\(H_ \infty\) control design in bilinear systems: A tensor formal series approach (Q1333845)

This paper is devoted to the study of \(H_ \infty\)-disturbance attenuation for finite-dimensional, autonomous bilinear systems \[ \dot x= Ax+ uBx+ Cd \] with \(L^ 2\)-control \(u(\cdot)\) and with \(L^ 2\)- disturbance \(d(\cdot)\). The problem is: Find the optimal control, which in the case of the ``worst'' disturbance, achieves some quadratic disturbance index of attenuation below a given level \(\gamma\). This problem is treated as a minimax problem of optimal control and is studied using dynamic games methods. The Bellman-Isaacs equation technique is used. First, some necessary and sufficient conditions of optimality are proved. Next, using the technique of tensor product and formal power series of S. P. Banks, the corresponding Bellman-Isaacs equation is solved and the corresponding optimal control and the ``worst'' disturbance are determined.