Constrained LQR Problems in Elliptic Distributed Control Systems with Point Observations--- on Convergence Rates
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In this paper, we continue to study the (bound constrained) linear-quadratic regulator problem in distributed boundary control systems governed by the elliptic equation with point observations. Due to the appearance of singularities in the problem, the traditional Galerkin variational method that leads to an adjoint system is not desirable, and the classical Lagrangian multiplier method is not reliable for providing numerical results. A characterization formula of the optimal control and its singularity decomposition formula are derived in [Z. Ding, L. Ji, and J. Zhou, SIAM J. Control Optim., 34 (1996), pp. 264-294; Z. Ding and J. Zhou, Appl. Math. Optim., to appear] by using the boundary integral equation and potential theory coupled with a variational inequality in a Banach space setting. Based on the characterization formula, a conditioned gradient projection method (CGPM) has been proposed in [Z. Ding and J. Zhou, Appl. Math. Optim., to appear]. Numerical experiment has shown that CGPM is efficient and also insensitive to the partition number of the boundary. In this paper, we estimate the rate of convergence for CGPM. First it is proved that for N = 2, CGPM converges exponentially in the L2 norm, and for N = 3, CGPM converges subexponentially in the Lp norm. Then, under a reasonable condition, it is proved that for N = 2, CGPM converges uniformly exponentially, and for N = 3, CGPM converges uniformly subexponentially.
