Asymptotic exactness of parameter-dependent lyapunov functions: An error bound and exactness verification

This paper provides an approximate approach to a robust semidefinite programming problem with a functional variable and shows its asymptotic exactness. This problem covers a variety of control problems including a robust stability/performance analysis with a parameter-dependent Lyapunov function. In the proposed approach, an approximate semidefinite programming problem is constructed based on the division of the set of parameter values. This approach is asymptotically exact in the sense that, as the resolution of the division becomes higher, the optimal value of the constructed approximate problem converges to that of the original problem. Our convergence analysis is quantitative. In particular, this paper gives an a priori upper bound on the discrepancy between the optimal values of the two problems. Moreover, it discusses how to verify that an optimal solution of the approximate problem is actually optimal also for the original problem.