{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:41:20Z","timestamp":1760150480605,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,11,23]],"date-time":"2023-11-23T00:00:00Z","timestamp":1700697600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We discuss the optimal control formulation for enhancement and denoising of satellite multiband images and propose to take it in the form of an L1 control problem for a quasi-linear parabolic equation with a nonlocal p[u] Laplacian and with a cost functional of a tracking type. The main characteristic features of the considered parabolic problem is that the variable exponent p(t,x) and the diffusion anisotropic tensor D(t,x) are not predefined well a priori; instead, these characteristics nonlocally depend on the form of the solution of this problem (i.e., pu=p(t,x,u) and Du=D(t,x,u)). We prove the existence of optimal pairs with sparse L1 controls used for the indirect approach and a special family of approximation problems.<\/jats:p>","DOI":"10.3390\/axioms12121073","type":"journal-article","created":{"date-parts":[[2023,11,23]],"date-time":"2023-11-23T11:31:25Z","timestamp":1700739085000},"page":"1073","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Optimal Sparse Control Formulation for Reconstruction of Noise-Affected Images"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1593-0510","authenticated-orcid":false,"given":"Peter","family":"Kogut","sequence":"first","affiliation":[{"name":"Department of Mathematical Analysis and Optimization, Oles Honchar Dnipro National University, Gagarin av. 72, 49010 Dnipro, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-0431-0895","authenticated-orcid":false,"given":"Yaroslav","family":"Kohut","sequence":"additional","affiliation":[{"name":"Department of Mathematical Analysis and Optimization, Oles Honchar Dnipro National University, Gagarin av. 72, 49010 Dnipro, Ukraine"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"025006","DOI":"10.1088\/1361-6420\/ab5178","article-title":"A denoising model adapted for impulse and Gaussian noises using a constrained-PDE","volume":"2","author":"Afraites","year":"2020","journal-title":"Inverse Probl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"827","DOI":"10.3934\/ipi.2022001","article-title":"A non-convex denoising model for impulse and Gaussian noise mixture removing using bi-level parameter identification","volume":"16","author":"Afraites","year":"2022","journal-title":"Inverse Probl. 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