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).

Effects of pitch area-restrictions on tactical behavior, physical and physiological performances in soccer large-sided games

The aim of this study was to identify how pitch area-restrictions affects the tactical behavior, physical and physiological performances of players during soccer large-sided games. A 10 vs. 9 large-sided game was performed under three experimental conditions: (i) restricted-spacing, the pitch was divided into specific areas where players were assigned and they should not leave it; (ii) contiguous-spacing, the pitch was divided into specific areas where the players were only allowed to move to a neighboring one; (iii) free-spacing, the players had no restrictions in space occupation. The positional data were used to compute players’ spatial exploration index and also the distance, coefficient of variation, approximate entropy and frequency of near-in-phase displacements synchronization of players’ dyads formed by the outfield teammates. Players’ physical and physiological performances were assessed by the distance covered at different speed categories, game pace and heart rate. Most likely higher values were found in players’ spatial exploration index under free-spacing conditions. The synchronization between dyads’ displacements showed higher values for contiguous-spacing and free-spacing conditions. In contrast, for the jogging and running intensity zones, restricted-spacing demanded a moderate effect and most likely decrease compared to other scenarios (~20-50% to jogging and ~60-90% to running). Overall, the effects of limiting players’ spatial exploration greatly impaired the co-adaptation between teammates’ positioning while decreasing the physical and physiological performances. These results allow for a better understanding of players’ decision-making process according to specific task rules and can be relevant to enrich practice task design, such that coaches acknowledge the differential effect by using specific pitch-position areas restrictions.