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In this paper, we consider, as space of signals, the usual Banach space of <jats:inline-formula><jats:alternatives><jats:tex-math>$${L^{p}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mi>p<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> functions, or the space of distributions spanned by <jats:inline-formula><jats:alternatives><jats:tex-math>$${L^{p}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mi>L<\/mml:mi>\n                    <mml:mi>p<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> functions and by their distributional derivatives, of any order (input spaces which include signals with not necessarily left-bounded support), we give a systematic theoretical analysis of the existence, uniqueness and invertibility of continuous linear time-invariant input\u2013output stable systems (both causal and non-causal ones) associated with the differential equation and, in case of invertibility, we characterize the continuous inverse system. We also give necessary and sufficient conditions for causality. As an application, we consider the problem of finding a suitable <jats:italic>almost inverse<\/jats:italic> of a causal continuous linear time-invariant input\u2013output stable <jats:italic>non-invertible<\/jats:italic> system, defined on the space of finite-energy functions, associated with a simple differential equation.<\/jats:p>","DOI":"10.1007\/s00034-021-01689-7","type":"journal-article","created":{"date-parts":[[2021,3,18]],"date-time":"2021-03-18T21:44:51Z","timestamp":1616103891000},"page":"4301-4345","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Continuous LTI Input\u2013Output Stable Systems on $${L^{p}(\\mathbb {R})}$$ and $${\\mathscr {D'}_{L^{p}}(\\mathbb {R})}$$ Associated with Differential Equations: Existence, Invertibility Conditions and Inversion"],"prefix":"10.1007","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0077-5020","authenticated-orcid":false,"given":"M.","family":"Ciampa","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,3,18]]},"reference":[{"key":"1689_CR1","volume-title":"Linear Systems","author":"PJ Antsaklis","year":"2006","unstructured":"P.J. 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