Control problems for semilinear second order equations with cosine families

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1 Jeong et al. Avances in Difference Equations 16) 16:15 DOI /s y R E S E A R C H Open Access Control problems for semilinear secon orer equations with cosine families Jin-Mun Jeong *, Eun-Young Ju an Seong-Ho Cho * Corresponence: jmjeong@pknu.ac.kr Department of Applie Mathematics, Pukyong National University, Busan, , Korea Abstract This paper aims to obtain the approximate controllability for the secon orer nonlinear control systems with a strongly cosine family an the associate with sine family. MSC: Primary 35Q53; seconary 93C3 Keywors: approximate controllability; regularity for solutions; cosine family; sine family 1 Introuction The first part of this paper gives some basic results on the regularity of solutions of the abstract semilinear secon orer initial value problem wt) = Awt)+Ft, w)+f t), < t T, t w) = x, t w) = y 1.1), in a Banach space X. Here, the nonlinear part is given by Ft, w)= kt s)g s, ws) ) s, where k belongs to L, T)ang :[,T] X X is a nonlinear mapping such that w gt, w) satisfies Lipschitz continuity. In 1.1) A is the infinitesimal generator of a strongly continuous cosine family Ct), t R.LetE be a subspace of all x X for which Ct)x is a once continuously ifferentiable function of t. In [1], when f : R X is continuously ifferentiable, x DA), y E, ank W 1,, T), the existence of a solution w L, T; DA)) W 1,, T; E) of1.1) foreach T > is given. Moreover, one establishe a variation of constant formula for solutions of the secon orer nonlinear system 1.1). The work presente in this paper, base on the regularity for solution of 1.1), investigates necessary an sufficient conitions for the approximate controllability for 1.1)with the strict range conition on B even though the system 1.1) contains unboune principal operators an the convolution nonlinear term, which is a more flexible necessary assumption than the one in []. 16 Jeong et al. This article is istribute uner the terms of the Creative Commons Attribution 4. International License which permits unrestricte use, istribution, an reprouction in any meium, provie you give appropriate creit to the original authors) an the source, provie a link to the Creative Commons license, an inicate if changes were mae.

2 Jeongetal. Avances in Difference Equations 16) 16:15 Page of 13 Wewillmakeuseofsomeofthebasicieasfromcosinefamilyreferreto[3, 4]anthe regular properties for solutions in [1, 5] foraiscussionofthecontrolresults. In[6, 7] a one-imensional nonlinear hyperbolic equation of convolution type which is nonlinear in the partial ifferential equation part an linear in the hereitary part is treate. As a secon part in this paper, we consier the approximate controllability for the nonlinear secon orer control system wt) = Awt)+Ft, w)+but), < t T, t w) = x, t w) = y, in a Banach space X where the controller B is a boune linear operator from some Banach space U to X. In[, 8, 9] the approximate controllability for 1.) was stuie uner the particular range conitions of the controller B epening on the time T. In Section 3 we establish to the approximate controllability for the secon orer nonlinear system 1.) uner a conition for the range of the controller B without the inequality conition inepenent to the time T, an we see that the necessary assumption is more flexible than the one in [, 9]. Finally, we give a simple example to which our main result can be applie. 1.) Preliminaries Let X be a Banach space. The norm of X is enote by. We start by introucing a strongly continuous cosine family an sine family in X. Definition.1 [1] LetCt)foreacht R be a boune linear operators in X. Ct) is calle a strongly continuous cosine family if the following conitions are satisfie: c1) Cs + t)+cs t)=cs)ct), for all s, t R,anC) = I; c) Ct)x is continuous in t on R for each fixe x X. Let A be the infinitesimal generator of a one parameter cosine family Ct)efineby Ax = t C)x. Then we enow it with the omain DA)={x X : t Ct)x is continuous} enowe with the norm { } x DA) = x + sup t Ct)x : t R + Ax. We shall also make use of the set { } E = x X : Ct)x is continuous t with the norm { x E = x + sup t Ct)x }. : t R

3 Jeongetal. Avances in Difference Equations 16) 16:15 Page 3 of 13 It is well known that DA)anE with the given norms are Banach spaces. Let St), t R, be the one parameter family of operators in X efine by St)x = Cs)xs, x X, t R..1) The following basic results on cosine an sine families are from Propositions.1 an. of [1]. Lemma.1 Let Ct)t R) be a strongly continuous cosine family in X. The following are true: c3) C t)=ct) for all t R; c4) Cs) an St) commute for all s, t R; c5) St)x is continuous in t on R for each fixe x X; c6) there exist constants K 1 an ω such that Ct) Ke ω t for all t R, an St1 ) St ) t1 K e ω s s for all t 1, t R; t c7) if x E, then St)x DA) an Ct)x = St)Ax = ASt)x = t t St)x, moreover, if x DA), then Ct)x DA) an Ct)x = ACt)x = Ct)Ax; t c9) if x X an r, s R, then s s ) Sτ)xτ DA) an A Sτ)xτ = Cs)x Cr)x. r r First, we consier the following linear equation: wt) = Awt)+f t), t, t w) = x, ẇ) = y..) The following results are crucial in iscussing regular problem for the linear case for a proof see [1]). Proposition.1 Let f : R X be continuously ifferentiable, x DA), y E. Then the mil solution wt) of.) represente by wt)=ct)x + St)y + St s)f s) s, t R,

4 Jeongetal. Avances in Difference Equations 16) 16:15 Page 4 of 13 belongs to L, T; DA)) W 1,, T; E), an we see that there exists a positive constant C 1 such that, for any T >, w L,T;DA)) C 1 1+ x DA) + y E + f W 1,,T;X))..3) If f is continuously ifferentiable an x, y ) DA) E,itiseasilyshownthatw is continuously ifferentiable an satisfies ẇt)=ast)x + Ct)y + Ct s)f s) s, t R. Here, we note that if w is a solution of.) inaninterval[,t 1 + t ]witht 1, t >.Then for t [, t 1 + t ], we have wt)=ct t 1 )wt 1 )+St t 1 )ẇt 1 )+ { = Ct t 1 ) Ct 1 )x + St 1 )y + 1 { + St t 1 ) ASt 1 )x + Ct 1 )y + + St s)f s) s t 1 = Ct)x + St)y + St s)f s) s t 1 } St 1 τ)f τ) τ 1 St s)f s) s; here, we use the following relations, for all s, t R: an Cs + t)+cs t)=cs)ct), Ss + t)= Ss)Ct)+St)Cs), Cs + t)= Ct)Cs) St)Ss), Cs + t) Ct s)=ast)ss), } Ct 1 τ)f τ) τ St)ASs)=ASt)Ss)= 1 Ct + s) 1 Ct s)=ct + s) Ct)Cs). This means the mapping t wt 1 + t) is a solution of.) in[,t 1 + t ] with initial ata wt 1 ), ẇt 1 )) DA) E. From now on, we introuce the regularity of solutions of the abstract semilinear secon orer initial value problem 1.1)inaBanachspaceX. We make the following assumptions. The nonlinear mapping g from [, T] DA)toX is such that t gt, w)ismeasurable an gt, w 1 ) gt, w ) DA) L w 1 w,.4) gt,)=, for a positive constant L.

5 Jeongetal. Avances in Difference Equations 16) 16:15 Page 5 of 13 For w L, T; DA)), we set Ft, w)= kt s)g s, ws) ) s, where k belongs to L, T). We will seek a mil solution of 1.1), that is, a solution of the integral equation wt)=ct)x + St)y + St s) { Fs, w)+f s) } s. Lemma. If w L, T; DA)) for any T >,then F, w) L, T; X) an F, w) L,T;X) L k L,T) T w L,T;DA)). Moreover, let w 1, w L, T; DA)). Then we have F, w 1 ) F, w ) L,T;X) L k L,T) T w1 w L,T;DA)). Proof By using the Höler inequality an.4), it is easily shown that F, w) T L,T;X) k L,T) kt s)g s, ws) ) s t T L ws) s t L k L,T) T w L,T;DA)). By a similar argument, the secon paragraph is obtaine. Now, as in Theorem 3.1 of [1], we give a norm estimation of the solution of 1.1) an establish the global existence of solutions with the ai of norm estimations. Proposition. Suppose that the assumption.4) are satisfie. If f : R Xiscontinuously ifferentiable, x DA), y E, an k W 1,, T), then the solution w of 1.1) exists an is unique in L, T; DA)) W 1,, T; E) for T >,an there exists a constant C 3 epening on T such that w L,T;DA)) C 3 1+ x DA) + y E + f W 1,,T;X))..5) 3 Approximate controllability In this section, we eal with the approximate controllability for the semilinear secon orer control system wt) = Awt)+Ft, w)+but), < t T, t w) = x, t w) = y 3.1), in a Banach space X where the controller B is a boune linear operator from some Banach space U to X,whereU is another Banach space. Assume the following.

6 Jeongetal. Avances in Difference Equations 16) 16:15 Page 6 of 13 Assumption G) The nonlinear mapping g :[,T] X X is such that t gt, w) is measurable an gt, w 1 ) gt, w ) L w 1 w, kt) M, 3.) for a positive constant L. Here,weremarkthatsincetheAssumptionG) is a more general conition than.4), the equation of 3.1), written wt)=ct)x + St)y + St s) { Fs, w)+bus) } s, 3.3) belongs to L, T; DA)) W 1,, T; E)forT >. Given a strongly continuous cosine family Ct) t R), we efine linear boune operators Ĉ an Ŝ mapping L, T; X)intoX by Ĉp = T CT t)pt) t, Ŝp = T ST t)pt) t, for p ) L, T; X)anSt)istheassociatesinefamilyofCt). We efine the reachable sets for the system 3.1) as follows. Definition 3.1 Let wt; F, u) be a solution of the 3.1) associate with nonlinear term F an control u at the time t.then R T F)= { wt; F, u), ẇt; F, u) ) : u L, T; U) } X = X X, R T ) = { wt;,u), ẇt;,u) ) : u L, T; U) } X. The nonempty subset R T F) inx consisting of all terminal states of 3.1) is calle the reachable sets at the time T of the system 3.1). The set R T ) is one of the linear cases where F. Definition 3. The system 3.1) is sai to be approximately controllable on the interval [, T]if R T F)=X, where R T F) istheclosureofr T F) inx,thatis,foranyɛ >, x DA) anȳ E there exists a control u L, T; U)suchthat x CT)x ST)y ŜF, w) ŜBu < ɛ, ȳ AST)x CT)y ĈF, w) ĈBu < ɛ. We introuce the following hypothesis.

7 Jeongetal. Avances in Difference Equations 16) 16:15 Page 7 of 13 Assumption B) For any ε >anp L, T; X), there exists a u L, T; U)suchthat Ĉp ĈBu < ε, Bu L,t;X) q 1 p L,t;X), t T, where q 1 is a constant inepenent of p. We remark that from the relations between the cosine an sine families, the operator Ŝ also satisfies the conition B), thatis,foranyε >anp L, T; X) thereexistsa u L, T; U)suchthat Ŝp ŜBu < ε, Bu L,t;X) q 1 p L,t;X), t T. For the sake of simplicity we assume that the sine family St)isbouneasinc6): St) Kt), t. Here, we may consier the following inequality: Kt) ω 1 Ke ωt 1). Lemma 3.1 Let the Assumption G) be satisfie. If u 1 an u are in L, T; U), then we have wt; F, u1 ) wt; F, u ) KT)e KT)MLT T Bu1 Bu L,t;X) for t T. Proof From the Assumption G),itfollowsthat,for t T, wt; F, u1 ) wt; F, u ) KT) t Bu 1 s) Bu s) L,t;X) + KT)MLt ws; F, u1 ) ws; F, u ) s, where L is the constant in the Assumption G). Therefore, by using Gronwall s inequality this lemma follows. For the approximate controllability for the linear equation, we recall the following necessary lemma before proving the main theorem. Lemma 3. Let the Assumption G) be satisfie. Then we have R T ) = X. Proof Let x DA), ȳ E. Putting η 1 = x CT)x ST)y DA), η = ȳ AST)x CT)y E,

8 Jeongetal. Avances in Difference Equations 16) 16:15 Page 8 of 13 then there exists some p C 1 [, T]:X)suchthat η 1 = T ST t)pt) t, η = T CT t)pt) t, for instance, take pt)={ct T)+St T)}η /T.ByhypothesisB) there exists a function u L, T; U)suchthat x CT)x ST)y ŜBu < ɛ, ȳ AST)x CT)y ĈBu < ɛ. The enseness of the omain DA) inx implies the approximate controllability of the corresponing linear system. Theorem 3.1 Let the Assumptions G) an B) be satisfie. Then the system 3.1) is approximately controllable on [, T], T >. Proof We will show that DA) E R T F), i.e., forgivenε >anξ T, ξ T ) DA) E there exists u L, T; U)suchthat ξ T wt; F, u) < ε, 3.4) ξ T ẇt; F, u) < ε. 3.5) Since ξ T, ξ T ) DA) E,thereexistsap L, T; X)suchthat Ŝp = ξ T CT)x ST)y, Ĉp = ξ T AST)x CT)y. Let u 1 L, T; U) be arbitrary fixe. Then by the Assumption B), thereexistsu L, T; U)suchthat Ŝ p F, w ; F, u 1 ) )) ŜBu < ε 4, Ĉ p F, w ; F, u 1 ) )) ĈBu < ε 4. Hence, we have ξt CT)x ST)y ŜF, w ; F, u 1 ) ) ŜBu < ε 4, ξ T AST)x CT)y ĈF, w ; F, u 1 ) ) ĈBu < ε ) Moreover, by the Assumption B), we can also choose v L, T; U)suchthat Ŝ F, w ; F, u ) ) F, w ; F, u 1 ) )) ŜBv < ε 8, Ĉ F, w ; F, u ) ) F, w ; F, u 1 ) )) ĈBv < ε 8, 3.7)

9 Jeongetal. Avances in Difference Equations 16) 16:15 Page 9 of 13 an also Bv L,t;X) q 1 F, w ; F, u 1 ) ) F, w ; F, u ) ) L,t;X) for t T. From now, we will only prove 3.4), while the proof of 3.5) is similar. In view of Lemma 3.1 an the Assumption B),wehave { Bv L,t;X) q 1 F τ, wτ; F, u ) ) F τ, wτ; F, u 1 ) ) τ { q 1 ML q 1 MLKT)e KT)MLT T { q 1 MLKT)e KT)MLT T wτ; F, u ) wτ; F, u 1 ) s τ } 1 } 1 } 1 Bu Bu 1 L,s;X) s τ 1 s τ) 1 Bu Bu 1 L,t;X) t ) 1 = q 1 MLKT)e KT)MLT T Bu Bu 1 L,t;X). Put u 3 = u v.wechoosev 3 such that Ŝ F, w ; F, u 3 ) ) F, w ; F, u ) )) ŜBv 3 < ε 8, Bv 3 L,t;X) q 1 F, w ; F, u 3 ) ) F, w ; F, u ) ) L,t;X) for t T.Thus,wehave Bv 3 L,t;X) { q 1 F τ, wτ; F, u3 ) ) F τ, wτ; F, u ) ) τ { q 1 ML q 1 MLKT)e KT)MLT T { q 1 MLKT)e KT)MLT T { ws; F, u 3 ) ws; F, u ) s τ q 1 MLKT)e KT)MLT T ) { q 1 MLKT)e KT)MLT T ) } 1 } 1 Bu 3 Bu L,s:X) s τ } 1 Bv L,s;X) s τ } 1 s Bu Bu 1 L,s;X) s τ } 1 s s τ ) 1 Bu Bu 1 L,t;X) ) q 1 MLKT)e KT)MLT t 4 ) 1 T Bu Bu 1 4 L,t;X).

10 Jeongetal.Avances in Difference Equations 16) 16:15 Page 1 of 13 By proceeing this process, an from the equality Bun u n+1 ) L,t;X) = Bv n L,t;X) q 1 MLKT) Te KT)MLT ) n 1 t n 4 n ) ) 1 Bu Bu 1 L,t;X) q1 MLKT) ) Te KT)MLT t n 1 1 = Bu Bu 1 L n 1)!,t;X), we obtain Bu n+1 Bu n L,T;X) n=1 q1 MLKT) Te KT)MLT t n= ) n 1 n! Bu Bu 1 L,T;X) <. Thus, there exists u L, T; X)suchthat lim Bu n = u in L, T; X). 3.8) n Combining 3.6)an3.7), we have ξt CT)x ST)y ŜF, w F, u ) ) ŜBu 3 = ξt CT)x ST)y ŜF, w F, u 1 ) ) ŜBu + ŜBv Ŝ [ F, w ; F, u ) ) F, w ; F, u 1 ) )] 1 < + 1 ) ε. 3 If we etermine v n L, T; U)suchthat Ŝ F w ; F, u n ) ) F w ; F, u n 1 ) )) ŜBv n < ε, n+1 then putting u n+1 = u n v n,wehave ξ T ST)g ŜF, z ; g, f, u n ) ) ŜΦu n+1 1 < ) ε, n =1,,... n+1 Therefore, for any ε > there exists an integer N such that ŜBu N+1 ŜBu N < ε,

11 Jeongetal.Avances in Difference Equations 16) 16:15 Page 11 of 13 an hence ξt CT)x ST)y ŜF, w ; F, u N ) ) ŜBu N ξ T CT)x ST)y ŜF, w ; F, u N ) ) ŜBu N+1 + ŜBu N+1 ŜBu N 1 < ) ε + ε N+1 ε. By a similar metho, we also obtain ξ T AST)x CT)y ĈF, w ; F, u N ) ) ĈBu N ε. Thus, as N tens to infinity, the system 3.1)is approximatelycontrollableon [,T]. Example We consier the following partial ifferential equation: wt,x) = Awt, x)+ft, w)+but), < t,<x < π, t wt,)=wt, π)=, t R, w, x)=x x), t w, x)=y x), < x < π. 3.9) Let X = L [, π]; R). If e n x)= π sin nx,then{e n : n =1,...} is an orthonormal base for X. The operator A : X X is efine by Awx)=w x), where DA)={w X : w, ẇ are absolutely continuous, ẅ X, w) = wπ)=}.then Aw = n w, e n )e n, w DA), n=1 an A is the infinitesimal generator of a strongly continuous cosine family Ct), t R, in X given by Ct)w = cos ntw, e n )e n, w X. n=1 Let us for g 1 t, x, w, p), p R m, assume that there is a continuous ρt, r):r R R + an a real constant 1 γ such that g1) g 1 t, x,,)=, g) g 1 t, x, w, p) g 1 t, x, w, q) ρt, w ) p q, g3) g 1 t, x, w 1, p) g 1 t, x, w, p) ρt, w 1 + w ) w 1 w. Let gt, w)x = g 1 t, x, w, Dw).

12 Jeongetal.Avances in Difference Equations 16) 16:15 Page 1 of 13 Then noting that gt, w 1 ) gt, w ), + g1 t, x, w 1, Dw 1 ) g 1 t, x, w, Dw ) u it follows from g1)-g3) that g 1 t, x, w 1, Dw 1 ) g 1 t, x, w, Dw ) u, gt, w1 ) gt, w ), L w 1 DA), y DA) ) w1 w DA), where L w 1 DA), w DA) ) is a constant epening on w 1 DA) an w DA).Weset Ft, w)= kt s)g s, ws) ) s, where k belongs to L, T). Let U = X,<α < T, an efine the intercept controller operator B α on L, T; X)by, t < α, B α ut)= ut), α t T, for u L, T; X)see[1]). For a given p L, T; X) let us choose a control function u satisfying, t < α, ut)= pt)+ α T α Ct α α t α))p t α)), α t T. T α T α Then u L, T; X)anŜp = ŜB α u.from B α u L,T;X) = u L α,t;x) α)) α L p L α,t;x) + Ke ωt p T α α,t;x) ) T α 1+Ke ωt p α L,T;X), it follows that the controller B α satisfies Assumption B). Therefore, from Theorem 3.1, we see that the nonlinear system given by 3.9) is approximately controllable on [, T]. Competing interests The authors eclare that they have no competing interests. Authors contributions All authors have mae equal contributions. All authors rea an approve the final manuscript.

13 Jeongetal.Avances in Difference Equations 16) 16:15 Page 13 of 13 Acknowlegements This research was supporte by Basic Science Research Program through the National research Founation of Korea NRF) fune by the Ministry of Eucation, Science an Technology 15R1D1A1A9593). Receive: 8 December 15 Accepte: 3 May 16 References 1. Jeong, JM, Hwang, HJ: Regularity for solutions of secon orer semilinear equations with cosine families. Sylwan J. 1589), ). Park, JY, Han, HK, Kwun, YC: Approximate controllability of secon orer integroifferntial systems. Inian J. Pure Appl. Math. 99), ) 3. Travis, CC, Webb, GF: Cosine families an abstract nonlinear secon orer ifferential equations. Acta. Math. 3, ) 4. Travis, CC, Webb, GF: An abstract secon orer semilinear Volterra integroifferential equation. SIAM J. Math. Anal. 1), ) 5. Jeong, JM, Kwun, YC, Park, JY: Regularity for solutions of nonlinear evolution equations with nonlinear perturbations. Comput. Math. Appl. 43, ) 6. Maccamy, RC: A moel for one-imensional, nonlinear viscoelasticity. Q. Appl. Math. 35, ) 7. Maccamy, RC: An integro-ifferential equation with applications in heat flow. Q. Appl. Math. 35, ) 8. Jeong, JM, Kwun, YC, Park, JY: Approximate controllability for semilinear retare functional ifferential equations. J. Dyn. Control Syst. 53), ) 9. Zhou, HX: Approximate controllability for a class of semilinear abstract equations. SIAM J. Control Optim. 1, ) 1. Naito, K: Controllability of semilinear control systems ominate by the linear part. SIAM J. Control Optim. 5, )