, . . . , F L ( j)) to become obtained as answer of

, . . . , F L ( j)) to become obtained as answer of (16c). So
, . . . , F L ( j)) to be obtained as option of (16c). So, the TVPs solved by Pk (, 1 k L are interconnected by way of (16c). To facilitate the statement on the principal outcome of this section, we rewrite (16c) within a compact form as: d ( P1 (t j ), . . . , PL (t j ), j)F( j) = -d ( P1 (t j ), . . . , PL (t j )), (19)T T exactly where F( j) = (F1 ( j) F2 ( j) . . . F T ( j)) T and the matrices d ( P1 (t j ), . . . , PL (t j ), j) and L d ( P1 (t j ), . . . , PL (t j )) are obtained applying the block elements of (16c).2.3. Sampled Information Nash Equilibrium Strategy Initially we derive a necessary and adequate condition for the existence of an equilibrium tactic of form (9) for the LQ differential game offered by the controlled system (5), the overall performance criteria (7) and the set with the admissible tactics U sd . To this finish we adapt the argument applied inside the proof of ([22], Theorem 4). We prove: Theorem 1. Below the assumption H. the following are equivalent: (i) the LQ differential game defined by the dynamical program controlled by impulses (5), the overall performance criteria (7) as well as the class with the admissible tactics of form (9) includes a Nash equilibrium technique uk ( j) = Fk ( j) (t j ), 0 j N – 1, 1 k L. (20)Mathematics 2021, 9,7 of(ii)the TVP with constraints (16) features a answer ( P1 (, P2 (, . . . , PL (; F1 (, F2 (, . . . , F L () defined on the complete interval [t0 , t f ] and satisfying the situations below for 0 j N – 1: d ( P1 (t j ), . . . , PL (t j ), j)d ( P1 (t j ), . . . , PL (t j ), j) d ( P1 (t j ), . . . , PL (t j )) = = d ( P1 (t j ), . . . , PL (t j )).(21)If situation (21) holds, then the feedback matrices of a Nash equilibrium Boc-Cystamine Purity & Documentation approach of type (9) will be the matrix elements of your answer in the TVP (16) and are provided by T T L (F1 ( j) F2 ( j) . . . F T ( j))T = -d ( P1 (t j ), . . . , PL (t j ), j) d ( P1 (t j ), . . . , PL (t j )), 0 j N – 1.- T The minimal value from the expense from the k-th player is 0 Pk (t0 ) 0 .(22)Proof. From (14) and Remarks 1 and two(a), 1 can see that a approach of type (9) defines a Nash equilibrium approach for the linear differential game described by the controlled technique (five), the performance criteria (7) (or equivalently (13)) if and only if for every single 1 k L the optimal control trouble described by the controlled system d (t) = A (t)dt + C (t)dw(t), t j t t j+1 (t+ ) = A[-k] ( j) (t j ) + Bdk uk ( j), j = 0, 1, . . . , N – 1, j ( t0 ) = 0 R and also the quadratic functionaltf n+m(23a) (23b) (23c),J[-k] (t0 , 0 ; uk ) =E[ (t f )Gk (t f ) +t0 N -TT (t)Mk (t)dt]+j =T E[ T (t j )M[-k] ( j) (t j ) + uk ( j)Rkk ( j)uk ( j)],(24)has an optimal control inside a state feedback kind. The controlled technique (23) and the performance criterion (24) are obtained substituting u ( j) = F ( j) (t j ), 1 k, L, = k in (5) and (7), respectively. A[-k] and M[-k] are computed as in (17) and (18), respectively, i ( j ). but with Fi ( j) replaced by F To UNC6934 Cancer receive important and sufficient conditions for the existence of your optimal control inside a linear state feedback type we employ the results proved in [20]. Initially, notice that inside the case of your optimal handle trouble (23)24), the TVP (16a), (16b), (16d) plays the part with the TVP (19)23) from [20]. Applying Theorem 3 in [20] inside the case with the optimal control trouble described by (23) and (24) we deduce that the existence of your Nash equilibrium approach from the kind (9) for the differential game described by the controlled method (5), the.