C...

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and Plitt, K.J

Bock, H.G. and Plitt, K.J. (1984). A multiple shooting al- gorithm for direct solution of optimal control problems. IF AC Proceedings Volumes, 17(2), 1603–1608

work page 1984

[2] [2]

Bordalba, R., Schoels, T., Ros, L., Porta, J.M., and Diehl, M. (2022). Direct collocation methods for trajectory Fig. 2.||F(z)|| 2 versus number of solver iterations for both min and Fischer-Burmeister NCP functions. Regular- ization parameterδwas chosen experimentally. For min function:δ= 1×10 −1. For Fischer-Burmeister function:δ= 1×10 −2 . optimization...

work page 2022

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(2018).Applied optimal control: optimiza- tion, estimation and control

Bryson, A.E. (2018).Applied optimal control: optimiza- tion, estimation and control. Routledge

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and Gerdts, M

Chen, J. and Gerdts, M. (2011). Numerical solution of control-state constrained optimal control problems with an inexact smoothing newton method.IMA journal of numerical analysis, 31(4), 1598–1624

work page 2011

[5] [5]

Chow, G.C. (1976). Control methods for macroeconomic policy analysis.The American Economic Review, 66(2), 340–345

work page 1976

[6] [6]

Petersen, C. (2022). Efficient trajectory optimization for constrained spacecraft attitude maneuvers.Journal of Guidance, Control, and Dynamics, 45(4), 638–650

work page 2022

[7] [7]

Gerdts, M. (2008). Global convergence of a nonsmooth newton method for control-state constrained optimal control problems.SIAM Journal on Optimization, 19(1), 326–350

work page 2008

[8] [8]

Gong, Q., Fahroo, F., and Ross, I.M. (2008). Spectral al- gorithm for pseudospectral methods in optimal control. Journal of Guidance, Control, and Dynamics, 31(3), 460–471

work page 2008

[9] [9]

Hauser, J. (2002). A projection operator approach to the optimization of trajectory functionals.IF AC Proceed- ings Volumes, 35(1), 377–382

work page 2002

[10] [10]

and Saccon, A

Hauser, J. and Saccon, A. (2006). A barrier function method for the optimization of trajectory functionals with constraints.Proceedings of the IEEE Conference on Decision and Control, 864–869

work page 2006

[11] [11]

Hindmarsh, A.C., Brown, P.N., Grant, K.E., Lee, S.L., Ser- ban, R., Shumaker, D.E., and Woodward, C.S. (2005). Sundials: Suite of nonlinear and differential/algebraic equation solvers.ACM Transactions on Mathematical Software (TOMS), 31(3), 363–396. Hinterm¨ uller, M. and Stadler, G. (2003). A semi-smooth newton method for constrained linear-quadratic con...

work page 2005

[12] [12]

Kelly, M. (2017). An introduction to trajectory optimiza- tion: How to do your own direct collocation.SIAM review, 59(4), 849–904

work page 2017

[13] [13]

Kopp, R.E. (1962). Pontryagin maximum principle. In Mathematics in Science and Engineering, volume 5, 255–279. Elsevier

work page 1962

[14] [14]

Liao-McPherson, D., Huang, M., and Kolmanovsky, I. (2018). A regularized and smoothed Fischer–Burmeister method for quadratic programming with applications to model predictive control.IEEE Transactions on Automatic Control, 64(7), 2937–2944

work page 2018

[15] [15]

and Kolmanovsky, I

Liao-McPherson, D. and Kolmanovsky, I. (2020). Fbstab: A proximally stabilized semismooth algorithm for con- vex quadratic programming.Automatica, 113, 108801. Løwenstein, K.F., Bernardini, D., and Patrinos, P. (2024). Qpalm-ocp: A newton-type proximal augmented la- grangian solver tailored for quadratic programs arising in model predictive control.IEEE C...

work page 2020

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(1988).Numerical solution of optimal control problems with state constraints by sequential quadratic programming in function space

Machielsen, K.C. (1988).Numerical solution of optimal control problems with state constraints by sequential quadratic programming in function space. Centrum voor wiskunde en informatica

work page 1988

[17] [17]

Malanowski, K. (2003). On normality of lagrange mul- tipliers for state constrained optimal control problems. Optimization, 52(1), 75–91

work page 2003

[18] [18]

Mifflin, R. (1977). Semismooth and semiconvex functions in constrained optimization.SIAM Journal on Control and Optimization, 15(6), 959–972

work page 1977

[19] [19]

Rao, A.V. (2014). Trajectory optimization: a survey. In Optimization and optimal control in automotive sys- tems, 3–21. Springer

work page 2014

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Schiela, A. (2009). Barrier methods for optimal control problems with state constraints.SIAM Journal on Optimization, 20(2), 1002–1031

work page 2009

[21] [21]

and G¨ unther, A

Schiela, A. and G¨ unther, A. (2011). An interior point algorithm with inexact step computation in function space for state constrained optimal control.Numerische Mathematik, 119(2), 373–407

work page 2011

[22] [22]

and Wollner, W

Schiela, A. and Wollner, W. (2011). Barrier methods for optimal control problems with convex nonlinear gradi- ent state constraints.SIAM Journal on Optimization, 21(1), 269–286

work page 2011

[23] [23]

Shao, J., Naris, M., Hauser, J., and Nicotra, M.M. (2024). Solving quantum optimal control problems us- ing projection-operator-based newton steps.Physical Review A, 109(1), 012609

work page 2024

[24] [24]

Skibik, T., Liao-McPherson, D., and Nicotra, M.M. (2022). A terminal set feasibility governor for linear model predictive control.IEEE Transactions on Automatic Control, 68(8), 5089–5095

work page 2022

[25] [25]

and Qi, L

Sun, D. and Qi, L. (1999). On ncp-functions.Computa- tional Optimization and Applications, 13(1), 201–220

work page 1999