What do you exactly mean ? Are you speaking of shaping urban space for people becoming elder in cities ("aging in cities") ? Or, do you consider the city itself to age, with "aging sites" developing inside "without barriers" ? I don't see precisely what you mean with these "barriers". Is it the idea that cities should be allowed to extend without barriers but in an optimal planed way ? Or would urban planing have to take account of people aging in the city to shape urban space accordingly by appropriate extensions without barriers ?
I am interested in shaping, creating places in the urban space for eleder people, but not only "aging in cities" or "Design for All" but theory of city without architectural barriers in the urban space, is that theory exists?
To my incomplete knowledge on the subject, there is no real theory of city without architectural barriers for elderly. Rather, there are policies, strategies, practices based on studies as referred to hereafter.
There is the research project OPUS (Older People's Use of Unfamiliar Space) which concluded that urban planning should include a smooth transition between walk, drive and public transport and take account of how elderly people use these transport means. You can find some info in enclosed slides in pdf "Older People's Use of Unfamiliar Space" (2013). There is also a website "OPUS : New Dynamics Ageing" which has been developed from OPUS results (see link below).
You probably know the WHO permanent web page for their newly founded Global Network of Age-Friendly Cities and Communities (GNAFCC). Follow the link "who.int" below. There is also the research project "UrbAging - Planning and designing the urban space for an ageing society” (2009) on 2 Swiss cities (see enclosed pdf "ris_ist...). See also the "Swissinfo" article of 2012 in the joint link. Another project is the enclosed 2010 Austrian "sALTo" project for Vienna ("Urban Planning to Support Seniors"). And the approach for Barcelona in Spain (2013) : see the link to the University of Barcelona web page.
In addition, it is perhaps worth quoting 2 books on "Urban planning" as they contain chapters on elderly people :
"Urban Planning: An Introduction" (by Chris Couch) (2016) : pp. 209-211 for elderly people
"Handbook on Transport and Urban Planning in the Developed World" (publhed by Michiel C.J. Bliemer,Corinne Mulley,Claudine J. Moutou) (2015) : see Part I -3 History and theory of urban transport planning (pp. 36-53) and Part V -20 City Logistics (pp. 375-385)
Finally for curiosity (and for the records) there is the old US study on "Planning and an Aging Population" (1961) (see joint pdf).
I'd also like to mention a Dutch study checking the possibility of segregating elderly people in senior cities (2009) : probably not a good idea (see enclosed pdf).
Alzheimer’s Society (2013). Building Dementia-Friendly Communities: A Priority for Everyone. London: Alzheimer’s Society. Retrieved from https://www.alzheimers.org.uk/site/scripts/download.php?type=downloads&fileID=1916
Building Futures and RIBA (2013). Silver Linings: The Active Third Age and the City. London: Building Futures and the Royal Institute of British Architects (RIBA). Retrieved from https://www.architecture.com/Files/RIBAHoldings/PolicyAndInternationalRelations/BuildingFutures/Projects/SilverLinings.pdf
Oxford Institute for Sustainable Development (2004), Neighborhoods for Life’ – Designing dementia friendly outdoor environments. Oxford: Oxford Institute for Sustainable Development. Retrieved from http://www.housinglin.org.uk/_library/Resources/Housing/Support_materials/Other_reports_and_guidance/Neighbourhoods_for_Life_Findings_Leaflet.pdf
WHO (2007a). Checklist of Essential Features of Age-friendly Cities. Geneva: World Health Organization (WHO). Retrieved from http://www.who.int/ageing/publications/Age_friendly_cities_checklist.pdf
WHO (2007b). Global Age-friendly Cities: A Guide. Geneva: World Health Organization (WHO). Retrieved from http://www.who.int/ageing/publications/Global_age_friendly_cities_Guide_English.pdf
You might also be interested in the work and publications of Pr Michel Poulain (University of Tallinn) : demography, care to elderly, Blue Zones,.... Here is the link to his RG profile :