Robust Control with Gradient Uncertainty

Qian Qi

REVIEW 4 major objections 4 minor 12 references

Robust Control with Gradient Uncertainty

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A robust-control framework treats the value function gradient itself as an adversarial target, yielding a well-posed PDE and showing that any gradient uncertainty destroys the quadratic structure of linear-quadratic control.

desk verdict A genuinely new formulation of robust control that targets gradient uncertainty, but the central game is never defined and the key LQ proof is flawed; the idea is worth a serious referee, not acceptance as-is. read the letter →

arxiv 2507.15082 v1 pith:MJDWCACU submitted 2025-07-20 cs.LG cs.AImath.OC

Qian Qi This is my paper · ORCID

classification cs.LGcs.AImath.OC MSC 49L2549N7093E20

keywords robustcontrolgradientuncertaintyHamilton-Jacobi-Bellman-Isaacsequationviscositysolutionslinear-quadraticperturbationanalysisactor-criticreinforcementlearningzero-sumdynamicgames

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks how a controller should act when it is uncertain not only about the model driving its system, but also about the marginal value of its own state. It answers by adding a second adversarial perturbation, chosen pointwise from a ball of radius $\epsilon$, to the gradient of the value function inside a standard robust zero-sum game. That leads to a new dynamic programming equation, the GU-HJBI equation, whose viscosity solutions are shown to exist and be unique under uniform ellipticity. In the linear-quadratic case the paper proves the classical quadratic value function ansatz fails for any $\epsilon>0$, and a perturbation expansion shows the correction is non-polynomial, making the optimal control law nonlinear. The paper then converts the theory into a practical regularizer for actor-critic reinforcement learning and reports that it stabilizes training on a pendulum task.

What carries the argument

The load-bearing object is the inner adversary's maximization in the reduced GU-HJBI equation, namely $G(x,u,p)=\sup_{\lVert\delta\rVert\le\epsilon}\big[(p+\delta)^T f(x,u)+\frac{\eta}{2}\lVert\sigma(x,u)^T(p+\delta)\rVert^2\big]$. This combines the drift sensitivity vector $v=f+\eta\sigma\sigma^T p$ with a norm penalty, and its expansion $G=p^T f+\frac{\eta}{2}\lVert\sigma^T p\rVert^2+\epsilon\lVert v\rVert+O(\epsilon^2)$ is what breaks the quadratic ansatz: the norm term is not a polynomial in $x$. The comparison principle is carried by the doubling-of-variables argument, with uniform ellipticity controlling the Hessian terms, and the perturbation analysis is carried by the linearized Lyapunov operator $\mathcal{L}_{\text{lin}}[W]=(\nabla W)^T A_{\text{eff},0}x+\frac{1}{2}\mathrm{Tr}(\Sigma\Sigma^T D^2 W)$.

What would settle it

Compute, on a fine grid, the viscosity solution of (11) for the scalar LQ problem with $\epsilon>0$ and compare it with the dynamic-programming value of an explicitly discretized version of the proposed zero-sum game; any discrepancy would show that equation (10) is not the game's dynamic programming equation. As a lighter check, test Proposition 5.2 by evaluating the second derivative of the numerically computed value function: a constant Hessian away from the origin would refute the claimed non-quadratic structure.

Watch

Extended reading notes

Core claim

The central claim is that ambiguity about the value function gradient is a distinct, tractable form of uncertainty that changes the structure of robust control. The paper formulates a zero-sum game in which Nature chooses both a drift perturbation $h$ and a gradient perturbation $\delta$, and argues that its dynamic programming form is the GU-HJBI equation (10). The paper proves a comparison principle and existence for this equation's viscosity solutions under a uniform ellipticity condition (Theorems 4.3 and 4.5). In the linear-quadratic setting it proves Proposition 5.2: for any $\epsilon>0$ with non-degenerate data, the value function cannot be quadratic, because the perturbation Hamiltonian contributes a non-polynomial source term of the form $\epsilon \lVert A_{\text{eff},0}x\rVert$. The first-order correction $V_1$ then solves a linear PDE whose Feynman-Kac solution is generically non-polynomial, and the control correction $u_1 = -R^{-1}B^{T}\nabla V_1$ is consequently nonlinear.

Load-bearing premise

The load-bearing premise is that equation (10) is the dynamic programming equation of a well-defined zero-sum game with pointwise gradient perturbations, yet the paper never formally defines the game, its strategies, or a dynamic programming principle, so without that link the equation lacks the game-theoretic interpretation that motivates it.

Editorial extensions

If this is right

Even in the simplest linear-quadratic control problem, any nonzero gradient uncertainty $\epsilon$ makes the value function non-quadratic, so Riccati-equation methods cannot directly solve it; approximations must handle the non-polynomial correction.
For small $\epsilon$, the value function correction solves a linear PDE with a norm source term, giving a concrete, numerically solvable path to compute robust policies beyond the standard LQ regulator.
In reinforcement learning, penalizing the actor with the norm of the drift sensitivity $f+\eta\sigma\sigma^T\nabla_x Q$ stabilizes training and reduces variance, as demonstrated in the Pendulum-v1 experiments.
Changing the geometry of the gradient-uncertainty set to $\ell^2$, $\ell^\infty$, or a Mahalanobis ellipsoid changes the penalty to the corresponding dual norm, giving modelers a way to encode prior knowledge about gradient errors.
The framework extends the robust-control interpretation beyond relative entropy: the gradient perturbation attacks the agent's internal valuation, not the external model.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves the game-theoretic underpinning of (10) implicit; supplying a rigorous dynamic programming principle would let the GU-HJBI equation serve as the definition of gradient-uncertainty robustness in its own right.
The non-polynomial norm source suggests value functions under gradient uncertainty can have kinks; numerical solvers and neural approximators that include nonsmooth features may track the solution better than smooth-only bases.
GURAC's penalty is a principled relative of gradient clipping and trust-region updates; a natural test is whether the regularization weight $\lambda_R$ should scale with an estimate of the critic's gradient error rather than being a fixed hyperparameter.
The same construction applies to financial hedging where Greeks are approximate: adversarial perturbations of the Greeks would lead to nonlinear hedging strategies of the same kind.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Qi's paper has a genuinely novel idea: treat the agent's uncertainty about the value-function gradient as an adversary that perturbs the gradient inside the HJBI Hamiltonian. That produces a new PDE (GU-HJBI) and a plausible story for why function-approximation RL is unstable. The paper also does some things well: the comparison principle under uniform ellipticity is a standard adaptation but carefully stated, the numerical solutions of the linearized first-order PDE look consistent, and the GURAC algorithm is a concrete, sensible translation of the idea to actor-critic.

The problems are all in the load-bearing layer. First, the paper never defines the zero-sum game it claims to solve. Eq. (10) is asserted as the "dynamic programming form" but there is no payoff functional, no admissible strategy spaces, no information structure for the gradient perturbation, and no proof of a dynamic programming principle. Theorem 4.5 even refers to "the value function defined by the underlying game" when no game has been defined. This is not cosmetic: with the gradient perturbation entering as a shift of the value gradient inside the supremum, the equation is not a standard differential-game DPP, and the interpretation of the perturbation as gradient uncertainty depends on a game that isn't specified. Second, the proof of Proposition 5.2 substitutes the control that minimizes only the quadratic part of the objective; that is not the infimizing control, so the equality in (15) is an inequality, and the contradiction doesn't follow. The claim may be true, but the proof as written is wrong. Third, the numerical studies solve the approximate equation (13), not the full GU-HJBI, so they validate the perturbation expansion, not the central PDE. The RL experiments are described in enough detail to be plausible but not enough to be reproduced without code.

So: interesting idea, honest exposition, but the formal core isn't established. The missing DPP and the flawed LQ proof are fixable, but they are central. I'd send it to review because the idea deserves careful referee time, but I'd expect major revision or rejection in current form. If the author can define a proper game and prove a DPP (or state the PDE as the primary object and demote the game interpretation), and fix the LQ proof, the core could be salvageable.

Referee Report

4 major / 4 minor

Summary. The paper proposes a robust control framework in which an adversary perturbs both the system dynamics and the gradient of the value function, leading to a new PDE called the Hamilton-Jacobi-Bellman-Isaacs Equation with Gradient Uncertainty (GU-HJBI). After deriving this equation heuristically, the paper claims well-posedness via viscosity-comparison and Perron-type existence theorems, proves that the quadratic value-function ansatz fails in the linear-quadratic case, gives a formal perturbation expansion and numerical solutions, and finally proposes a GURAC-TD3 reinforcement-learning algorithm with an empirical study on Pendulum-v1.

Significance. The problem is well motivated, and the GU-HJBI equation is an interesting object: if the game-theoretic identification, well-posedness, and LQ non-quadraticity were rigorously established, the paper would be a meaningful extension of robust control and a useful conceptual bridge to RL. The manuscript is self-contained in notation and includes concrete numerical studies (Section 6) and an empirical comparison of GURAC-TD3 against TD3 (Section 8), which are strengths; the derivations are also largely self-contained rather than circular. However, the load-bearing derivation of the dynamic programming equation is asserted rather than proved, the LQ proof applies only to a substituted control in an approximate equation, and the well-posedness theorems do not cover the LQ setting used for the main structural claim. The contribution is therefore presently suggestive rather than established.

major comments (4)

[Section 3, Eq. (10)] The central PDE is introduced as the 'dynamic programming form' of a zero-sum game in which the adversary chooses both h and δ, but no game is actually defined: there is no payoff functional, no admissible strategy space for δ (state-dependent? adapted? with full knowledge of ∇V?), and no dynamic programming principle is stated or proved. Because the gradient perturbation enters the Hamiltonian as p+δ = ∇V(x)+δ, V itself appears inside the supremum, so the standard differential-game DPP used for Eqs. (5)-(8) cannot be invoked by analogy. Theorem 4.5 then asserts, without derivation, that 'the value function defined by the underlying game is this unique solution.' This is the load-bearing link between the game interpretation and all downstream results (Eq. (13), Section 5, Section 8), and it remains unsupported.
[Section 5.1, Proposition 5.2 and Eq. (15)] The proof substitutes u*(x) = -R^{-1}B^T P x, which minimizes only the quadratic part of the objective, into the approximate equation, and observes that the result contains ϵ||Mx||. This does not establish the proposition, because the infimum in Eq. (11) ranges over all controls; a nonlinear control could in principle compensate the norm term, and the proof gives no argument that it cannot. Moreover, Eq. (15) is the first-order approximation (13), not the full GU-HJBI equation (11), so the claim that the quadratic ansatz fails for the full equation for any ϵ>0 is not proved.
[Section 4, Theorems 4.3 and 4.5; Appendix C] The comparison-principle proof is only a sketch: the key continuity step G(x,u,px)-G(y,u,py)→0 is asserted rather than proved, and the treatment of the semifinal jet estimates is abbreviated. More importantly, Theorem 4.5 requires bounded viscosity sub-/supersolutions, while Assumption 2.1's growth condition on L is violated by the quadratic LQ cost x^T Qx used in Section 5; constant barrier functions cannot control an unbounded running cost on all of R^n. In addition, Section 5 sets U = R^k, whereas Assumption 2.1 requires a compact control set, so the viscosity results of Section 4 are not formally applicable to the LQ case on which the main structural claim rests.
[Section 5.2, Eq. (18) and Appendix G] The perturbation expansion is formal. The source term ||Aeff,0 x|| is not differentiable at x = 0, so the Implicit Function Theorem justification in Appendix G cannot hold in the C^2 or Hölder spaces where the linearized operator Llin is elliptic; likewise u1(x) = -R^{-1}B^T∇V1(x) requires differentiability of V1, which has not been established for a PDE with a non-smooth source. Consequently, the conclusion that the optimal control law is nonlinear, and the numerical solutions of Eq. (20), should be regarded as heuristic rather than proven.

minor comments (4)

[Section 6, Figure 1] The high-frequency oscillations in the reported control law are attributed to numerical artifacts; a convergence study under mesh refinement would materially strengthen the claim that the nonlinear character is genuine rather than an artifact.
[Section 8, Algorithm 1] The GURAC penalty requires estimates of the environment dynamics f and diffusion σ, and the paper states that these are 'assumed known' for Pendulum-v1; this is a strong modeling assumption that should be stated as a limitation of the method, not as an implementation detail.
[Throughout] There are repeated notation slips, e.g., 'D2V V' in Appendix E and in the definition of the generator in Section 2.1; these should be corrected to D^2 V or to a single consistent symbol for the Hessian argument.
[Throughout] No code, data, or hyperparameter ranges are provided for the GURAC experiments, so the empirical claims in Section 8.4 cannot be independently reproduced or checked for statistical sensitivity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the GU-HJBI derivation and LQ analysis are self-contained; the undefined-game identification is a rigor gap, not a circular reduction.

full rationale

The paper's central derivation is self-contained rather than circular. The new GU-HJBI equation (10) is introduced as a PDE, then Proposition 3.2 reduces it algebraically to (11) by completing the square in h; Proposition 3.3 obtains the small-epsilon expansion from convexity and the compact ball, with no fitted parameter being renamed as a prediction. The LQ analysis in Section 5 works directly from that expansion: Proposition 5.2 shows no quadratic solution can satisfy the approximate equation when epsilon > 0 because the norm term ||Mx|| is non-polynomial, and the first-order correction V1 solves a linear Lyapunov PDE whose non-polynomial source is the same norm by construction. Section 6 solves those same PDEs on grids, so it is a numerical consistency check rather than an external validation; it does not fit parameters to data and then call the fit a prediction. The three self-citations (Qi 2025a,b,c) appear only as related-work pointers for function approximation and PDE solvers; none carries the comparison principle, existence proof, LQ result, or the GU-HJBI equation, so they are not load-bearing. The main weakness is a completeness gap, not circularity: the zero-sum game behind Eq. (10) is never formally defined, no dynamic programming principle is proved, and Theorem 4.5's sketch asserts 'It can also be shown that the value function defined by the underlying game is this unique solution' without specifying that game. That assertion is unsubstantiated, but it does not reduce Eq. (10) to its own inputs by construction. Consequently no circular step is exhibited.

Assumptions & free parameters 3 free parameters · 8 assumptions · 1 invented entities

The paper's central equation rests on user-chosen parameters (epsilon, eta) and on modeling assumptions about the adversary's knowledge and the validity of the dynamic programming principle. No free parameters are fitted to data to make the theory match a target result; the parameters are inputs.

free parameters (3)

epsilon (gradient uncertainty radius) = 0.5 (1D example), 0.1 (2D), lambda_R=0.01 in GURAC
User-chosen magnitude of the adversarial gradient perturbation; not derived from data or theory.
eta (model uncertainty parameter) = 0.2 (1D), 0.1 (2D and GURAC)
User-chosen strength of the drift perturbation penalty; standard in robust control, chosen by hand.
lambda_R (GURAC regularization weight) = 0.01
Tunable hyperparameter in the RL algorithm, corresponding to epsilon in the theory.

assumptions (8)

standard math Viscosity solution theory and Perron's method for existence and uniqueness
Used in Section 4 as background for nonlinear second-order PDEs.
standard math Ito calculus and Girsanov theorem for the drift perturbation h
Used in Section 2.2 to derive the standard HJBI penalty.
domain assumption Definition 3.1: gradient perturbations lie in the l2 ball of radius epsilon
Modeling choice for the uncertainty set; alternative geometries are considered in Section 7.
domain assumption Assumption 4.2: uniform ellipticity sigma sigma^T >= nu I
Stated as critical for the comparison principle proof.
domain assumption Assumption 5.1: stabilizability and detectability for the LQ problem
Standard assumptions ensuring a unique stabilizing Riccati solution.
ad hoc to paper The dynamic programming principle for Eq. (10) holds
No proof is provided that the value function of the proposed game satisfies Eq. (10).
ad hoc to paper The adversary observes the state and nominal gradient and chooses delta pointwise
Core modeling assumption of the paper; not derived from physical or statistical principles.
ad hoc to paper For GURAC, the environment dynamics f and diffusion sigma are known for computing the penalty
Assumed in Algorithm 1, Step 17; unrealistic for fully model-free RL.

invented entities (1)

Gradient perturbation delta
purpose: Models the agent's uncertainty about its own value-function gradient and gives the adversary an additional lever.
A theoretical construct; no direct measurable prediction outside the model. It is not a physical quantity.

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Cite this review

Pith. "Pith review of Robust Control with Gradient Uncertainty." pith.science (2026). https://pith.science/paper/MJDWCACU

@misc{pith2026250715082,
  author       = {Pith},
  title        = {Pith review of: Robust Control with Gradient Uncertainty},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MJDWCACU}},
  note         = {Machine review of arXiv:2507.15082}
}

read the original abstract

We introduce a novel extension to robust control theory that explicitly addresses uncertainty in the value function's gradient, a form of uncertainty endemic to applications like reinforcement learning where value functions are approximated. We formulate a zero-sum dynamic game where an adversary perturbs both system dynamics and the value function gradient, leading to a new, highly nonlinear partial differential equation: the Hamilton-Jacobi-Bellman-Isaacs Equation with Gradient Uncertainty (GU-HJBI). We establish its well-posedness by proving a comparison principle for its viscosity solutions under a uniform ellipticity condition. Our analysis of the linear-quadratic (LQ) case yields a key insight: we prove that the classical quadratic value function assumption fails for any non-zero gradient uncertainty, fundamentally altering the problem structure. A formal perturbation analysis characterizes the non-polynomial correction to the value function and the resulting nonlinearity of the optimal control law, which we validate with numerical studies. Finally, we bridge theory to practice by proposing a novel Gradient-Uncertainty-Robust Actor-Critic (GURAC) algorithm, accompanied by an empirical study demonstrating its effectiveness in stabilizing training. This work provides a new direction for robust control, holding significant implications for fields where function approximation is common, including reinforcement learning and computational finance.

Figures

Figures reproduced from arXiv: 2507.15082 by the authors.

**Figure 2.** Sensitivity Analysis for the 1D LQ problem. (Left) Increasing the model uncertainty [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** Numerical results for the 2D LQ problem. (Left) A contour plot of the value function [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Empirical results on the Pendulum-v1 environment, averaged over 10 random seeds. (a) Learning curves (mean ± one std. dev.) show that GURAC-TD3 exhibits a significantly more stable performance trajectory compared to the baseline TD3, supporting H1. (b) Performance degr…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

(px, X) ∈ ¯J 2,+u(x)

work page
[2]

Furthermore, the matricesX and Y satisfy the crucial inequality: − 1 γ + ∥A∥2 I 0 0 I ! ≤ X 0 0 −Y ! ≤ A + γA2 where A = α I −I −I I !

(py, Y) ∈ ¯J 2,−v(y) where ¯J 2,+ and ¯J 2,− are the second-order super- and subjets, and the gradients are given by the derivatives of the penalization terms: px := ∇x α 2 ∥x − y∥2 + β 2 ∥x∥2 = α(x − y) + βx, py := −∇y − α 2 ∥x − y∥2 − β 2 ∥y∥2 = α(x − y) − βy. Furthermore, the matricesX and Y satisfy the crucial inequality: − 1 γ + ∥A∥2 I 0 0 I ! ≤ X 0 ...

work page
[3]

From Ishii’s Lemma, we haveX ≤ Y, which meansX − Y is a negative semidefinite matrix

work page
[4]

The trace of a product of a positive definite matrix and a negative semidefinite matrix is non-positive

FromtheUniformEllipticity(Assumption4.2), wehave a(y, uδc) = σ(y, uδc)σ(y, uδc)T ≥ νI for some ν >0. The trace of a product of a positive definite matrix and a negative semidefinite matrix is non-positive. Thus, Tr(a(y, uδc)(X − Y )) ≤ 0. Taking the limit of the entire inequality, we have: ρM = lim ρ(u(x) − v(y)) ≤ 0 + 0 + 0 = 0. Since we assumed ρ > 0, t...

work page
[5]

By the Cauchy-Schwarz inequality,vT δ ≤ ∥v∥2 ∥δ∥2

ℓ2 Uncertainty (∆(2) ϵ = {δ | ∥δ∥2 ≤ ϵ}): The problem is to findsup∥δ∥2≤ϵ vT δ. By the Cauchy-Schwarz inequality,vT δ ≤ ∥v∥2 ∥δ∥2. Since ∥δ∥2 ≤ ϵ, we havevT δ ≤ ϵ ∥v∥2. Equality is achieved when δ is aligned withv, specifically forδ∗ = ϵ v ∥v∥2 (if v ̸= 0). Thus, sup ∥δ∥2≤ϵ vT δ = ϵ ∥v∥2 . 29

work page
[6]

This is equivalent to maximizingPn i=1 viδi subject to |δi| ≤ϵ for alli

ℓ∞ Uncertainty (∆(∞) ϵ = {δ | ∥δ∥∞ ≤ ϵ}): The problem is to findsup∥δ∥∞≤ϵ vT δ. This is equivalent to maximizingPn i=1 viδi subject to |δi| ≤ϵ for alli. To maximize this sum, we should choose each δi to have the same sign asvi and the maximum possible magnitude,ϵ. Therefore, the optimal perturbation isδ∗ i = ϵ · sgn(vi). The maximum value is: nX i=1 vi(ϵ ...

work page
[7]

Since M is positive definite, the constraint set is a compact ellipsoid

Quadratic Form (M) Uncertainty (∆(M ) ϵ = {δ | δT M δ≤ ϵ2}): The problem is to solve the convex optimization problem: max δ∈Rn vT δ subject to δT M δ≤ ϵ2. Since M is positive definite, the constraint set is a compact ellipsoid. The maximizer must lie on the boundary, so the constraint is active: δT M δ= ϵ2. We use the method of Lagrange multipliers. The L...

work page
[8]

From L(x, u): uT 1 Ru1

work page

Show all 12 references

[9]

From pT (Ax + Bu): pT 1 Bu1
[10]

From η 2 ΣT p 2: η 2 pT 1 Sp1. 32
[11]

The ϵ2 coefficient is vT 0 v1 ∥v0∥

From the expansion of ϵ ∥v(ϵ)∥: Let v1 := Bu1 + ηSp 1. The ϵ2 coefficient is vT 0 v1 ∥v0∥. (A detailed Taylor expansion is shown in the thought process). Summing these gives the initial expression for the source term: H2(x) = uT 1 Ru1 + pT 1 Bu1 + η 2 pT 1 Sp1 + vT 0 (Bu1 + ηS...
[12]

Thus, we imposed homogeneous Neumann boundary conditions: V ′ 1(−L) = V ′ 1(L) = 0

Since aef f,0 < 0, this suggests thatV ′ 1(x)approaches zero for large|x|. Thus, we imposed homogeneous Neumann boundary conditions: V ′ 1(−L) = V ′ 1(L) = 0. This results in a tridiagonal system of linear equations which is solved directly. The gradientV ′ 1(x) used for the c...

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

(px, X) ∈ ¯J 2,+u(x)

work page

[2] [2]

Furthermore, the matricesX and Y satisfy the crucial inequality: − 1 γ + ∥A∥2 I 0 0 I ! ≤ X 0 0 −Y ! ≤ A + γA2 where A = α I −I −I I !

(py, Y) ∈ ¯J 2,−v(y) where ¯J 2,+ and ¯J 2,− are the second-order super- and subjets, and the gradients are given by the derivatives of the penalization terms: px := ∇x α 2 ∥x − y∥2 + β 2 ∥x∥2 = α(x − y) + βx, py := −∇y − α 2 ∥x − y∥2 − β 2 ∥y∥2 = α(x − y) − βy. Furthermore, the matricesX and Y satisfy the crucial inequality: − 1 γ + ∥A∥2 I 0 0 I ! ≤ X 0 ...

work page

[3] [3]

From Ishii’s Lemma, we haveX ≤ Y, which meansX − Y is a negative semidefinite matrix

work page

[4] [4]

The trace of a product of a positive definite matrix and a negative semidefinite matrix is non-positive

FromtheUniformEllipticity(Assumption4.2), wehave a(y, uδc) = σ(y, uδc)σ(y, uδc)T ≥ νI for some ν >0. The trace of a product of a positive definite matrix and a negative semidefinite matrix is non-positive. Thus, Tr(a(y, uδc)(X − Y )) ≤ 0. Taking the limit of the entire inequality, we have: ρM = lim ρ(u(x) − v(y)) ≤ 0 + 0 + 0 = 0. Since we assumed ρ > 0, t...

work page

[5] [5]

By the Cauchy-Schwarz inequality,vT δ ≤ ∥v∥2 ∥δ∥2

ℓ2 Uncertainty (∆(2) ϵ = {δ | ∥δ∥2 ≤ ϵ}): The problem is to findsup∥δ∥2≤ϵ vT δ. By the Cauchy-Schwarz inequality,vT δ ≤ ∥v∥2 ∥δ∥2. Since ∥δ∥2 ≤ ϵ, we havevT δ ≤ ϵ ∥v∥2. Equality is achieved when δ is aligned withv, specifically forδ∗ = ϵ v ∥v∥2 (if v ̸= 0). Thus, sup ∥δ∥2≤ϵ vT δ = ϵ ∥v∥2 . 29

work page

[6] [6]

This is equivalent to maximizingPn i=1 viδi subject to |δi| ≤ϵ for alli

ℓ∞ Uncertainty (∆(∞) ϵ = {δ | ∥δ∥∞ ≤ ϵ}): The problem is to findsup∥δ∥∞≤ϵ vT δ. This is equivalent to maximizingPn i=1 viδi subject to |δi| ≤ϵ for alli. To maximize this sum, we should choose each δi to have the same sign asvi and the maximum possible magnitude,ϵ. Therefore, the optimal perturbation isδ∗ i = ϵ · sgn(vi). The maximum value is: nX i=1 vi(ϵ ...

work page

[7] [7]

Since M is positive definite, the constraint set is a compact ellipsoid

Quadratic Form (M) Uncertainty (∆(M ) ϵ = {δ | δT M δ≤ ϵ2}): The problem is to solve the convex optimization problem: max δ∈Rn vT δ subject to δT M δ≤ ϵ2. Since M is positive definite, the constraint set is a compact ellipsoid. The maximizer must lie on the boundary, so the constraint is active: δT M δ= ϵ2. We use the method of Lagrange multipliers. The L...

work page

[8] [8]

From L(x, u): uT 1 Ru1

work page