Optimal control problems on time scales described by Volterra integral equations on time scales

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1 Artigo Originl DOI: / X18004 Ciênci e Ntur, Snt Mri v.38 n.2, 2016, Mi.- Ago. p Revist do Centro de Ciêncis Nturis e Exts - UFSM ISSN impress: ISSN on-line: X Optiml control problems on time scles described by Volterr integrl equtions on time scles Problems de controle ótimo em escls temporis descritos por equções integris de Volterr em escls temporis Iguer Sntos iguerluis@mt.feis.unesp.br Abstrct In this work we consider two clss of optiml control problems on time scles described by Volterr integrl equtions on time scles. We hve estblished the existence of solutions for these two clss of optiml control problems. Keywords: Optiml control, Volterr integrl equtions, time scles. Resumo Neste trblho nós considermos dus clsses de problems de controle ótimo em escls temporis descritos por equções integris de Volterr em escls temporis. Nós estbelecemos existênci de soluções pr esss dus clsses de problems de controle ótimo. Plvrs-chve: Controle ótimo, equções integris de Volterr, escls temporis. Recebido: 14/05/2015 Aceito: 10/02/2015

2 Ciênci e Ntur v.38 n.2, 2016, p Introduction Results of existence of optiml controls for optiml control problems on time scles re to be found in [Sntos e Silv (2014), Theorem 5.2], [Liu et l. (2011), Theorem 4.3], [Peng et l. (2011), Theorem 4.3], [Zhn et l. (2012), Theorem 4.1] nd [Zhn e Wei (2009), Theorem 4.2]. Using the theorems [dos Sntos (2015), Theorem 3.3] nd [dos Sntos (2015), Theorem 3.4], we give sufficient conditions for the existence of optiml controls to optiml control problems on time scles described by Volterr integrl equtions on time scles. To the best of our knowledge, the optiml control problems of system governed by the Volterr integrl equtions on time scles hve not been considered in the literture. 2 Preliminries A time scle T is nonempty closed subset of rel numbers. Here we use n rbitrry bounded time scle T such tht = min T < b = mx T. Define the forwrd jump opertor σ : T T by σ(t) =inf{s T : s > t} where inf = sup T. Let c,d T be such tht c d. Given function f : T R, we indicte the Riemnn -integrl of f on [c,d] T by d c f (s) s. For functions f : T R n the integrtion is considered componentwise. We study optiml control problems on time scles described by the following Volterr integrl equtions on time scles x(t) =u(t)+ g(t,s,x(s)) s (1) x(t) =u(t)+ g(t,s,x(σ(s))) s (2) where x : T R n is the unknown function, g : T T R n R n nd f : T R n re given functions, nd t T. The existence of continuous solutions to Eqs. (1) nd (2) cn be found in Kulik e Tisdell (2008). Below we stte the convergence results of solutions used to estblish the existence of optiml controls to optiml control problems on time scles described by Volterr integrl equtions on time scles. Theorem 2.1 (dos Sntos (2015)). Let {g k } be sequence of continuous functions with g k : T T R n R n stisfying g k (t,s,x) C(1 + x ) on its domin. Suppose tht { f k } is sequence of uniformly bounded nd equicontinuous functions with f k : T R n nd f k f 0. Suppose lso 1. g k (t,s,x) g(t,s,x) on T T R n ; 2. there exists L > 0 such tht g k (t,s,x) g k (t,s,y) L x y for ech k; 3. for ech k, ψ k (t) is solution of t T; ψ k (t) = f k (t)+ g k (t,s,ψ k (s)) s, 4. given ɛ > 0 nd M > 0, there exists δ > 0 such tht g k (t,s,x) g k (t 1,s,x) ɛ t t 1 when s T, t t 1 < δ, t,t 1 T, x M nd k is n integer. Then there is subsequence {ψ kj } {ψ k } nd function ψ : T R n such tht ψ kj ψ 0, nd ψ stisfies ψ(t) = f (t)+ g(t,s,ψ(s)) s

3 742 Sntos : Optiml control problems described by Volterr integrl equtions... Theorem 2.2 (dos Sntos (2015)). Tke sequence of continuous functions g k : T T R n R n such tht g k (t,s,x) C(1 + x ) for ll (t,s,x) T T R n. Consider sequence of uniformly bounded nd equicontinuous functions f k : T R n with f k f 0. Consider lso the hypotheses 1. g k (t,s,x) g(t,s,x) on T T R n ; 2. there exists L > 0 such tht g k (t,s,x) g k (t,s,y) L x y for ech k; 3. for ech k, ψ k (t) is solution of ψ k (t) = f k (t)+ t T; g k (t,s,ψ k (σ(s))) s, 4. for ech ɛ > 0 nd M > 0, there exists δ > 0 such tht g k (t,s,x) g k (t 1,s,x) ɛ t t 1 whenever s T, t t 1 < δ, t,t 1 T, x M nd k is n integer. If C(b ) < 1, then there is subsequence {ψ kj } {ψ k } nd function ψ : T R n such tht ψ kj ψ 0, nd ψ stisfies for some constnt K > 0. Consider the following optiml control problem (P 1 ): min h(x(),x(b)) on ll pirs (x,u) C(T,R n ) U, such tht { x(t) =u(t)+ g(t,s,x(s)) s, t T (x(),x(b)) E where E R 2n is closed set nd h : R n R n R is lower semicontinuous function. We sy tht (x,u) C(T,R n ) U is n dmissible process for (P 1 ), if the pir (x,u) stisfies the Eq. (1) nd x obeys the condition (x(),x(b)) E. A process ( x,ū) is clled n optiml process for (P 1 ), if it is n dmissible process for (P 1 ) tht stisfies h( x(), x(b)) h(x(),x(b)) for ll dmissible process (x,u) of (P 1 ). Theorem 3.1. Let g : T T R n R n be continuous function stisfying g(t,s,x) C(1 + x ) on its domin. Suppose tht 1. there exists L > 0 such tht g(t,s,x) g(t,s,y) L x y ; ψ(t) = f (t)+ g(t,s,ψ(σ(s))) s 2. given ɛ > 0 nd M > 0, there exists δ > 0 such tht g(t,s,x) g(t 1,s,x) ɛ t t 1 3 Optiml control problems on time scles We denote the set of ll continuous functions with domin T nd tking vlues on R n by C(T,R n ). Let U K, where K is the set of ll functions f : T R n stisfying f (t) f (s) K t s, t,s T when s T, t t 1 < δ, t,t 1 T nd x M. Then (P 1 ) hs n optiml process. Proof. Denote by inf{p 1 } the gretest lower bound on h(x(),x(b)) over dmissible processes (x,u) of (P 1 ). Hence there exists sequence of dmissible processes (x i,u i ) of (P 1 ) obeying inf{p 1 } = lim h(x i (),x i (b)).

4 Ciênci e Ntur v.38 n.2, 2016, p Since {u i } is equicontinuous sequence, from Arzel-Ascoli s Theorem there exists subsequence of {u i }, we do not relbel, such tht u i ū 0. It follows from [dos Sntos (2015), Theorem 3.3] tht there is subsequence of {x i }, we do not relbel, such tht x i x 0. Moreover, the function x : T R n stisfies x(t) =ū(t)+ g(t,s, x(s)) s As (x i (),x i (b)) E nd E is closed set, we conclude tht ( x(), x(b)) E nd then ( x,ū) is n dmissible process for (P 1 ). We lso hve inf{p 1 } = lim h(x i (),x i (b)) = lim inf h(x i (),x i (b)) h( x(), x(b)) inf{p 1 } nd thence ( x,ū) is n optiml process for (P 1 ). Now consider the following optiml control problem (P 2 ): min h(x(),x(b)) on ll pirs (x,u) C(T,R n ) U, such tht { x(t) =u(t)+ g(t,s,x(σ(s))) s, (x(),x(b)) E. t T Below we get the existence of solutions to (P 2 ). Theorem 3.2. Assume tht g : T T R n R n is continuous function stisfying g(t,s,x) C(1 + x ) on its domin. We lso suppose tht 1. there exists L > 0 such tht g(t,s,x) g(t,s,y) L x y ; 2. for ech ɛ > 0 nd M > 0, there exists δ > 0 such tht g(t,s,x) g(t 1,s,x) ɛ t t 1 whenever s T, t t 1 < δ, t,t 1 T nd x M. If C(b ) < 1, then (P 2 ) hs n optiml process. Proof. Let inf{p 2 } denote the gretest lower bound on h(x(),x(b)) over dmissible processes (x,u) of (P 2 ). Thence there exists sequence of dmissible processes (x i,u i ) of (P 2 ) obeying inf{p 2 } = lim h(x i (),x i (b)). As {u i } is equicontinuous sequence, it follows from Arzel-Ascoli s Theorem tht there exists subsequence of {u i }, we do not relbel, such tht u i ū 0. From [dos Sntos (2015), Theorem 3.4] there exists subsequence of {x i }, we do not relbel, such tht x i x 0. Furthermore, the pir ( x,ū) stisfies the eqution x(t) =ū(t)+ g(t,s, x(σ(s))) s Since (x i (),x i (b)) E nd E is closed set, we hve lim(x i (),x i (b)) = ( x(), x(b)) E nd then ( x,ū) is n dmissible process for (P 2 ). Finlly, since inf{p 2 } = lim h(x i (),x i (b)) = lim inf h(x i (),x i (b)) h( x(), x(b)) inf{p 2 } we conclude tht ( x,ū) is n optiml process for (P 2 ).

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