Ateiwi, Ali MahmudVoladymyrivna Komashynsk, İrya2024-07-122024-07-122009Ateiwi, A. M. ve Komashynsk Voladymyrivna, I. (2009). The existence of the optimal control of systems with quadratic quantity criterium. Maltepe Üniversitesi. s. 86.9.78605E+12https://www.maltepe.edu.tr/Content/Media/CkEditor/03012019014112056-AbstractBookICMS2009Istanbul.pdf#page=76https://hdl.handle.net/20.500.12415/2299Consider optimal control problem in R n with quadratic in control quality criterium: dx dt = f(x, t)B(t)u (1) x(s) = y I(s, y, u) = ?(?, x(?)) + Z ? s [?(t, x(t)) + (N(t)u(t), u(t))] dt ? inf (2) Here t ? [0, T], x ? R n, Q0 = (0, T) × R n, Q is bounded sub domain of Q0 with the boundary ?Q. We assume that : 1) The functions ?(t, x) and ?(t, x) are nonnegative, smoth in their arguments in Q¯, morover, ?? ?x is Lipshitz in x in Q¯ (Q¯ is the closure of Q) 2) f(t, x) is smooth in Q¯ and ?f ?x is Lipshitz in x in Q¯. 3) n × m is dimensional matrix B(t) is smooth in t in Q¯. 4) m × m is dimensional matrix N(t) is positive definite in Q¯. and smooth in t The bellman’s equation of the problem (1) , (2) is ?V ?t + µ f(t, x), ?V ?t ¶ + ?(t, x) ? 1 4 µ B(t)N ?1 (t)B ? (t) ?V ?t , ?V ?t ¶ = 0 With the boundary condition. THEOREM 1. If the hyper surface ?Q is correctly embedded into R n+1, and the conditions (1)-(4) hold , then the boundary value problem (7), (8) has the unique solution in Q, which is continuous together with it’s partial derivative up to the second order.enCC0 1.0 Universalinfo:eu-repo/semantics/openAccessConference Object8786