This paper deals with the regulator problem of a distributed-parameter system described by a parabolic or hyperbolic linear partial differential equation. The Riccati-type partial differential equation or the infinite-dimensional two point boundary value problem must be solved to obtain the solution
u0 of the regulator problem of a distributed-parameter system. It is very difficult to calculate the solution of such a complex problem. Therefore, it is reasonable to approximate the distributed-parameter system with a lumped-parameter system having
n main modes. By solving the regulator problem of the lumped-parameter system, an approximate solution
un will be obtained instead of
u0. This authors' idea is an extension of Modal Control Method proposed by Gould to the regulator problem of a distributed-parameter system. But if
J(u) is the performance index of the regulator problem and
n tends to infinity, the convergence of
J(un) to
J(u0) is equivalent to the convergence of
un to
u0 in an appropriate topology. Then it must be investigated whether
un converges to
u0 and in what topology this convergence is assured.
The authors show that
un converges to
u0 in the mean square, and furthermore it converges uniformly if
J(u) contains the derivative of the control input with time.
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