(1 , 2 , . . . , L ) : k are arbitrary sequences defined as in (11).(11)(12)Mathematics 2021, 9,5 ofsd Remark
(1 , 2 , . . . , L ) : k are arbitrary sequences defined as in (11).(11)(12)Mathematics 2021, 9,five ofsd Remark 1. By (9) and (ten), there is a a single to one correspondence between the sets U sd and U . sd is often identified together with the sequence = ( (0), (1), . . . , ( N – 1)) of its Each uk from Uk k k k k feedback matrices.Depending on this remark we can rewrite the efficiency criterion (7) as:tfJk (t0 , 0 ; 1 , 2 , . . . , L ) =E[ T (t f )Gk (t f ) +t0 N -T (t)Mk (t)dt]+j =E[ T (t j )Fit ( j)Rki ( j)Fi ( j) (t j )],i =L(13)sd for all (1 , two , . . . , L ) U . Similarly to Definition 1, 1 can define a Nash equilibrium approach for the LQ differential game described by the controlled program (five), the efficiency criteria (13) and sd the class of admissible strategies U described by (12).Definition two. The L-tuple of admissible tactics (1 , 2 , . . . , L ) is mentioned to achieve a Nash equilibrium for the differential game described by the controlled program (five), the price function (13), sd sd plus the class of the admissible techniques U , if for all (1 , 2 , . . . , L ) U , we have Jk (t0 , 0 ; 1 , 2 , . . . , L ) Jk (t0 , 0 ; 1 , two , . . . ., k-1 , k , k+1 , . . . , L ). Remark two. (a) Based on the Remark 1 we could infer that if (1 , 2 , . . . , L ) is definitely an equilibrium approach in the sense with the Definition two, then (u1 , u2 , . . . , u L ) provided by (9) making use of the matrix elements of k , supplies an equilibrium strategy for the LQ differential game described by (five), (7) as well as the household of admissible approaches U sd . Amongst the feedback matrices from (9) some possess the form: Fk ( j) = ( Fk ( j) 0m k m ), (15) (14)(b)exactly where Fk ( j) Rmk . Hence, some admissible strategies (9) are of form (four). Therefore, in the event the feedback matrices on the Nash equilibrium tactic (1 , 2 , . . . , L ) possess the structure provided in (15), then the method of sort (9) with these feedback matrices provide the Nash equilibrium strategy for the LQ differential game described by (1), (2) and (four). To acquire explicit formulae for the feedback matrices of a Nash equilibrium tactic of sort (9) (or, equivalently (11), (12)), we make use of the following technique of matrix linear differential equations (MLDEs) with jumps and algebraic constraints:Mathematics 2021, 9,six of- Pk (t) = A T Pk (t) + Pk (t)A + C T Pk (t)C + Mk , t j t t j+T T Pk (t- ) = A[-k] ( j) Pk (t j )A[-k] ( j) – A[-k] ( j) Pk (t j )Bdk j T T Rkk ( j) + Bdk Pk (t j )Bdk ) Bdk Pk (t j )A[-k] ( j) + M[-k] ( j) k -1 i =1 T T Bdk Pk (t j )Bdi Fi ( j) + (Rkk ( j) + Bdk Pk (t j )Bdk )Fk ( j) L(16a)(16b)+i = k +T T Bdk Pk (t j )Bdi Fi ( j) = -Bdk Pk (t j )Ad(16c) (16d)Pk (t- ) = Gk , N exactly where we’ve denoted A[-k] ( j) = Ad + and M[-k] ( j) =k = 1, . . . , L,i =1,i =k LLBdi Fi ( j)(17)i =1,i =kFiT ( j)Rki ( j)Fi ( j),(18)though the superscript denotes the generalized Fluorescent-labeled Recombinant Proteins site inverse of a matrix. Remark three. A answer of the terminal value difficulty (TVP) with algebraic constraints (16) is really a 2L-uple in the kind ( P1 (, P2 (, . . . , PL (; F1 (, F2 (, . . . , F L () exactly where, for each 1 k L, Pk ( is really a remedy of your TVP (16a), (16b), (16d) and Fk ( j) Rmk n+m) , 0 j N – 1. Around the 2-Hydroxydocosanoic acid MedChemExpress interval [t N -1 , t N ], Pk ( is the remedy of the TVP described by the perturbed Lyapunovtype equation from (16a) and the terminal worth provided in (16d). On every interval [t j-1 , t j ), j N – 1, the terminal value Pk (t- ) of Pk ( is computed via (16b) together with (17) and (18) j offered that (F1 ( j), F2 ( j).
