Alec Kriebel

Failure of strong quadratic growth in unregularized sparse control of a critical wave equation

By Alec Kriebel, independent researcher.
Manuscript dated · Version 1.0 · Unrefereed preprint

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Abstract

We give an explicit counterexample to retaining ordinary critical-cone strict positivity and strong control-L² quadratic growth after removing quadratic control regularization from the energy-critical quintic-wave problem of Kunisch and Meinlschmidt. On the three-dimensional unit ball, zero is the unique global optimizer for a sufficiently small positive pointwise control cap. Its ordinary critical cone is {0}, so the strict condition is vacuous, but feasible controls converging in L∞ have objective gap o(‖u‖²L²). Nearby radial terminal targets admit global optimizers, and every such optimizer has L² displacement divided by target displacement tending to infinity. Local L¹ quadratic growth survives. The construction realizes the established concentration and noncoercivity mechanism in the actual nonlinear source model, using classical wave estimates.

The idea in plain English

In optimal control, a control is an input chosen to make a system behave as desired. This paper studies a nonlinear wave equation, where the objective balances a desired final wave profile against penalties on the state and on the amount of control used.

A useful stability property says that moving away from the best control raises the objective by at least a fixed multiple of the squared distance. In this example, doing nothing is the unique global optimum, but increasingly short, small pulses have an objective penalty much smaller than their squared L² size. Uniqueness therefore does not supply that stronger growth property.

The striking feature is that the ordinary critical-cone test still holds: its cone contains only zero, so there are no nonzero directions on which the strict test can fail. The concentrating pulses reveal what this test misses. A weaker L¹ quadratic-growth estimate survives, and the paper also bounds how optimizers respond to nearby terminal targets.

The precise result

The model is the real defocusing quintic wave equation yₜₜ − Δy + y⁵ = u on the three-dimensional unit ball B, with zero position and velocity initially, homogeneous Dirichlet boundary conditions and final time 1. Let H=L²(B). Controls lie in L²(0,1;H) and obey ‖u(t)‖H ≤ ρ almost everywhere in time. There is ρ*>0 such that the conclusions hold for every fixed 0<ρ≤ρ* and every fixed finite ν≥0.

The objective Jd,ν retains one half of the squared terminal-position tracking error, one quarter of the fourth power of the L⁴(0,1;L¹²(B)) state norm, and the L¹(0,1;H) sparse-control penalty: γ=β₁=1. Only the quadratic control penalty is removed, β₂=0. The optional terminal-velocity term is ν‖yₜ(1)‖²H⁻¹(B)/2.

For target d=πe, with e the normalized first radial Dirichlet eigenfunction, zero is the unique global optimizer over all admissible controls. With A=‖u‖L¹(0,1;H), the objective gap Δ(u)=Jπe,ν(u)−Jπe,ν(0) satisfies Δ(u)≥A³/(12ρ²). The ordinary critical cone is {0}, so the strict substitute second-variation condition is vacuous. The cap threshold and this lower bound can be chosen independently of ν.

The actual nonlinear feasible pulses uₕ=he·1{|t−1/2|<h}, for 0<h<min(ρ,1/2), satisfy ‖uₕ‖L∞(0,1;H)=h, ‖uₕ‖²L²(0,1;H)=2h³ and Δ(uₕ)=(π²/3+2/π²)h⁴+O(h⁶). Strong control-L² quadratic growth therefore fails in both L² and L∞ neighborhoods of zero.

Global optimizers exist for every terminal target d∈H in this fixed small-cap setting. Along dε=(π+ε)e, as ε decreases to zero through positive values, every choice of global optimizers satisfies ‖uε‖L²/ε→∞. No uniqueness of perturbed optimizers is required.

Corollary 3 retains local weaker-norm growth: Δ(u)≥‖u‖²L¹/(4π²) for every admissible control with sufficiently small L¹ norm. For an arbitrary nearby target πe+w, put ε=‖w‖H. Every global optimizer, compared with the fixed zero optimizer, satisfies ‖u‖L¹=O(ε) and ‖u‖L²=O(√ε). The constants and this upper-bound neighborhood can be chosen uniformly over all finite ν≥0.

On the positive radial ray w=εe, every global optimizer satisfies ‖uε‖L¹=Θ(ε) and cε³ᐟ⁴≤‖uε‖L²≤C√ε for sufficiently small ε. The lower-comparison neighborhood may depend on the fixed ν. These bounds do not assert a sharp L² response exponent.

Why the result holds

Classical energy and Strichartz estimates make the nonlinear state close to the linear wave for small control mass: the state is O(‖u‖L¹), and the nonlinear remainder is O(‖u‖⁵L¹). This estimate is in a compatible energy-and-Strichartz state space; it does not assume that the nonlinear solution stays in one eigenmode.

The first eigenmode gives an explicit sine kernel for the terminal linear response. At the midpoint in time, the tracking gain exactly meets the sparse-control threshold. A cap-constrained moment estimate proves global optimality of zero, while pulses concentrating at that midpoint make the quadratic-growth quotient vanish.

The terminal tracking square also provides a weaker L¹ quadratic lower bound. A separate pulse comparison for perturbed targets yields the lower response bounds for every optimizer. These arguments use analytic estimates; the accompanying finite algebra checks are not a numerical solution of the nonlinear PDE.

Scope and limitations

This is a counterexample to retaining the ordinary critical-cone implication unchanged when the quadratic control penalty is removed. It does not rule out adapted sufficient conditions, enlarged cones, nondegeneracy assumptions or growth in weaker norms.

The ordinary critical cone is zero, so strict positivity there is vacuous. The result should not be described as a counterexample with a uniformly positive Hessian on every control direction.

The control cap is pointwise in time on the spatial L² norm. It is not a pointwise bound on |u(t,x)|. Here L¹ and L∞ mean time norms with values in L²(B), while control L² is the time-space L² norm. The state penalty remains present; “unregularized” refers to removing the quadratic control term.

The construction uses the three-dimensional unit ball, the defocusing quintic, zero initial data, a small positive cap and a particular target. It does not assert the same failure for every domain, every target, arbitrary large caps, Neumann conditions or boundary control.

The cap and base-target lower bounds can be chosen independently of ν. Upper target-response estimates and their neighborhood are uniform in finite ν; the radial lower-comparison neighborhood may depend on the fixed ν. No assertion concerns ν = ∞.

The response estimates compare optimizers with the fixed zero optimizer. They do not prove perturbed-optimizer uniqueness, a pairwise Lipschitz selection, L∞ convergence of perturbed optimizers, or a sharp exponent for the L² response.

The paper gives no certified numerical PDE constant or numerical value of the small-cap threshold. Its finite checks do not replace the analytic proof. The historical question and earlier mechanisms are attributed without a worldwide-firstness or present-day-openness claim.

The failure is specific to the selected target. For target d=0 in the same model, J0,ν(u)≥‖u‖L¹≥‖u‖²L²/ρ, so strong quadratic growth does hold. The theorem does not assert failure for all targets or all control caps.

Status and AI use

AI tools were used extensively in construction, derivation, computation, source comparison and writing. This is an unrefereed preprint; internal AI examinations are not external human peer review or proof-assistant certification.

The manuscript and Zenodo edition are dated 14 October 2026 and identify version 1.0. The Zenodo record was created on the same date. These dates identify this edition; they do not establish first priority.

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I welcome substantive collaboration from researchers in PDE-constrained optimization, critical wave equations and nonsmooth analysis on independent verification, comparison with prior results and development toward journal review. Substantive mathematical verification or a meaningful extension could support a joint submission. Authorship would reflect actual research contributions.

Authorship and responsibilities would be discussed in relation to the substantive contributions made.

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Cite this paper

Alec Kriebel. (2026). Failure of strong quadratic growth in unregularized sparse control of a critical wave equation (Version 1.0) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23358014

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Background and prior work