Regresar

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

Abstract:

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)= z 0, where z(t)∈ ℝn, u(t)∈ ℝm, A(t), B(t) are continuous matrices of dimensions n×n and n×m respectively, the control function u belongs to L 2=L2(0,τ; ℝm) and the nonlinear function f:[0,τ]× ℝn× ℝm→ ℝn is continuous and there are a,b,c∈ℝ and 1/2≤β<1 such that ∥f(t,z,u) ∥ℝn≤ a ∥ z∥ℝn+b par;u par; 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]. Moreover, we could exhibit a control steering the nonlinear system from an initial state z0 to a final state z1 at time τ>0. © 2014 Elsevier B.V. All rights reserved.