| We consider a $2\times 2$ system of parabolic equations with first and zeroth coupling and establish a Carleman estimate by extra data of only one component without data of initial values. Then we apply the Carleman estimate to inverse problems of determining some or all of the coefficients by observations in an arbitrary subdomain over a time interval of only one component and data of two components at a fixed positive time $\theta$ over the whole spatial domain. The main results are Lipschitz stability estimates for the inverse problems. For the Lipschitz stability, we have to assume some non-degeneracy condition at $\theta$ for the two components and for it, we can approximately control the two components of the $2 \times 2$ system by inputs to only one component. Such approximate controllability is proved also by our new Carleman estimate. Finally we establish a Carleman estimate for a $3\times 3$ system for parabolic equations with coupling of zeroth-order terms by one component to show the corresponding approximate controllability with a control to one component. |
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