Invariance and polynomial design of strategies in the linear-quadratic game

A new algorithm to solve the $\mathcal H^\infty$ control problem in the case of full information was presented. It combines the spectral and matrix methods. The polynomial Lur'e–Riccati operator was introduced. Parametrization of all solutions of the controlled plant equation by hidden variables was presented within the framework of the J. C. Willems behavioral approach. The kernel of the polynomial Lur'e–Riccati operator was decomposed into the direct sum of subspaces that are similar to the Jordan blocks. The saddle point of the linear-quadratic game which was found by V. A. Yakubovich in 1970 was shown to provide solution to the $\mathcal H^\infty$ control problem for a considerable class of controlled plants.