Pith Number

pith:JT365REX

pith:2026:JT365REXH46QTAESWNHD5U7CCE

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Quantitative Soft-to-Hard Terminal Constraint Convergence for the Heat Equation

Sung-Sik Kwon

Penalized formulations of the heat-equation control problem converge to the exact hard terminal constraint at explicit rates O(alpha to the power minus theta).

arxiv:2605.14060 v1 · 2026-05-13 · math.OC

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Claims

C1strongest claim

We establish explicit quantitative convergence estimates of order O(α^{-θ}), including the sharp O(1/α) rate under stronger modal summability assumptions on the terminal mismatch.

C2weakest assumption

Stronger modal summability assumptions on the terminal mismatch are required to obtain the sharp O(1/α) rate; without them the rate is slower.

C3one line summary

Penalized optimal controls for the heat equation converge to the hard-constrained solution at explicit rates O(alpha to the minus theta), with sharp O(1/alpha) under stronger assumptions on the terminal mismatch.

References

17 extracted · 17 resolved · 0 Pith anchors

[1] Bertsekas.Constrained Optimization and Lagrange Multiplier Meth- ods 1982

[2] Boundary control of semilinear elliptic equations with pointwise state constraints.SIAM Journal on Control and Optimization, 31(4):993–1006, 1993 1993

[3] Reachable states for the distributed control of the heat equation.Comptes Rendus 2022

[4] Evans.Partial Differential Equations, volume 19 ofGraduate Stud- ies in Mathematics 2010

[5] R. Glowinski and J.-L. Lions. Exact and approximate controllability for dis- tributed parameter systems.Acta Numerica, 3:269–378, 1994 1994

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-17T23:39:12.536052Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

4cf7eec4973f3d098092b34e3ed3e21118e50a2573825fcf7fd61ad21817fdd6

Aliases

arxiv: 2605.14060 · arxiv_version: 2605.14060v1 · doi: 10.48550/arxiv.2605.14060 · pith_short_12: JT365REXH46Q · pith_short_16: JT365REXH46QTAES · pith_short_8: JT365REX

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/JT365REXH46QTAESWNHD5U7CCE \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 4cf7eec4973f3d098092b34e3ed3e21118e50a2573825fcf7fd61ad21817fdd6

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "bb8afc2bc1276c53b43f70aa4509eb0a1c8e7360b61f1f37e5e68bedd86f2207",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.OC",
    "submitted_at": "2026-05-13T19:34:37Z",
    "title_canon_sha256": "91a6c76278cde93d7210d0d1a0aec1a933ca4b7a0aa3b8d08399246bb92b470b"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.14060",
    "kind": "arxiv",
    "version": 1
  }
}