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If no control is applied, the contaminated set <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Omega (t)\\subset V$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03a9<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>t<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>\u2282<\/mml:mo>\n                    <mml:mi>V<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> expands with unit speed in all directions. By implementing a control, a region of area <jats:italic>M<\/jats:italic> can be cleared up per unit time. Given an initial set <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Omega (0)=\\Omega _0\\subseteq V$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03a9<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mn>0<\/mml:mn>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>\u03a9<\/mml:mi>\n                      <mml:mn>0<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>\u2286<\/mml:mo>\n                    <mml:mi>V<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, three main problems are studied: (1) existence of an admissible strategy <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$t\\mapsto \\Omega (t)$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>t<\/mml:mi>\n                    <mml:mo>\u21a6<\/mml:mo>\n                    <mml:mi>\u03a9<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>t<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> which eradicates the contamination in finite time, so that <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\Omega (T)=\\emptyset $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03a9<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>T<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mi>\u2205<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> for some <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$T&gt;0$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>T<\/mml:mi>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. (2) Optimal strategies that achieve eradication in minimum time. (3) Strategies that minimize the average area of the contaminated set on a given time interval [0,\u00a0<jats:italic>T<\/jats:italic>]. For these optimization problems, a sufficient condition for optimality is proved, together with several necessary conditions. Based on these conditions, optimal set-valued motions <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$t\\mapsto \\Omega (t)$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>t<\/mml:mi>\n                    <mml:mo>\u21a6<\/mml:mo>\n                    <mml:mi>\u03a9<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>t<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> are explicitly constructed in a number of cases.<\/jats:p>","DOI":"10.1007\/s00032-025-00419-x","type":"journal-article","created":{"date-parts":[[2025,6,15]],"date-time":"2025-06-15T14:58:22Z","timestamp":1749999502000},"page":"263-329","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Optimally Controlled Moving Sets with Geographical Constraints"],"prefix":"10.1007","volume":"93","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2030-842X","authenticated-orcid":false,"given":"Alberto","family":"Bressan","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3555-5105","authenticated-orcid":false,"given":"Elsa M.","family":"Marchini","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4776-5022","authenticated-orcid":false,"given":"Vasile","family":"Staicu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,6,15]]},"reference":[{"key":"419_CR1","doi-asserted-by":"publisher","first-page":"2909","DOI":"10.1002\/mma.5560","volume":"42","author":"S Anita","year":"2019","unstructured":"Anita, S., Capasso, V., Dimitriu, G.: Regional control for a spatially structured malaria model. 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