The existence of the optimal control of systems with quadratic quantity criterium
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CitationAteiwi, A. M. ve Komashynsk Voladymyrivna, I. (2009). The existence of the optimal control of systems with quadratic quantity criterium. Maltepe Üniversitesi. s. 86.
Consider 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) = ϕ(τ, x(τ)) + Z τ s [Ψ(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 ϕ(t, x) and Ψ(t, x) are nonnegative, smoth in their arguments in Q¯, morover, ∂Ψ ∂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 ¶ + Ψ(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.
SourceInternational Conference of Mathematical Sciences
- Makale Koleksiyonu 
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