Existence of optimal solutions for the convex control problem

Let $A\in\mathbb{R}^{n \times n}$, $B\in\mathbb{R}^{n \times m}$, $T>0$, $\beta\in\\{1,2,\dots,m-1\\}$, and $\alpha_j\in(0,T]$ for $j=1,2,\dots,m$, as in Problem~

$$. **Existence conjecture.** For every such choice of data, optimal solutions of Problem~$$

exist. The existence of optimal solutions is used in the paper's preceding results and is said to follow by an argument similar to a cited lemma, but no proof is supplied.