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For this kind of systems we prove that if there are no minimizing abnormal extremals then the value function $S$ is subanalytic. Secondly we prove that if there exists an abnormal minimizer of corank 1 then the set of end-points of minimizers at cost fixed is tangent to a given hyperplane. We illustrate this situation in sub-Riemannian geometry."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0607424","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.OC","submitted_at":"2006-07-18T15:20:48Z","cross_cats_sorted":[],"title_canon_sha256":"e7725ff02d51be3e43d526dde05f3195a57e97ac4fb619bd7d864e8a8a80e82d","abstract_canon_sha256":"a80068ecaa0fbd9cd45c1214011beaf4a5f4150e1459f8694062089c1a1bc236"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:50.035024Z","signature_b64":"yfkzW52hoBBRktp27Lyjtw2+VZgwgdB22YvV2HJctcV/vRzZ8OiTcdWHG2g7XX/VXLa9IpoJJDjeVp2wN4mgCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f2f4adc177a26a4bed00d9b8d6e721b996835bc332d94eeaa2102485134eade","last_reissued_at":"2026-05-18T01:08:50.034392Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:50.034392Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some properties of the value function and its level sets for affine control systems with quadratic cost","license":"","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Emmanuel Tr\\'elat (IMB)","submitted_at":"2006-07-18T15:20:48Z","abstract_excerpt":"Let $T>0$ fixed. We consider the optimal control problem for analytic affine systems: $\\ds{\\dot{x}=f\\_0(x)+\\sum\\_{i=1}^m u\\_if\\_i(x)}$, with a cost of the form: $\\ds{C(u)=\\int\\_0^T \\sum\\_{i=1}^m u\\_i^2(t)dt}$. For this kind of systems we prove that if there are no minimizing abnormal extremals then the value function $S$ is subanalytic. Secondly we prove that if there exists an abnormal minimizer of corank 1 then the set of end-points of minimizers at cost fixed is tangent to a given hyperplane. 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