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Keywords: sensor selection LQ problem, uniqueness of solution, comparability of sensor Although it is recognized that linear control theory has been established mostly, but depending on realistic conditions there remain unsolved control problems. As an example we will discuss the sensor selection LQ problem in this paper. Recent sensor technology makes a process to arrange many low-priced sensors possible. In such case we can expect more effective real-time control by changing sensors properly. This problem will be an essential control problem, but it has not been discussed in the literature. This is because of the combinatorial property of the problem solution. In this paper, we first propose the concept of "the comparability of sensors" and yield a result for the uniqueness of optimal solution. Next, we extend the problem in the case of use of observer and yield a sufficient condition for existence of the unique solution.Let's consider the LQ problem with linear dynamic systemsand control performance indexwhere, x k ∈R n , u k ∈R m , y k ∈R l are the state vector, the input vector, the output vector respectively. The sensor matrixThen LQ problem with sensor selection is to determine optimal sensor sequence {C * k } and optimal input sequence {u * k } so as to minimize the control performance eq.(3) for the system in (1),(2). This problem is difficult to be solved because of the combinatory property. So, in this paper we discuss with the uniqueness of the solution and the comparability of sensor.By substituting (2) into (3), the control performance is rewritten asFollowing the solution of the usual LQ problem in the case of fixed sensor sequence {C k }, we assume the algorithm of the solution for this problem as follows.

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