Generalized $H_\infty$-optimal control as a trade-off between the$H_\infty$-optimal and $\gamma$-optimal controls

for the linear dynamic plant that is subjected to exogenous action and uncertainty generated by the unknown initial plant conditions, the level of perturbation suppression was determined as the greatest value of the ratio of the $L_2$-norm of the objective output to the quadratic root of the sum of the squared $L_2$-norm of the exogenous disturbance and norm of the initial state taken with a weight coefficient. Determination of the worst exogenous disturbance and the initial state maximizing this index was demonstrated. The role of the weight coefficient in the trade-off between the $H_\infty$-norm, that is, the level of suppression of the exogenous disturbance under the zero initial conditions, and the level $\gamma_0$ of suppression of the initial uncertainties, was clarified in the absence of exogenous disturbance. The generalized $H_\infty$-optimal laws of state and output control minimizing the chosen criterion were designed in terms of the linear matrix inequalities. The advantage of the generalized controller over the ordinary $H_\infty$-optimal controller was demonstrated by way of an example of the linear oscillator with unknown initial conditions.