I am reading this thesis (2006) by Daniel Wachsmuth on the optimal control of the unsteady Navier-Stokes equations. In Chapter 5: Stability of Optimal controls of this thesis, he use strong convexity to obtain the existence and uniqueness of the optimal control.
Theorem 5.3. Assume that ($\overline{y}, \overline{u},\overline{\lambda}$) satisfy the first-order necessary optimality condition and the coercivity condition (SSC). Then the perturbed linear-quadratic optimal control problem ($P_z$) admits a unique optimal control $u_z$.
Proof. Let us denote the Lagrangian associated to ($P_z$) by $\mathcal{L}^{(z)}$. Then it holds for all $y, u, \lambda$ $$\mathcal{L}^{(z)}_{vv}(y,u,\lambda)=\mathcal{L}_{vv}(\overline{y},\overline{u},\overline{\lambda}).$$ Now take two controls $u_1, u_2 \in \widetilde{U_{ad}}$ with associated solutions $y_1, y_2$ of (5.13b). Then the pair $(y_1 − y_2, u_1 − u_2)$ fits in the assumption of (SSC), and we find $$\mathcal{L}_{vv}(y,u,\lambda)[(y_1 − y_2, u_1 − u_2)^2]=\mathcal{L}_{vv}(\overline{y},\overline{u},\overline{\lambda})[(y_1 − y_2, u_1 − u_2)^2] \geq \delta \left\| u_1 - u_2\right\|_2^2.$$ Thus, the problem ($P_z$) is convex on the space of admissible controls $\widetilde{U_{ad}}$. Thus, the problem ($P_z$) as a linear-quadratic optimization problem with strongly convex objective functional is uniquely solvable.
The second-order sufficient condition ($SSC$) in the proof is given by $$\begin{cases}\text{There exist $\varepsilon>0$ and $\delta>0$ such that}\\ \mathcal{L}_{vv}(\overline{y},\overline{u},\overline{\lambda})[(w,h)^2] \geq \delta \left\| h\right\|_2^2\\ \text{for all $h \in \mathcal{T}_{U_{ad}}(\overline{u}), h_i=0$ on $Q_{\varepsilon,i},i=1,2$}\\ \text{and $w$ is the solution of the equation}\\ \begin{cases}w_t+Aw+B'(\overline{y}) = h\\w(0)=0\end{cases}\end{cases}$$
where $\mathcal{T}_{U_{ad}}(\overline{u})$ is the tangent cone of $U_{ad}$ at $\overline{u}$.
I have some questions on Wachsmuth's argument:
(1) Does the pair $(y_1 − y_2, u_1 − u_2)$ satisfies the condition ($SSC$)? It is not clear that whether $u_1-u_2 \in \mathcal{T}_{U_{ad}}(\overline{u})$ or not.
(2) Does strong convexity implies existence of optimal control as in his proof of Theorem 5.3?