On adaptive synchronization of Genesio-Tesi chaotic system with uncertain parameters (Q2461412)

The paper considers the Genesio-Tesi system \[ \dot{x}_m=y_m,\quad \dot{y}_m=z_m,\quad \dot{z}_m=-cx_m-by_m-az_m+x_m^2 \] and \(a>0\), \(b>0\), \(c>0\), \(ab<c\), displaying chaotic behavior. The paper aims to a synchronization between two such systems via adaptive control. The second system is a controlled one \[ \dot{x}_s=y_s+u_1,\quad \dot{y}_s=z_s+u_2,\quad \dot{z}_s=-cx_s-by_s-az_s+x_s^2+u_3. \] Introducing the state error \[ e_1=x_s-x_m,\quad e_2=y_s-y_m,\quad e_3=z_s-z_m \] an adaptive control is synthesized to ensure \[ \lim_{t\rightarrow\infty}\;| e(t)| =0 \] with \(e=(e_1,e_2,e_3)\). The control structure reads \[ u_1=-e_1-e_2-(x_s+x_m-c_1)e_3,\quad u_2=-e_2-(1-b_1)e_3\,\quad u_3=-(1-a_1)e_3. \] with \(a_1\), \(b_1\), \(c_1\) supposed unknown and updated according to \[ \dot{a}_1=z_me_3\;,\;\dot{b}_1=y_me_3\;,\;\dot{c}_1=z_me_3 \] The properties of the system are obtained using a quadratic Liapunov function. A numerical simulation example is given.