\atil(\r) \in \mathcal{E}_{\r}, \r \in \Omega_S&\\
&\hspace{-1.3cm}\left(\ahat(\r) \!\!+\!\! \atil(\r)\right)^H  \R  \left(\ahat(\r) \!\!+\!\! \atil(\r)\right) \geq (1-\delta)P, \forall \atil(\r) \in \mathcal{E}_{\r}, \r \in \Omega_T&\\
&\hspace{-1.3cm}\left(\ahat(\r) \!\!+\!\! \atil(\r)\right)^H  \R  \left(\ahat(\r) \!\!+\!\! \atil(\r)\right) \leq (1+\delta)P, \forall \atil(\r) \in \mathcal{E}_{\r}, \r \in \Omega_T&\\
&\hspace{-1cm}\R \in \mathcal{R},&
\end{aligned}$$ where $t$ is the gap between the desired power level set across $\Omega_T$ and power deposition in the healthy tissue $\Omega_S$, similar to . Note that we take into account every possible perturbation $\atil(\r) \in \mathcal{E}_{\r}$.

In contrast to the optimization problem which is a tractable convex problem, the robust problem is an SIP problem. For a given $\R$ in , there are infinite number of constraints in terms of $\atil(\r)$ to satisfy which makes the problem non-trivial. However, in the following theorem, extending the approach in [@14:Shariati2014c], we reformulate the robust power deposition problem as a convex SDP problem whose solution is the optimally robust covariance matrix.

\[theo:robustSDP\] The robust power deposition for an M-element transducer array with the probing signal covariance matrix $\R \in \mathcal{R}$ and the perturbation vector $\atil(\r) \in \mathcal{E}_{\r}$, i.e., the solution of , is given as a solution to the following SDP problem $$\label{eq:robustSDP}
\begin{aligned}
&\underset{\R,t,P,\beta_i,\beta_{j,1},\beta_{j,2}}{\textrm{max}} \hspace{.3cm} t \hspace{.5cm} \textrm{subject to}& \\
