Exact Boundary Controllability Of A Class Of Quasilinear Hyperbolic Equations

In this article, we consider the first order quasilinear hyperbolic systems of diagonal form which is composed of three equations andGiving the following initial condition and the final condition in the case that the diagonal variables are decoupled in the boundary conditions, that is: while (u,v,w) are suitably coupled on the right side of the system, by establishing the local exact boundary controllability for a kind of second order quasilinear hyperbolic systems, we obtain the one-sided exact boundary controllability of the equation (1) and two-sided exact boundary controllability with less controls of the equation (1). The main results as the following:Theorem2.1(One-sided control at x=L) Suppose λi,fi∈C1(i=1,2), λ3,f3∈C2, and (2)-(3) hold. Suppose furthermore that holds and For any given initial data(u0,v0,w0) and final data (u1,v1,w1) are all C1[O,L]×C1[0,L]×C2[0,L] vector functions with small C1×C1×C2norm,and for any boundary function h3(t)∈C2[O,T] with small C2norm,such that the conditions of C1compatibility are satisfied at the points(t,x)=(O,O) and (T,O)respectively,then there exist boundary control(h1(t),h2(t))∈C1[O,T]×C1[0,T] at x=L with small C1[0,T]×C1[0,T]norm, such that the mixed initial-boundary value problem(1),(4) and (6)-(7)admits a unique semi-global C1[0,T]×C1[0,T]×C2[0,T]solution(u,v,w)=(u(t,x),v(t,x),w(t,x))on the domain R(T)={(t,x)|0≤t≤T,0≤x≤L} with small norm,which satisfies exactly the final condition(5).Theorem2.2(Two-sided control with less control)Suppose that λi,fi∈C1(i=2,3),λ1,f1∈C2and(2)-(3)hold.Suppose furthermore that holds,and λ1≠λ2.Let For any given initial data(u0,v0,w0) and final data(u1,v1,w1) are all C2[0,L]×C1[0,L]×C1[0,L] vector functions with small C2[0,L]×C1[0,L]×C1[0,L] norms,and for any boundary function h1(t)∈C2[0,T]with small norm‖h1‖C2[0,T],such that the corre sponding conditions of C1compatibility are satisfied at the point(t,x)=(0,L)and (T,L) respectively,then there exist boundary control h2(t)∈C1[0,T]at x=L, and boundary contol,h3(t)∈C1[0,T]at x=0with small C1norm,such that the mixed initial-boundary value problem(1),(4)and(6)-(7)admits a unique semi-global C2×C1×C1solution (u,v,w)=(u(t,x),v(t,x),w(t,x))on the domain R(T) with small C2×C1×C1norm,which satisfies exactly the final condition (5).In what follows,we consider the first order quasilinear hyperbolic systems of diag-onal form which is composed of2n+1equations and Giving the following initial condition and the final condition the diagonal variables are decoupled in the boundary conditions,that is: Similarly,using the local exact boundary controllability for a kind of second order quasi-linear hyperbolic systems,we obtain the one-sided exact boundary controllability of the equations(8)and two-sided exact boundary controllability with less controls of the equations(8).The main results as the following:Theorem3.1(One-sided control at x=L)Suppose that λi,fi∈C1(i=1,…,n+1), μj,gj∈C2(j=1,…n),and(9)-(10)hold.Suppose furthermore that holds,and For any given initial data(u0,v0) and final data(u1,v1)are all C1[0,L]×C2[0,L] vector functions with small C1[0,L]×C2[0,L] norms,and for any boundary function Hj(t)∈C2[0,T](j=1,…,n) with small C2norms,such that the conditions of C1compatibility are satisfied at the points(t,x)=(0,0)and(T,0)respectively,then there exist boundary control hi(t)∈C1(i=1,…,n)at x=L with small C1[0,T]]norm,such that the mixed initial-boundary value problem(8),(11)and(13)-(14)admits a unique semi-global C1×C2solution(u,v)=(u(t,x),v(t,x))on the domain R(T)={(t,x)|0≤t≤T,0≤x≤L} with small C1×C2norm,which satisfies exactly the final condition (12). Theorem3.2(Two-sided control with less controls)Suppose that fi,λi∈C2(i=1,…,n),fn+1,λn+1,gj,μj∈C1(j=1,…,n),and(9)-(10)hold.Suppose furthermore that holds,and λi≠λn+1(i=1,…,n).Let For any given initial data(u0,v0)and final data(u1,v1) are all C2×C1vector functions with small C2×C1norms,and for any boundary function,hi(t)∈C2(i=1,…,n)with small C2norm,such that the corresponding conditions of C1compatibility are satisfied at the point(t,x)=(0,L)and(T,L)respectively,then there exist boundary control Hj(t)(j=1,…,n)with small C1norm at x=L,and boundary control,hn+1(t)at x=0with small C1norm,such that the mixed initial-boundary value problem(8),(11)and (13)-(14)admits a unique semi-global C2×C1solution(u,v)=(u(t,x),v(t,x))on the domain R(T)with small C2×C1norm,which satisfies exactly the final condition(12).