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arxiv: 2607.02339 · v1 · pith:IQPC3OWDnew · submitted 2026-07-02 · 🧮 math.OC

Sensitivity Analysis and Robust Optimal Control for Coupled Evolution Inclusions with State-Dependent Maximal Monotone Operators

Jinsheng Du , Boris Mordukhovich , Shengda Zeng This is my paper

Pith reviewed 2026-07-03 08:18 UTC · model grok-4.3

classification 🧮 math.OC
keywords evolution inclusionsmaximal monotone operatorsoptimal controlrobust controlsensitivity analysisBolza problemparameter uncertainty
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The pith

For Bolza-type optimization over solutions of coupled evolution inclusions with state-dependent maximal monotone operators, optimal pairs exist, the value function is continuous, and the optimal-solution map is upper semicontinuous.

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

The paper studies a class of strongly coupled nonsmooth systems with a semilinear evolution inclusion and a differential inclusion governed by state-dependent maximal monotone operators. It first establishes well-posedness, compactness, and Painlevé-Kuratowski continuity of the parameterized solution map. Building on this, it proves existence of optimal pairs for Bolza-type optimization, continuity of the value function, and upper semicontinuity of the optimal-solution map. It further shows existence results for fixed-parameter optimal control, simultaneous control-parameter design, min-max robust control, and Hurwicz-type compromise control under parameter uncertainty.

Core claim

The parameterized solution map for these coupled systems has well-posedness, compactness, and Painlevé-Kuratowski continuity properties. For Bolza-type optimization over the solution set, optimal pairs exist, the value function has continuity properties, and the optimal-solution map is upper semicontinuous. Existence results hold for fixed-parameter optimal control, simultaneous control-parameter design, min-max robust control, and Hurwicz-type compromise control under parameter uncertainty.

What carries the argument

The parameterized solution map of the coupled semilinear evolution inclusion and differential inclusion with state-dependent maximal monotone operators, which carries the well-posedness and continuity properties used for all optimization results.

If this is right

  • Optimal pairs exist for the Bolza-type optimization problem over the solution set.
  • The value function of the optimization problem is continuous.
  • The optimal-solution map is upper semicontinuous.
  • Existence holds for min-max robust control problems under parameter uncertainty.
  • Existence holds for Hurwicz-type compromise control under parameter uncertainty.

Where Pith is reading between the lines

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

  • These continuity properties may allow for numerical approximation schemes in practice.
  • The results could extend to other classes of nonsmooth dynamical systems beyond sweeping processes.
  • Parameter uncertainty handling suggests applications in uncertain environments like robotics or economics.

Load-bearing premise

The parameterized solution map possesses well-posedness, compactness, and Painlevé-Kuratowski continuity properties.

What would settle it

Finding a specific instance of the coupled system where no optimal pair exists for the Bolza problem or where the optimal-solution map fails to be upper semicontinuous despite the assumptions on the operators.

Figures

Figures reproduced from arXiv: 2607.02339 by Boris Mordukhovich, Jinsheng Du, Shengda Zeng.

Figure 1
Figure 1. Figure 1: Baseline simulation for Example 6.1: the ODE state x(t) (blue solid), sweeping state u(t) (red dash-dotted), and moving boundary β(t, u(t)) (yellow dashed). The sweeping trajectory alternates between boundary contact and interior motion while remaining feasible. In [PITH_FULL_IMAGE:figures/full_fig_p021_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Stability experiment for Example 6.1. Left: Relative error En versus ϵn ∈ [10−5 , 10−1 ] on log–log axes, together with a reference line of slope 1. Center and right: Perturbed states for ϵ ∈ {0.15, 0.08, 0.02} and the nominal trajectory ϵ = 0; the insets show deviations near active constraint intervals. III. Sensitivity of the Trajectory Cost for Example 6.1 For the same perturbation family, consider the … view at source ↗
Figure 3
Figure 3. Figure 3: shows the computed control and state response. The control remains within [−1, 1] and primarily attenuates x(t), with a smaller effect on u(t). The projected trajectory remains feasible. The objective decreases from Jnom ≈ 1.5293 to J ∗ ≈ 1.0121, a reduction of approximately 33.82%. 0 1 2 3 4 5 6 Time t -1 -0.5 0 0.5 1 C o ntrol In p ut w(t) Optimal Control Strategy Constraints '1 Optimal Control w $(t) 0 … view at source ↗
Figure 4
Figure 4. Figure 4: Joint design–control experiment for Example 6.1. Left: Optimized con￾trol. Center: Nominal and jointly optimized trajectories. Right: Feasibility relative to βη∗ (t, u∗ ). The computed values are ζ ∗ ≈ 0, η ∗ ≈ −0.15, and J ∗ ≈ 0.6945 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Robust-control experiment for Example 6.1. Left: Cost versus ζ at η = 0.15. Center: Cost versus η at ζ = 0.25. Right: Trajectories for the computed adverse pair (0.25, 0.15). Robust optimization reduces the adverse cost from approximately 1.8726 to 1.7718. VII. Hurwicz Compromise Control for Example 6.1 Finally, for α ∈ {0, 0.1, 0.5, 0.9, 1}, we minimize Hα(w) := α inf (ζ,η)∈P J (w, ζ, η) + (1 − α) sup (ζ,… view at source ↗
Figure 6
Figure 6. Figure 6: Hurwicz-control experiment for Example 6.1. Left: Best-case, worst-case, and Hurwicz values for α ∈ {0, 0.1, 0.5, 0.9, 1}. Right: Controls at representative optimism levels; the profiles for α = 0.9 and α = 1 are nearly indistinguishable. so H(Be)(i) holds with θ(t) = 1.2t and λ = 0.3 < 1. The standard hypomonotonicity estimate for normal cones to Lipschitz translations of a ball gives ⟨v1 − v2, u1 − u2⟩ ≥… view at source ↗
Figure 7
Figure 7. Figure 7: Baseline simulation for Example 6.2. Left: ODE states x1, x2 (dashed) and sweeping states u1, u2, u3 (solid). Right: Sweeping trajectory u(t), initial ball C(0, u(0)), terminal ball C(T, u(T)), and endpoint markers. 10!4 10!3 10!2 10!1 Perturbation Magnitude 0n 10!3 10!2 10!1 R elativ e Error En Stability Analysis Numerical Error En O(0) Reference 0 1 2 3 4 5 6 Time t -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x1(… view at source ↗
Figure 8
Figure 8. Figure 8: Stability experiment for Example 6.2. Top left: Relative error En versus ϵn ∈ [10−4 , 10−1 ] on log–log axes, with a slope-one reference. Other panels: Nominal and perturbed components for ϵ ∈ {0.15, 0.08, 0.02}; the insets show intervals of rapid variation [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: shows the computed control and trajectories. The objective decreases from Jnom ≈ 9.1690 to J ∗ ≈ 8.1512, a reduction of approximately 11.10%. Most of the improvement comes from reducing ∥x(t)∥; the effect on ∥u(t)∥ is indirect through the coupling. 0 1 2 3 4 5 6 Time t -1 -0.5 0 0.5 1 C o ntrol C o m p o n e nts Optimal Vector Control w $(t) w $ 1(t) w $ 2(t) Bounds '1 0 1 2 3 4 5 6 Time t 0 0.5 1 1.5 2 2.… view at source ↗
Figure 10
Figure 10. Figure 10: Joint design–control experiment for the reduced vector benchmark. Left: Op￾timized control components. Center: Uncontrolled and optimized state norms. Right: Sweeping components and their moving lower barriers. The computed values are Jnom ≈ 3.0544, J ∗ ≈ 1.4798, ζ ∗ ≈ 0, and η ∗ = −0.15. The uncertainty set is [0, 0.5] × [−0.2, 0.2] [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: reports a direct discrete min–max computation with a blockwise-constant open-loop control. The computed inner maximizer is (ζwc, ηwc) = (0, 0.2). At this pair, the nominal-controller cost is Jnom,wc ≈ 11.1412, whereas the robust controller gives Jrob,wc ≈ 10.5764, a reduction of approximately 5.07%. The cost curves cross near ζ = 0.03–0.04, after which the robust controller performs better. In this benchm… view at source ↗
read the original abstract

We consider a class of strongly coupled nonsmooth systems consisting of a semilinear evolution inclusion and a differential inclusion governed by state-dependent maximal monotone operators. Our main contributions are fourfold. First, we collect the well-posedness, compactness, and Painlev\'e--Kuratowski continuity properties of the parameterized solution map required for the subsequent optimization analysis. Second, for Bolza-type optimization over the solution set, we prove the existence of optimal pairs, establish continuity properties of the value function, and derive upper semicontinuity of the optimal-solution map. Third, we study fixed-parameter optimal control, simultaneous control-parameter design, min--max robust control, and Hurwicz-type compromise control under parameter uncertainty, and we establish existence results for each formulation. Fourth, we report numerical experiments for sweeping-type systems that illustrate the sensitivity and robustness phenomena predicted by the theory.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript considers strongly coupled nonsmooth systems consisting of a semilinear evolution inclusion and a differential inclusion governed by state-dependent maximal monotone operators. The authors collect well-posedness, compactness, and Painlevé-Kuratowski continuity properties of the parameterized solution map. For Bolza-type optimization over the solution set they prove existence of optimal pairs, continuity properties of the value function, and upper semicontinuity of the optimal-solution map. They establish existence results for fixed-parameter optimal control, simultaneous control-parameter design, min-max robust control, and Hurwicz-type compromise control under parameter uncertainty. Numerical experiments for sweeping-type systems illustrate the predicted sensitivity and robustness phenomena.

Significance. If the well-posedness and continuity properties of the solution map hold, the work supplies a coherent framework for sensitivity analysis and robust optimal control in nonsmooth coupled systems with state-dependent maximal monotone operators. The systematic treatment of four distinct control formulations under parameter uncertainty, together with the numerical illustrations for sweeping processes, adds concrete value to the literature on set-valued optimal control.

minor comments (3)
  1. [Abstract / §2] The abstract states that the well-posedness, compactness, and Painlevé-Kuratowski continuity properties are 'collected'; a short paragraph in §2 or §3 clarifying which of these properties are proved anew versus invoked from prior literature would improve readability.
  2. [Numerical experiments] In the numerical section, the reported trajectories for the sweeping-type systems should include a brief statement of the discretization scheme and step-size used, so that the observed sensitivity phenomena can be reproduced from the given data.
  3. [Introduction] Notation for the set-valued map and the parameter dependence is introduced gradually; a consolidated table of symbols at the end of the introduction would aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the constructive and positive assessment of our manuscript, including the recognition of its contributions to well-posedness, sensitivity analysis, and the four robust control formulations. We note the recommendation for minor revision and will address any editorial or minor points in the revised version.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper first collects well-posedness, compactness, and Painlevé-Kuratowski continuity of the parameterized solution map for the coupled semilinear evolution inclusion and state-dependent maximal monotone differential inclusion, relying on standard properties of maximal monotone operators and set-valued analysis. All subsequent Bolza optimization results, value-function continuity, upper semicontinuity of the optimal-solution map, and existence theorems for the four control formulations (fixed-parameter, simultaneous design, min-max robust, Hurwicz compromise) are derived conditionally from these map properties using standard arguments from optimal control and variational analysis. No load-bearing step reduces by construction to fitted inputs, self-definitional relations, or self-citation chains that render the central claims equivalent to their premises; the derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no information supplied on free parameters, background axioms, or new postulated entities.

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Reference graph

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