Rothe’s fixed point theorem and controllability of semilinear nonautonomous systems

In this paper we apply Rothe’s fixed point theorem to prove the controllability of the following semilinear system of ordinary differential equations {z′(t)=A(t)z(t)+B(t)u(t)+f(t,z(t),u(t)),t∈(0,τ],z(0)=z0, where z(t)∈Rnz(t)∈Rn, u(t)∈Rmu(t)∈Rm, A(t)A(t), B(t)B(t) are continuous matrices of dimensions n×nn×n and n×mn×m respectively, the control function uu belongs to L2=L2(0,τ;Rm)L2=L2(0,τ;Rm) and the nonlinear function f:[0,τ]×Rn×Rm→Rnf:[0,τ]×Rn×Rm→Rn is continuous and there are a,b,c∈Ra,b,c∈R and 12≤β<1 such that ‖f(t,z,u)‖Rn≤a‖z‖Rn+b‖u‖Rmβ+c,u∈Rm,z∈Rn. Under this condition we prove the following statement: if the linear ź(t)=A(t)z(t)+B(t)u(t) is controllable, then the semilinear system is also controllable on [0,τ][0,τ]. Moreover, we could exhibit a control steering the nonlinear system from an initial state z0z0 to a final state z1z1 at time τ>0τ>0.