functional , the exact knowledge of the structure of ${\boldsymbol{H}}$ can be exploited further in order to drastically reduce the weights ${\boldsymbol{w}}$ that are necessary.

\[lemma:Qexakt\] If the $l$-th row and $l$-th column of ${\boldsymbol{Q}}$ equals zero, then the

-   $l$-th column of ${\boldsymbol{h}}_{xr_i} \forall i \in \left\{0,\dots,N\right\}$,

-   $l$-th column of ${\boldsymbol{h}}_{ur_i} \forall i \in \left\{0,\dots,N\right\}$ and

-   $l$-th row and $l$-th column of ${\boldsymbol{h}}_{r_ir_j} \forall i,j \in \left\{0,\dots,N\right\}$

are all equal to zero. Thus, the number of non-redundant weights $L$ in ${\boldsymbol{H}}$ (corresponding to the weight vector ${\boldsymbol{w}}$) reduces to $$\begin{aligned}
\label{eq:L_sparse}
    L &= (n-q)(n-q+1)\left(\frac{N}{2}+1\right)+\frac{1}{2}n(n+1)\\ \nonumber&+\left(m+N(n-q)\right)(m+n)+\frac{1}{2}(N-2)(N-1)(n-q)^2,
    \end{aligned}$$ where ${\boldsymbol{h}}_{xr_0}={\boldsymbol{Q}}$ and ${\boldsymbol{h}}_{r_0r_0}=-{\boldsymbol{Q}}$ has been considered and $q$ denotes the number of rows and columns of ${\boldsymbol{Q}}$ that are both zero.

The rather technical proof directly follows from considering –.

Although Lemma \[lemma:Qexakt\] is of technical nature, sparsity of ${\boldsymbol{H}}$ is essential in order to implement efficient learning controllers as will be discussed in Section \[sectionsimulation\].

Based on the specific structure of the analytical solution of $Q_{0}$ derived in Theorem \[Theo:Q\], according to Lemma \[lemma:Qexakt\], $L$ quadratic basis functions are sufficient to parametrize $Q_{0}$ exactly.

Equation and Lemma \[lemma:Qexakt\] show that our proposed Q-function generalizes over reference trajectories as the 