What is Aim portfolio, multi-period optimisation?

Multi-period optimisation chooses trades by planning a sequence of future books under forecasts of returns, risks and costs, and executes the first (Boyd and co-authors). With quadratic costs $\frac\lambda2\Delta x^\top\Sigma\Delta x$ and signals whose predictive power decays at rates $\phi_k$, Gârleanu and Pedersen show that the optimal book trades each period a fraction $a/\lambda$ of the way from the current book to the aim portfolio $(\gamma\Sigma)^{-1}\sum_k B_k f_k/(1 + \phi_k a/\gamma)$, which weights slow signals more than fast ones, with $a$ the positive root of $a^2(1-\rho) + a(\gamma(1-\rho) + \lambda\rho) - \gamma\lambda(1-\rho) = 0$ and $\rho$ the discount rate.