How find this inequality minimum $\sum_{i=1}^{n}a^2_{i}-2\sum_{i=1}^{n-1}a_{i}a_{i+1}$

Let $n$ be a given positive integer, and let $a_{1},a_{2},\cdots,a_{n}\ge 0$ such that $a_{1}+a_{2}+\cdots+a_{n}=1$.Find this minimum value$$a^2_{1}+a^2_{2}+\cdots+a^2_{n}-2a_{1}a_{2}-2a_{2}a_{3}-2a_{3}a_{4}-\cdots-2a_{n-2}a_{n-1}-2a_{n-1}a_{n}.$$

I think maybe we can use this well known$$a^2_{1}+a^2_{2}+\cdots+a^2_{n}\ge a_{1}a_{2}+a_{2}a_{3}+\cdots+a_{n-1}a_{n}+a_{n}a_{1}?$$

But this problem is only$$-2a_{1}a_{2}-2a_{2}a_{3}-2a_{3}a_{4}-\cdots-2a_{n-2}a_{n-1}-2a_{n-1}a_{n}$$so I can't it.Thank you for help.

By the way:I don't know this problem have Someone research?if No,I think this is interesting problem.