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).

Exploring the potential of novel biomixtures and Lentinula edodes fungus for the degradation of selected pesticides. Evaluation for use in biobed systems

An approach to reduce the contamination of water sourceswith pesticides is the use of biopurificaction systems. The active core of these systems is the biomixture. The composition of biomixtures depends on the availability of local agro-industrial wastes and design should be adapted to every region. In Portugal, cork processing is generally regarded as environmentally friendly and would be interesting to find applications for its industry residues. In this work the potential use of different substrates in biomixtures, as cork (CBX); cork and straw, coat pine and LECA (Light Expanded Clay Aggregates), was tested on the degradation of terbuthylazine, difenoconazole, diflufenican and pendimethalin pesticides. Bioaugmentation strategies using the white-rot fungus Lentinula edodes inoculated into the CBX, was also assessed. The results obtained from this study clearly demonstrated the relevance of using natural biosorbents as cork residues to increase the capacity of pesticide dissipation in biomixtures for establishing biobeds. Furthermore, higher degradation of all the pesticides was achieved by use of bioaugmented biomixtures. Indeed, the biomixtures inoculated with L. edodes EL1were able to mineralize the selected xenobiotics, revelling that these white-rot fungi might be a suitable fungus for being used as inoculum sources in on-farm sustainable biopurification system, in order to increase its degradation efficiency. After 120 days, maximum degradation of terbuthylazine, difenoconazole, diflufenican and pendimethalin, of bioaugmented CBX, was 89.9%, 75.0%, 65.0% and 99.4%, respectively. The dominant metabolic route of terbuthylazine in biomixtures inoculated with L. edodes EL1 proceeded mainly via hydroxylation, towards production of terbuthylazine-hydroxy-2 metabolite. Finally, sorption process to cork by pesticides proved to be a reversible process,working cork as a mitigating factor reducing the toxicity to microorganisms in the biomixture, especially in the early stages.