\limsup_{k\to\infty} \int_0^T \dual{A \dot\bSigma_k + B^\star \dot\bu_k }{ \bT - \bSigma_k } \, \d t         \le         \int_0^T \dual{A \dot{\bSigma} + B^\star \dot{\bu} }{ \bT - \bSigma } \, \d t.$$ 