SUMMARYA new numerical method is developed for the boundary optimal control problems of the heat conduction equation in the present paper. When the boundary optimal control problem is solved by minimizing the objective function employing a conjugate-gradient method, the most crucial step is the determination of the gradient of objective function usually employing either the direct di erentiation method or the adjoint variable method. The direct di erentiation method is simple to implement and always yields accurate results, but consumes a large amount of computational time. Although the adjoint variable method is computationally very e cient, the adjoint variable does not have su cient regularity at the boundary for the boundary optimal control problems. As a result, a large numerical error is incurred in the evaluation of the gradient function, resulting in premature termination of the conjugate gradient iteration. In the present investigation, a new method is developed that circumvents this di culty with the adjoint variable method by introducing a partial di erential equation that describes the temporal and spatial dynamics of the control variable at the boundary. The present method is applied to the Neumann and Dirichlet boundary optimal control problems, respectively, and is found to solve the problems e ciently with su cient accuracy.

We consider problems of controlling the intensity of the Rayleigh-Be´nard convection by adjusting the heat flux distribution at the boundary while keeping the heat input the same. The Karhunen-Loe`ve Galerkin procedure is used to reduce the Boussinesq equation to a low dimensional dynamic model, which in turn is employed in a projected gradient method to yield the optimal heat flux distribution. The performance of the Karhunen-Loe`ve Galerkin procedure is assessed in comparison with the traditional technique employing the Boussinesq equation, and is found to be very accurate as well as efficient.

A method is developed for the recursive identification of thermal convection system governed by the Boussinesq equation using an extended Kalman filter. A computationally feasible Kalman filter is constructed by reducing the Boussinesq equation to a small number of ordinary differential equations by means of the Karhunen-Loe`ve Galerkin procedure which is a type of Galerkin method employing the empirical eigenfunctions of the Karhunen-Loe`ve decomposition. Employing the Kalman filter constructed by using the reduced order model, the thermal convection induced by a spatially varying heat flux at the bottom is identified recursively by using either the Boussinesq equation or the reduced order model itself. The recursive identification technique developed in the present work is found to yield accurate results for thermal convection even with approximate covariance equation and noisy measurements. It is also shown that a reasonably accurate and computationally feasible method of recursive identification can be constructed even with a relatively inaccurate reduced order model.

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SUMMARYAn inverse radiation problem is investigated where the spatially varying heat transfer coe cient h(z) and the absorption coe cient Ä in the radiant cooler are estimated simultaneously from temperature measurements. The inverse radiation problem is solved through the minimization of a performance function, which is expressed by the sum of square residuals between calculated and observed temperature, using the conjugate gradient method. The gradient of the performance function is evaluated by means of the improved adjoint variable method that can take care of both the function estimation and the parameter estimation e ciently. The present method is found to estimate h(z) and Ä with reasonable accuracy even with noisy temperature measurements.