A pointwise constrained version of the Liapunov convexity theorem for vectorial linear first-order control systems

We generalize the Liapunov convexity theorem's version for vectorial control systems driven by linear ODEs of first-order p = 1 , in any dimension d ∈ N , by including a pointwise state-constraint. More precisely, given a x ‾ ( ⋅ ) ∈ W p , 1 ( [ a , b ] , R d ) solving the convexified p-th order differential inclusion L p x ‾ ( t ) ∈ co { u 0 ( t ) , u 1 ( t ) , … , u m ( t ) } a.e., consider the general problem consisting in finding bang-bang solutions (i.e. L p x ˆ ( t ) ∈ { u 0 ( t ) , u 1 ( t ) , … , u m ( t ) } a.e.) under the same boundary-data, x ˆ ( k ) ( a ) = x ‾ ( k ) ( a ) & x ˆ ( k ) ( b ) = x ‾ ( k ) ( b ) ( k = 0 , 1 , … , p − 1 ); but restricted, moreover, by a pointwise state constraint of the type 〈 x ˆ ( t ) , ω 〉 ≤ 〈 x ‾ ( t ) , ω 〉 ∀ t ∈ [ a , b ] (e.g. ω = ( 1 , 0 , … , 0 ) yielding x ˆ 1 ( t ) ≤ x ‾ 1 ( t ) ). Previous results in the scalar d = 1 case were the pioneering Amar & Cellina paper (dealing with L p x ( ⋅ ) = x ′ ( ⋅ ) ), followed by Cerf & Mariconda results, who solved the general case of linear differential operators L p of order p ≥ 2 with C 0 ( [ a , b ] ) -coefficients. This paper is dedicated to: focus on the missing case p = 1 , i.e. using L p x ( ⋅ ) = x ′ ( ⋅ ) + A ( ⋅ ) x ( ⋅ ) ; generalize the dimension of x ( ⋅ ) , from the scalar case d = 1 to the vectorial d ∈ N case; weaken the coefficients, from continuous to integrable, so that A ( ⋅ ) now becomes a d × d -integrable matrix; and allow the directional vector ω to become a moving AC function ω ( ⋅ ) . Previous vectorial results had constant ω, no matrix (i.e. A ( ⋅ ) ≡ 0 ) and considered: constant control-vertices (Amar & Mariconda) and, more recently, integrable control-vertices (ourselves).

Characterizing Human Cysticercosis In Portugal

Abstract text Introduction: Cysticercosis results from the ingestion Taenia solium eggs directly by faecal-oral route or contaminated food or water. While, still considered a leading cause of acquired epilepsy in developed countries, this zoonosis has been controlled or eradicated in industrialized countries due to significant improvements in sanitation, pig rearing and slaughterhouse control systems. Objectives: the health burden of human cysticercosis in Portugal. Material and Metodes: We developed a retrospective study on human neurocysticercosis (NCC) hospitalisations based on the national database resulting from National Health Service (NHS) hospital episodes except those of Madeira and Azores Islands. Results: Between 2006 and 2013 there were 357 hospitalized NCC cases in Portugal. Annual frequency of cases between 2006-2013 kept stable (mean 45). NCC was most frequent in those aged 25-34 years (59; 16,5%) and those >75 years (65; 18,2%). Overall, mean age was 47,3 years (median age 45, standard deviation 41,1, mode 28) and 176 cases were in males (49,3%); no significant differences were observed between age and gender (t-student, p>0,05). In Norte Region cases tended to be older than in Lisboa and Vale do Tejo Region. Conclusions: The Directorate-General of Health established the National Observatory of Cysticercosis and Teniiasis which will define criteria for NCC cases monitoring and surveillance (hospitalized and non-hospitalized cases).